Chapter 6: The Black Scholes Option Pricing Model

Transcription

1 The Black Scholes Option Pricing Model 6-1 Chapter 6: The Black Scholes Option Pricing Model

2 The Black Scholes Option Pricing Model 6-2 Differential Equation A common model for stock prices is the geometric Brownian motion ds t = µs t dt +σs t dw t (6.1) Equivalently returns follow general Brownian motion ds t S t = µdt +σdw t Drift µ reflects the current expectations of the returns. Volatility σ reflects the standard deviation around that drift.

3 The Black Scholes Option Pricing Model 6-3 The idea of Black-Scholes (BS) was to construct a portfolio from stocks and bonds that yields the same return as a portfolio consisting only of an option. This so called hedge portfolio has the same cash flow in T as the option, and thus must have the same price.

4 The Black Scholes Option Pricing Model 6-4 In contrast to the hedge portfolios introduced in the first sections of this class, the balance of stocks and calls is adapted continuously. We shall see that the relation Value of hedge portfolio = Value of option portfolio yields a partial DEQ for the value of the call.

5 The Black Scholes Option Pricing Model 6-5 Pricing a call option on a stock There are two equivalent strategies: 1. Portfolio A: Call option with strike K and maturity T Portfolio B: n t = n(s t,t) stocks and m t = m(s t,t) zero bonds with nominal value B T = 1 2. Portfolio A: One stock and n t = n(s t,t) calls short with strike K and maturity T Portfolio B: m t = m(s t,t) zero bonds with nominal value B T = 1

6 The Black Scholes Option Pricing Model 6-6 As the proof of the BS pricing formula is essential in the theory of finance, we want to give three proofs of it. The first is more lengthy but allows for a convenient understanding of the portfolio adjustments. The second one is more condensed and more elegant. With the result, it will be possible to value derivatives of arbitrary payoff functions. A third proof is given in the technical appendix using martingale techniques.

7 The Black Scholes Option Pricing Model 6-7 Proposition 6.1 (BS I) Let S t be an asset governed by a geometric Brownian motion, and let FI be a financial instrument (derivative) on S t expiring in T. Let T be the exercise time and T = T if FI is not exercised. The value of FI at time t T is given by the function F(S t,t).

8 The Black Scholes Option Pricing Model There exists a portfolio in S t and zero bonds B t duplicating FI, i.e. it generates the same payoff in T as FI and has the same time T -value as FI. 2. The value F(S,t) fulfills the BS-DEQ: F(S,t) t rf(s,t)+bs F(S,t) S σ2 S 2 2 F(S,t) S 2 = 0 (6.2) for t T, where b is cost of carry, r is risk free interest rate.

9 The Black Scholes Option Pricing Model 6-9 Proof: W.l.o.g. we assume that the object is a stock with continuous dividend d and costs of carry b = r d. We construct a (dynamic) hedge portfolio V consisting of n t = n(s t,t) stock m t = m(s t,t) bonds (with B T = 1) such that the financial instrument F and the hedge portfolio have the same value in T : V(S T,T ) = F(S T,T ).

10 The Black Scholes Option Pricing Model 6-10 The value of hedge portfolio is V t = V(S t,t) = n t S t +m t B t. Here B t = B T e r(t t) = e r(t T) How does it change in a small time interval dt? dv t = V t+dt V t, dn t = n t+dt n t dv t = {n t+dt S t+dt n t S t } + {m t+dt B t+dt m t B t } (6.3)

11 The Black Scholes Option Pricing Model 6-11 Note that the first term equals: n t+dt (S t+dt S t )+n t+dt S t n t S t ={n t+dt n t }(S t+dt S t ) +n t (S t+dt S t )+{n t+dt n t }S t +n t S t n t S t which is just dn t ds t +n t ds t +S t dn t = dn t (ds t +S t )+n t ds t Hence equation (6.3) can be written as : dv t = dn t (S t +ds t )+n t ds t +dm t (B t +db t )+m t db t. (6.4)

12 The Black Scholes Option Pricing Model 6-12 Apply now Itô s Lemma dg(s t,t) = g t dt + g S ds t g S 2σ2 S 2 tdt to g = n t and g = m t. Furthermore, use (ds t ) 2 = (µs t dt +σs t dw t ) 2 = σ 2 St(dW 2 t ) 2 = σ 2 Stdt 2 + O(dt) db t = rb t dt and exploit that ds t dt and (dt) 2 are O(dt).

13 The Black Scholes Option Pricing Model 6-13 dv t = ( nt t dt + n t S ds t ) 2 n t σ 2 S 2 S tdt 2 n t ds t +( mt t dt + m t S ds t S t + n t S σ2 Stdt+ 2 2 m t S 2 σ 2 S 2 tdt ) B t +m t rb t dt The assumption of no cash flow up to T means that all payments and costs in dt (these are all terms in the above equation, except n t ds t and m t rb t dt = m t db t ) are neutralized by payments and costs of the object i.e. d n t S t dt = (r b)n t S t dt.

