Deriving an Option PDE with a Stochastic Short Rate

Summary

The document poses a derivation question for an option whose value depends on time, a stochastic short rate, and the stock price. It compares two proposed routes: constructing a hedge with the stock and a zero-coupon bond, then using the bond’s pricing equation; or applying Itô’s lemma to the discounted option value and requiring its drift to vanish. This frames the key modeling issue as how to account for risk from both the stock and the rate when deriving a pricing equation.

No answers or derivation are included, so the document does not establish that the approaches are equivalent or state the assumptions under which either method applies. In particular, it leaves unspecified the rate dynamics, the bond model, market completeness, and the relevant risk premia. The material is therefore useful as a prompt identifying two common PDE derivation approaches, but it is not a complete recipe or a worked solution.

Key ideas

• The option value is modeled as depending on time, the short rate, and the underlying stock price.
• One proposed PDE derivation hedges stock and interest-rate exposure using the stock and a zero-coupon bond.
• Another proposed route applies Itô’s lemma to the discounted option value and sets its drift to zero.
• The document does not provide a solution or establish when the two approaches agree.
• Rate dynamics and assumptions about hedging and market completeness remain unspecified.

Tags

# Finding a PDE for an option $V(t,r(t),S(t))$


# Finding a PDE for an option $V(t,r(t),S(t))$












I have 2 approaches in my mind for finding a pde of an option that depends both on the short rate as well as the stock price- $V(t,r(t),S(t)$. Are these equivalent?

- Find a hedging portfolio by shorting delta of the stock and delta ratio of the option to a ZCB. So my portfolio has the underlying stock, a ZCB, and the money market. What is left must be riskless and I get a PDE. The ZCB has its own PDE (say Hull White) which I can use that to manipulate and obtain the final PDE without having a ZCB price in it.

- Use Ito's Lemma and write that $exp(-∫r(u)du) V(t,r(t),S(t))$ is a martingale (where limits of the integral are from 0 to t.) and thus put its drift equal to 0.

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