Option Pricing. Stefan Ankirchner. January 20, Brownian motion and Stochastic Calculus

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1 Option Pricing Stefan Ankirchner January 2, The Binomial Model 2 Brownian motion and Stochastic Calculus We next recall some basic results from Stochastic Calculus. We do not prove most of the results. Proofs can be found e.g. in Chapter 4, [4]. 2.1 Brownian motion Recall the definition of a Brownian motion. Definition 2.1. A stochastic process (W t ) t with continuous paths is called a Brownian motion if the initial value is given by W =, for any = t t 1 t n, the increments W t1 W t,, W tn W tn 1 are independent, for any s < t the increment (W t W s ) is normally ( ) distributed with mean zero and variance t s, i.e. W t W s N, t s. Figure 2.1 shows a realization of a Brownian path. By the very definition of a BM, the variance of an increment is given by the length of the increment, i.e. var(w t W s ) = t s. The standard deviation is given by the square-root of the time length: st. deviation of (W t W s ) = t s. We next show that a BM has non-vanishing quadratic variation. Let Π : = t < t 1 < < t n = t be a finite partition of [, t]. We call the mesh size of Π. Π := max 1 i n t i t i 1 1

2 Figure 1: Realization of a BM Definition 2.2. A stochastic process X : Ω R + R is of finite quadratic variation if there exists an increasing process [X, X] such that P lim X ti X ti 1 2 = [X, X]. Π Π Remark: ˆ P lim n X n = X means: X n converges to X in probability ˆ [X, X] is called quadratic variation of X Observe that a Brownian motion W satisfies E Π W ti W ti 1 2 = t. Indeed, W is of finite quadratic variation: Theorem 2.3. P lim Π Π W t i W ti 1 2 = t. Sketch of the proof. Let Π : = t < t 1 < < t n = t with t i = i nt. By the CLT, for large n the random variable ( Wti W ti 1 2 n) t N := n i=1 n std( Wti W ti 1 2 ) is approximately N (, 1)-distributed. This means that n ( Wti W 2) ti 1 = t + n std( W ti W 2 ti 1 )N i=1 = t + n 2 t n N = t + 2 t n N, (1) 2

3 where in the second equality we use that a N (, σ 2 )-distributed r.v. X satisfies var(x 2 ) = 2σ 4. Thus, for n large we have n W ti W ti 1 2 t. i=1 Equation (1) indicates that the standard deviation of the sum of squared increments should decrease by factor 2 if we increase the number of time points by factor 4. We verify this with by simulation: Simulate 5 paths, and let s n denote the empirical standard deviation of Q n = n i=1 W t i W ti 1 2. Then n s n s n /s n Here is the source code of a Matlab program that estimates the quadratic variation: npaths = 5; msteps = 1; t = 1 ; dt = t / msteps ; x = s q r t ( dt )* randn ( npaths, msteps ) ; qv = sum( x. ˆ 2, 2 ) ; estd=std ( qv ) ; h i s t ( qv, 5 ) ; t i t l e ( s t r c a t ( sum o f quadratic increments s t. dev :,... num2str ( estd ), \ newline number o f time s t e p s =,... num2str ( msteps ), \ newline number o f s i m u l a t i o n s =,... num2str ( npaths ) ) ) ; Recall that the trajectories of a Brownian motion R + t W t (ω) are nowhere differentiable, for almost all ω Ω. One can show that the quadratic variation of a differentiable function f : [, t] R is zero. Since the quadratic variation of the Brownian motion is non-zero, the paths can not be differentiable. Finally recall that a Brownian motion is a martingale. We will prove this later. Before we collect some definitions from martingale theory. 3

4 2.2 Martingales We first recall the definition of conditional expectations. Let X : Ω R be an integrable random variable on a probability space (Ω, F, P ), and let G be a σ-field such that G F. Lemma 2.4. There exists a unique r.v. Y such that Y is G-measurable, E[ZX] = E[ZY ] for any G-measurable bounded r.v. Z. Definition 2.5. The random variable Y of Lemma 2.4 is referred to as the conditional expectation of X relative to G, and is usually denoted by E[X G]. An alternative definition. Suppose that X : Ω R is a square-integrable random variable, i.e. E(X 2 ) <. In this case the conditional expectation E[X G] is the unique G-measurable r.v. that minimizes the mean square error, i.e. E[(X E[X G]) 2 ] = min E[(X Z is G measurable Z)2 ]. In other words, E[X G] is the G-measurable r.v. that is closest to X in a least squares sense; one also says that E[X G] is the L 2 projection of X onto the set of G-measurable r.v. s. Lemma 2.6 (Properties of conditional expectations). Let X and Y be integrable random variables. Then we have ˆ Linearity: for any α, β R E[αX + βy G] = αe[x G] + βe[y G] ˆ Monotonicity: If X Y, then E[X G] E[Y G], P -a.s. ˆ E[E[X G]] = E[X] ˆ E[X G] = X if X is G-measurable ˆ Tower property: if H G, then E[E[X G] H] = E[X H] (the coarser information prevails). ˆ E[X G] = E[X] if X is independent of G Proof. Stochastics lecture. 4

5 if A family of σ-fields (F t ), indexed by R + (think of time), is called a filtration F s F t whenever s < t. A stochastic process X : Ω R + R is said to be adapted to the filtration (F t ) if for any t we have that X t is F t measurable. In the remainder we will only work with the filtration generated by a Brownian motion W. The filtration generated by W is essentially the smallest filtration (F t ) such that for all t W t is F t measurable. Mathematical definition: F t = σ(w s : s t) P -null sets in F. Definition 2.7. Let (F t ) be a filtration. A stochastic process M : Ω R + R is called martingale wrt (F t ) if ˆ M is adapted to (F t ), ˆ M t is integrable for all t, ˆ E[M t F s ] = M s, for all s t. Theorem 2.8. A Brownian motion W is a martingale wrt the filtration generated by W. Proof. Let s t. Then E[W t F s ] = E[(W t W s ) + W s F s ] = E[W t W s F s ] + E[W s F s ] = E[W t W s ] + W s = W s. 2.3 The Ito integral Let (W t ) be a Brownian motion and T R +. Aim: We want to give meaning to integrals of the form T where is adapted to the filtration generated by W. (t)dw t, (2) 5

6 For differentiable functions g : R + R one usually defines T (t)dg t := T (t)g (t)dt. Since the paths of W are not differentiable we have to choose another way to define (2). We first define the integral for so-called buy-and-hold strategies. Definition 2.9. An adapted process is called buy-and-hold strategy if there exists a finite partition Π : = t < t 1 < < t n = T of [, T ] such that is constant on each subinterval [t j, t j+1 [, j n 1. Interpretation: (t) = # asset shares in a portfolio at time t. The portfolio is only adjusted at trading times t, t 1,..., t n. Adaptedness implies that the strategy is non-anticipating: (t) depends only on the information available at time t. In mathematics buy-and-hold strategies are called simple integrands. (t) t 1 t 2 t 3 t 4 t 5 t What are the gains from a buy-and-hold strategy? Suppose the price of an asset evolves according to a Brownian motion W. Let t [, T ] and k such that t k t t k+1. Then the gain up to t is given by k 1 I t = (t k )(W t W tk ) + (t j )(W tj+1 W tj ). (3) The process I t is called simple Ito integral process of wrt W. Notation: I t = t (s)dw s. j= Lemma 2.1. The simple integral process (3) is a martingale. Lemma 2.11 (Ito isometry). For all t [, T ] we have E[I 2 t ] = E t 2 (u)du. 6

7 Lemma The quadratic variation [I, I] of the simple integral I t = t (s)dw s satisfies [I, I] t = t 2 (u)du. Next: Ito integral for continuously varying trading strategies. Definition By a general integrand we mean any process (t) such that (1) is adapted, (2) is square-integrable, i.e. E T 2 (t)dt <, (3) there exists a sequence of simple integrands n such that lim n E T ( n (t) (t)) 2 dt =. Remark Assumption (3) is not very restrictive. Let be a general integrand, and ( n ) an approximating sequence of simple integrands. Then t n(s)dw s is already defined for all n, and the Ito isometry for simple Ito integrals implies ( ) 2 T T lim E ( n (t) m (t))dw t = lim E ( n (t) m (t)) 2 dt n,m n,m =. Consequently, T n(t)dw t is a Cauchy sequence in L 2 (Ω), and possesses a limit denoted by T (t)dw t. (4) Definition: The random variable in (4) is called Ito integral of wrt W. For any time t T we can analogously define the Ito integral t (s)dw s. Remark We can further generalize the Ito integral. Indeed, it can be defined for any adapted process satisfying T 2 (s)ds <, P -a.s. The Ito integral process I t = t (s)dw s has the following properties. ˆ Continuity: there exists a version of the Ito integral process I t such that the paths t I t (ω) are continuous for all ω. ˆ Adaptivity: I t is F t -measurable, ˆ Linearity: if J t = t Γ(s)dW s is another Ito integral, then I t + J t = t ( (s) + Γ(s))dW s, 7

8 ˆ Martingale property: I t is a martingale, ˆ Ito isometry: E(I 2 t ) = E t 2 (s)ds, ˆ Quadratic variation: [I, I] t = t 2 (s)ds. We next recall the Ito formula. Theorem Let f : R + R R be once continuously differentiable in t, and twice continuously differentiable in x. Then T f(t, W T ) = f(, W ) T f xx (t, W t )dt. Sketch of the proof. Proof for f(x) = 1 2 x2. T f t (t, W t )dt + f x (t, W t )dw t f(y) f(x) = f (x)(y x) f (x)(y x) 2 = x(y x) (y x)2. Let Π : = t < t 1 < < t n = T be a partition. Then f(w T ) f(w ) = = n 1 (f(w tj+1 ) f(w tj )) j= n 1 W tj (W tj+1 W tj ) + 1 n 1 (W tj+1 W tj ) 2. 2 j= j= Note that n 1 j= W t j (W tj+1 W tj ) is a simple integral. Thus, letting Π, we get f(w T ) f(w ) = General case: use Taylor s formula. T W t dw t T. Exercise 1. Let (W t ) be a Brownian motion and (F t ) the filtration generated by W t. Use Ito s formula to check that the following processes are martingales wrt (F t ): σ2 σwt 1) Geometric Brownian motion: M t = e 2 t, 2) N t = e 1 2 t cos(w t ). 8

9 Definition An Ito process X t is a stochastic process of the form X t = x + t (s)dw s + t where x R, is a general integrand, and Θ satisfies T Properties of Ito processes: Θ(s)ds, (5) Θ(s) ds <, a.s. ˆ Uniqueness: Suppose that X t satisfies (5) and additionally X t = y + t Γ(s)dW s + t β(s)ds, for all t [, T ]. Then x = y, and (a.s.) Γ =, and β = Θ. ˆ Quadratic variation: [X, X] t = t 2 (s)ds. Definition Let X be as in (5). The integral of a process Γ wrt X is defined as t Γ(s)dX s = t Γ(s) (s)dw s + t provided the two integrals on the RHS are well defined. Γ(s)Θ(s)ds, Theorem Let f : R + R R be once continuously differentiable in t, and twice continuously differentiable in x. Then T f(t, X T ) = f(, X ) T T f t (t, X t )dt + f x (t, X t )dx t f xx (t, X t )d[x, X] t. Example 2.2. Let α and σ be two bounded general integrands. Consider the Ito process t t ( ) X t = σ s dw s + α s σ2 s ds. (6) 2 Frequently, we use the differential notation (that has no precise mathematical meaning) ( ) dx t = σ t dw t + α t σ2 t dt. 2 Suppose that the price of an asset is given by S t = S e Xt. 9

10 The Ito formula implies, with f(x) = S e x, ds t = df(x t ) = S e Xt dx t S e Xt d[x, X] t = S t dx t S td[x, X] t ( ) = S t σ t dw t + S t α t σ2 t dt σ2 t S t d t = S t σ t dw t + S t α t dt. We observe that S solves the stochastic differential equation ds t = S t σ t dw t + S t α t dt. Theorem Let X t and Y t be two Ito processes with X t = x + Y t = y + Then the product XY satisfies t t α(s)ds + γ(s)ds + t t β(s)dw s δ(s)dw s. d(xy ) t = X t dy t + Y t dx t + dx t dy t. Question: What does dx t dy t stand for? Heuristics: dx t dy t = (α(t)dt + β(t)dw t )(γ(t)dt + δ(t)dw t ) Calculating with dt and dw t terms: dw t dt dw t dt dt = (α(t)γ(t)dt dt) + (α(t)δ(t)dt dw t ) Using the calculus rules from the table we get +(γ(t)β(t)dt dw t ) + (β(t)δ(t)dw t dw t ) dx t dy t = β(t)δ(t)dt. We can now rewrite the product formula. The product XY satisfies d(xy ) t = X t dy t + Y t dx t + dx t dy t = (X t γ(t) + Y t α(t) + β(t)δ(t))dt + (X t δ(t) + Y t β(t))dw t. 1

11 Sketch of the proof of Theorem Proof for X t = αt + βw t and Y t = γt + δw t, where α, β, γ and δ R. Notice that xy x y = y (x x ) + x (y y ) + (x x )(y y ). (Remark: You can derive this formula also with the Taylor approximation.) Let t i = i nt be the equidistant partition of [, t]. Then X t Y t X Y = n 1 (X ti+1 Y ti+1 X ti Y ti ) i= = i Y ti (X ti+1 X ti ) + X ti (Y ti+1 Y ti ) (7) + i (X ti+1 X ti )(Y ti+1 Y ti ). The first term in (7) satisfies Y ti (X ti+1 X ti ) = i i Notice that and lim n Y ti α(t i+1 t i ) = i P lim n Y ti α(t i+1 t i ) + i t Y s αds, Y ti β(w ti+1 W ti ) = i t Y ti β(w ti+1 W ti ) P a.s., Y s βdw s. Therefore, P lim n i Y t i (X ti+1 X ti ) = t Y sαds + t Y sβdw s = t Y sdx s. Similarly, one can show P lim n i X t i (Y ti+1 Y ti ) = t X sdy s. Next we consider i (X t i+1 X ti )(Y ti+1 Y ti ). Note that (X ti+1 X ti )(Y ti+1 Y ti ) = A(n) + B(n) + C(n) + D(n), where i A(n) = i B(n) = i C(n) = i D(n) = i α(t i+1 t i )γ(t i+1 t i ) α(t i+1 t i )δ(w ti+1 W ti ) γ(t i+1 t i )β(w ti+1 W ti ) β(w ti+1 W ti )δ(w ti+1 W ti ). 11