14 The Black Scholes Option Pricing Model 6-14 This yields: n t (r b)s t dt = + ( nt t dt + n t S ds t ( mt t dt + m t S ds t n ) t S 2 σ2 Stdt 2 S t + n t S σ2 Stdt 2 2 m ) t S 2 σ2 Stdt 2 B t

15 The Black Scholes Option Pricing Model 6-15 Insert ds t = µs t dt +σs t dw t and order stochastic (dw) and non-stochastic components (dt): {( nt t + n t S µs t n t S 2 σ2 S 2 t +n t (b r)s t }dt + ) S t + n t S σ2 St 2 + ( nt t S t + m t S B t ( mt t + m t S µs t+ 1 2 ) σs t dw t = 0 2 m t S 2 σ2 S 2 t ) B t

16 The Black Scholes Option Pricing Model 6-16 It follows that the stochastic and riskless part of this equation must equal zero: ( nt t + n t S µs t ) n t 2 S 2 σ2 St 2 S t + n t S σ2 St 2 and + = 0 ( mt t + m t S µs t m t S 2 σ2 S 2 t ) B t +n t (b r)s t n t S S t + m t S B t = 0 (6.5)

17 The Black Scholes Option Pricing Model 6-17 Combining equations we obtain: ( mt t ( nt t Differentiate (6.5) w.r.t. S: 2 n t S 2 t 2 m t S 2 σ2 S 2 t ) S t + n t S σ2 St+ 2 ) B t +n t (b r)s t = 0 (6.6) σ 2 S 2 t Insert into equation (6.6): 2 n t S 2 S t + 2 m t S 2 B t = n t S (6.7) n t t S t + m t t B t n t S σ2 S 2 t +n t (b r)s t = 0 (6.8)

18 The Black Scholes Option Pricing Model 6-18 By definition of V we have: Differentiate w.r.t. t: m t t m t = Bt 1 (V t n t S t ) (6.9) V t = n t S t +m t B t (6.10) ( V = rbt 1 (V t n t S t )+Bt 1 t n ) t t S t Insert into equation (6.8) and obtain: 1 2 σ2 S 2 t n t S + V t +n tbs t rv t = 0 (6.11)

19 The Black Scholes Option Pricing Model 6-19 Differentiate (6.9) w.r.t. S and observe that B t / S = 0: This yields together with (6.5): m ( t V S = B 1 t S n t n ) t S S t n t = V S, which is the so called Delta. It yields the number of stocks in the hedge portfolio.

20 The Black Scholes Option Pricing Model 6-20 Since m t = V t n t S t B t we may construct the desired duplication portfolio if we know V t = V(S t,t). Insert this into (6.11) and we obtain the DEQ of BS V(S,t) t rv(s,t)+bs V(S,t) S σ2 S 2 2 V(S,t) S 2 = 0, (6.12) The function V(S t,t) follows this DEQ and has at time T the same pay-off as the financial instrument F(S T,T ), i.e. V(S T,T ) = F(S T,T ). (6.13)

21 The Black Scholes Option Pricing Model 6-21 Check solution The function V K,T (S t,τ) = S t e (b r)τ Ke rτ with τ = T t for the value of a forward contract fulfills the DEQ (6.12) and satisfies V(S T,0) = S T K. Check: V t = S te (b r)(t t) (r b) Ke r(t t) r V S = e(b r)τ 2 V S 2 = 0 (6.14)

22 The Black Scholes Option Pricing Model 6-22 Plug into (6.12): Se (b r)(t t) (r b) Ke r(t t) r rv +bse (b r)τ +0 = bse (b r)τ rv+rv bse (b r)τ = 0. When one tries to follow the hedging strategy in Proposition 6.1 in practice, accumulating transaction costs may diminish the quality of the hedge.

23 The Black Scholes Option Pricing Model 6-23 F Options Investment in a foreign currency is comparable with a stock with continuous dividend payments, that corresponds to the riskfree interest rate r f in the foreign country. Under the assumption that the F rate follows a geometric Brownian motion, the value V of a financial instrument also satisfies the BS DEQ (6.12) with d = r f and b = r r f. Empirical observations indicate, however, that F rates are not very well described by geometric Brownian motion. BS option theory should be applied carefully.

24 The Black Scholes Option Pricing Model 6-24 Proposition 6.2 (BS II) Let the processes for a stock and a bond be described by the following equations: ds t = µs t dt +σs t dw t db t = rb t dt, then every financial instrument (derivative) of the form f(s T ) may be duplicated by a dynamic portfolio V t such that:

25 The Black Scholes Option Pricing Model the duplication portfolio has the same cash flow as f(s T ), i.e: f(s T ) = V T = F(S T,T), (6.15) 2. a function F(S,t) exists, which fulfills V t = F(S t,t) and: F(S,t) t rf(s,t)+rs F(S,t) S σ2 S 2 2 F(S,t) S 2 = 0, (6.16)

26 The Black Scholes Option Pricing Model 6-26 Proof: Let n t and m t describe the allocations in stock and bond. We look for a self financing portfolio which replicates the claim f(s T ). Self financing means that no funds are required after the initial investment. If such a portfolio exists, then its initial value is the unique price of the derivative. The definition of the portfolio and the self financing condition imply: V t = n t S t +m t B t (6.17) dv t = n t ds t +m t db t = n t ds t +m t rb t dt (6.18)

27 The Black Scholes Option Pricing Model 6-27 Assume now that V t = F(S t,t) for a sufficiently smooth function F(, ), so that we can apply Itô calculus. Then: dv t = df(s t,t) = F S ds t + ( F S 2σ2 St 2 + F t By the self financing condition (6.18) we obtain: n t = F(S t,t) S. ) dt. (6.19)

28 The Black Scholes Option Pricing Model 6-28 Furthermore, from (6.17) we have: m t = {F(S t,t) n t S t }B 1 t = { F(S t,t) F(S } t,t) S t Bt 1, S and from equation (6.18) we can rewrite the self financing condition in the form: dv t = df(s t,t) = F S ds t + ( F F ) S S t rdt. (6.20) Finally, equating (6.19) and (6.20) we obtain (6.16).

