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6.2.1 The Black-Scholes equation for American options $ \spadesuit $

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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 $ D_0$ , the random walk for the underlying price increment (3.3.1#eq.1) is modified to

$\displaystyle \frac{dS}{S} = (\mu-D_0) dt +\sigma dW(t)$ (6.2.1#eq.1)

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

$\displaystyle \Pi=V -\Delta S$ (6.2.1#eq.2)

$\displaystyle d\Pi=dV -\Delta (dS + D_0S dt)$ (6.2.1#eq.3)

Using Itô's lemma (3.3.2#eq.2) to calculate the stochastic increment in the option value $ dV$ as a function of the underlying, the random component is again eliminated by continuously re-hedging the portfolio with a number $ \Delta=\partial V/\partial S$ 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

$\displaystyle d\Pi=\left[ \frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2 S^...
...l^2 V}{\partial S^2} -D_0 S\frac{\partial V}{\partial S} \right] dt \le r\Pi dt$ (6.2.1#eq.4)

which leads directly to Black-Scholes equation for American options

$\displaystyle \frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} +(r-D_0)S \frac{\partial V}{\partial S} -rV \le 0$ (6.2.1#eq.5)

In addition to the usual boundary and terminal condition $ V(S,T)=\Lambda(S)$ , this inequality (known in mathematics as an obstacle problem) must be supplemented by the free boundary condition $ V(S,t)\ge \Lambda(S), \forall t$ . 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 $ u(x,\tau)$

$\displaystyle \frac{\partial u}{\partial \tau} -\frac{\partial^2 u}{\partial x^2} \ge 0$ (6.2.1#eq.6)

keeping in mind that the solution has to satisfy the corresponding free-boundary condition of the form $ u(x,\tau)\ge c(x,\tau)$ .

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