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### 6.2.1 The Black-Scholes equation for American options

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Following the same procedure as in chapter 3, the Black-Scholes model is first extended to account for the possibility of exercising the American option anytime up to the expiry. Allowing for a continuous dividend payment at a rate , the random walk for the underlying price increment (3.3.1#eq.1) is modified to

 (6.2.1#eq.1)

Create a portfolio combining an American option with a number of shares and store the earnings from a dividend yield. The initial value and the incremental change are

 (6.2.1#eq.2)

 (6.2.1#eq.3)

Using Itô's lemma (3.3.2#eq.2) to calculate the stochastic increment in the option value as a function of the underlying, the random component is again eliminated by continuously re-hedging the portfolio with a number of shares. No arbitrage arguments show that without taking any risk, the portfolio can at most earn the risk-free return of the spot rate. Because an American option can be exercised any time until it expires, the incremental change in the portfolio value satisfies the inequality

 (6.2.1#eq.4)

which leads directly to Black-Scholes equation for American options

 (6.2.1#eq.5)

In addition to the usual boundary and terminal condition , this inequality (known in mathematics as an obstacle problem) must be supplemented by the free boundary condition . Apart from that, the Black-Scholes equation is the same for European options paying a dividend with the strict equality here replaced by an inequality. The same change of variables (4.3.1#eq.1,4.3.1#eq.4) can therefore be used to transform the problem to log-normal variables

 (6.2.1#eq.6)

keeping in mind that the solution has to satisfy the corresponding free-boundary condition of the form .

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