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Appendix 27A: An Alternative Method to Derive The Black-Scholes Option Pricing Model (Related to 27.5) By Cheng Few Lee Joseph Finnerty John Lee Alice C Lee Donald Wort Appendix 27A: An Alternative Method to Derive The Black-Scholes Option Pricing Model • • • • 2 Perhaps it is unclear why it is assumed that investors have riskneutral preferences when the usual assumption in finance courses is that investors are risk averse. It is feasible to make this simplistic assumption because investors are able to create riskless portfolios by combining call options with their underlying securities. Since the creation of a riskless hedge places no restrictions on investor preferences other than nonsatiation, the valuation of the option and its underlying asset will be independent of investor risk preferences. Therefore, a call option will trade at the same price in riskneutral economy as it will in a risk-averse or risk-preferent economy. Appendix 27A.1: Assumptions and the Present Value of the Expected Terminal Option Price By Cheng Few Lee Joseph Finnerty John Lee Alice C Lee Donald Wort Appendix 27A.1: Assumptions and the Present Value of the Expected Terminal Option Price • • • In the risk-neutral assumptions of Cox and Ross (1976) and Rubinstein (1976), today’s option price can be determined by discounting the expected value of the terminal option price by the riskless rate of interest. Today’s call option price is: (27A.1) C exp rt Max St X ,0 where C the market value of the call option; r riskless rate of interest; t time to expiration; St the market value of the underlying stock at time t; and X exercise or striking price. 4 Appendix 27A.1: Assumptions and the Present Value of the Expected Terminal Option Price • Assuming that the call option expires in the money, then the present value of the expected terminal option is equal to the present value of the difference between the expected terminal stock price and the exercise price as: C exp rt E Max St X , 0 (27A.2) exp rt x St X h St dSt where h St is the log-normal density function of St • To evaluate the integral in (27A.2), rewrite it as the difference between two integrals: C exp rt St h St dSt X h St dSt (27A.3) x x Ex St exp rt X exp rt 1 H X Ex St the partial expectation of St , truncated from below at x; and H X the probability that St X 5 Appendix 27A.1: Assumptions and the Present Value of the Expected Terminal Option Price • The terminal stock price can be rewritten as the product of the current price and the t-period log-normally distributed price ratio. So, Equation (27A.3) can also be rewritten as: St St dSt St dSt (27A.4) C exp rt S g X x s g S S x s S S S S S exp rt Ex S t S X X exp rt 1 G S where S g t S S Ex S t S log normal density function of St S ; the partial expextation of St S , truncated from below at x S ; X G the probability that St S X S . S 6 Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price By Cheng Few Lee Joseph Finnerty John Lee Alice C Lee Donald Wort Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price • • The right-hand side of Equation (27A.4) is evaluated by considering the two integrals separately. The first integral, S exp rt Ex S St S , can be solved by assuming the return on the underlying stock follows a stationary random walk: St exp Kt (27A.5) S S ln t S • 8 Kt Following Garven (1986), it can be transformed into a density function of a normally distributed variable Kt according to the relationship St S exp Kt as: (27A.6) St St g f Kt S S Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price • These transformations allow the first integral in Equation (27A.4) to be rewritten as: St S exp rt Ex S S exp rt f Kt exp Kt t dK ln x S S 12 2 2 f Kt 2 K t exp 12 Kt K t K2 • Substitution yields: St S exp