12 We consider the last four terms separately. We have ( ) 2 t lim A(n) = lim nαγ =. n n n By the Cauchy-Schwarz Inequality, i a ib i ( i a2 i ) 1 2 ( i b2 i ) 1 2, which yields α(t i+1 t i )δ(w ti+1 W ti ) i ( ) 1 ( ) α 2 (t i+1 t i ) 2 δ 2 (W ti+1 W ti ) 2 i Recall that P lim n ( i δ2 (W ti+1 W ti ) 2) 1 2 = δ t. Since i lim n ( i α 2 ( t n )2 ) 1 2 = lim n α 2 t n =, we have P lim n B(n) =. Similarly one can show that P lim n C(n) =. Finally observe that P lim n D(n) = P lim n βδ i (W ti+1 W ti ) 2 = βδt. Since t dx sdy s = βδt, we have shown the result. 2.4 Stochastic Differential Equations Let µ : R + R R and σ : R + R R be measurable functions. An equation of the form dx t = µ(t, X t )dt + β(t, X t )dw t, X = x, (8) is called stochastic differential equation (SDE) with initial condition X = x R. A solution of the SDE (8) is an Ito process such that X t = X + t µ(s, X s )ds + t β(s, X s )dw s. Theorem Suppose that there exists a constant K such that for all t R + and x, y R µ(t, x) µ(t, y) + σ(t, x) σ(t, y) K x y, µ(t, x) + σ(t, x) K(1 + x ). Then there exists a unique solution X of (8). The uniqueness of the solution means that if X and Y are two solutions, then P -a.s, for all t R +, X t = Y t. 12

13 Example 2.23 (Interest rate in the Vasicek model). In the Vasicek interest rate model the interest rate is assumed to satisfy where α, β and σ are positive reals. The solution of (9) is given by dr t = (α βr t )dt + σw t, R = r, (9) Indeed, let R t = e βt r + α t β (1 e βt ) + σe βt e βs dw s. f(t, x) = e βt r + α β (1 e βt ) + σe βt x, and X t = t eβs dw s. By applying Ito s formula for Ito processes to f(t, X t ) one obtains (9). Exercise 2 (Inverse of a geometric Brownian motion). Let dx t = µx t dt + σx t dw t, X = x >. (1) σ2 σwt+(µ Recall that X t = xe 2 )t is the solution of (1). Show that also Y t = 1 X t solves a linear SDE, i.e. an SDE of the form with c 1, c 2 R. dy t = c 1 Y t dt + c 2 Y t dw t, Y = 1 x >. 2.5 Multi-dimensional Ito formula Consider two Ito processes X t and Y t. Theorem Let f : R + R 2 R be once continuously differentiable in t, and twice continuously differentiable in x R 2. Then f(t, X T, Y T ) = f(, X, Y ) + T + + T T f y (t, X t, Y t )dy t T f t (t, X t, Y t )dt + T f xy (t, X t, Y t )dx t dy t f x (t, X t, Y t )dx t f xx (t, X t, Y t )dx t dx t T f yy (t, X t, Y t )dy t dy t Definition A stochastic process W t = (W 1 t, W 2 t,..., W d t ) is called a d- dimensional Brownian motion if for all 1 k d the process W k t is a 1-dim Brownian motion, and 13

14 W 1,..., W d are independent. Lemma For k j we have dw k dw j =. Sketch of the proof. Consider the mean and the variance of (Wt k i+1 Wt k i )(W j t i+1 W j t i ) and let Π. Π Corollary Let f : R + R d R be once continuously differentiable in t, and twice continuously differentiable in x R d. Then T d T f(t, W T ) = f(, W ) + f t (t, W t )dt + f xi (t, W t )dwt i i=1 + 1 d T f xix 2 i (t, W t )dt i=1 Consider two Ito processes driven by a 2-dim Brownian motion X t = x + Y t = y + t t α s ds + δ s ds + t t β s dw 1 s + η s dw 1 s + t t γ s dw 2 s, ζ s dw 2 s. Let f : R + R 2 R be once continuously differentiable in t, and twice continuously differentiable in x R 2. Then f(t, X T, Y T ) f(, X, Y ) T [ = + f t (t, X t, Y t ) + α t f x (t, X t, Y t ) + δ t f y (t, X t, Y t ) (β2 t + γt 2 )f xx (t, X t, Y t ) + + +(β t η t + γ t ζ t )f xy (t, X t, Y t ) + 1 ] 2 (η2 t + ζt 2 )f yy (t, X t, Y t ) dt T T [β t f x (t, X t, Y t ) + η t f y (t, X t, Y t )]dw 1 t [γ t f x (t, X t, Y t ) + ζ t f y (t, X t, Y t )]dw 2 t. 3 The Black Scholes model 3.1 The model setup Throughout this section let (Ω, F, P ) a probability space and W a Brownian motion. Let (F t ) be the filtration generated by W, completed by the P -null sets. 14

15 Consider a financial market with two assets. One asset is risky (think of a stock) and its price satisfies the dynamics ds t = µs t dt + σs t dw t, S = x R, (11) where σ > is the volatility and µ R the drift rate. The second asset is non-risky (think of a zero coupon bond), and its value is described by the ODE ds t = rs t dt, S = 1. (12) St can also be interpreted as the future value of 1 Euro. r is the yearly interest rate with continuous compounding. Note that (11) implies that S t = xe σwt+(µ σ2 /2)t, and (12) implies St = e rt. A trading strategy is a pair of adapted processes (, η) such that the integrals t (u)ds u and t η(u)ds u are defined. Interpretation: ˆ (t) = shares of risky asset in the portfolio at time t ˆ η(t) = bonds in the portfolio at time t The time t value of the portfolio consisting of (t) shares and η(t) bonds is V t = (t)s t + η(t)s t. A portfolio strategy (, η) is called self-financing if dv t = (t)ds t + η(t)ds t, i.e. if the instantaneous changes in the portfolio value are fully determined by the price changes of the two assets. Let V t be the value process associated to a strategy (, η). Then Thus, if (, η) is self-financing, then η(t) = 1 St (V t (t)s t ) dv t = (t)ds t + (V t (t)s t )r dt. (13) One can show that (V t ) solves the SDE (13) iff (, η) is self-financing. Notice that (V t (t)s t )r dt is the interest earning on the cash position. A self-financing portfolio strategy (, η), with associated value process (V t ), is called admissible if there exists c R such that for all t [, T ] V t c, P a.s. (14) In the following we consider only admissible strategies. By imposing Condition (14), we exclude self-financing strategies that are arbitrage opportunities, e.g. doubling strategies. A doubling strategy is best explained in a binomial model with infinite time horizon. 15

16 Example 3.1. Consider the binomial model with up factor u = 2, down factor d = 1 2 and r =. Consider the following strategy: At time invest 1 Euro in the risky asset. If the price increases in the first step, then stop trading. If the price decreases, then buy additional shares for 2 Euros. If the price increases in the second step, then stop trading. If the price decreases, then buy additional shares for 4 Euros. Continue by doubling the Euros invested until the price increases for the first time. The profit of the strategy is 1 Euro, with probability one. Doubling strategies as in Example 3.1 can also be constructed in the Black- Scholes model with finite time horizon. 3.2 Replicating portfolios Consider an option with expiration date T and payoff h(s T ), where h : R + R +. We say that the option is replicable if there exists an admissible portfolio (, η) with associated value process (V t ) such that V T = h(s T ). Remark 3.2. One can actually prove that any option is replicable in the BS model. Theorem 3.3. At time t, the only arbitrage free price of a replicable option is V t, the value of the replicating portfolio at time t. Proof. Suppose that at time t the market price M t is different from V t. Then the market admits arbitrage: ˆ if M t < V t, then buy the option, sell the replicating portfolio. ˆ if M t > V t, then sell the option, buy the replicating portfolio. In both cases you make at risk-free profit of M t V t e r(t t) at time T. Let C = (S T K) + be a call on the risky asset with strike K and expiration date T. Let (V t ) be the value process of a replicating portfolio. Assume that V t is a function only of time and the asset price, i.e. there exists c : [, T ] R + R such that V t = c(t, S t ). (15) Moreover, assume that the function c is once continuously differentiable in t, and twice continuously differentiable in x. 16

17 Remark 3.4. One can actually prove that there is a function c such that (15) holds true, and that c C 1,2 ([, T ) R > ). We next show that V t = c(t, S t ) is an Ito process. From Ito s formula for Ito processes, see Theorem 2.19, we get that the arbitrage free price of the call satisfies d(c(t, S t )) = c x (t, S t )ds t c xx(t, S t )d[s, S] t (16) + c t (t, S t )dt = c x (t, S t )µs t dt + c x (t, S t )σs t dw t c xx(t, S t )S 2 t σ 2 dt + c t (t, S t )dt = c x (t, S t )σs t dw t [ + c x (t, S t )µs t + 1 ] 2 c xx(t, S t )St 2 σ 2 + c t (t, S t ) dt. The replicating self-financing portfolio dynamics are given by dv t = (t)ds t + (V t (t)s t )r dt = (t)σs t dw t + (t)µs t dt + (V t (t)s t )r dt. The arbitrage free price satisfies c(t, S t ) = V t. Thus d(c(t, S t )) = (t)σs t dw t + (t)µs t dt + (c(t, S t ) (t)s t )r dt. (17) We have two representations of c(t, S t ) as an Ito process: (16) and (17). But the representation is unique (see Chapter 2). Matching the coefficients yields and Equation (18) implies c x (t, S t )σs t = (t)σs t (18) c x (t, S t )µs t c xx(t, S t )S 2 t σ 2 + c t (t, S t ) = (t)µs t + (c(t, S t ) (t)s t )r. (19) (t) = c x (t, S t ). Using this in Equation (19) yields (note that the µ terms drop out) 1 2 c xx(t, S t )S 2 t σ 2 + c t (t, S t ) = (c(t, S t ) c x (t, S t )S t )r, which is only satisfied if c(t, x) solves the PDE c t (t, x) + rxc x (t, x) σ2 x 2 c xx (t, x) rc(t, x) =, (2) 17

18 for all t [, T ) and x R >. We assume that the portfolio replicates the call option payoff. Therefore c(t, S T ) = V T = (S T K) +. To sum up, the function c(t, x) has to solve the PDE c t (t, x) + rxc x (t, x) σ2 x 2 c xx (t, x) rc(t, x) =, (21) with terminal condition c(t, x) = (x K) +. (22) Definition. The PDE (63) with terminal condition (22) is called Black-Scholes PDE. We next revert the line of reasoning. So far we have shown: If a self-financing portfolio with value V t = c(t, S t ) replicates the option, then c(t, x) solves the PDE (necessary condition). Question: Does the strategy obtained, (t) = c x (t, S t ) (23) η(t) = e rt (c(t, S t ) c x (t, S t )S t ) (24) indeed replicate the option (sufficient condition)? The answer is yes if we choose a nice solution of (63)! We show later that a solution of the Black-Scholes PDE is given by c(t, x) = xφ(d 1 ) Ke r(t t) Φ(d 2 ). (25) Here Φ denotes the standard normal distribution function Φ(x) = 1 x e y2 2 dy, 2π and We have the following result. d 1 = log ( ) x K + (r + σ 2 2 )(T t) σ, (26) T t d 2 = d 1 σ T t. (27) Proposition 3.5. Let c(t, x) be defined as in (25). Then the pair (, η), defined in (23) and (24), is an admissible strategy replicating the call (S T K) +. As a consequence we obtain that if the underlying is traded at S at time t, then the time t arbitrage free price of a European call with strike K and maturity T is given by BS call(s, K, T t, σ, r) = SΦ(d 1 ) Ke r(t t) Φ(d 2 ), (28) where d 1 and d 2 are defined in (26) and (27). Equation (28) is called Black- Scholes formula. 18

19 3.3 Solving PDEs with the Feynman-Kac formula Question: How can we prove that the function c(t, x) = xφ(d 1 ) Ke r(t t) Φ(d 2 ), (29) (see Eq. (25)) is indeed a solution of the Black-Scholes PDE? ˆ 1. way: Check it directly. ˆ 2. way: Use the so-called Feynman-Kac formula. Why using Feynman-Kac? ˆ The Feynman-Kac formula allows to solve a general class of PDEs ˆ Recipe for deriving pricing formulas for many option types Consider an SDE of the form dx s = µ(s, X s )ds + β(s, X s )dw s, (3) where µ : R + R R and β : R + R R are nice. Notation: given a function h : R R we denote by E t,x [h(x T )] the expectation of h(x T ), where X is the solution of (3) with initial condition X t = x. Whenever we write E t,x [h(x T )], we implicitly assume that the expectation is defined. Theorem 3.6 (Feynman-Kac). Fix T >. Let h : R R be Borel-measurable and suppose that u(t, x) = E t,x (h(x T )) is defined and finite for all (t, x) [, T ] R >. Then u(t, x) solves the PDE with terminal condition u t (t, x) + µ(t, x)u x (t, x) β2 (t, x)u xx (t, x) =, u(t, x) = h(x). We need also a slightly different version, called discounted Feynman-Kac formula. Consider again an SDE of the form Our ultimate recipe will be based on: dx s = µ(s, X s )ds + β(s, X s )dw s. (31) Theorem 3.7 (Discounted Feynman-Kac). Let T >, and h : R R be such that v(t, x) = e r(t t) E t,x (h(x T )) is defined and finite for all (t, x) [, T ] R >. Then v(t, x) solves the PDE v t (t, x) + µ(t, x)v x (t, x) β2 (t, x)v xx (t, x) rv(t, x) =, with terminal condition v(t, x) = h(x). 19

20 3.4 Solving the Black-Scholes PDE We derive the BS formula with Feynman-Kac. Recall the Black-Scholes PDE c t (t, x) + rxc x (t, x) σ2 x 2 c xx (t, x) rc(t, x) =, (32) with terminal condition c(t, x) = (x K) +. By Theorem 3.7 a solution of (32) is given by where The solution of c(t, x) = e r(t t) E t,x [(X T K) + ] dx s = rx s ds + σx s dw s. dx s = rx s ds + σx s dw s with initial condition X t = x is given by σ2 σ(ws Wt)+(r X s = xe 2 )(s t), s t. Notice that X s is lognormally distributed, i.e. log(x s ) is normally distributed with ˆ mean: log(x) + (r σ2 2 )(s t), ˆ variance: σ 2 (s t). Thus, if N is N (, 1)-distributed, then log(x) + (r σ2 2 )(s t) + σ s t N has the same distribution as log(x s ). Therefore, c(t, x) = e r(t t) E t,x [(X T K) + ] = e r(t t) 1 σ2 log(x)+(r [e 2 )(T t)+σ T ty K] + e y2 2 dy. 2π From this we obtain the Black-Scholes formula Proof. c(t, x) = xφ(d 1 ) e r(t t) KΦ(d 2 ). 2

