Hedging Derivation
The original derivation by Black and Scholes is based on a dynamic hedging argument. In order to apply this argument, we will have to make the following assumptions:
- We can sell any amount of stock at any time, even fractional parts
- We can…
Now we will create a portfolio which involves options (in some way), stocks, and which is riskless. We note that if the stock S changes with some very small \delta S, then the option C will change with approximately \delta C = C_S \delta S, where C_S = \frac{\partial C}{\partial S}. Therefore, our strategy will be to own one option and to keep selling continuously C_S parts of the stock. Our portfolio P can be expressed via the equation:
P(S, t) = C(S, t) - C_S SSince we have hedged out all the risk, the value of the portfolio must increase like a risk-free bond. Applying Ito\’s formula, we get:
dP = C_S \mu Sdt + \frac{dC}{dt}dt + \frac{\sigma^2S^2}{2}\frac{\partial^2 C}{\partial S^2}dt - C_S mu S dt = \left(\frac{dC}{dt} + \frac{\sigma^2 S^2}{2}\frac{\partial^2 C}{\partial S^2}\right) dt = r P dt.Therefore, combining (ref{1}) and (ref{2}), we get the famous Black-Scholes equation:
begin{align}
rC – r frac{partial C}{partial S} S = frac{dC}{dt} + frac{sigma^2 S^2}{2}frac{partial^2 c}{partial S^2}.
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