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Deriving a Pricing PDE for a Stochastic-Rate Floorlet

Article Quant Q&A · Author: Andrei
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Summary

The document asks how to price a floorlet whose payoff depends on a linearly compounded rate derived from a bond price. It specifies a risk-neutral model in which the short rate and a second state variable, sigma, both evolve stochastically. The proposed approach is to express the derivative value as a function of time and those two state variables, then apply the multidimensional Itô formula to derive its pricing equation.

The response points to the standard Heston PDE derivation as a template for organizing the drift, diffusion, and cross-derivative terms. It does not work through the resulting PDE, give boundary or terminal conditions, or explain how the bond price inside the payoff is computed under the stated model. Consequently, this is guidance on a derivation method rather than a complete pricing solution; applying it requires careful treatment of the model’s two correlated or independent Brownian drivers and the bond-price dependence of the payoff.

Key ideas

  • Represent the derivative value as a function of time, the short rate, and the volatility state.
  • Apply multidimensional Itô calculus to obtain the generator and pricing PDE.
  • A Heston-style PDE derivation can serve as a structural reference.
  • The response does not provide the full equation or the conditions needed to solve it.

Tags

Fixed incomeDerivatives pricingVolatility
Full text
# What is the PDE for this interest rate derivative?


# What is the PDE for this interest rate derivative?












We have the following model for the short rate $r_t$under $\mathbb{Q}$:

$$dr_t=(2\%-r_t)dt+\sqrt{r_t+\sigma_t}dW^1_t\\d\sigma_t=(5\%-\sigma_t)dt+\sqrt{\sigma_t}dW^2_t$$

What is the PDE of which the solution gives the price of the floorlet with the following payoff in $t=1$:

$$X=0.5\bigg[ 2.5\%-L(0.5,1) \bigg]^+$$

where $L(0.5,1)=(P(0.5,1)^{-1}-1)\frac{1}{0.5}$ is the linearly compounded rate from $0.5$ to $1$.

## Answer by quant_son (score 1)

https://quant.stackexchange.com/a/74394

You can work out the specific PDE by applying the multi-dimensional Itô formula to the (sufficiently smooth) value function $\Pi_t = f(t, r, \sigma)$.

Follow the steps from the regular Heston PDE derivation, e.g. like this section 2 https://www.frouah.com/finance%20notes/The%20Heston%20model%20short%20version.pdf

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.