21 3.5 Risk-neutral pricing - part I Question: How is the auxiliary process X t linked to our asset price process S t? Notice that σ2 σwt+(µ ˆ S t = xe 2 )t σ2 σwt+(r ˆ X t = xe 2 )t In differential form: ds t = µs t dt + σs t dw t, dx t = rx t dt + σx t dw t. Note that e rt X t is a martingale. Moreover, X t has the same mean rate of return as the riskless asset St. Indeed, the mean rate of return of X on [, t] is given by ( ) Xt X E = e rt 1, X which is the same as the rate of return of S. We confirm the pricing principle observed already in the binomial model: Modify the drift rate such that the mean rate of return is equal to riskless rate of return. Then the fair option value is the discounted expectation of the option s payoff. Outlook: There exists an equivalent prob. measure Q such that the distribution of S t under Q coincides with the distribution of X t under P. Q is called risk-neutral probability measure, since the discounted price process e rt S t is a martingale with respect to Q. Alternatively, Q is called pricing measure. 3.6 Other options Following the same steps, we can derive the pricing formula for European puts. 1) Assume that a put is replicable and that the put value / replicating portfolio value is equal to p(t, S t ). 2) p(t, S t ) is an Ito process with 2 decompositions: [ dp(t, S t ) = p x (t, S t )σs t dw t + p x (t, S t )µs t + 1 ] 2 p xx(t, S t )St 2 σ 2 + p t (t, S t ) dt, dp(t, S t ) = (t)σs t dw t + (t)µs t dt + (p(t, S t ) (t)s t )rdt. 3) Matching the coefficients: p(t, x) has to solve the PDE p t (t, x) + rxp x (t, x) σ2 x 2 p xx (t, x) rp(t, x) =, with terminal condition p(t, x) = (K x) +. 21

22 4) Solving the PDE with Discounted Feynman-Kac: where p(t, x) = e r(t t) E t,x [(K X T ) + ] dx s = rx s ds + σx s dw s. Since X T is lognormally distributed, this yields that the arbitrage free price of a put is given by BS put(s, K, T t, σ, r) = Ke r(t t) Φ( d 2 ) SΦ( d 1 ). We formulate a general recipe for deriving pricing formulas. 1. Assume that the value function is of the form f(t, S t ) 2. Write f(t, S t ) as an Ito process 1) first by applying Ito s formula to f(t, S t ) 2) second by using the self-financing condition 3. Match the Ito process coefficients and derive a PDE for f(t, x) 4. Solve the PDE with Discounted Feynman-Kac The recipe works for many option types, e.g. for options with underlyings paying dividends and multi-asset options. 4 Including dividend payments in the Black-Scholes model In Section 3 we have assumed that the underlying asset does not pay any dividends. How do the Black-Scholes option pricing formulas change if we allow for dividend payments? We distinguish between two types of dividend payments: ˆ continuously paying dividends ˆ lump payments of dividends 4.1 Continuous dividend payments Consider a stock, or more realistically a basket of stocks, ˆ with value S t at time t, ˆ providing continuous dividend payments at a constant rate q [, 1). 22

23 The latter means that holding Γ shares between t and t + dt entails a dividend yield of ΓqS t dt. Continuously payed dividends have an impact on the ˆ price dynamics: The dividend payments reduce the stock value. The dynamics are given by ds t = µs t dt + σs t dw t qs t dt. ˆ self-financing condition: Let (, η) be a trading strategy, and V t = (t)s t + η(t)st the portfolio value. The portfolio is self-financing if dv t = (t)ds t + η(t)ds t + (t)qs t dt. Consider a European option with payoff h(s T ) at expiration T. We follow the steps of our recipe to derive the arbitrage free pricing formula. 1) Assume that the option is replicable and that the time t fair value / replicating portfolio value is equal to f(t, S t ). 2) Applying Ito s formula to f(t, S t ) implies df(t, S t ) = f t (t, S t )dt + f x (t, S t )(µ q)s t dt + f x (t, S t )σs t dw t The self-financing condition yields f xx(t, S t )S 2 t σ 2 dt. (33) df(t, S t ) = (t)µs t dt + (t)σs t dw t + (f(t, S t ) t S t )rdt. (34) 3) By matching Equations (33) and (34), it must holds that (t) = f x (t, S t ), and f t (t, S t ) + f x (t, S t )(µ q)s t f xx(t, S t )St 2 σ 2 = f x (t, S t )µs t + (f(t, S t ) t S t )r. Thus f(t, x) has to satisfy the PDE f t + (r q)xf x σ2 x 2 f xx r f =, with terminal condition f(t, x) = h(x). 23

24 4) Applying discounted Feynman-Kac we obtain the solution where f(t, x) = e r(t t) E t,x [h(x T )] dx s = (r q)x s ds + σx s dw s. Remark 4.1. Note that the drift term of (X t ) depends on q. The next proposition provides the arbitrage free price of a call option. Proposition 4.2. Consider a European call, with strike K and maturity T, on an asset paying continuously dividends at a constant rate q. The time t arbitrage free price is given by where Proof. BS call(s, K, T t, σ, r, q) = e q(t t) SΦ(d 1 ) Ke r(t t) Φ(d 2 ), d 1 = log ( ) S K + (r q + σ 2 2 )(T t) σ, T t d 2 = d 1 σ T t. Notice that we get the same call option value for ˆ a stock traded at e q(t t) S t, with no dividend payments, ˆ a stock traded at S t, with a continuous dividend yield at a rate of q. Explanation: ˆ Stock price is reduced by the dividend payments. ˆ Lower growth rate is equivalent to starting from a lower stock price. ˆ Dividend payments don t influence the value of a replicating portfolio. 4.2 Dividend payments at finitely many times In this subsection we assume that there are n dividend payments up to expiration T. The dividend payment dates are known in advance, and given by < t 1 < < t n < T. Moreover, a j (, 1) is the dividend as a percentage of the stock price at time t j, i.e. dividend payment at t j = a j S tj. 24

25 The stock price jumps at payment dates: S(t j ) = (1 a j )S(t j ). Between the payment dates the price dynamics satisfy ds t = µs t dt + σs t dw t. Notice that the stock price at time T is given by S T = S { Π n j=1 (1 a j ) } e σw T +(µ σ 2 /2)(T t). Therefore, at time T the stock price coincides with the price of an asset satisfying the SDE d S t = µ S t dt + σ S t dw t, with initial condition S = S Π n j=1 (1 a j). Consider a portfolio with a position of stock at time t j. The dividend payments entail a dividend yield of (t j )a j S(t j ) (35) and the marked-to-market value of the stock position declines by (t j )(S(t j ) S(t j )) = (t j )a j S(t j ). Thus the total portfolio value remains unchanged. Consider a European option with payoff h(s T ) at expiration T. We follow our recipe to derive the pricing formula. For simplicity assume that there is only one dividend payment. Consider first the situation after the dividend payment date t 1. Suppose that the value of a portfolio replicating the option is given by f(t, S t ) for all t [t 1, T ]. As before, we obtain that ˆ f satisfies the Black-Scholes PDE on [t 1, T ] ˆ f(t, x) = e r(t t) E t,x (h(x T )), for t [t 1, T ], where dx t = rx t dt + σx t dw t. What happens at the dividend payment date? Denote by g(t, S t ) the value of the replicating portfolio before t 1. Since the portfolio value remains unchanged at t 1, we have Thus g(t, x) has to satisfy g(t 1, S t1 ) = f(t 1, S t1 ) = f(t 1, (1 a 1 )S t1 ). g(t 1, x) = f(t 1, (1 a 1 )x). 25

26 Before the dividend payment date, on [, t 1 ], we need to replicate f(t 1, (1 a 1 )S t1 ). One can show that g(t, x) satisfies the Black-Scholes PDE with terminal condition g(t 1, x) = f(t 1, (1 a 1 )x). By applying Discounted Feynman-Kac we derive for all t [, t 1 ] where dx t = rx t dt + σx t dw t. Observe that for t t 1 g(t, x) = e r(t1 t) E t,x [f(t 1, (1 a 1 )X t1 )], g(t, x) = e r(t1 t) E t,x [f(t 1, (1 a 1 )X t1 )] = e r(t1 t) E t,(1 a1)x[f(t 1, X t1 )] = e r(t1 t) E t,(1 a1)x[e r(t t1) E t1,x t1 (h(x T ))] = e r(t t) E t,(1 a1)x[h(x T )]. Notice that g(t, x) is also the value of an option on an asset with a price of (1 a 1 )x at time t, no dividend payments up to T. For the price of a call with strike K and expiration date T we get where S (1 a 1 )Φ(d 1 ) Ke r(t t) Φ(d 2 ), d 1 = log ( ) S K + log(1 a1 ) + (r + σ2 2 )(T t) σ, T t d 2 = d 1 σ T t. Notice that the call value coincides with the price of a call on a non-dividend paying asset with current price reduced by the factor (1 a 1 ). It is straightforward to generalize the price formula to more than one payment. The next proposition provides the price of a call if n payments up to expiration are expected. Proposition 4.3. Consider a stock paying dividends of a j S tj at several times t 1 < < t n between t and T. The arbitrage free price of a call option on this stock, with strike K and maturity T, is given by where S Π n j=1(1 a j ) Φ(d 1 ) Ke r(t t) Φ(d 2 ), d 1 = log ( ) S n K + j=1 log(1 a j) + (r + σ2 2 )(T t) σ, T t d 2 = d 1 σ T t. 26

27 Notice that we get the same call option value for ˆ a stock traded at Π n j=1 (1 a j) S, with no dividend payments, ˆ a stock traded at S, with dividend payments a 1,..., a n up to expiration T. 4.3 Conclusion We have the following rule of thumb for pricing European options with dividends, whether we model dividend payments as lump payments or as a continuous dividend payment stream: Reduce the starting price by the dividend payments up to maturity, and then use pricing formulas for non-dividend paying assets. 5 Properties of Plain Vanilla Options & Implied Vola This chapter is partly based on material from J. C. Hull: Options, Futures, and other Derivatives, Prentice-Hall, New York. 5.1 Non-BS-specific properties Factors affecting option prices ˆ current price of the underlying S ˆ volatility σ ˆ time to maturity T ˆ strike price K ˆ interest rate r ˆ dividend payments The following table (from Chapter 9 in [2]) depicts the dependencies of option prices on the model parameters. variable European European American American call put call put current price strike ttm vola int. rate dividends 27

28 The next proposition provides bounds on European call resp. put options. Proposition 5.1. Let C denote the price of a European call on a non-dividend paying asset. Then it must hold ˆ upper bound: C S ˆ lower bound: Proof. Proof of (36): At time, set up two portfolios: Portfolio (A): one call, and e rt K bonds, Portfolio (B): one share of the underlying. portfolio value at value at T (A) C + e rt K max(s T, K) (B) S S T C S Ke rt. (36) Since at T, the value of (A) exceeds the value of (B), it must hold: C + e rt K S. Else the market admits arbitrage. Proposition 5.2. Let P denote the price of a European put on a non-dividend paying asset. Then it must hold ˆ upper bound: P e rt K ˆ lower bound: P e rt K S. (37) Proof. Proof of (37): At time, set up the two portfolios: Portfolio (A): one put, and one share of the underlying, Portfolio (B): e rt K bonds. portfolio value at value at T (A) P + S max(s T, K) (B) e rt K K Since at T, the value of (A) exceeds the value of (B), it must hold: P + S e rt K. Else the market admits arbitrage. We next derive the Put-call parity for non-dividend paying assets. Proposition 5.3. (Put-call parity for non-dividend paying assets) Suppose that a European call option with the strike K and expiration T is traded at a market price of C. Then the only arbitrage free price for a put with the same strike K and expiration T is given by P = C S + e rt K. (38) 28

29 Proof. 1. Suppose first that P > C S + e rt K. Define δ := P (C S + e rt K) >, and set up the following portfolio: asset put call underlying Bonds position e rt K + δ Then, the portfolio value at is zero, and the portfolio value at T is δe rt. Thus the market admits arbitrage. 2. Next suppose that P < C S + e rt K. Define δ := (C S + e rt K) P >, and set up the following portfolio: asset put call underlying Bonds position e rt K + δ Then, the portfolio value at is zero, and the portfolio value at T is δe rt. Again the market admits arbitrage. Therefore P = C S + e rt K is the unique arbitrage free price for the put. By using the Put-call parity, we can derive the BS put formula from the BS call formula (and vice versa). Indeed, BS put(s, K, T t, σ, r) = r(t t) BS call(s, K, T t, σ, r) S + Ke = SΦ(d 1 ) Ke r(t t) r(t t) Φ(d 2 ) S + Ke 5.2 Implied vola = Ke r(t t) (1 Φ(d 2 )) S(1 Φ(d 1 )) = Ke r(t t) Φ( d 2 ) SΦ( d 1 ). For simplicity, from now on we consider only options on non-dividend paying assets. Definition: Implied vola = vola for which the Black-Scholes price coincides with the market price. Let C M be the market price of an actively traded call option. Then σ imp is the vola such that BS call(s, K, T t, σ imp, r) = C M. The next propositions show that the prices of Plain Vanilla Options are strictly monotone increasing in σ. Therefore the implied vola is uniquely defined. Definition 5.4. The derivative of an option value wrt the volatility is called vega and is denoted by V. Proposition 5.5. The vega of a call is given by where ϕ(x) = 1 2π e x2 2. We have V > for t < T. V = Sϕ(d 1 ) T t, (39) 29

30 Proof. Observe that Note that C σ = S Φ(d 1) σ e r(t t) K Φ(d 2) σ = Sϕ(d 1 ) d 1 σ e r(t t) Kϕ(d 2 ) d 2 σ. d 1 σ d 2 σ = d 2 σ = d 1 σ and ϕ(d 2 ) = ϕ(d 1 )e d1σ T t 1 2 σ2 (T t) = ϕ(d 1 ) S K er(t t). From this we get V = Sϕ(d 1 ) T t. Proposition 5.6. The vega of a put is given by V = Sϕ(d 1 ) T t. Remark 5.7. Notice that the vega of a put = vega of the corresponding call. Moreover, V > for t < T. 5.3 How to find the implied vola? Question: Is there an explicit expression for σ imp satisfying BS call(s, K, T t, σ imp, r) = C M? NO! σ imp can only be found numerically. We discuss two methods: ˆ bisection method ˆ Newton s method Let C M be an observed market price of a call with expiration T and strike K Bisection algorithm for finding the implied vola 1. step: Find first an upper bound for the vola: s i g h i g h =. 3 ; b s p r i c e = C a l l e [ S,K,T t, r, s i g h i g h ] ; while (C M > b s p r i c e ) s i g h i g h = s i g h i g h +. 3 ; b s p r i c e = C a l l e [ S,K,T t, r, s i g h i g h ] ; end 3

31 2. step: Bisect the interval [, sig high] until we are closer to σ imp than a given ε >. Start with the lower bound sig low = ; Choose an accuracy level at which you stop the searching algorithm, say ε =.1. Initial distance = Call e[s,k,t-t,r,sig high] - C M. Then use a while loop: while ( d i s t a n c e > accuracy ) s i g a v e r a g e = ( s i g l o w + s i g h i g h ) / 2 ; b s p r i c e = C a l l e [ S,K,T t, r, s i g a v e r a g e ] ; i f b s p r i c e > C M s i g h i g h = s i g a v e r a g e ; e l s e s i g l o w = s i g a v e r a g e ; end d i s t a n c e = abs ( b s p r i c e C M ) ; end Newton s method Newton s method allows to find the roots of a differentiable function f : R R. ˆ Start with an initial guess x. ˆ Follow the tangent of f in x until you hit the x-axis. ˆ The value x 1 where the tangent crosses the x-axis satisfies ˆ Note that x 1 = x f(x) f (x ). f(x ) + f (x )(x 1 x ) =. ˆ Next follow the tangent of f in x 1 until you hit the x-axis. ˆ... Under some nice conditions, the sequence defined via the iteration formula x n+1 = x n f(x n) f (x n ) converges to a root of f. We now apply Newton s method for finding the implied vola. Let f(σ) = C M BS call(s, K, T t, σ, r), and σ n+1 = σ n f(σ n) f (σ n ). Remark: Notice that f is equal to the vega V of the call. 31

