Pricing Calls with a Stochastic Short Rate

Summary

The document poses a risk-neutral pricing problem for a European call when the short rate follows a Brownian process and is driven by the same Brownian motion as the stock. It asks whether the usual Black–Scholes formula can be applied using the average rate over the option’s life, and whether an explicit price can be obtained.

No solution or supporting evidence is provided, so the key learning value is recognizing that stochastic rates and shared rate–equity shocks complicate the standard Black–Scholes setup. The proposed average-rate substitution is a question, not an established method. Any pricing formula would depend on the joint dynamics and assumptions in the model; the document does not resolve those details or provide a derivation.

Key ideas

• The document asks how to price a European call when the short rate is stochastic.
• The stock and short rate are driven by the same Brownian motion, creating dependence between them.
• It questions whether replacing the stochastic rate with its time average in Black–Scholes gives a valid price.
• No formula, derivation, or answer is included.

Tags

# Black & Scholes with stochastic interest rate


# Black & Scholes with stochastic interest rate












Consider the following model

$$\begin{cases} dS_t=r_tS_tdt+\sigma S_tdW_t, \\ dr_t=adt+\eta dW_t\\ \end{cases} $$ where $W$ is a Brownian motion and $\sigma, a ,b, \eta$ are positive constants.

I have to find a formula of the price of a call option:

$$E \left[ e^{-\int_0^T r_s ds}(S_T-K)^+ \right]. $$

Is it $$BS(S_0,K,-\frac{1}{T}\int_0^T r_s ds,T,\sigma) \quad ?$$

Can I obtain a more explicit formula?

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