Black-Scholes Equation April 15, 2008

1 Contents Options Black Scholes PDE Solution Method

2 Derivatives There are many financial instruments, stocks, bonds,… A derivative is a financial instrument whose value is derived from the value of some other instrument(s) Forward contract: an agreement now to receive a specified goods at a future time and at a specified price. Swap: agreement to exchange certain commodities Option:

3 Options European Option: At a pre-specified time, called the expiry or expiration date, T, the holder of the option has the right, but not the obligation, to exchange a pre- specified asset, called the underlying asset, S, at a pre- specified price, called the strike, K. European Call Option: Buy the asset, pay the strike. European Put Option: Sell the asset, receive the strike. American Option: exercise can take place any time before the expiration date.

4 Option Payoff at T Payoff DiagramPayout Function

5 Why PG&E cares about Option Value Serving the load: Load varies from forecast, option gives one the flexibility to cover the shortfalls. Hedging risks: Reduces the risks of load and prices. Reducing cost: Less expensive way to serve the load.

6 Option Value The value of the option depends on T, K, risk-free discount rate r, and assumptions on S. Bachelier’s assumption (1900): change in S is a random walk. Black-Scholes assumed where  is called the drift,  is the volatility, both constants, and dX is a normally distributed random variable with mean 0 and variance dt.

7 Black-Scholes Equation With the above assumptions, Black and Scholes derived that, for any option whose value V(S,t) depending on S and t only, V(S,t) satisfies The terminal conditions for Call and Put are (1) The boundary conditions for Call and Put are

8 Black-Scholes Solution (2) To solve equation (1), we make the following change of variables: This leads to the following

9 Black-Scholes Solution To further simplify the equation, let We get (3) Choosing Equation (2) becomes the heat equation

10 Black-Scholes Solution (4) The heat equation has the solution where For Call, we have Substituting the initial condition in (4) yields the value of the call option.

11 Black-Scholes Solution (5) Finally, where

12 Black-Scholes Solution Using the simple, distribution free relation the value of put can be found as where h=1 if it is a call, h=-1 if it is a put. Combining with equation (5) we can write

13 American Option For American Option, the same differential equation, terminal condition and boundary conditions hold To permit early exercise, we impose Analytical solution?

14 References 1.F. Black, M. Scholes, The pricing of options and corporate liabilities, Jour. Political Economy, vol. 81 (1973), pp P. Wilmott, S. Howison, J. Dewynne, The Mathematics of Financial Derivatives, A Student Introduction, Cambridge University Press, 1995.