29 The Black Scholes Option Pricing Model 6-29 A solution to the DEQ (6.16) with boundary condition (6.15) provides a complete description to the price and hedging strategy of a derivative f(s T ). It can be shown (homework) that the solution takes the form: V t = F(S t,t) = e r(t t) + f [ { S t exp σ T tx + whenever the above integral is finite. (r 12 σ2 ) }] e x2 2 (T t) dx, 2π

30 The Black Scholes Option Pricing Model 6-30 Black-Scholes (BS) Formula for European Options An American option can be exercised anytime till the expiration date. A European option can only be exercised on the expiration date. Theoretically and practically European options and American options have different prices. This is because e.g. in case of a put or in case of a call on dividend paying there is a positive probability of exercising prior to expiration.

31 The Black Scholes Option Pricing Model 6-31 Call prices BS DEQ for a call C = V in (6.12) rc bs C(S,t) S 1 2 σ2 S 2 2 C(S,t) S 2 = C(S,t) t, 0 t T, 0 < S <, (6.21) C(S,T) = max{0,s K}, 0 < S <, (6.22)

32 The Black Scholes Option Pricing Model 6-32 C(0,t) = 0, lim C(S,t) S = 0, 0 t T. (6.23) S The first part in (6.23) follows from the fact that 0 is an absorbing state for the geometric Brownian motion: S t = 0, t < T, implies S T = 0, and the call is not exercised. The second part in (6.23) can be motivated by the fact that the probability for S T < K is small if S t K in t already. Hence the call is exercised and C T = S T K S T.

33 The Black Scholes Option Pricing Model 6-33 Transformation of the BS DEQ: There exists an analytic solution based on theory for parabolic DEQ with boundary conditions. Multiply (6.21) with 2 and put σ 2 α = 2r, β = 2b and τ = T t. σ 2 σ 2 Moreover define: C(S,T τ) = e rτ g(u,v), with v = σ 2 (β 1) 2τ 2, u = (β 1)log S K +v

34 The Black Scholes Option Pricing Model 6-34 A transformed version of the DEQ is then: 2 g u 2 = g v, 0 v σ2 (β 1) 2T 2 = v, (6.24) < u <, with boundary conditions g 0 (u) def = g(u,0) = K max{0,e u β 1 1}, < u < Such DEQ occurs in physics (as the heat equation).

35 The Black Scholes Option Pricing Model 6-35 The solution is: g(u,v) = 1 2 πv g 0(ξ)e (ξ u) 2 4v dξ Transforming back yields: C(S,T) = e rτ g(u,v) = e rτ 1 2 πv g 0(ξ)e (ξ u) 2 4v dξ

36 The Black Scholes Option Pricing Model 6-36 By substituting ξ = (β 1)log y K, the boundary condition is in its original form is recovered: max{0,y K}. Replace (u,v) by the corresponding expressions in S and τ: C(S,τ) = e rτ 0 max{0,y K} 1 σ τy 2π e [logy (logs+(b σ2 2 )τ)]2 2σ 2 τ dy

37 The Black Scholes Option Pricing Model 6-37 Interpretation: C(S,τ) a discounted expectation of the payoff function taken with respect to log-normal distribution, the risk neutral distribution: C(S,τ) = e rτ E[max{0,S K}] = e rτ E(S τ K)1(S τ > 0)1(S τ K > 0) (6.25) 1(S τ K > 0) = { 1 if Sτ K > 0 0 if S τ K < 0

38 The Black Scholes Option Pricing Model 6-38 Why? E[g(x)] = g(x)f(x)dx 1 withf(x) = σ 2πτ e andg(x) = max(0,x K) [logx {logst+(b σ2 2 )τ}2 ] 2σ 2 τ Where the expected value is calculated for a rv logy N( µ, σ 2 ) with µ = logs t +(b σ2 2 )τ and σ2 = σ 2 τ.

39 The Black Scholes Option Pricing Model 6-39 We transform the formula for C(S,t) further: C(S,τ) = e (r b)τ SΦ(y +σ τ) e rτ KΦ(y) (6.26) y = log S K +(b σ2 2 )τ σ τ Here Φ describes the cdf of a standard normal rv. Φ(y) = 1 2π y 1 y 2 e z2 2 dz = π e z2 dz.

40 The Black Scholes Option Pricing Model 6-40 Interpretation of the BS formula: C(S,τ) = e (r b)τ SΦ(y +σ τ) e rτ KΦ(y), (6.27) the first term (SΦ(y +σ τ), for r = b) represents the value of the stock in case the option is exercised for S > K. the second term e rτ KΦ(y) represents the discounted value of the strike price.