32 5.3.3 Bisections versus Newton Pros Cons ˆ Bisection method: very robust works for any kind of option ˆ Newton: very rapid ˆ Bisection method: slower than Newton ˆ Newton: you need to know the derivative of the function Example: Implied vola of Exxon Mobile Corp. Data: NYSE closing prices of Exxon Mobile Corp. on November 8, 213 (source: Implied volas of calls expiring on December 21, 213 strike call price implied vola Other parameters: ttm = 3/252, S = 92.73, r = The Greeks Definition 5.8. The derivative of an option value wrt the price of the underlying is called Delta and is denoted by. Why the Delta is useful: ˆ it measures the option s sensitivity wrt to the underlying s price; ˆ it tells you the # shares of the underlying you have to keep in your replicating portfolio. 32

33 The Delta of a European call is defined by = BS call(s, K, T t, σ, r). S The Delta tells you by how much the call value changes approximately if the underling s price increases by e1. Proposition 5.9. The Delta of a call is given by where Φ(x) = 1 x 2π e y2 2 dy and = Φ(d 1 ), (4) d 1 = log ( ) S K + (r + σ 2 /2)(T t) σ. T t The Delta of a European put is = BS put(s,k,t t,σ,r) S. Proposition 5.1. The Delta of a put is given by where Φ(x) = 1 x 2π e y2 2 dy and = Φ(d 1 ) 1, (41) d 1 = log ( ) S K + (r + σ 2 /2)(T t) σ. T t Recall: = # shares of the underlying in the replicating portfolio. How frequently do you we have to adjust this position? The Gamma tells you, roughly, how often. Definition The derivative of an option s Delta wrt the price of the underlying is called Gamma and is denoted by Γ. Note that the Gamma is the second partial derivative of the option value wrt to the underlying Γ = S = 2 (option value) S 2. Proposition The Gamma of a call is given by where ϕ(x) = 1 2π e x Γ = ϕ(d 1 ) Sσ T t, (42) Proposition The Gamma of a put is equal to the Gamma of the corresponding call, namely 1 Γ = ϕ(d 1 ) Sσ T t. (43) 33

34 Remark If the call resp. put is at the money, you have to readjust the replicating portfolio more frequently. How sensitive is an option value wrt to decreasing time-to- Question: maturity? Definition The derivative of an option wrt time t is called Theta and is denoted by Θ. Note that the Theta is the negative partial derivative wrt timeto-maturity. Θ = (option value) t = (option value). ttm Proposition The Theta of a call is given by Θ = Sσϕ(d 1) 2 T t Kre r(t t) Φ(d 2 ) (44) where ϕ is the density and Φ is the distribution function of the standard normal distribution. Proof. Hint: first show that ϕ(d 2 )Ke r(t t) = ϕ(d 1 )S. Remark Notice that ˆ the Θ of a call is always negative ˆ Θ is highest for ATM calls Proposition The Theta of a put is given by Θ(put) = Sσϕ(d 1) 2 T t + Kre r(t t) Φ( d 2 ). Remark Observe the similarity with the Theta of a call Θ(call) = Sσϕ(d 1) 2 T t Kre r(t t) Φ(d 2 ). The formulas for Greeks of options on dividend paying stock look very similar. You can find a list in Section of [2]. 34

35 5.5 Greeks of a portfolio Consider a portfolio of n different options on a stock. Let q i be the quantity of option i in the portfolio. The Delta of the whole portfolio is the sum of the Deltas of each option position, i.e. (portfolio) = n q i (option i). i=1 Similarly, the Γ, V,... of the portfolio is the sum of the Γs, Vs,... of each option position. Definition 5.2. A portfolio is called delta neutral if its Delta is zero. And a portfolio is called gamma neutral if its Gamma is zero. Remark. ˆ One stock share has a Delta of 1 and a Gamma of. By buying / selling shares you change the portfolio s Delta, but not the Gamma. ˆ How to make a portfolio delta-gamma-neutral: first buy / sell options to make the portfolio gamma neutral, then buy / sell shares of the underlying to make the portfolio delta neutral. Recall the Black-Scholes PDE: v t + rxv x σ2 x 2 v xx rv =. We may rewrite the PDE in terms of the Greeks as follows: If a portfolio is delta neutral, then Θ + rx σ2 x 2 Γ rv =. Θ σ2 x 2 Γ = rv, (45) where v is the portfolio value. Equation (45) shows that in delta neutral portfolios the Θ is closely linked to the Γ. 6 Solving the Black Scholes PDE numerically with finite differences In this chapter we explain how one can solve a PDE numerically with a so-called finite difference scheme. We illustrate the method with the BS PDE. Further reading: Chapter 19 in [2]. 35

36 Recall the BS PDE f t + rsf S + σ2 2 S2 f SS rf =. The main idea is to choose grid points in [, T ] R +, the domain of f, and then to calculate the solution f on the grid points via a backward recursion. Grid on the domain [, T ] R + : ˆ choose S, the length of a price step ˆ choose a maximal price S max, at least 3 times the strike; ˆ let N S N such that N S S = S max ˆ choose t, the length of a time step ˆ let N T N such that N T t = T Notation for the value of f(t, S) at the grid points: f(i, j) = f(i t, j S), i N T, j N S. Discrete terminal condition The terminal values f(n T, ) are determined by the option payoff at expiration, e.g. ˆ for a call: f(n T, j) = (j S K) +, ˆ for a put: f(n T, j) = (K j S) +, where j N S. For the values before expiration we use a backward recursion. 6.1 Implicit FD scheme Black Scholes PDE f t + rsf S + σ2 2 S2 f SS rf =. Approximate the differential quotients as follows: ˆ f t f(i+1,j) f(i,j) t, ˆ f S f(i,j+1) f(i,j 1) 2 S, ˆ f SS f(i,j+1) f(i,j) S f(i,j) f(i,j 1) S S = f(i,j+1) 2f(i,j)+f(i,j 1) ( S) 2. 36

37 Using the difference quotients approximations the Black Scholes PDE becomes f(i + 1, j) f(i, j) t f(i, j + 1) f(i, j 1) + rj S 2 S + σ2 2 j2 2 f(i, j + 1) 2f(i, j) + f(i, j 1) ( S) ( S) 2 rf(i, j) =. Putting f(i + 1, ) to the LHS, and f(i, ) to the RHS yields where f(i + 1, j) = α j f(i, j 1) + β j f(i, j) + γ j f(i, j + 1), (46) α j = 1 2 t ( rj σ 2 j 2) β j = 1 + σ 2 j 2 t + r t γ j = 1 2 t ( rj + σ 2 j 2). We can rewrite the family of Equations (46) as matrix equations. Indeed, let β 1 γ 1 α 2 β 2 γ 2 A =... (47) α NS 2 β NS 2 γ NS 2 α NS 1 β NS 1 Fix i {,..., N T 1}. Then the family of (46), with j {1,..., N S 1}, is equivalent to the matrix equation f(i + 1, 1) α 1 f(i, ) f(i, 1) f(i + 1, 2) A. =.. (48) f(i, N S 1) f(i + 1, N S 2) f(i + 1, N S 1) γ NS 1f(i, N S ) We solve the matrix equations (48) via a backward recursion, starting with i = N T 1. To solve the matrix equation for time step i {,..., N T 1}, we need to know ˆ (Input) f(i + 1, ),..., f(i + 1, N S ), f(i, ) and f(i, N S ), and we get ˆ (Output) f(i, 1),..., f(i, N S 1). In order to solve the matrix equations one need to specify the boundary conditions, i.e. the values of f for i = N T and for j {, N S }. The boundary 37

38 conditions depend on the option considered. We next derive the boundary conditions for a put option (American or European). The boundary condition at expiration for a put option with strike K is given by f(n T, j) = (K j S) +. In order to derive the remaining boundary conditions notice that ˆ if S =, then the American put value K and the European put value e r(t t) K ˆ if S = S max 3K, then the put value Therefore we choose as boundary conditions f(i, ) = K (American put) f(i, ) = e rt K (European put) f(i, S N ) =. Algorithm 6.1. (Implicit scheme for pricing European puts) 1) Define f(i, j) on the boundary 2) Find the (N S 1)-dimensional vector y satisfying A y = where A is defined as in (47). Set f(n T, 1) α 1 f(n T 1, ) f(n T, 2). f(n T, N S 2) f(n T, N S 1) γ NS 1f(N T 1, N S ) f(n T 1, 1). f(n T 1, N S 1) 3) Then find the (N S 1)-dimensional vector y satisfying 4)... A y = f(n T 1, 1) α 1 f(n T 2, ) f(n T 1, 2). f(n T 1, N S 2) f(n T 1, N S 1) γ NS 1f(N T 2, N S ) Algorithm 6.2. (Implicit scheme for pricing American puts) 1) Define f(i, j) on the boundary = y. 38

39 2) Find the (N S 1)-dimensional vector y satisfying f(n T, 1) α 1 f(n T 1, ) f(n T, 2) A y =. f(n T, N S 2) f(n T, N S 1) γ NS 1f(N T 1, N S ) where A is defined as in (47). 3) Compare with the value if the put is exercised immediately: f(n T 1, j) = max(y(j), (K j S) + ) 4) Then find the (N S 1)-dimensional vector y satisfying f(n T 1, 1) α 1 f(n T 2, ) f(n T 1, 2) A y =. f(n T 1, N S 2) f(n T 1, N S 1) γ NS 1f(N T 2, N S ) 5) Compare with the value if the put is exercised immediately: 6)... f(n T 2, j) = max(y(j), (K j S) + ) Remark 6.3. The numerical method is called implicit FD scheme because one has to solve equations of the form A y = b, (49) with matrix A and vector b given. To solve (49) with MATLAB simply write y = A\b. 6.2 Explicit FD scheme Black Scholes PDE f t + rsf S + σ2 2 S2 f SS rf =. Approximate the differential quotients as follows: ˆ f t f(i+1,j) f(i,j) t. ˆ f S f(i+1,j+1) f(i+1,j 1) 2 S ˆ f SS f(i+1,j+1) 2f(i+1,j)+f(i+1,j 1) ( S) 2 39

40 Replacing the derivatives in the BS PDE with the finite difference quotients yields f(i + 1, j) f(i, j) t f(i + 1, j + 1) f(i + 1, j 1) + rj S 2 S + σ2 2 j2 2 f(i + 1, j + 1) 2f(i + 1, j) + f(i + 1, j 1) ( S) ( S) 2 rf(i, j) =. By putting f(i, ) to the LHS, and f(i + 1, ) to the RHS we get f(i, j) = a j f(i + 1, j 1) + b j f(i + 1, j) + c j f(i + 1, j + 1), (5) where a j = b j = c j = ( 1 2 rj t + 1 ) 2 σ2 j 2 t r t r t (1 σ2 j 2 t) r t ( 1 2 rj t σ2 j 2 t ). One can rewrite (5) as a matrix equation with b 1 c 1 a 2 b 2 c 2 M =... a NS 2 b NS 2 c NS 2 a NS 1 b NS 1 (51) Algorithm 6.4. (Explicit scheme) 1) Define f(i, j) on the boundary 2) Compute the (N S 1)-dimensional vector f(n T, 1) y = M. + f(n T, N S 1) 3)... where M is defined as in (51). Set a 1 f(n T, ). c NS 1f(N T, N S ) f(n T 1, 1). f(n T 1, N S 1) = y. 4

41 Remark 6.5. (Explicit vs implicit scheme) ˆ The explicit scheme is stable only if t ( S)2 2. ˆ The implicit scheme is stable for any choice of t and S. ˆ The Crank-Nicholson method combines the implicit and the explicit scheme. 7 Girsanov s theorem and risk neutral pricing Let (W t ) be a Brownian motion on a probability space (Ω, F, P ). We denote by (F t ) the filtration generated by (W t ) and completed by the P -null sets. Let (ϑ t ) be an adapted, bounded process. Notice that the integral t ϑ sdw s is defined. We set ( t M t := exp ϑ s dw s 1 2 t ) ϑ 2 sds. Observe that M t is a martingale and satisfies the SDE dm t = ϑ t M t dw t. We define a new probability measure Q on F T via Q(A) := E [1 A M T ], A F T. Exercise 3. Show that Q is indeed a probability measure. Theorem 7.1 (Girsanov). Let Q be the probability measure defined by Q(A) = E [1 A M T ]. Then W t := W t is a Brownian motion wrt Q. t Remark 7.2. Theorem 7.1 implies that ϑ s ds, t T, ˆ W t W s and W s W u are Q-independent if u < s < t, ˆ W t W s is N (, t s)-distributed under Q. One can use Girsanov s theorem in order to eliminate drifts. To explain this, suppose that we are given a Brownian motion with drift µ R, X t = W t + µt, t T. We are looking for a measure Q such that (X t ) is a Q-Brownian motion. Define Q(A) := E [1 A M T ], with M T = e µw T 1 2 µ2t. 41

42 By Girsanov s theorem X t = W t + µt, t [, T ], is a Q-Brownian motion. Recall from Section 3 that the BS price of a European call on a non-dividend paying stock is given by where BS call(x, K, T, σ, r) = e rt E,x [(X T K) + ], (52) dx t = rx t dt + σx t dw t. In the BS model the underlying s price satisfies the dynamics ds t = µs t dt + σs t dw t. We next aim at finding a probability measure Q such that the process distribution of S t under Q = process distribution of X t under P. We have to find a measure Q with respect to which the process W t + µ r σ t is a Q-Brownian motion. The measure doing the job is Q(A) := E [1 A M T ], where M T = e µ r σ W T 1 (µ r) 2 2 σ 2 T. Then, by Girsanov s theorem, W Q t := W t + µ r σ t is a Q-Brownian motion. Moreover, observe that ds t = µs t dt + σs t dw t = µs t dt + σs t d(w Q t µ r σ t) = rs t dt + σs t dw Q t. We have that the dynamics of (S t ) under Q coincide with the dynamics of (X t ) under P. Same dynamics imply the same distribution: Proposition 7.3. Let X = S. Then the distribution of the process (S t ) under Q coincides with the distribution of (X t ) under P. Proof. One can appeal to a general result on solutions of SDEs guaranteeing that the distribution of (S t ) under Q is equal to the distribution of (X t ) under P. We can check the result also directly. Notice that S t = S e σw Q t +(r σ2 2 )t, σ2 σwt+(r X t = S e 2 )t. Since any Brownian motion has the same process distribution, we obtain the result. 42