rt Ex S S 1 2 2 S exp rt 2 K t 2 exp Kt exp 12 Kt K t K2 t t dK ln x S (27A.7) • 9 Equation (27A.7)’s integrand can be simplified by adding the terms in the two exponents, multiplying and dividing the result by exp 12 K2 t Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price First, expand the term Kt K t and factor out t so that: 2 exp Kt exp 12 Kt K t K2 t Next, factor out t so: exp Kt exp 12 t K 2 2K K K2 K2 2 Now combine the two exponents: exp 12 t ( K 2 2 K K K2 2 K2 K ) K2 Now, multiply and divide this result byexp 12 K2 t to get: exp 12 t ( K 2 2 K K K2 2 K2 K K4 K4 ) K2 Next, rearrange and combine terms to get: 2 2 exp t K K K K4 2 K K2 K2 1 2 exp K 1 2 10 2 K t exp 1 2 Kt K 2 K t 2 K2 t (27A.8) Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price In Equation (27A.8), exp t E S S K 1 2 2 K t Therefore, Equation (27A.7) becomes: (27A.9) S S exp rt Ex S t S 2 1 2 St 2 2 1 S E exp rt 2 K t exp ( Kt ) exp 2 Kt K K t K2 t ln x S S • • Since the equilibrium rate of return in a risk-neutral economy is the riskless rate: S S E t exp rt S exp rt exp rt S S So Equation (27A.9) becomes St (27A.10) S exp rt Ex S S S 2 t 2 K 11 1 2 exp Kt K ln x S 1 2 2 K t 2 K2 t t dK Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price • To complete the simplification of this part of the Black–Scholes formula, define a standard normal random variable y: y Kt K K2 t K2 t1 2 Kt K K2 t K t1 2 y, • The lower limit of integration becomes: • Further simplify the integrand by noting that the assumption of a risk-neural economy implies: ln x S K K2 t K t1 2 exp K 12 K2 t exp rt , • 12 t dK K t1 2 dy Hence, K K 12 K2 t rt 12 K2 t r 12 K2 t Appendix 27A.2: Present Value of the Partial Expectation of the Terminal Stock Price • The lower limit of integration is now: ln S x r 12 K2 t K t1 2 d1 • • Substituting this into Equation (27A.10) and making the transformation to y yields: 2 12 St S exp rt Ex S S exp 12 y 2 dy d1 S Since y is a standard normal random variable (distribution is symmetric around zero) the limits of integration can be exchanged: S S exp rt Ex S t S 13 d1 1 2 S exp y 2 S N d1 2 12 dy (27A.11) Appendix 27A.3: Present Value of the Exercise Price under Uncertainty By Cheng Few Lee Joseph Finnerty John Lee Alice C Lee Donald Wort Appendix 27A.3: Present Value of the Exercise Price under Uncertainty • Start by making the logarithmic transformation: • The differential can be written as: S d t exp Kt t dK S Therefore, X exp rt 1 G X S X exp rt f Kt t dK ln X S • S ln t S X exp rt 2 t 2 K • 1 2 ln X Kt 2 exp 12 Kt K t K2 t t dK S The integrand is now simplified by following the same procedure used in simplifying the previous integral. Define a standard normal random variable Z: Kt K t Z 12 Kt 15 (27A.12) Appendix 27A.3: Present Value of the Exercise Price under Uncertainty • Making the transformation from Kt to Z means the lower limit of integration becomes: ln X S K t K t1 2 • Again, note that the assumption of a risk-neutral economy implies: 2 exp K 12 K t exp rt • Taking the natural logarithm of both sides yields: • K 12 K2 t rt , K t r 12 K2 t Therefore, the lower limit of integration becomes: ln S x r 12 K2 t K t1 2 16 d1 K t1 2 d 2 Appendix 27A.3: Present Value of the Exercise Price under Uncertainty • Substitution yields: x exp rt 1 G X S x exp rt exp 12 Z 2 d2 x exp rt exp 12 Z 2 d2 dZ x exp rt N d 2 (27A.13) • Substituting, Equations (27A.11) and (27A.13) into Equation (27A.4) completes the derivation of the Black–Scholes formula: (27A.14) C S N d1 X exp rt N d2 • This appendix provides a simple derivation of the Black–Scholes call-option pricing formula. Under an assumption of risk neutrality the Black–Scholes formula was derived using only differential and integral calculus and a basic knowledge of normal and log-normal distributions. • 17 2 12 12 2 dZ