41 The Black Scholes Option Pricing Model 6-41 S, where C S Value of stock in hedge portfolio is C S C hedge ratio. Differentiate (6.27) w.r.t. S: is the S = Φ(y +σ τ). The first term stands for the capital invested in stock, the second for the capital invested in zerobonds. If S K then the value of C S e rτ K If S = 0 then C(0,τ) = 0. The value of a perpetual European put (τ = ) is zero

42 The Black Scholes Option Pricing Model 6-42 Put prices With the help of the put-call parity P(S,τ) = C(S,τ) Se (r b)τ +Ke rτ (6.28) the value of a European put option is given by: P(S,τ) = e rτ KΦ( y) e (r b)τ SΦ( y σ τ) (6.29) SFEPutCall

43 The Black Scholes Option Pricing Model C(S,tau) C(S,tau) S S Figure 1: Black-Scholes prices for the European call option C(S,τ) for different values of times to maturity τ = 0.6 and r = 0.1 and strike price K = 100. Left figure σ = 0.15, right figure σ = 0.3. SFEbsprices

44 The Black Scholes Option Pricing Model 6-44 Stock Price Path & Call Price Path (σ = 0,3) 140 S t t C(S,t) t Figure2:Black-ScholespriceC(S,τ)asafunctionofS t, whichismodelled as a geometric Brownian motion. Upper panel: sample path of the price process of the underlying S, lower panel: Black-Scholes prices C(S,τ) for strike K = 100, r = 0.05 and expiry at T = 1 where the initial value of the underlying is taken from the above sample path. SFEbsbm

45 The Black Scholes Option Pricing Model 6-45 Heat-transfer equation of physics In physics, Green s function of heat equation for temperature G is: with boundary condition: G τ = 1 G 2 σ2 2 x 2 G(x,0) = δ(x x 0 )

46 The Black Scholes Option Pricing Model 6-46 Here, we seek the temperature distribution G(x,t;x 0 ) in a one dimension rod when t > 0. And with a Dirac delta function δ(x x 0 ) as an initial condition, which means that, the initial temperature is infinitely large at the point x 0, and 0 at the other point. The solution to this problem is the heat kernel (also called Green funcion) to describe the change of heat distribution over time.

47 The Black Scholes Option Pricing Model 6-47 If we apply Green s formula which is given by: G(x,τ;x 0 ) = 1 2σ 2 πτ exp{ (x x 0) 2 2σ 2 } τ to the value of a European call C(S,t) satisfying following PDE: C t σ2 S 2 2 C S 2 +rs C S rc = 0 (6.30) and satisfying the final condition C(S,T) = max(s K,0), then we get for the price of the call option: C(S,τ) = SΦ Φ { log(s/k)+(r + σ 2 2 τ) σ τ { log(s/k)+(r σ 2 2 τ) } σ τ } e rτ K

48 The Black Scholes Option Pricing Model 6-48 Numerical Approximation Important are good approximations to the normal cdf! Edgeworth Expansion: (a) Φ(y) 1 (a 1 t +a 2 t 2 +a 3 t 3 )e y2 /2, with 1 t = 1+by, b = , a 1 = , a 2 = , a 3 = The approximation error is independent of y and of order O(10 5 ). SFENormalApprox1

49 The Black Scholes Option Pricing Model 6-49 Edgeworth Expansion with higher accuracy: (b) second approximation Φ(y) 1 (a 1 t +a 2 t 2 +a 3 t 3 +a 4 t 4 +a 5 t 5 )e y2 /2, with t = 1 1+by, b = , a 1 = , a 2 = , a 3 = , a 4 = , a 5 = Error of approximation: O(10 7 ) SFENormalApprox2

50 The Black Scholes Option Pricing Model 6-50 (c) another approximation is: Φ(y) 1 1 2(a 1 t +a 2 t 2 +a 3 t 3 +a 4 t 4 +a 5 t 5 ) 8, with a 1 = , a 2 = , a 3 = , a 4 = , a 5 = Error of approximation: O(10 5 ) SFENormalApprox3

51 The Black Scholes Option Pricing Model 6-51 (d) A Taylor expansion yields Φ(y) ( ) y y3 2π 1! y5 2!2 2 5 y7 3! = ( 1) n y 2n+1 2π n!2 n (2n+1) n=0 Here the approximation error depends on the order of the expansion. SFENormalApprox4

52 The Black Scholes Option Pricing Model 6-52 Comparison of Approximation approximation of normal distribution x norm-a norm-b norm-c norm-d iter Table 1: Several approximations to the normal distribution

53 The Black Scholes Option Pricing Model 6-53 Simulation In many situations we are unable to compute the derivative price analytically. The paths of the underlying need to be simulated. The performance of these simulations depends decisively on the quality of random numbers used. No random number generator in common software packages is satisfactory in every respect.

54 The Black Scholes Option Pricing Model 6-54 Konrad Zuse s Z3 Figure 3: Konrad Zuse with a rebuilt Z3 in 1961

55 The Black Scholes Option Pricing Model 6-56 Linear congruent generator Choose N 0 (seed) with (a,b,m) define U i U[0,1] pseudo random N i = (an i 1 +b)modm (6.31) U i = N i /M (6.32) Arrange the N i in random vectors of m-triples (N i,n i+1,...,n i+m 1 ) Problem: (U i,...,u i+m 1 ) [0,1] lie on a (m 1) dimensional hyperplane.