43 Remark 7.4. Notice that the discounted price process e rt S t is a Q-martingale. In the following we refer to the probability measure Q as the equivalent risk neutral measure or martingale measure. Sometimes we also refer to is as the pricing measure. Let us now come back to option prices. The BS price of a call (see (52)) satisfies BS call(x, K, T, σ, r) = e rt E[(X T K) + ] = e rt E Q [(S T K) + ]. Observation: Instead of using an new process X, we can simply change the probability measure for determining the arbitrage free price of the call. Pricing paradigm: Let Q be the risk neutral measure, i.e. the probability measure under which the discounted price of every tradable asset is a Q-martingale. The arbitrage free value of an option written on an asset is the expected discounted payoff wrt Q. As we will see in the next chapters, the pricing rule remains valid if we have ˆ more than one risky asset, ˆ more than one risk-neutral measure. 8 Currency options This chapter is partly based on Chapter 17 in [1]. 8.1 A foreign and domestic currency model Denote by X t the exchange rate of the domestic currency, say e, wrt a foreign currency, say CNY : X t = units of the domestic currency units of the foreign currency We suppose that the exchange rate evolves according to a geometric BM dx t = X t α X dt + X t σ X dw t. Furthermore suppose that we have two interest rates (continuously compouded): ˆ domestic interest rate: r d ˆ foreign interest rate: r f 43

44 The corresponding riskless asset resp. bond prices B d and B f satisfy db d t = r d B d t dt, db f t = r f B f t dt. What is the arbitrage free price of a currency option, i.e. an option of the type h(x T ), with h : R + R? To give an example: if h(x) = (x K) +, then the option is a currency call. Consider a portfolio of domestic and foreign bonds with ˆ η(t) = # domestic bonds at time t, ˆ ξ(t) = # foreign bonds at time t. The e value of the portfolio at time t is given by V t = η(t)b d t + ξ(t)b f t X t. (53) We next derive the self-financing condition of the portfolio. Suppose the bond positions are constant equal to (η, ξ) between t and t +. Then the value of a self-financing portfolio changes by V t+ V t = η(b d t+ B d t ) + ξ(b f t+ X t+ B f t X t ). Letting, implies that a self-financing portfolio satisfies the SDE dv t = η(t)db d t + ξ(t)d(b f X) t. From the product formula for Ito processes we know that d(b f X) t = B f t dx t + X t db f t + db f t dx t = B f t dx t + X t db f t, and therefore the self-financing condition simplifies to From (53) we have dv t = η(t)db d t + ξ(t)b f t dx t + ξ(t)x t db f t. (54) η(t) = (V t ξ(t)b f t X t ) 1 Bt d. Therefore, the self-financing condition can be rewritten as dv t = (V t ξ(t)b f t X t ) 1 Bt d dbt d + ξ(t)b f t dx t + ξ(t)x t db f t = (V t ξ(t)b f t X t )r d dt + ξ(t)b f t dx t + ξ(t)x t B f t r f dt. 44

45 8.2 Currency option prices via PDEs Having derived the self-financing condition, we can use our 4 steps recipe for deriving option price formulas. 1) Assume that the option is replicable and that its value at time t [, T ] is equal to v(t, X t ), where v C 1,2 ([, T ) R > ). 2) v(t, X t ) is an Ito process with 2 decompositions: dv(t, X t ) = (v(t, X t ) ξ(t)b f t X t )r d dt + ξ(t)b f t dx t + ξ(t)x t B f t r f dt dv(t, X t ) = v t (t, X t )dt + v x (t, X t )dx t v xx(t, X t )dx t dx t. 3) Matching the coefficients yields first ξ(t) = v x(t, X t ) B f, t and then that v(t, x) has to solve the PDE v t (t, x) + (r d r f )xv x (t, x) σ2 Xx 2 v xx (t, x) r d v(t, x) =. (55) 4) Theorem 3.7 (discounted Feynman-Kac) shows that a solution of the PDE (55) is given by v(t, x) = e r d(t t) E t,x [h(y T )], (56) where dy t = (r d r f )Y t dt + σ X Y t dw t. A currency call gives the owner the right to buy a unit of a foreign currency at a price of K e. The call s payoff (in e ) is given by (X T K) +. Consequently, the arbitrage free price of a currency call is given by GK call(x, K, T t, σ X, r d, r f ) = e r d(t t) E t,x [(Y T K) + ]. (57) Proposition 8.1. (Garman-Kohlhagen formula) The price of a currency call with strike K e satisfies GK call(x, K, T t, σ X, r d, r f ) = e r f (T t) xφ(d 1 ) e r d(t t) KΦ(d 2 ), where Proof. This follows from (57). d 1 = log ( x K ) + (rd r f + σ2 X 2 )(T t) σ X T t, d 2 = d 1 σ X T t. 45

46 Remark 8.2. The arbitrage free price of a currency put option is given by GK put = e r d(t t) KΦ( d 2 ) e r f (T t) xφ( d 1 ). Notice that a foreign currency option has the same value as an option on a stock with price X t and paying continuously dividends at a rate of r f. If r f increases, then the future value of one foreign currency unit decreases, and hence the call option value decreases, too. 8.3 Risk neutral pricing approach We next discuss the risk neutral approach for deriving currency option prices. We start with the following observation. Lemma 8.3 (cf. Lemma 17.1 in [1]). The possibility of buying a foreign currency, and investing it at an interest of r f, is equivalent to the possibility of investing in a domestic asset with price process The dynamics of B f satisfy Proof. B f t = X t B f t. d B f t = B f t (α X + r f )dt + B f t σ X dw t. According to the risk neutral pricing paradigma, the arbitrage free price of an option is its expected discouted payoff under the risk neutral measure. Lemma 8.4. The risk neutral probability measure Q is given by [ ( Q(A) = E 1 A exp α X (r d r f ) σ X W T (α X (r d r f )) 2 2σ 2 X )] T. The process W Q t = W t + α X (r d r f ) σ X t is a Q-Brownian motion, and the dynamics of B f satisfy d B f t = r d Bf t dt + B f t σ X dw Q t. Proof. Corollary 8.5. The arbitrage free price of a currency call with strike K and expiration date T is given by GK call(x, K, T t, σ X, r d, r f ) = e r d(t t) E Q t,x[(x T K) + ]. (58) 46

47 We next show that the two formulas (57) and (58) are equivalent. First recall that with Y = x. Moreover X satisfies dy t = (r d r f )Y t dt + σ X Y t dw t, (59) dx t = α X X t dt + σ X X t dw t, = (r d r f )X t dt + σ X X t dw Q t, X = x. (6) The dynamics of (X t ) under Q coincide with the dynamics of (Y t ) under P. Consequently, the distribution of (X t ) under Q coincides with the distribution of (Y t ) under P. Therefore, the formulas (57) and (58) are consistent. Notice that Lemma 8.4 implies that B f t = B f eσ X W Q t +(r d+σ 2 X /2)t. In particular, e r dt Bf t is a Q-martingale. We can thus refine the pricing paradigm: All domestic assets, discounted by B d t, have to be Q-martingales wrt the risk-neutral measure (= pricing measure) Q. Proposition 8.6. (Put-call parity for currency options with strike K and expiration date T ) We have Proof. GK put = GK call e r f (T t) x + e r d(t t) K. In the remainder of this chapter we aim at deriving pricing formulas for so-called quanto options. To this end we recall the some results on correlated Brownian motions. 8.4 Correlated Brownian motions and Lévy s theorem Definition 8.7. Let V t and W t be two Brownian motions. We say that the correlation between V and W is ρ [ 1, 1] if for all t cov(v t, W t ) std(v t )std(w t ) = ρ. One can easily construct correlated Brownian motions by using a 2 independent Brownian motions. Let ρ [ 1, 1] and (W 1, W 2 ) be two independent Brownian motions. Then V t = ρw 1 t + 1 ρ 2 W 2 t a Brownian motion and the correlation between V and W 1 is ρ. The covariation between two Ito processes X t and Y t satisfies, provided it exists, t dx s dy s = P lim (X ti X ti 1 )(Y ti Y ti 1 ), Π where Π is the mesh size of a finite partition Π of [, t]. Π 47

48 Lemma 8.8. Let V and W be two Brownian motions. The correlation between V and W is equal to ρ [ 1, 1] iff for all t. t dv s dw s = ρ t, Proof. Theorem 8.9. (Lévy s theorem) Let M t, t, be a continuous martingale starting in zero, i.e. M =. If [M, M] t = t for all t, then M t is a Brownian motion. Exercise 4. Use Lévy s theorem to show that ρwt ρ 2 Wt 2 motion. is a Brownian 8.5 Quanto options In this section we derive prices for the following quanto options: foreign call, struck in foreign currency foreign call, struck in domestic currency domestic call, struck in foreign currency We first study a foreign call struck in foreign currency. following assumptions: We make the ˆ domestic currency: Euro ˆ foreign currency: $ ˆ domestic interest rate (continuously compouded): r d ˆ foreign interest rate (continuously compouded): r f ˆ exchange rate e / $ satifies dx t = X t α X dt + X t σ X dw X t. Suppose that the price in $ of a US stock is ds f t = S f µ f dt + S f σ f dw f t. What is the value, in Euros, of a call on the stock S f with expiration date T and strike K $? 48

49 Proposition 8.1. The e value at time t of a foreign call, struck in foreign currency, is given by C(x, s f, K, T t, σ f, r f ) = xs f Φ(d 1 ) xe r f (T t) KΦ(d 2 ), where x is the e / $ exchange rate at time t, and d 1 = log ( s f K d 2 = d 1 σ f T t. ) + (rf + σ2 f 2 )(T t), σ f T t Proof. By the Black-Scholes formula, the $ value of the call (S f T K)+ is given by c f (t, s f ) = s f Φ(d 1 ) e r f (T t) KΦ(d 2 ). By multiplying the $ value with the exchange rate x we get the Euro value. Next we consider a foreign call struck in domestic currency. More precisely, we aim at finding the arbitrage free price, in e, of a call on S f with strike K e, i.e. of a European option to buy the foreign stock at time T, by paying e K. Payoff of the call = (X T S f T K)+ Again let the e / $ exchange rate satisfy dx t = X t α X dt + X t σ X dw X t, (in e). and suppose that the correlation between W X and W f is ρ. The value of the foreign stock in domestic units is given by S f t = X t S f t. The product formula for Ito processes implies d S f t = S f t (µ f + α X + ρσ f σ X )dt + S f t (σ f dw f t + σdw X t ) = S f t (µ f + α X + ρσ f σ X )dt + σ S f t dz t, ( σf 2 + σ2 X + 2ρσ f σ X and Z = 1 σ σ X Wt W ). Notice that Z where σ = + σ f W f t is a Brownian motion (Levy s theorem). Think of S f as a new domestic asset. The risk-neutral dynamics of S f are given by d S f t = S f t r d dt + S f σdb t, (61) where B = Z + α+µ f +ρσ X σ f r d σ is a BM with respect to the measure Q with density ( dq dp = exp µ f + α X + ρσ X σ f r Z T (µ f + α X + ρσ X σ f r) 2 ) σ 2 σ 2 T. 49

50 Proposition The e value of a foreign call, struck in domestic currency, is given by C( s f, K, T t, σ, r d ) = s f Φ(d 1 ) e r d(t t) KΦ(d 2 ), where σ = d 1 = σf 2 + σ2 X + 2ρσ f σ X, ( ) sf ln K + (r d + σ2 2 )(T t) σ, T t d 2 = d 1 σ T t. Proof. According to the risk neutral pricing paradigma, the arbitrage free price of the call is given by C( s f, K, T t, σ, r d ) = e r d(t t) E Q t, s f [( S f T K)+ ]. The result follows now by a straightforward calculation. Finally, we turn to a domestic call struck in foreign currency. Suppose that a domestic stock has the price dynamics ds d t = S d t µ d dt + S d t σ d dw d t. We consider a call on S d with strike K $, i.e. a European option to buy the domestic stock at time T, by paying $ K. Payoff of the call = (S d T X T K) + (in e). Proposition The e value of a domestic call, struck in foreign currency, is given by C(x, s d, K, T t, σ Y, r f ) = s d Φ(d 1 ) xe r f (T t) KΦ(d 2 ), with d 1 = ln ( ) sd /x K + (r f + σ2 Y 2 )(T t) σ Y T t d 2 = d 1 σ Y T t where σ Y = σ 2 d + σ2 X 2ρσ dσ X and ρ is the correlation between W X and W d. Proof. 5

51 9 Barrier Options This chapter is based on Section 7.3 in [4]. Definition 9.1. A barrier option is an option where the right to exercise depends on whether the underlying crosses a certain barrier level before expiration. One distinguishes two types: ˆ knock-out options: the right to exercise is lost if the barrier is crossed. The option becomes worthless. ˆ knock-in options: the right to exercise is obtained if the barrier is crossed. Remark 9.2. Barrier options have smaller premiums, i.e. they are cheaper. Example 9.3. A down and out (D&O) call is European call that is knocked out (you also say deactivated ) if the underlying crosses a barrier L before expiration. L is smaller than the present asset value S. The payoff of the D&O call is given by C D&O T = { (ST K) +, if S t > L for all t T,, if S t L for at least one t T. We categorize barrier options with respect to the following items: ˆ standard option if active: call or put ˆ barrier level in relation to current asset price: down or up ˆ knock-in or knock-out Consequently there are 2 3 = 8 standard types of barrier options. Call Put Up Down Up Down In Out In Out In Out In Out By following the first 3 steps of our recipe one can derive pricing PDEs for barrier options. We will do so first for knock out calls. 9.1 Knock out calls We first introduce some notation. As before let S t, t, be the underlying s price process. For simplicity we make the assumptions of the Black Scholes model. Besides we denote by m t = min u t S u the minimum price between time and t. The payoff of the D&O call satisfies C D&O T = (S T K) + 1 {mt >L}. Notice that the value of the D&O call at time t depends on S t and m t. However, under the assumption that there has been no knock out prior to t, the value depends only on S t! D&O call: Rolling the steps 51

52 1) Assume that the D&O call is replicable. Denote by v(t, x) the time t option value / replicating portfolio value under the assumption that the barrier has not been attained before t and that S t = x. 2) v(t, S t ) is an Ito process. Ito s formula implies [ dv(t, S t ) = v x (t, S t )σs t dw t + v x (t, S t )µs t + 1 ] 2 v xx(t, S t )St 2 σ 2 + v t (t, S t ) dt. The self-financing condition yields dv(t, S t ) = (t)ds t + (v(t, S t ) (t)s t )rdt = (t)σs t dw t + (t)µs t dt + (v(t, S t ) (t)s t )rdt. [ dv(t, S t ) = v x (t, S t )σs t dw t + v x (t, S t )µs t + 1 ] 2 v xx(t, S t )St 2 σ 2 + v t (t, S t ) dt, dv(t, S t ) = (t)σs t dw t + (t)µs t dt + (v(t, S t ) (t)s t )rdt. 3) Matching the coefficients yields (t) = v x (t, S t ) and that v(t, x) has to satisfy the PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) =. We next determine the boundary conditions. Note that at the knock out the D&O call is worthless, i.e. v(t, L) =, t T. If there is no knock-out prior to expiration, then (S T K) +. Thus v(t, x) = (x K) +, x > L. One can solve the PDE numerically. The boundary conditions for solving the PDE with a finite difference scheme: v(t, L) =, t T, v(t, x) = (x K) +, x > L, v(t, S max ) S max e r(t t) K, t T. The next theorem summarizes our findings. Theorem 9.4. (See Thm in [4].) Let v(t, x) be the D&O call value at time t under the assumption that the call has not been knocked out before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) =, (t, x) [, T ) [L, ), with boundary conditions v(t, L) =, t T, v(t, x) = (x K) +, x > L. 52

