The Black- Scholes Equation Chapter 21 The Black- Scholes Equation

Differential Equations and Valuation Under Uncertainty An example of transformation of a valuation equation into a differential equation: stock valuation Consider the familiar stock valuation setup which can be rewritten as Dividing by h and letting h  0 We obtain the differential equation that describes the evolution of the stock price over time to generate an appropriate rate of return Change in stock price Cash payout Return on Stock Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

Differential Equations and Valuation Under Uncertainty (cont’d) Dividend-paying stocks If the value of the stock at time T is , that is the terminal boundary condition is , the solution is Bonds Let S(t) represent the price of a zero-coupon bond that pays $1 at T The general solution to this differential equation is S(t)=Ae-r(T-t) With the terminal boundary condition S(T)=$1, the particular solution for the bond value is S(t)= $1 x e-r(T-t) Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

The Black-Scholes Equation Consider the problem of creating a riskless hedge for an option position through trading in shares and bonds Assume that the stock price follows geometric Brownian motion where a is the expected return in the stock, s is the stock’s volatility, and d is the continuous dividend yield If we invest W in these bonds, the change in value of the bond is dW = rWdt Let I denote the total investment in the option, N stocks, and W in the risk-free bonds so that the total investment is zero I = V (S, t) + NS + W =0 Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

The Black-Scholes Equation (cont’d) Applying Itô’s Lemma Since the option D is VS , set N = –VS ; this will make W = VS S – V Also, dI = 0, because with zero investment, the return should also be zero. Substituting all these and dividing by dt This is the Black-Scholes PDE for any contingent claim, assuming Underlying asset follows const. volatility geometric Brownian motion Underlying asset pays a continuous proportional dividend at d rate The contingent claim itself pays no dividend The interest rate is fixed with equal borrowing and lending rates There are no transaction costs Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

The Black-Scholes Equation (cont’d) The call price formula satisfies the Black-Scholes PDE using the appropriate boundary condition. Similarly, it can be shown that the put pricing formula, and the formulas for asset-or-nothing options, cash-or-nothing options, and gap options satisfy the Black-Scholes PDE using the appropriate boundary condition in each case Black-Scholes PDE can be generalized to the case when one uses equilibrium expected return on the underlying asset instead of the risk-free return Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

The Black-Scholes Equation (cont’d) When the underlying asset is not an investment asset where the dividend yield d in the original PDE is replaced by , the “lease rate” of the asset is the difference between the equilibrium expected return and the actual expected return m on the noninvestment asset Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

Risk-Neutral Pricing The original Black-Scholes PDE does not contain the expected stock return a, but only the risk-free rate. The following version of the Black-Scholes equation indicates that the option appreciates on average at the risk-free rate Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

Risk-Neutral Pricing (cont’d) Clearly related to the above equation is the following equation For the risk-neutral process the above equation becomes the Kolmogorov backward equation Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

Changing the Numeraire What happens when the number of options received at expiration is random? Currency translation: a cash flow originating in yen can be valued in yen, or in some other currency (more on this in Chapter 22) Quantity uncertainty: an agricultural producer who wants to insure production of an entire field must hedge total revenue rather than quantity alone All-or-nothing options: they can be structured either to pay cash if a certain event occurs or in shares. The value of an all-or-nothing option is the first term in the BS formula, which can be viewed as a risk-neutral probability with a change in numeraire Copyright © 2006 Pearson Addison-Wesley. All rights reserved.