56 The Black Scholes Option Pricing Model 6-57 Analysis for m = 2 N i = (an i 1 +b)modm = an i 1 +b km for km an i 1 +b < (k +1)M For all z 0,z 1 : z 0 N i 1 +z 1 N i = z 0 N i 1 +z 1 (an i 1 +b km) = N i 1 (z 0 +az 1 )+z 1 b z 1 km z 0 +az 1 = M(N i 1 M z 1k)+z 1 b Hence : z 0 U i 1 +z 1 U i = c +z 1 bm 1 (6.33)

57 The Black Scholes Option Pricing Model 6-58 Example N i = 2N i 1 mod11 a = 2,b = 0,M = U i U i-1 Figure 5: The scatterplot of U i 1 vs. U i The question is how to choose (a,b,m). SFErangen1

58 The Black Scholes Option Pricing Model 6-59 Example N i = 1229N i 1 mod U i U i-1 Figure 6: The scatterplot of U i 1 vs. U i SFErangen2

59 Figure 7: The scatterplot of U i 2 vs. U i 1 vs. U i SFErandu Y The Black Scholes Option Pricing Model 6-60 A famous example is RANDU (the official IBM U[0,1] generator for years ) N i = an i 1 modm,a = ,M = 2 31 Make a scatterplot and rotate:

60 The Black Scholes Option Pricing Model 6-61 Fibonacci generators The Fibonacci sequences: N i+1 = N i +N i 1 mod M The ratio of consecutive random numbers converges to the golden ratio lagged Fibonacci N i+1 = N i ν +N i µ modm Example U i = (U i 17 U i 5 ) if U i < 0 then U i = U i +1.0

61 The Black Scholes Option Pricing Model 6-62 Notre-Dame and the golden ratio Figure 8: Application of the golden ratio in the design of the cathedral Notre-Dame in Paris.

62 The Black Scholes Option Pricing Model 6-63 Fibonacci generator Repeat: ζ = U i U j if ζ < 0: ζ = ζ +1 U i = ζ i = i 1 j = j 1 if i = 0: i = 17 if j = 0: j = 17

63 The Black Scholes Option Pricing Model U i U i-1 Figure 9: The scatterplot of U i 1 vs. U i SFEfibonacci

64 The Black Scholes Option Pricing Model 6-65 Inversion method How to generate i F? i = F 1 (U i ) Proof: P( i x) = P { F 1 (U i ) x } = P{U i F(x)} = F(x) Problem : F 1 is often hard to calculate numerically.

65 The Black Scholes Option Pricing Model 6-66 Transformation methods Example Exponential distribution f Y (y) = λe λy I(y 0) Define y = h(x) = λ 1 logx, x > 0 h 1 (y) = e λy for y 0 U[0,1] leads to Y exp(λ) f Y (y) = f { h 1 (y) } dh 1 (y) dy = ( λ)e λy = λe λy

66 The Black Scholes Option Pricing Model 6-67 Box-Muller method Unit square S = [0,1] 2, f (x) = 1 uniform y 1 = y 2 = h 1 (x) = Jacobian = 2logx1 cos2πx 2 = h 1 (x 1,x 2 ) 2logx2 sin2πx 2 = h 2 (x 1,x 2 ) { x1 = exp { 1 2 (y2 1 +y2 2 )} x 2 = (2π) 1 arctany 2 /y 1 ) det ( x1 y 1 x 1 y 2 x 2 y 1 x 2 y 2

67 The Black Scholes Option Pricing Model 6-68 Here: { x 1 = exp 1 } y 1 2 (y2 1 +y2) 2 ( y 1 ) { x 1 = exp 1 } y 2 2 (y2 1 +y2) 2 ( y 2 ) x 2 = 1 ( ) 1 y2 y 1 2π 1+y2 2/y2 1 y1 2 x 2 = 1 ( ) 1 1 y 2 2π 1+y2 2/y2 1 y 1 Jacobian = 1 { 2π exp 1 }( 2 (y2 1 +y2) 2 1 y 1 = 1 2π exp Hence (Y 1,Y 2 ) N(0,I 2 ) { 1 } 2 (y2 1 +y2) 2 1+y 2 2 /y2 1 1 y 1 y y 2 2 /y2 1 y2 2 y1 2 )

68 The Black Scholes Option Pricing Model 6-69 Algorithm Box-Muller 1. U 1 U[0,1], U 2 U[0,1] 2. θ = 2πU 2, ρ = 2logU 1 3. Z 1 = ρcosθ is N(0,1) Z 2 = ρsinθ is N(0,1) Variant of Marsaglia avoids the calculation of the trigonometric functions.

69 The Black Scholes Option Pricing Model Z Z 1 Figure 10: The scatterplot of (Z 1, Z 2 ) SFEbmuller

70 The Black Scholes Option Pricing Model 6-71 Variant of Marsaglia Method Generate V 1,V 2 U[ 1,1]. Accept (V 1,V 2 ) if V 2 1 +V2 2 < 1 ie. (V 1,V 2 ) uniform on a unit circle with density π 1. The transformation ( 1 2 ) = ( V 2 1 +V2 2 2π 1 arctanv 2 /V 1 to S = [0,1] 2 gives ( 1, 2 ) U(S). Since cos2π 1 = V 1 V 2 1 +V2 2 and sin2π 2 = ) V 2 V 2 1 +V2 2 hold, there arises no need for an evaluation of the trigonometric functions.

71 The Black Scholes Option Pricing Model 6-72 Marsaglia Method 1. U 1,U 2 U[0,1], V i = 2U i 1 with W = V 2 1 +V2 2 < 1 2. Z 1 = V 1 2log(W)/W N(0,1) Z 2 = V 2 2log(W)/W N(0,1)

72 The Black Scholes Option Pricing Model 6-73 Risk Management with hedge strategies Example current time t 6 weeks term T 26 weeks time of expiration T t 20 weeks = interest rate r 0.05 annual volatility of the stock σ 0.20 current stock price S t 98 EUR strike price K 100 EUR Table 2: data of example

73 The Black Scholes Option Pricing Model 6-74 A bank sells calls on a dividend free asset for EUR. BS option price is EUR, which is approximately EUR. Hence the bank has sold the call too expensive. SFEBSCopt2

74 The Black Scholes Option Pricing Model 6-75 Naked position If S T rises to 120 EUR the call is exercised and the bank has to deliver shares at the strike price K = 100 EUR. In order to cover the underlying position the bank has to buy shares at the market price S T = 120 EUR. The bank loses (S T K) = EUR. This is much higher than the premium of EUR, and the loss is EUR. If S T < K the option will not be exercised. The bank has gained EUR.

