Martingale Pricing of a European Call and the Black–Scholes PDE

Summary

The document poses a problem: represent the value of a European call as the conditional expectation of its terminal payoff under a stock-price model driven by Brownian motion. The payoff is the positive part of the difference between the terminal stock price and the strike, and the user asks how to derive both the value process and the Black–Scholes partial differential equation.

No derivation or solution is supplied, so there is no method, evidence, or discussion of assumptions beyond the stated stochastic stock model and option payoff. In particular, it does not specify a risk-free rate or explain the measure and conditions needed to connect the conditional expectation to the PDE. It is useful as a clear statement of a mathematical problem, but does not itself teach the solution.

Key ideas

• The option payoff is the positive part of terminal stock price minus strike.
• The value process is posed as the conditional expectation of that payoff given current information.
• The stock price is modeled with a time-varying volatility and Brownian motion.
• The document asks for a derivation but provides no answer or supporting evidence.

Tags

# Martingale representation of European option


# Martingale representation of European option












Let stock price $S$ satisfy

$$S(t)=S(0)e^{(\int_0^t\sigma(s)dB_s-\frac{1}{2}\int_0^t\sigma(s)^2ds)}$$

I want to calculate the Martingale representation $V(t)=E(F|F_t)$ of European option with strike price $M$ and maturity $T$ which is given by

$$F=(S(T)-M)^+$$

How to find the solution and the Black-Scholes PDE?

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