Black–Scholes PDE and Replication with Time-Varying Parameters

Summary

The document poses a derivation problem for a European call in a Black–Scholes setting with deterministic, time-varying interest rates, drift, and volatility. It presents the option value as a risk-neutral expectation of the terminal payoff and asks how to obtain the corresponding partial differential equation and replicating strategy.

No derivation or answer is included, so the document does not establish the PDE or hedge explicitly. Its useful focus is the connection between risk-neutral valuation, the pricing equation, and replication when model coefficients vary over time. The setup also refers to the stock as discounted while writing a stock process with drift, so readers should check the numeraire and price conventions before deriving results. The document provides no numerical example, proof, or discussion of assumptions beyond deterministic bounded coefficients.

Key ideas

• The setup considers a European call with deterministic time-varying rates and volatility.
• It expresses the option value as an expectation under a changed probability measure.
• The central question is how to derive the pricing PDE and replicating hedge from that expectation.
• The document provides no solution, so the PDE and strategy are left unresolved.
• The discounted-stock convention should be clarified before applying the model.

Tags

# Finding the PDE and replicating strategy of a european contigent claim


# Finding the PDE and replicating strategy of a european contigent claim












Suppose that we have the Black and Scholes model where the interest rate and the volatility are time varying:

$dB(t)=r(t)B(t)dt$ and

$dS(t)=S(t)b(t)dt+S(t)\sigma(t)dW(t), S(0)=s>0$

where $r,b,\sigma$ deterministic, positive and bounded.

I know that the price of european call option with strike price $K$ is given by

$\tilde{V}(t)=U(t,S(t))$ and

$U(t,S(t)=\tilde{E}[max[S(0)e^{\int_t^T \sigma(s)d\tilde{W}(s)-\dfrac{1}{2}\int_t^T \sigma^2(s)ds}-Ke^{-\int_t^T r(s)ds}],0]$

where $\tilde{E}$ the expected value under the new probability measure $\tilde{P}$ and $\tilde{W}$ is a wiener process. $S(t)$ here is the discounted price of the stock.

Can someone help me how to find the PDE and the replicating strategy of the european call option?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.