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Black-Scholes Equation April 15, 2008

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1 Contents Options Black Scholes PDE Solution Method

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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:

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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.

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4 Option Payoff at T Payoff DiagramPayout Function

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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.

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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.

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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

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

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

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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.

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11 Black-Scholes Solution (5) Finally, where

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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

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

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14 References 1.F. Black, M. Scholes, The pricing of options and corporate liabilities, Jour. Political Economy, vol. 81 (1973), pp. 637-654. 2.P. Wilmott, S. Howison, J. Dewynne, The Mathematics of Financial Derivatives, A Student Introduction, Cambridge University Press, 1995.

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