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