75 The Black Scholes Option Pricing Model 6-76 Covered position Immediately after the sale of the call the bank buys stock at S t = 98 EUR S t = EUR If S T > K the stock is delivered at K. Without considering interest payments the gain is roughly EUR from the sale of the options. If S T < K, eg. S T = 80 EUR, the option will not be exercised. The bank has to sell the stock at the market for EUR. The bank loses EUR, which is again more than the gain of EUR from the sale of the options.

76 The Black Scholes Option Pricing Model 6-77 Both risk management strategies are unsatisfying since the costs vary between 0 and large values. According to BS the average cost should be EUR and a perfect hedge should eliminate the randomness and should just create these costs.

77 The Black Scholes Option Pricing Model 6-78 Stop-Loss strategy The bank that issues the calls changes to a naked position if S t < K changes to a covered position if S t > K, i.e. the stocks to be delivered in case of exercise are bought as soon as S t is bigger than K.

78 The Black Scholes Option Pricing Model 6-79 All purchases after t > 0 are done at price K. In T either no stocks (S T < K) or stocks bought at K are hold. Hence costs of this strategy occur only if S 0 > K. Cost of stop-loss strategy: max{s 0 K, 0}. The cost of hedging are therefore equal to the intrinsic value at issuance. If interest rates were zero, it is clear that these cost are smaller than the BS price C(S 0,T). Arbitrage seems possible by selling an option and hedging with the stop-loss strategy.

79 The Black Scholes Option Pricing Model 6-80 Problems of this strategy: going short and long in the stocks creates transaction costs the long position in stock before T creates losses in interest in practice sales and purchases are not possible at price K. When the stock increases the price is K +δ, when the stock decreases the price is K δ, δ > 0 in practice sales and acquisitions are done in t time units. The bigger t, the bigger is δ and the smaller are transaction costs.

80 The Black Scholes Option Pricing Model 6-81 Table 3, taken from Hull (2000), shows results of a simulation of the stop-loss strategy with M = 1000 sample paths. Costs Λ m, m = 1,...,M are recorded and the variance ˆυ 2 Λ = 1 M M Λ m 1 M M Λ j 2 m=1 j=1 is computed.

81 The Black Scholes Option Pricing Model t (weeks) L Table 3: Performance of the stop loss strategy 1 4 We measure the risk from this strategy by dividing the standard deviation of the costs by the call price: ˆυ Λ 2 L = C(S 0,T). A perfect hedge has L = 0.

82 The Black Scholes Option Pricing Model 6-83 Delta Hedging In order to reduce the risk associated with option trading more complex hedging strategies than those considered so far are applied. One possibility is to try to make the value of the portfolio for small time intervals as insensitive as possible to small changes in the price of the underlying stock. This is called delta hedging.

83 The Black Scholes Option Pricing Model 6-84 The Delta of a call (also called hedge ratio) is the derivative of the option price wrt. the underlying = C S or = C S The delta of a stock is = S/ S = 1.

84 The Black Scholes Option Pricing Model 6-85 A forward contract on a dividend free stock has the forward price V(S t,τ) = S K e rτ (see proposition 2.1), hence the Delta of a forward contract is = V/ S = 1. Therefore stocks and forward contracts are interchangeable in -hedging.

85 The Black Scholes Option Pricing Model 6-86 Example A bank sells calls at C = 10 EUR/stock with S 0 = 100 EUR. Suppose the delta of the call is = 0.4. In order to delta hedge the option position the bank buys = 800 shares. If e.g. the share price rises by S = 1 EUR, i.e. the total value of shares held rises by 800 EUR, then the value of a call on 1 share rises by C = S = 0.4 EUR. The total value of all calls sold therefore rises by 800 EUR. Gains and losses are perfectly balanced in this case. The whole portfolio has = 0, i.e. is a delta neutral position.

86 The Black Scholes Option Pricing Model 6-87 The delta neutral positions have only a short time horizon because the Delta of an option depends, among others, on time and stock price. In practice, the portfolio has to be rebalanced frequently in order to adapt to the changing environment. Example Suppose that for the above example the stock rises to 110 EUR and the delta changes to = 0.5. To obtain a delta neutral position it is necessary to buy ( ) = 200 stocks.

87 The Black Scholes Option Pricing Model 6-88 Improving Risk Management: Delta Hedging Delta Hedging is at the heart of the Black-Scholes proof At t 0, sell a call and immediately buy stocks, i.e. make the portfolio Delta-neutral As changes with S t, τ and σ, Delta-neutrality only holds for a short period of time To achieve perfect hedge: constant rebalancing, which is called Dynamic Delta Hedging In contrast to Stop-Loss Strategy, the investor never stays passive

88 The Black Scholes Option Pricing Model 6-89 Delta Hedging Logic The Logic of Delta Hedging Y DELTA Step n Figure 11: Logic of the Delta Hedging Strategy SFEDeltaHedgingLogic

89 The Black Scholes Option Pricing Model 6-90 Delta Hedging Dependencies Delta vs S(t) Delta : Delta with contant tau S(t) Delta over time Delta Step n Figure 12: Dependence of Delta on Asset and Steps SFEDeltaHedgingLogic

