Black-Scholes Option Pricing

Transcription

1 Black-Scholes Option Pricing The pricing kernel furnishes an alternate derivation of the Black-Scholes formula for the price of a call option. Arbitrage is again the foundation for the theory. 1

2 Risk-Free Asset and Stock The risk-free asset has the return r dt; a one-dollar investment at t is worth 1 + r dt at t + dt. The stock has the return ds t =(r + µ) dt + σ dz t ; s t a one-dollar investment at t is worth 1 +(r + µ) dt + σ dz t at t + dt. 2

3 Stochastic Discount Factor Let p t denote a stochastic discount factor. For an asset with price q t and future payments d t, p t q t = t E t (p τ d τ ) dτ, the present discounted value of the future payments. 3

4 Then p t q t = p t d t dt + E t (p τ d τ ) dτ t+dt ( ) = p t d t dt + E t p τ d τ dτ t+dt = p t d t dt + E t (p t+dt q t+dt ). 4

5 Stock Pricing For the stochastic discount factor to price the stock, p t s t = E t (p t+dt s t+dt ). Hence p t s t = E t (p t+dt s t+dt ) = E t [(p t + dp t )(s t + ds t )] = E t (p t s t + s t dp t + p t ds t + dp t ds t ). 5

6 Dividing by p t s t and cancelling gives 0 = E t ( dpt p t + ds t s t + dp t p t ds t s t ). (1) 6

7 Risk-Free Asset Pricing For the stochastic discount factor to price the risk-free asset, so p t = E t [p t+dt (1 + r dt)], p t = E t [(p t + dp t )(1 + r dt)] = E t (p t + p t r dt + dp t ), since the second-order term is zero. Dividing by p t and cancelling gives ( 0 = E t r dt + dp ) t. (2) 7 p t

8 Pricing Kernel The pricing kernel p t is a stochastic discount factor of the form dp t p t = a dt + b dz t, the span of the returns on the risk-free asset and the stock. 8

9 By (2), for the pricing kernel to price the risk-free asset requires a = r. 9

10 By (1), for the pricing kernel to price the stock requires ( dpt 0 = E t + ds t + dp ) t ds t p t s t p t s t = E t {( r dt + b dz t )+[(r + µ) dt + σ dz t ] +( r dt + b dz t )[(r + µ) dt + σ dz t ]} = E t [ r dt +(r + µ) dt + bσ dt], so b = µ/σ. 10

11 Thus the pricing kernel follows the stochastic differential equation dp t p t = r dt µ σ dz t. For the initial condition p 0 = 1, the solution is [ ln p t = r 1 ( µ ) ] 2 t µ 2 σ σ z t. 11

12 Arbitrage Following Black and Scholes, assume that the call price c t is a function of the stock price. Then its return lies in the span of the returns of the stock price and the risk-free asset. The absence of arbitrage then requires that the return on the call can be priced by the pricing kernel for the stock and the risk-free asset. 12

13 Black-Scholes Partial Differential Equation If c t = c(s t,τ) (τ is the time to expiration), by the second-order Taylor series expansion dc t = c τ dt + c s ds t c ss (ds t ) 2 = c τ dt + c s s t [(r + µ) dt + σ dz t ] c sss 2 t [(r + µ) dt + σ dz t ] 2. For the pricing kernel to price the call, 0 = E t ( dpt p t + dc t c t 13 + dp t p t dc t c t ).

14 Hence [ ( 0 = E t r dt µ ) σ dz t + dc ( t + r dt µ ) ] c t σ dz dct t. c t [ = { r + 1ct c τ + c s s t (r + µ)+ 1 2 c ssst 2 σ 2 µ ]} σ c ss t σ which yields the Black-Scholes partial differential equation 0 = rc t c τ + c s s t r c sss 2 t σ 2. dt, (Here c t is the call price at time t,butc τ is the partial derivative of the price with respect to τ.) 14

15 Present Discounted Value Equivalently, the call price is the present value of its exercise value at expiration, using the pricing kernel as the stochastic discount factor. Theorem 1 Here c(s 0,t)=E 0 [p t c(s t,0)]. (3) c(s t,0)=max[s t x,0], the value at expiration with striking price x. 15

16 Computation of the Expected Value Since s t = s 0 exp [(r + µ 12 ] )t σ 2 + σz t, therefore s t x for z t 1 [ ln(x/s 0 ) (r + µ 12 ) σ σ 2 ] t := z. (4) 16

17 From our previous work, [ p t s t = s 0 exp 1 ( 2 [ p t x = xe rt exp 1 2 σ µ ) 2 ( t + σ µ ] )z t σ σ ( µ σ (5) ) ] 2 µ t σ z t. (6) We calculate the expected value of these expressions over the range (4). 17

18 The probability density function of z t is ( 1 exp 1 ) 2πt 2 z2 t /t. When one integrates to find the expectations, the quadratic in z t combines with the terms linear in z t in the exponentials (5)-(6) to form a quadratic. This quadratic is again a normal probability density function, still with variance t, but the mean is non-zero. 18

19 E 0 (p t s t ) over z t z { = s 0 exp z 1 2πt exp = s 0 = s 0 F z [ 1 2 ( σ µ ) 2 ( t + σ µ ) σ σ ( 1 ) } 2 z2 /t dz { 1 exp 1 2πt 2 ) {[( σ µ σ ] z [ ( z σ µ ) ] } 2 t /t dz σ ] t z / } t in which F is the cumulative distribution function for a normal with mean zero and variance one. 19

20 Substituting for z gives E 0 (p t s t ) over z t z {( = s 0 F σ µ ) t σ [ + ln(s 0 /x)+ (r + µ 12 ) σ 2 = s 0 F ] t /σ } t {[ ln(s 0 /x)+ (r + 12 ) ] σ 2 t /σ } t. Here µ has cancelled out! 20

21 E 0 (p t x) over z t z { = xe rt exp z 1 2πt exp [ 1 ( µ ) ] 2 µ t 2 σ σ z ( 1 ) } 2 z2 /t dz [ = xe rt 1 exp 1 (z + µ ) ] 2 z 2πt 2 σ t /t = xe rt F [( µ ) σ t z / ] t. dz 21

22 Substituting for z gives E 0 (p t x) over z t z { = xe rt F µ [ t + ln(s 0 /x)+ (r + µ 12 ) σ σ 2 {[ = xe rt F ln(s 0 /x)+ (r 12 ) ] σ 2 t /σ } t. Again µ has cancelled out! ] t /σ } t 22

23 Black-Scholes Formula The price of the call option is the difference in the two present discounted values. Theorem 2 (Black-Scholes) The price of the call option is E 0 (p t max[s t x,0]) {[ = s 0 F ln(s 0 /x)+ (r + 12 ) σ 2 xe rt F ] t /σ } t {[ ln(s 0 /x)+ (r 12 ) ] σ 2 t /σ } t. 23