53 We next derive the pricing PDE for an up and out (U&O) call. Let U > K be an upper barrier level. We denote by M t = max u t S u maximum of the price between time and t. The payoff of the U&O call satifies C U&O T = (S T K) + 1 {MT <U}. A derivation similar to the one of the previous section entails the following result for U&O calls. Theorem 9.5 (see Thm in [4]). Let v(t, x) be the time t U&O call value under the assumption that it has not been knocked out before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) =, (t, x) [, T ) [, U], with boundary conditions v(t, U) =, t T, v(t, ) =, t T, v(t, x) = (x K) +, x < U. Remark 9.6. Notice that v(t, x) is not continuous in (T, U) if U > K! One can derive explicit formulas for the price of knock out calls within the BS model. We next explain how to do so for the U&O call. By the risk neutral pricing paradigma (or by applying the discouted Feynman Kac formula up to the knock out time) we get that the value of the U&O call at time t = is given by C U&O = e rt E Q [(S T K) + 1 {MT <U}], where Q is the risk neutral measure. The joint Q-distribution of (M T, S T ) is known. Therefore the value of the U&O call can be calculated explicitly. Recall that S satisfies the dynamics ds t = rs t dt + σs t dw Q t, where W Q is a Brownian motion under the risk neutral measure Q. Let α = 1 σ2 σ (r 2 ) and Ŵt = αt + W Q t. Note that S T = S e σw Q T = S e σŵt, +(r σ2 2 )T and S T K if and only if ŴT k := 1 σ log ( K S ). 53

54 Then Now let M T denote the maximum M T = max Ŵ u. u T max S u = max S e σŵu = S e σ M T, u T u T and the barrier U is hit iff M T b := 1 σ log ( U S ). The time t = value of an U&O call is thus given by C U&O = e rt E Q [(S e σŵt K) + 1 { MT <b} ] = e rt E Q [(S e σŵt K) 1 { MT ]. (62) <b,ŵt k} The next proposition describes the joint distribution of the BM with drift Ŵ and its running maximum M. Proposition 9.7 (see Thm in Shreve). The density of the joint distribution of ( M T, ŴT ) under Q is given by f(m, w) = 2(2m w) T 2πT eαw 1 2 α2 T 1 2T (2m w)2, w m, m, and f(m, w) = for other values of m and w. With the proposition we can derive explicit pricing formula for U&O calls. From (62) we obtain C U&O = e rt b b (S e σw 2(2m w) K) k w + T 2πT eαw 1 With some tedious calculations one can show that ( C U&O = S [Φ (δ + T, S )) ( Φ (δ + T, S )) ] K U [ ( e rt K Φ (δ T, S )) ( Φ (δ T, S where U ( ) 2r [ S σ 2 Φ U +e rt K K ( (δ + T, ( ) 1 2r [ S σ 2 Φ U U 2 KS ( (δ T, )) Φ U 2 KS δ ± (T, s) = 1 [ σ log s + (r ± 12 ) ] T σ2 T. For a proof we refer to Shreve [4], Chapter 7. 2 α2 T 1 2T (2m w)2 dmdw. ))] U ))] (δ + (T, US )) ))] Φ (δ (T, US, 54

55 9.2 Knock in calls As before let L be a lower barrier, U an upper barrier, M t = max u t S u and m t = min u t S u. The payoff of an up and in (U&I) call with strike K is given by C U&I T = (S T K) + 1 {MT U}, and the payoff of a down and in (D&I) call, with strike K, by C D&I T = (S T K) + 1 {mt L}. Recall that BS call(s, K, T t, σ, r) denotes the price of a Plain Vanilla Call at time t, where S is the current price of the underlying,... Theorem 9.8. Let v(t, x) be the time t U&I call value under the assumption that it has not been activated before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) =, (t, x) [, T ) [, U], (63) with boundary conditions v(t, U) = BS call(u, K, T t, σ, r) t T, v(t, ) = t T, v(t, x) = x < U. Theorem 9.9. Let v(t, x) be the time t D&I call value under the assumption that it has not been activated before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE (63) with boundary conditions 9.3 In-Out Parity v(t, L) = BS call(l, K, T t, σ, r) t T, v(t, x) = x > L. Let C T = (S T K) + be the payoff of a Plain Vanilla Call. And let CT U&O and CT U&I be the payoff of barrier calls with strike K and barrier U > S. Notice that C T = C U&O T + C U&I T. Thus the price of the U&I call at time satisfies C U&I = C C U&O. Similarly, we have an in-out parity for down options: C D&I = C C D&O. 55

56 9.4 Double barrier options The payoff of a double knock out call, with lower barrier L and upper barrier U, is given by DKOC = (S T K) + 1 {min t T S t>l, max t T S t<u}. The correponding put has the payoff DKOP = (K S T ) + 1 {min t T S t>l, max t T S t<u}. Theorem 9.1. Let v(t, x) be the time t DKOC value under the assumption that it has not been knocked out before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 x 2 v xx (t, x) rv(t, x) =, (t, x) [, T ) [L, U], with boundary conditions v(t, L) =, t T, v(t, U) =, t T, 9.5 Parisian barrier options v(t, x) = (x K) +, L < x < U. For standard barrier options the option trigger only depends on a single crossing of the barrier by the underlying price process. The counterparty may manipulate the underlying for a short time such that the barrier option is knocked-out resp. knocked in. So-called Parisian barrier options require the knock-out / knock-in condition to be satisfied for a certain time, and thus prevent the counterparty to influence prices. There are 2 types of Parisian barrier options: ˆ Standard Parisian barrier option: option is knocked out if the underlying asset value stays consecutively below the barrier for a time longer than some pre specified time window d before the maturity date. ˆ Cumulative Parisian barrier option: option is knocked out if the underlying asset value spends until maturity in total d units of time below the barrier. For pricing of Parisian barrier options one usually uses Monte-Carlo methods (see next Chapter). 56

57 1 Asian Options and Monte Carlo Methods The payoff of an Asian option depends on the average of the underlying asset price process. Example 1.1. The payoff of an Asian call with strike K and expiration date T is given by ( + 1 T S t dt K). (64) T Asian options provide the holder with an insurance against average price changes. Note that the counterparty has no incentive to influence prices at expiration. 1.1 Pricing PDEs We next aim at deriving a pricing PDE for Asian options by following the first 3 steps of our recipe. For simplicity we consider the Asian call from (64) and make the assumptions of the BS model. First observe that the Asian call s value at time t depends not only the price of the underlying, but also of the integral up to time t. Y t = t S u du, 1) Assume that the Asian call is replicable. Denote by v(t, x, y) the time t option value / replicating portfolio value under the assumption that S t = x and Y t = y. 2) v(t, S t, Y t ) is an Ito process. Ito s formula implies dv(t, S t, Y t ) = v t (t, S t, Y t )dt + v x (t, S t, Y t )ds t v xx(t, S t, Y t )d[s, S] t + v y (t, S t, Y t )dy t [ = v t (t, S t, Y t ) + µs t v x (t, S t, Y t ) + 1 ] 2 σ2 St 2 v xx (t, S t, Y t ) + S t v y (t, S t, Y t ) dt +σs t v x (t, S t, Y t )dw t. The self-financing condition yields dv(t, S t, Y t ) = (t)ds t + (v(t, S t, Y t ) (t)s t )rdt = (t)σs t dw t + (t)µs t dt + (v(t, S t, Y t ) (t)s t )rdt. 3) Matching the coefficients yields (t) = v x (t, S t, Y t ) and that v(t, x, y) has to satisfy the PDE v t (t, x, y) + rxv x (t, x, y) σ2 x 2 v xx (t, x, y) + xv y (t, x, y) rv(t, x) =. 57

58 Unfortunately the Feynman-Kac formula does not lead to an explicit solution. This is because the distribution of the average of an exponential Brownian motion is not known. One has to draw on numerical methods for calculating Asian call prices. Theorem 1.2 (see Thm in Shreve). Let v(t, x, y) be the time t Asian call value under the assumption that S t = x and Y t = y. Then v(t, x, y) satisfies the PDE v t (t, x, y) + rxv x (t, x, y) σ2 x 2 v xx (t, x, y) + xv y (t, x, y) rv(t, x) =, t < T, x, y R, and the boundary conditions v(t,, y) = e r(t t) ( y T K)+, t T, lim v(t, x, y) =, x, t T, y v(t, x, y) = ( y T K)+, x, y R. We next show that one can characterize the Asian call s price with a simpler PDE that possesses only one state variable. To this end consider the dynamics of the risky and non-risky asset ds t = rs t dt + σs t dw Q t, S = x R, db t = rb t dt, B = 1, where W Q t is a Brownian motion with respect to the risk neutral measure Q. Let (γ t, η t ) be a trading strategy (see Ch. 3) where γ t = 1 ( 1 e r(t t)). rt Let V t = γ t S t + η t B t be the associated value process. We assume that (γ t, η t ) is self-financing, i.e. dv t = γ t ds t + (V t γ t S t )rdt (65) Besides, assume that the portfolio s initial value is given by V = 1 rt (1 e rt )S e rt K. Proposition 1.3. The value of the portfolio at expiration is given by V T = 1 T T S u du K. (66) Proof. 58

59 Observe that the payoff of the Asian call satisfies ( ) + 1 T S u du K = V + T T. Thus the value of the Asian call at time t is given by A t = e r(t t) E Q [ V + ] T F t. Notice that Z t = e rt S t S new measure as follows: is a martingale with Z = 1. Thus we may define a Girsanov s theorem implies that Q S (A) = E Q [1 A Z T ], A F T. W S t = W Q t σt is a Brownian motion wrt Q S. A lemma from Stochastic Analysis says that for any random variable X we have E Q [X F t ] = E QS [ Z t Z T X F T ] Thus the value of the Asian call satisfies A t = e r(t t) E Q [ V + T Ft ] = e r(t t) E QS [ Zt Z T V + T = S t E QS [ 1 S T V + T ] Ft = S t E QS [ max( V T S T, ) F t Lemma 1.4. The dynamics of Y t = Vt S t Proof. satisfy ] Ft ]. dy t = σ(γ t Y t )dw S t. (67) Notice that Y is a Q S -martingale. Next assume that there exists a function g C 1,2 ([, T ) R) such that [ g(t, Y t ) = E QS max( V T, ) ] F t = E [ QS max(y T, ) ] F t. S T 59

60 An application of Ito s formula yields d(g(t, Y t )) = g t (t, Y t )dt + g y (t, Y t )dy t g yy(t, Y t )d[y, Y ] t = [g t (t, Y t ) σ2 (γ t Y t ) 2 g yy (t, Y t )]dt +d(martingale). Since g(t, Y t ) is a martingale, the dt-term is equal to zero. Therefore g t (t, Y t ) σ2 (γ t Y t ) 2 g yy (t, Y t ) =. In other words, g(t, y) satisfies the PDE g t (t, y) σ2 (γ t y) 2 g yy (t, y) =. (68) Theorem 1.5 (see Thm in Shreve). The time t arbitrage free price of ) + an Asian call with payoff S udu K is given by ( 1 T T A t = S t g ( t, V ) t, S t where g(t, y) solves (68) and satisfies the boundary condition Proof. g(t, y) = max(, y), y R, lim g(t, y) y =, t T, lim [g(t, y) y] y =, t T. 1.2 Pricing standard calls with Monte Carlo To illustrate the Monte Carlo method, we first look at a standard European call. Recall some assumptions and facts of the Black-Scholes model: The risky asset price satisfies the SDE ds t = rs t dt + σs t dw Q t, S = x R, where W Q t is a Brownian motion with respect to the risk neutral measure Q. The value at time of a call option value with strike K and expiration date T is given by C = e rt E Q [(S(T ) K) + ]. Recall that S(T ) is lognormally distributed. I.e. if Z is N (, 1)-distributed, then S(T ) d = S e σ T Z+(r σ2 2 )T. 6

61 Let Z 1, Z 2,... be a sequence of independent and N (, 1)-distributed random variables. Define S i (T ) = S e σ T Z i+(r σ2 2 )T. The law of large numbers implies that 1 lim n n n (S i (T ) K) + = E Q [(S(T ) K) + ], i=1 (Q-a.s.) The MC algorithm for pricing calls can be summarized as follows: For all i = 1,..., n generate an independent sample of Z i calculate S i (T ) = S e σ T Z i+(r σ2 2 )T set C i = (S i (T ) K) + The call value is approximately given by e rt 1 n n i=1 C i. Matlab source code (Call MC.m) S = 5 ; K = 5 ; sigma =. 3 ; T = 1 ; r =. 5 ; nsim = 1; z = randn ( nsim, 1 ) ; S = S*exp ( sigma* s q r t (T)* z + ( r sigma ˆ2/2)*T) ; C = max(, S K) ; c a l l = exp( r*t)*sum(c)/ nsim 1.3 General MC algorithm Here is the MC algorithm for approximating the expectation E[X], where X is a square-integrable random variable: generate independent samples X i, distributed as X calculate the average 1 n n i=1 X i By the central limit theorem, for large n we have n i=1 (X i E[X]) n std(x) N (, 1). 61

62 In other words, the approximation error satisfies ( ) 1 n X i E[X] N (, var(x) ) n n i=1 Remark: The average approximation error is the lower ˆ the higher the number of simulations, ˆ the lower the st. deviation of the underlying random variable. Confidence intervals of the MC approximation Let µ n = 1 n n i=1 X i. The strong law of large numbers guarantees Denote by lim µ n = E[X]. n ŝ n = 1 n (X i µ n) 2 n 1 i=1 the sample standard deviation. Moreover, let u(x 1,..., X n ) = 1.96 ŝn n + µ n and v(x 1,..., X n ) = 1.96 ŝn n + µ n. Then [u(x 1,..., X n ), v(x 1,..., X n )] is the 95% confidence interval for E[X], i.e. Proof of (69): P (u(x 1,..., X n ) E(X) v(x 1,..., X n )) 95% (69) 1.4 Pricing Asian options with MC Recall: the payoff of an Asian call is given by ( + 1 T S t dt K). T We next explain how to price the option with the Monte-Carlo method. We first need to generate independent samples of T S tdt. Since the distribution of T S tdt is not explicitly known, we have to simulate the whole path t S t. To this end ˆ let t i = i nt be an equi-distant partition of [, T ] ˆ = T n step length 62