90 The Black Scholes Option Pricing Model 6-91 Simulation: Parameters Current time t 6 weeks Maturity T 26 weeks Time to maturity τ = T t 20 weeks = Continuous annual interest rate r 0.05 Annualized stock volatility σ 0.20 Current stock price S t 98 EUR Exercise price K 100 EUR Table 4: The parameters of the simulation

91 The Black Scholes Option Pricing Model 6-92 Simulation: Results I The calculation of L, the remaining risk of the portfolio, as a function of t yields: t (weeks) /2 1/4 L Table 5: Performance of the Delta Hedging Strategy SFEDeltaHedging

92 The Black Scholes Option Pricing Model 6-93 Simulation: Results II StockPaths with Strike Price Costs of Delta Hedging Stockprice(t) Costs Steps (n) Delta T (weeks) Figure 13: Three stock paths and the cost function of all paths SFEDeltaHedging

93 The Black Scholes Option Pricing Model 6-94 Testing the Strategy: Does Delta Hedging Eliminate Risk? L 0 as t 0, the strategy constitutes a perfect hedge if the portfolio is continuously rebalanced This is an intuitive result, and also one behind a derivation of the Black-Scholes differential equation. It is similar to the idea of a duplicating portfolio.

94 The Black Scholes Option Pricing Model 6-95 Assumptions and Drawdowns Continuously rebalancing a portfolio is impossible Extremely costly in terms of transaction costs Trade-off must be decided upon by the book manager Recall the formula for costs L = ˆν 2 Λ C(S 0,T) Is this the only measure of risk?

95 The Black Scholes Option Pricing Model 6-96 BS Delta The BS formula was derived via a dynamic hedge portfolio argument. The BS Delta is: = C S = P S where: = { e (r b)τ SΦ(y +σ } τ) e rτ KΦ(y) S = Φ(y +σ τ) = Φ(y +σ τ) 1, y = log S K +(b σ2 2 )τ σ τ

96 The Black Scholes Option Pricing Model 6-97 Figure 14: Delta as function of stock prices (right axis) and time of expiration (left axis). SFEdelta

97 The Black Scholes Option Pricing Model 6-98 Properties of Delta: For increasing S the approaches 1. For decreasing S the approaches 0. If the option is ITM, the writer of the option should cover the risk by holding stocks in sufficient size. If the option is OTM, the writer of the option does not need to hold stocks in too large quantities. The probability that an OTM option will be exercised and an ITM option will not be exercised at maturity increases with τ.

98 The Black Scholes Option Pricing Model 6-99 The following table shows the performance of the hedging as a function of t. The limit t 0 yields the riskless BS portfolio strategy 1 1 t (weeks) L Table 6: Performance of Delta hedging Linearity Portfolios are linear. If a portfolio consists of w 1,...,w m stocks the delta of the portfolio is: m p = w j j j=1

99 The Black Scholes Option Pricing Model Example A portfolio of USD F options consists of bought calls (long position) with K = 1.70 EUR and τ = 4 months. The delta is 1 = sold calls (short position) with K = 1.75 EUR and τ = 6 months. The delta is 2 = sold puts (short position) with K = 1.75 EUR and τ = 3 months. The delta is 3 = 0.51 The delta of the portfolio is p = = The portfolio is delta neutral when USD are sold.

100 The Black Scholes Option Pricing Model Gamma and Theta Delta hedging: C is locally approximated by a linear function in S. Should t not be short, this is not adequate any more. A more accurate approximation can be considered. Taylor expansion of C as a function of S and t: C = C(S + S, t + t) C(S,t) = C S S + C t t C S 2( S)2 + O( t) S is of order t hence the dominant term is C S. If we consider terms of order t: C S +Θ t Γ( S)2

101 The Black Scholes Option Pricing Model Again: C S +Θ t Γ( S)2 Here Θ = C t is the Theta and Γ = 2 C S 2 the Gamma of the option. Theta is also called time decay. From BS formula: Θ = σs 2 τ ϕ(y +σ τ) rke rτ Φ(y) and Γ = 1 σs τ ϕ(y +σ τ) where y = log S K +(b σ2 2 )τ σ τ

102 The Black Scholes Option Pricing Model Gamma hedging consists of buying or selling derivatives. However, buying or selling further derivatives makes the portfolio value even more sensitive to changes in the stock price. As delta is constant for stocks and futures: Γ = 0. Therefore stocks and future contracts can be used to make a gamma neutral portfolio delta neutral.

103 The Black Scholes Option Pricing Model Example A portfolio of USD options and USD is neutral with Γ = On the future market a USD-call with B = 0.52 and Γ B = 1.20 is offered. The portfolio will be Γ-neutral by adding Γ/Γ B = calls. The is now: B = The -neutral position may be obtained by shorting USD from the portfolio. This will not change the Γ.

104 The Black Scholes Option Pricing Model Figure 15: Gamma as a function of stock price (right axis) and time of expiration (left axis). SFEgamma

105 The Black Scholes Option Pricing Model Figure 16: Theta as a function of stock price (right axis) and time of expiration (left axis). SFEtheta

106 The Black Scholes Option Pricing Model Time decay If and Γ are both zero the value of the portfolio changes essentially with Θ = C/ t. The parameter Θ is for most options negative: An option loses in value as it approaches the delivery date, even if all other parameters remained constant. From BS: rv = Θ+ 1 2 σ2 S 2 Γ where V is the value of the portfolio. Hence Θ and Γ are related in a straightforward way. Consequently, Θ can be used instead of Γ to gamma hedge a delta neutral portfolio.

