Step 1: Constructing the Hedging Portfolio
We define the hedging portfolio \( \Pi \) as:
\( \Pi = V(S, t) - \Delta S \) (equation 1)
Where \( V(S, t) \) is the value of the call option as a function of the stock price \( S \) and time \( t \), and \( \Delta \) is the number of shares held in the portfolio.
The instantaneous change in the portfolio value is given by:
\( d\Pi = dV - \Delta dS \) (equation 2)
Step 3: Stock Price Dynamics
The stock price follows a stochastic process described by:
\( dS = \mu S dt + \sigma S dW \) (equation 3)
- \( \mu S dt \): Drift term, representing the deterministic component of the stock's return.
- \( \sigma S dW \): Diffusion term, capturing the random fluctuations in the stock price, where \( W \) is Brownian motion.
Using Itô's Lemma , the differential of the option value \( V \) is given by:
\( dV = \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} (dS)^2 \) (equation 4)
- \( \frac{\partial V}{\partial t} \): Theta, the rate of change of \( V \) with respect to time.
- \( \frac{\partial V}{\partial S} \): Delta, the sensitivity of \( V \) to changes in \( S \).
- \( \frac{\partial^2 V}{\partial S^2} \): Gamma, the convexity of \( V \) with respect to \( S \).
Substituting \( dS \) from equation (3) into \( (dS)^2 \), we find:
\( (dS)^2 = (\mu S dt + \sigma S dW)^2 \)
Simplifying using stochastic calculus rules :
- \( (\mu S dt)^2 = 0 \), since \( (dt)^2 = 0 \).
- \( 2(\mu S dt)(\sigma S dW) = 0 \), since \( dt \cdot dW = 0 \).
- \( (\sigma S dW)^2 = \sigma^2 S^2 dt \), using \( (dW)^2 = dt \).
Thus, the result is:
\( (dS)^2 = \sigma^2 S^2 dt \) (equation 5)
Using equation (5), the differential of \( V \) becomes:
\( dV = \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} dt \) (equation 6)
Substituting \( dV \) from equation (6) into equation (2), we have:
\( d\Pi = \left( \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \right) dt + \left( \frac{\partial V}{\partial S} - \Delta \right) dS \)
Setting \( \Delta = \frac{\partial V}{\partial S} \) ensures that the stochastic term \( \left( \frac{\partial V}{\partial S} - \Delta \right) dS \) vanishes, leaving:
\( d\Pi = \left( \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \right) dt \) (equation 7)
Since the portfolio is risk-free, its change must equal the risk-free rate:
\( d\Pi = r \Pi dt \)
Substituting \( \Pi = V - \Delta S \) and \( \Delta = \frac{\partial V}{\partial S} \):
\( \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} = r V - r S \frac{\partial V}{\partial S} \)
Rearranging terms, we obtain the Black-Scholes PDE:
\( \frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - r V = 0 \)
In this equation, we identify key terms:
- \( \frac{\partial V}{\partial t} \): This represents the time value of the option, showing how the option's value decreases as it approaches maturity.
- \( r S \frac{\partial V}{\partial S} \): This term corresponds to the sensitivity of the option to the stock price (Delta) adjusted by the risk-free rate.
- \( \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \): This captures the impact of volatility and convexity (Gamma) on the option's value.
- \( -r V \): This accounts for the discounting effect at the risk-free rate.
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