63 ˆ let Z 1,..., Z n be independent and N (, 1)-distributed Random walk construction of a Brownian motion: Set W = and W ti = W ti 1 + Z i, for all 1 i n. Then the discrete sequence W, W t1,..., W tn exactly coincides with a Brownian motion at the points t, t 1,..., t n. A geometric Brownian motion with drift r is given by σ2 σwt+(r S t = S e 2 )t. (7) If W, W t1,..., W tn is a discrete approximation of the Brownian motion W, then S ti = S e σwt σ2 +(r i 2 )ti is a discrete approximation of the GBM with drift (7). Matlab source code for generating price paths (GBM pathsgenerator.m) T = 1 ; sigma =. 3 ; r =. 5 ; S = 5 ; s t e p s = 256; dt = T/ s t e p s ; nsim = 1; % Simulation o f nsim Brownian paths z = randn ( nsim, s t e p s ) ; w = s q r t ( dt )*cumsum( z, 2 ) ; w = [ z e r o s ( nsim, 1 ), w ] ;... For pricing Asian calls with MC use the discrete-time approximation of the continuous-time average 1 T S t dt 1 n S ti. T n + 1 MC algorithm with N samples: 1) For all 1 k N generate a sample price path S k, S k t 1,..., S k t n i= 2) Compute for all 1 k N ( C k 1 n = St k n + 1 i K 3) Set C = e rt 1 N N k=1 Ck. i= ) + 63

64 11 Options on Futures This chapter is partly based on Chapter 5 & 16 in [2] Forward contracts Definition of forward contracts from Wikipedia: A forward contract, or simply a forward, is a contract between two parties to buy or sell an asset at a specified future time at a price agreed today. In a forward contract one specifies ˆ the asset to be delivered, ˆ the quantity, ˆ the delivery date, ˆ the delivery price. The buyer of the contract is said to be long in the asset, and the seller is said to be short. The delivery price is usually chosen such that it does not cost anything to enter the forward contract. This particular price is called forward price. This means that at the time where a forward contract is entered, the forward price is equal to the delivery price. Notice that between the two parties making a forward contract there is only a cash resp. asset flow at the delivery date, but not at the contract date. There are many different types of assets underlying forward contracts. We distinguish between investment assets and consumption assets. Examples for investment assets are stocks, bonds, gold. Examples for consumption assets are commodities (e.g. crude oil, copper,...), orange juice,... The forward price of an investment asset can be determined from its spot price and other observable market factors. Let ˆ S = spot price of the asset, ˆ T = delivery date, ˆ r = risk free rate (continuous compounding), ˆ F = forward price. Theorem Consider an investment asset providing no additional income (e.g. a stock paying no dividends, or a zero-coupon bond). The only arbitrage free forward price for the asset to be delivered at T is F = e rt S. (71) 64

65 Constructive proof. portfolio: Suppose first that F > e rt S. Set up the following forward contract underl. asset cash position S The portfolio value at time is zero. Indeed, V = + S S =. Moreover, the portfolio value at time T satisfies V T = (S T F ) + S T e rt S = F e rt S >, and hence it is positive with probability one. Thus the market admits arbitrage. Next suppose that F < e rt S. Set up the following portfolio: forward contract underl. asset cash position S The portfolio value at time is given by V = S + S =, and the portfolio value at time T is positive with probability one: V T = (S T F ) S T + e rt S = e rt S F >. Again the market admits arbitrage. Therefore F = e rt S is the unique arbitrage free price. Proof of (71) via risk neutral pricing. A forward can be seen as a derivative with payoff S T K, where K is the delivery price. According to the pricing principle, the arbitrage free price is the expected payoff with respect to the risk neutral measure Q. As it does not cost anything to enter a forward contract, in an arbitrage free market is must hold true that E Q (S T F ) =. (72) Under Q, the discounted asset price is a martingale; hence E Q (e rt S T ) = S. Therefore, with (72), F = E Q (S T ) = e rt E Q (e rt S T ) = e rt S, which entails (71). We next determine the forward price of dividend paying stocks. Theorem Consider a dividend paying stock, and let I be the discounted value of all dividends paid up to T. The only arbitrage free forward price with delivery at T is F = e rt (S I). (73) 65

66 Proof. Suppose first that F > e rt (S I) and consider the portfolio forward contract underl. asset cash position S The portfolio value at time is zero, and at T it satisfies V T = (S T F ) + S T e rt S + e rt I = F e rt (S I) >, and hence the market admits arbitrage. The case F < e rt (S I) can be treated similarly Futures contracts Like a forward, a futures contract (or simply a futures) is an agreement to buy/sell an asset at a future time at a price specified already today. The main difference to forwards is that futures are exchange-traded, whereas forwards are traded OTC (over-the-counter). The next table summarizes some stylized differences between forwards and futures. futures exchange traded highly standardized margin payments no counterparty risk The margining mechanism forwards OTC traded tailor-made no margining counterparty risk Anyone trading futures has to set up a margin account at the exchange. At the moment a futures contract is entered, an initial margin has to be deposited. At the end of every trading day the futures position is marked-to-market, and the margin account is adjusted accordingly. The precise mechanism is best explained with an example: Suppose that on July 4, 211, an investor buys one gold futures at a price of 15$/oz. Delivery is December 211, and the contract size is 1 ounces. Suppose that the exchange requires an initial margin of 2$. Assume that prices fall after the investor has entered the contract, and that the futures closes at 1492$/oz on July 4. The value of the investor s position has declined by ( ) $ 1oz = 8$. oz The margin account is reduced by 8$, and has a new balance of 12$. Suppose that on July 5 prices soar, and the futures closes at 152$/oz. Then the exchange transfers ( ) $ 1oz = 28$ oz to the margin account, having then a new balance of 4$. 66

67 The account is adjusted like this every day up to delivery... The investor can withdraw from the margin account the cash exceeding the initial margin, but has to make sure that a minimum maintenance margin is always set. We next turn to futures prices. Definition The futures price is the delivery price of a traded futures contract (note that it does not cost anything to enter a futures). Do futures prices coincide with forward prices? Under the assumptions that the interest rate r is constant, and that there is no default risk, the futures price coincides with the forward price. In particular, the futures price of an investment asset with no additional income is given by e rt S. Remark Under stochastic interest rate r(t), t, the futures price Fut and the forward price For, with delivery T, are given by Fut = E Q [S T ] For = S E Q [ T e r(s) ds] (see e.g. Ch.5 in Shreve: Stoch. Calculus for Finance). The futures price is a martingale with respect to the risk neutral measure. Theorem Consider a futures on an investment asset without additional income. Then the futures price is a martingale wrt the risk neutral measure Q. Proof. Remark If interest rates are stochastic, then in general forward prices are not martingales, whereas futures prices are Futures Options Definition The owner of an option on a futures contract, or simply a futures option, has the right, but not the obligation, to enter a futures contract at a specific future date. A call is the option to buy the futures, and a put is the option to sell it. The option strike price is the cash the option buyer has to give to the option seller, provided the option is exercised. Examples are treasury bond futures options, crude oil futures options. See CBOT for many further examples. Question: Why options on futures and not on the spot itself? In order to describe the option value we introduce some notation. Let ˆ T = expiration date of the option ( the delivery date of the underlying futures), 67

68 ˆ F T = futures price at T, ˆ K = strike price. If the option is a call, then the payoff resp. value at expiration is given by (F T K) +. For a put the payoff is (K F T ) +. We next derive the self-financing condition of futures portfolios. To this end consider a portfolio consisting of ξ futures contracts and η bonds at time t. Denote by St the bond price and by V t the portfolio value at t. After any margin payment the futures position has value zero. Therefore, V t = ηs t. Let t+δ be the next trading day. By how much does the portfolio value change? Let X = X t+δ X t be the futures price change. Then the margin account is adjusted by ξ X. The bond position earns approximately an interest of rδv t. (Notice that rδ (e rδ 1) for small delta.) To sum up, the portfolio value change satisfies Letting δ, we obtain V t+δ V t ξ X + rδv t. dv t = ξ(t)dx t + rv t dt. (74) Equation (74) is the self-financing condition for futures portfolios. Remark The solution of the SDE (74) is given by t ) V t = e (V rt + ξ(s)e rs dx s. Theorem 11.9 (Put-call parity for futures options). Suppose that a European call on a futures, with strike K and expiration T, is traded at a market price of C. Then the only arbitrage free price for a put with the same strike K and expiration T is given by where F is the current futures price. Proof. P = C e rt F + e rt K, (75) In a famous model by Black (see F. Black. The pricing of Commodity Contracts, Journal of Financial Economics, 3 (1976)) the futures price is assumed to satisfy the dynamics df t = σf t dw t, where W is a BM with respect to the risk-neutral measure Q. In the following we derive the pricing PDE for a futures option with payoff h : R R +. The option is a call if h(x) = (x K) +, and a put if h(x) = (K x) +. We use the 4 step recipe. 68

69 1) Assume that the futures option is replicable and that its value at time t [, T ] is equal to v(t, F t ), where v C 1,2. 2) v(t, F t ) is an Ito process with 2 decompositions: From the self-financing condition we have and from Ito s formula dv(t, F t ) = ξ(t)df t + r v(t, F t )dt = ξ(t)σf t dw t + r v(t, F t )dt, dv(t, F t ) = v t (t, F t )dt + v f (t, F t )df t v ff (t, F t )df t df t = v t (t, F t )dt + v f (t, F t )σf t dw t v ff (t, F t )σ 2 F 2 t dt. 3) Matching the coefficients yields first ξ(t) = v f (t, F t ), and then that v(t, f) has to solve the PDE v t (t, f) σ2 f 2 v ff (t, f) r v(t, f) =, (76) with terminal condition v(t, x) = h(x). 4) Discounted Feynman-Kac shows that the solution of the PDE (76) is given by v(t, x) = e r(t t) E t,x [h(f T )]. (77) From this we can derive the following pricing formula for futures call and put options. Theorem Assume a futures price of F t = f at time t. Then the time t arbitrage free price of a European call futures option, with strike K e and expiration date T, is given by Black76 call(f, K, T t, σ X, r) = e r(t t) fφ(d 1 ) e r(t t) KΦ(d 2 ), where d 1 = ( ) log f K + σ2 X 2 (T t) σ, T t d 2 = d 1 σ T t. Proof. The formula follows from (77) by using that F T is lognormally distributed under Q. 69

70 Theorem The arbitrage free price of a European put futures option is given by Black76 put(f, K, T t, σ, r) = e r(t t) KΦ( d 2 ) e r(t t) fφ( d 1 ). Remark Notice that Black76 call(f, K, τ, σ, r) = BS call(fe rτ, K, τ, σ, r). 12 Local volatility models Recall that the Black Scholes implied volatility usually changes with the strike and with time to maturity. The volatility surface is the graph of the implied volatility as a function of time to maturity and moneyness Local volatility and transition densities One way to make model prices consistent with market prices for Plain Vanilla options is to assume that the volatility is a function of time and the underlying price: ds t = rs t dt + S t σ(t, S t )dw Q t. (78) Definition The mapping (t, x) σ(t, x) is called local volatility function. We will see that ˆ European call / put prices uniquely determine the local volatility function, ˆ our pricing PDEs for American and exotic options remain almost the same: only the constant vola has to be replaced with the local volatility function. Suppose that the market call prices C(T, K) are known for all possible expiration dates T > and strike prices K. Then, under some nice conditions, the local volatility function satisfies σ(t, K) = 2 C T + rk C K K 2 2 C K 2. (79) Equation (79) is called Dupire s formula. For the proof we use the transition density of the price process S t. Let ϕ(, x; T, y) be the probability density of S T at y conditional to S = x. This means that for any nice B R P,x (S T B) = ϕ(, x; T, y)dy. B Definition ϕ(, x; T, y) is called transition density of the process (S t ) t R+. 7

71 Notation: In the following we fix the initial condition (, x) and simply write ϕ(t, y) = ϕ(, x; T, y). Theorem 12.3 (Kolmogorov forward equation). Suppose that S t satisfies ds t = rs t dt + S t σ(t, S t )dw Q t, S = x. Then the transistion density ϕ(t, y) of S t satisfies the PDE T ϕ(t, y) = y (ryϕ(t, y)) + 1 ( y 2 2 y 2 σ 2 (T, y)ϕ(t, y) ). Remark The Kolmogorov forward equation is sometimes also called Fokker-Planck equation. The price of a call option expiring at T and struck at K satisfies 2 C(T, K) = e rt E Q [ (S T K) +] = e rt (y K) + ϕ(t, y)dy Observe that the partial derivatives satisfy R = e rt (y K)ϕ(T, y)dy. K C(T, K) = rc(t, K)+e rt (y K) ϕ(t, y)dy (8) T K T C(T, K) = e rt ϕ(t, y)dy (81) K K 2 K 2 C(T, K) = e rt ϕ(t, K). (82) With the Kolmogorov forward equation we get C(T, K) = rc(t, K) + e rt (y K) ϕ(t, y)dy T K T [ = rc(t, K) e rt (y K) (ryϕ(t, y)) K y 1 2 ( y 2 2 y 2 σ 2 (T, y)ϕ(t, y) ) ] dy. (83) The following conditions guarantee that Dupire s formula holds true. ( A1) lim y (y K) y y 2 σ 2 (T, y)ϕ(t, y) ) =, A2) lim y ( y 2 σ 2 (T, y)ϕ(t, y) ) =, A3) lim y (y K)ryϕ(T, y) =. 71

72 Theorem 12.5 (Dupire). Suppose that A1) A3) hold true. Then the local volatility function is given by C C T + rk K σ(t, K) = 2. K 2 2 C K 2 Proof Pricing American and exotic options In this subsection we explain how local volatility models can be used for pricing American or exotic options that are not actively traded. The basic procedure is as follows: First derive the local volatility function from standard European options. Second use the local vola function in the pricing PDE for the American or exotic option considered, and solve the PDE numerically. Example 12.6 (Pricing D&O call with a local volatility model). Assume that ds t = rs t dt + S t σ(t, S t )dw Q t, S = x. and that σ(t, x) has been calibrated from market prices of standard calls or puts. Let v(t, x) be the time t D&O call value under the assumption that it has not been knocked out before t and that S t = x. Then v(t, x) satisfies the Black- Scholes PDE v t (t, x) + rxv x (t, x) σ2 (t, x)x 2 v xx (t, x) rv(t, x) =, with boundary conditions v(t, L) =, t T, v(t, x) = (x K) +, x > L. Dupire s formula requires market prices for all strikes and maturities. In reality we have market prices for only a finite number of Plain Vanilla Calls. There are several ways to determine the local vola function in practice. One idea is to use a time-space parameterization of the local vola function, and to choose the parameters such that the model prices for European calls & puts are as close as possible to the real market prices. A simple parameterization of the local vola function is given by Let σ(t, y) = a + a 1 y + a 2 y 2 + a 3 t + a 4 t 2 + a 5 ty. (84) T 1 T m be the maturities of traded calls. K i,1,..., K i,n(i) strikes of traded calls expiring at T i, 1 i m. 72