107 The Black Scholes Option Pricing Model Rho and Vega The BS approach assumes constant volatility σ. Empirical evidence shows that this assumption is questionable. The Vega is: V = C σ Stock and futures have V = 0. Traded options have to be used to vega hedge a portfolio. Since a vega neutral portfolio is not necessarily delta neutral two distinct options have to be involved to achieve simultaneously V = 0 and Γ = 0.

108 The Black Scholes Option Pricing Model BS yields: V = S τϕ(y +σ τ) The Black Scholes formula was derived under the assumption of a constant volatility. It is therefore actually not justified to compute the derivative with respect to σ. However, the above formula for V is quite similar to a equation following on from a more general stochastic volatility model. The equation for V can be used as an approximation to the real vega.

109 The Black Scholes Option Pricing Model Figure 17: Vega as function of stock price (left axis) and time of expiration (right axis). SFEvega

110 The Black Scholes Option Pricing Model The Rho of an option is the derivative w.r.t. changes in the interest rate: ρ = C r BS yields for a call on a dividend free stock: ρ = K τ e rτ Φ(y) where y = log S K +(b σ2 2 )τ σ τ

111 The Black Scholes Option Pricing Model Volga and Vanna Volga and Vanna display the sensitivity of the volatility vega to changes in this volatility and in the stock price. Volga Volga is defined as the second derivative of the option price with respect to the volatility: Volga = V σ = 2 C 2 σ. The BS formula yields: Volga = S τ y(y +σ τ) σ ϕ ( y +σ τ ).

112 The Black Scholes Option Pricing Model Figure 18: Volga as a function of stock price (right axis) and time to maturity (left axis). SFEvolga

113 The Black Scholes Option Pricing Model Vanna The effect of changes in the stock price S on the volatility vega is given by vanna: Vanna = V S = 2 C σ S. Vanna, derived from the BS formula for a call option, is given by: ( ) Vanna = τ + 1 σ ϕ ( y +σ τ ).

114 The Black Scholes Option Pricing Model Figure 19: Vanna as a function of stock price (right axis) and time to maturity (left axis). SFEvanna

115 The Black Scholes Option Pricing Model Historical and implied volatility Historical volatility is an estimator for σ: S 0,...,S n stock prices at times 0, t,2 t,...,n t. Returns are R t = log S t S t 1, t = 1,...,n independent normal rv s, if the stock is modelled as a geometric Brownian motion. Hence, v = Var(R t ) = σ 2 t 1 n ˆv = (R t R n ) 2 (unbiased estimator of v) n 1 t=1 R n = 1 n R t n t=1

116 The Black Scholes Option Pricing Model The rv (n 1)ˆv/v is χ 2 n 1. Hence [ ) ] (ˆv v 2 [ 1 E = v (n 1) 2 Var (n 1)ˆv ] v = 2 n 1 since v = σ 2 t: ˆσ = ˆv/ t (ˆσ is the estimator from the sample) ( ) 1 E[ˆσ] = σ +O n it follows that the larger n the less the estimation will be biased [ ) ] (ˆσ σ 2 ( ) 1 1 E = σ 2(n 1) +O n (relative mean-squared error)

117 The Black Scholes Option Pricing Model Choice of t: t corresponds to 1 day since in most cases one considers daily quotations. For calendar days t = which however is not reasonable since volatility decreases over weekends. One better uses t = since the number of trading days is approximately 252.

118 The Black Scholes Option Pricing Model Choice of n Theoretically ˆσ becomes more and more reliable. In practice though σ is not constant. As a compromise one calculates ˆσ for the last 90 or 180 days.

119 The Black Scholes Option Pricing Model Implied volatility The unknown parameter in the BS formula is the standard deviation of the underlying stock. The implied volatility σ I is often used as the estimate of the standard deviation, which is calculated by: S Φ(y +σ I τ) e rτ K Φ(y) = C B with y = 1 σ I τ {log SK + ( r σ2 I 2 ) } τ. Unfortunately this equation has no closed form solution which means the equation must be solved numerically.

120 The Black Scholes Option Pricing Model Figure 20: Implied Volatility of the DA-Option on 29th of May 2005 SFEVolSurfPlot

121 The Black Scholes Option Pricing Model Example Two options of a stock are on the market. One of them is ATM and has σ I1 = The other is ITM and has σ I2 = ATM the dependence of option price and volatility is very strong, i.e. the market price of the first option tells us more about the σ, (σ I1 is the better approximation to σ). In an approximation to the true σ the first volatility should obtain higher weight. For example: σ = 0.8 σ I σ I2

122 The Black Scholes Option Pricing Model Realized volatility Let Y t = logs t be the logarithmic stock price; R t = Y( t) Y { (t 1)}, t = 1,2,...,n the returns over. The variance of R t : Var(R t ) = σ 2 (t). V(t) = t 0 σ2 (u)du is the integrated variance

123 The Black Scholes Option Pricing Model Using the entire past of Y(t), estimate the integrated variance V(t) by means of quadratic variation [Y](t): {Y} t = M j=1 [ Y { (t 1) + j M } Y { (t 1) + (j 1) M }] 2, M denotes intraday observations during each day for a 24-hour market, daily realized volatility based on 5-minute underlying returns is defined as the sum of 288 intra-day squared 5-minute returns, taken day by day.