73 C(T i, K i,j ) = market price of the traded call with expiration T i and strike K i,j. Assume that the local vola function satisfies (84). For a given set of parameters (a,..., a 5 ) one can calculate the model call prices Ĉ i,j = e rti K i,j (y K i,j )ϕ(, x; T i, y)dy for all 1 i m and 1 j n(i). The quadratic distance to the market prices is given by error(a, a 1, a 2, a 3, a 4, a 5 ) = n(i) m (C(T i, K i,j ) Ĉi,j) 2. i=1 j=1 With a search algorithm one can find (a,..., a 5) such that error(a,..., a 5) min error(a,..., a 5 ). (a,...,a 5) The local volatility function is then approximately given by σ(t, y) = a + a 1y + a 2y 2 + a 3t + a 4t 2 + a 5ty Local vola from BS implied vola Definition We denote by σ BS (T, K) the Black Scholes (BS) implied vola for a call with strike K and expiration date T. Note that the market call price C(T, K) satisfies C(T, K) = BS call(s, K, T, σ BS (T, K), r). We show next that the local volatility function σ(t, K) is uniquely determined by the BS implied volatility function σ BS (T, K). Instead of parameterizing and calibrating the local vola function directly, one can therefore proceed as follows: ˆ parameterize the BS implied vola function, ˆ calibrate the BS implied vola function to market prices, ˆ calculate the local vola function from the calibrated BS implied vola function. We need some new variables: ( ˆ log moneyness y = log ) K e rt S ˆ implied total variance w(t, y) = T σ 2 BS (T, ert S e y ) 73

74 Lemma The local volatility function satisfies = σ 2 (T, e rt S e y ) (85) 1 y w w y (T, y) w 2 w T (T, y) y (T, y) ( w + y2 w )( w y )2 (T, y) Sketch of the proof. a) We first write the BS call price as a function of log moneyness and implied total variance w C BS (y, w) := BS call(s, e rt S e y, T, T, r). Observe that C BS (y, w) = S [ Φ( y w + The partial derivatives of C BS satisfy 2 C BS y 2 2 C BS w 2 (y, w) = 2 C BS (y, w) y w = (y, w) CBS (y, w) = y w 2 ) ey Φ( y ] w w 2 ) b) We next write the market call price as a function of T and log moneyness: Dupire s formula implies C(T, y) := C(T, e rt S e y ). 1 2 σ2 (T, e rt S e y )( 2 C y 2 C y )(T, y) = C T (T, y). (86) c) Finally observe that by definition we have C(T, y) = C BS (y, w(y, T )). Now one can rewrite Dupire s formula in terms of C BS and then derive (85) Objections There are several issues for which local volatility models are criticized: ˆ bad statistical properties: the local volatility functions change considerably over time, e.g. parameters usually change from one week to the next. 74

75 bad prediction properties. See Dumas, Fleming, Whaley. Implied Volatility Functions: Empirical Tests. Journal of Finance, ˆ The local volatility model predicts the smile/skew to move in the opposite direction as the underlying; in reality, both move in the same direction. See Hagan, Kumar, Lesniewski, Woodward. Managing Smile Risk. Wilmott Magazine, 22. ˆ hedging based on volatility models is inconsistent ˆ ad hoc model; no economic explanation of the local volatility function 13 Pricing options with Least-Squares Monte- Carlo The Least-Squares Monte-Carlo (LSMC) method is a powerful tool for pricing options with a complex payoff structure, in particular options with American exercise features. There are several versions of the algorithm. We discuss mainly the version of Longstaff and Schwartz [3], which is probably the most prominent one. We first illustrate the algorithm with a simple example, we then formalize it to a general setting and finally we discuss some more advanced applications An illustrating example In this subsection we explain how one can use the LSMC algorithm for pricing a put option of American type. Assume a current underlying s price of S = 1 and a strike of K = 1. Suppose that the put option can be exercised at the discrete times t = {, 1, 2, 3}, with t = 3 being the expiration date. Let r =.2 be the interest rate, with continuous compounding, per time step. Note that e r.982. The next table contains ten independent samples of the asset price. Price paths path t= t=1 t=2 t= ,18 1,61 9, ,11 13,39 12, ,5 9,5 11, ,56 8,41 8, ,41 1,95 1, ,9 9,94 11, ,84 7,88 8, ,22 11,11 9, ,75 12,54 13, ,86 9,64 9,92 75

76 The LSMC algorithm allows to derive an estimate of the put option price by using the 1 samples. We start by considering the option s payoff at t = 3, provided it has not been exercised before. Payoff vector at t = 3 path t=3 1, , ,1 8,35 9 1,8 The next table depicts the payoff if the option is exercised at time 2. Paths in-the-money at t = 2 path t= ,5 4 1,59 5 6,6 7 2, ,36 Notice that for 5 paths the option is in-the-money at time t = 2. We next use a regression for estimating the continuation value, the expected option value if we do not exercise the option at t = 2 (and before). We then compare the continuation value with the immediate exercise value and decide whether to exercise the option at t = 2. We denote by X the 5-dimensional vector of stock prices that are in-themoney at t = 2. Moreover let Y be the vector of corresponding discounted payoffs at t = 3, i.e. X = 9, 5 8, 41 9, 94 7, 88 9, 64 and Y = e r 1, 88 1, 1, 8 Suppose that the continuation value is a function of the underlying s price at. 76

77 t = 2. We approximate it with a polynomial of degree 2. To this end we regress the vector Y onto the vector with entries 1, and the vectors X and X 2. The regression coefficients are α = 43, 45, α 1 = 1, 77 and α 2 =, 64. By setting C i = α + α 1 X i + α 2 Xi 2 we obtain approximate continuation values for the five in-the-money paths, shown in the following table. path exercise continuation 1 2 3,5,44 4 1,59 1,32 5 6,6 -,34 7 2,12 1, ,36,21 For every in-the-money path the continuation value is smaller than the immediate exercise value, and, therefore, the option should be exercised at t = 2. The next matrix keeps track of the optimal stopping time and the corresponding option payoff, provided the option has not been exercised before t = 2. Payoff matrix for t 2 path t=2 t=3 1,79 2 3,5-4 1,59-5 6,6-7 2,12-8,35 9 1,36 - We now turn to t = 1. The next table depicts the payoff if the option is exercised at time 1. Paths in-the-money at t = 1 77

78 path t= , , ,86 Let X be the price vector of the three paths that are in-the-money at t = 1. And let Y be the discounted payoff vector if the option is not exercised. We have X = 9, 56 7, 84 9, 86 and Y = e r 1, 59 e r 2, 12 e r, 36 Regressing Y onto 1, X and X 2 we get α = 133, 46, α 1 = 31, 71 and α 2 = 1, 84. The approximate continuation values are depicted in the next table. path exercise continuation ,44 1, ,16 2, ,14,35 It is only optimal to exercise path # 7 at time t = 1. For the other paths it is optimal not to stop at t = 1. To sum up, we get the following payoff matrix. Payoff matrix for t 1. 78

79 path t=1 t=2 t=3 1,79 2 3, ,59-5 6,6-7 2, , ,36 - Since the option is at-the-money at t =, it is not optimal to exercise at t =. One obtains an estimate of the option value by calculating the average payoff along all paths. option value i=1 pv of payoff along path i = 1 ( e 3r, e 2r e 2r, 36 ) 1 =, Formal algorithm for American options In this subsection we formalize the LSMC for pricing American put options. Let T R + be the option s expiration date, and denote by h(x) = (K x) + the payoff function. We denote by S be the price process of the underlying. Let (F t ) be the filtration generated by S. We assume that S is non-negative Markov process. The LSMC algorithm requires samples of the price process S. Suppose we have M N samples of the price along the discrete time points t i = i N T, with N N and i {, 1,..., N}. We denote sample j by Ŝj. In particular, Ŝj i is the price value of sample j at time t i. We assume that the option can only be exercised at the discrete times t i. The basis for the algorithm is the set of simulated price paths. For every price path the algorithm determines an optimal exercise time and the corresponding payoff. To achieve this, it proceeds via a backward recursion. In every recursion step the option s inner value is compared to the continuation value. This allows to keep track, along every path, of the optimal exercise time, conditional to that the option has not been exercised yet. The precise mathematical definition of the continuation value at time t n is C n = esssup {E[S τ F tn ] τ is an (F t ) stopping time with values in {t n+1,..., t N }}. Suppose that there exists an optimal stopping time τ with values in {t n+1,..., t N } such that E[S τ F t ] = C n. Since S is a Markov process, there exists a function R + R +, denoted by x C n (x), such that E[S τ F t ] = C n (S t ). 79

80 Notice that by the least-squares property of conditional expectations, for any function g : R + R we have E (S τ C n (S tn )) 2 E (S τ g(s tn )) 2. (87) The algorithm approximates the continuation value function C n (x) with a linear combination of basis functions f, f 1,..., f b. The coefficients of the basis functions are determined with an ordinary linear regression. More precisely, the regression determines the coefficients α, α1,..., αb that minimize the quadratic error ( 2 M b Ŝ j τ (j) αkf k n)) (Ŝj. j=1 k= Because of least-squares property (cf. (87)), the linear combination b k= α k f k(x) provides an approximation of the continuation value function C n (x). The choice of basis function influences the accuracy of the algorithm. The next two examples give standard choices. Example 13.1 (Polynomials). Frequently one chooses polynomials as basis functions: f (x) = 1, f 1 (x) = x, f 2 (x) = x 2, etc. Example 13.2 (Weighted Laguerre polynomials). For some applications it is helpful to damped the basis functions. Longstaff and Schwarz [3] use the following weighted Laguerre polynomials: f (x) = e x/2, f 1 (x) = e x/2 (x 1), f 2 (x) = e x/2 (x 2 /2 2x + 1), etc. Steps of the algorithm 1) Initialize a payoff vector p by setting for all samples j = 1,..., M. 2) Recursion step n + 1 n p(j) = h(ŝj N ) Denote by X the price vector at time t n, i.e. X(j) = Ŝj n, j {1,..., M}. Approximate the continuation value with basis functions f,..., f b. More precisely, regress the vector p onto the vectors f (X), f 1 (X),..., f b (X). Recall that the regression determines the coefficients α, α 1,..., α b that minimize the quadratic error ( M p(j) j=1 2 b αkf k n)) (Ŝj. k= 8

81 Calculate the approximate continuation values C j n = α f (X) α bf b (X), j {1,..., M}. Update the payoff vector by setting { p(j) = h(ŝj n), if h(ŝj n) > e rt/n Cn j e rt/n p(j), else. 3) Final step: an estimate of the option price is given by put value 1 M M p(j). j=1 Matlab source code We next provide an implementation of the algorithm in Matlab. The implementation includes a generation of sample paths. The prices are geometric Brownian motions, possibly shifted by a dividend payment rate. The implementation uses the antithetic variates method for reducing the variance of the Monte-Carlo estimate. Moreover, the algorithm allows to price put and call options. %%%%%%%%%%%%%%% parameters %%%%%%%%%%%%%%%%% S=1; %spot p r i c e K=1; %s t r i k e T=1; %option maturity r =.6; dividend =; v =.2; %v o l a t i l i t y n s i m u l a t i o n s =2; n f i x i n g s =5; CallPutFlag = p ; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nsteps=n f i x i n g s ; n s i m u l a t i o n s=f l o o r ( n s i m u l a t i o n s / 2 ) ; n s i m u l a t i o n s=n s i m u l a t i o n s * 2 ; i f CallPutFlag== c, z =1; e l s e z= 1; end ; smat=z e r o s ( nsimulations, nsteps ) ; dt=t/( nsteps 1); 81

82 smat (:,1)= S ; d r i f t =(( r dividend) v ˆ2/2)* dt ; vsqrdt=v* dt ˆ. 5 ; randmat1=randn ( n s i m u l a t i o n s /2, nsteps 1); randmat2= randmat1 ; randmat=[randmat1 ; randmat2 ] ; smat ( :, 2 : nsteps)=exp ( d r i f t+vsqrdt *randmat ) ; smat=cumprod ( smat, 2 ) ; pvec = z e r o s ( nsimulations, 1 ) ; % i n i t i a l v a l u e s f o r p a y o f f v e c t o r pvec = max( z *( smat ( :, nsteps) K), ) ; p=z e r o s ( 4, 1 ) ; %c o e f f o f b a s i s f u n c t i o n s i n n e r v a l u e = z e r o s ( nsimulations, 1 ) ; c o n t i v a l u e = z e r o s ( nsimulations, 1 ) ; % r e c u r s i o n t i c f o r k=nsteps 1: 1:1, end toc i n n e r v a l u e = max( z *( smat ( :, k) K), ) ; % s e l e c t the in the money paths idx=f i n d ( innervalue >); X=smat ( idx, k ) ; Y=pvec ( idx ) ; % r e g r e s s i o n regrmat =[ ones ( s i z e (X, 1 ), 1 ),X,X. ˆ 2,X. ˆ 3 ] ; p=regrmat \Y; c o n t i v a l u e ( idx)=p(1)+p (2)*X+p (3)*X.ˆ2+ p (4)*X. ˆ 3 ; %Reset the p a y o f f optimal e x e r c i s e time pvec = exp( r* dt )* pvec ; f o r j =1: n s i m u l a t i o n s i f ( i n n e r v a l u e ( j ) > exp( r* dt )* c o n t i v a l u e ( j ) ) pvec ( j )= i n n e r v a l u e ( j ) ; end end 82

83 a m o p t i o n p r i c e =1/ n s i m u l a t i o n s *sum( pvec ) Modifications of the algorithm ˆ Getting rid of the bias The payoff vector and the continuation value vector are determined from the same set of samples, and therefore the payoff vector is biased. The bias decreases as the number of samples increases. If one uses independent samples for estimating the continuation value, then the payoff vector is not biased. The algorithm then looks as follows. 1) As above 2) As above, but save the regression coefficients α(n),..., αb (n) in every time step n {, 1,..., n 1}. 3) Generate new sample paths S j i, 1 j M, i N. Calculate for every time n < N the continuation values C j n = α f ( S j n) α bf b ( S j n), j {1,..., M}. Define recursively (n + 1 n) the payoff vector by setting { h( p(j) = S n), j if h( S n) j > e rt/n Cn j e rt/n p(j), else. 4) Final step: an estimate of the option price is given by put value 1 M M p(j). j=1 ˆ Iterative approximation of the value functions 1) Define the terminal value function vector v N by setting for all samples j = 1,..., M. v N (j) = h(ŝj N ) 2) Backward recursion for n = N 1,..., 1: Let X(j) = Ŝj n, for j {1,..., M}. Choose some basis functions f,..., f b, and regress the vector v n+1 onto the vectors f (X), f 1 (X),..., f b (X). Calculate the approximate continuation values C j n = α f (X) α bf b (X), j {1,..., M}. Set ( ) v n (j) = max h(ŝj n), e rt/n v n+1 (j). 83

84 3) An estimate of the option price is given by 1 M put value h(s ), v 1 (j). M References [1] Tomas Björk. Arbitrage Theory in Continuous Time. Oxford University Press, [2] J. C. Hull. Options, Futures, and Other Derivatives. Prentice Hall, Seventh edition, 28. [3] Francis A. Longstaff and Eduardo S. Schwartz. Valuing american options by simulation: A simple least-squares approach. Review of Financial Studies, 14(1): , 21. [4] Steven E. Shreve. Stochastic calculus for finance. II. Springer Finance. Springer-Verlag, New York, 24. Continuous-time models. j=1 84