options - Other numerraire choices when applying Feynman Kac

all of the books and notes I have seen on the Feynman Kac formula mostly applied to Risk neutral measure, i.e. different interest rate models, stochastic volatility, etc. I think risk neutral measure can be replaced with any other measure associated with a traded numerraire $N(t)$ such that $$\frac{V(t)}{N(t)}=\mathbb{E}_t^N\left[\frac{V(T)}{N(T)}\right]$$ So what came to my mind is annuity measure and swaption price or forward measure and cap price. However, I could not find any references on those PDEs. Can someone point me to some references or provide different measure examples and how PDE is derived in that case. It would be especially useful if the example is a "real application" one and can be seen in practice pricing financial instruments.

Answer

Assuming that you

• Have an (or a set of) SDE(s) describing the dynamics of an asset $X$, with $t$-value $X_t$;


• Define $V$ as a claim contingent on the asset $X$, with $t$-value $V_t$;


• Define $N$ as a claim that may but need not be contingent on the asset $X$, with $t$-value $N_t$;


• Define a probability measure $\mathbb{Q}^N$ associated to the asset $N$ such that $$ \frac{V_t}{N_t}=\mathbb{E}_t^{\mathbb{Q}^N}\left[\frac{V_T}{N_T}\right] $$ hence $N$ is regarded as a numéraire.

then the pricing PDE directly follows from the measure you've just defined: just use Ito's lemma to impose that the process $V_t/N_t$ should be a $\mathbb{Q}^N$-martingale (martingale representation theorem). Typically, with simple diffusion processes, this means writing that the finite variation part (drift) should be zero(*).

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[Example]

Let the $t$-value of an underlying asset $X$ be driven by the following SDE (diffusion) $$ dX_t = \mu(t,X_t) dt + \sigma(t,X_t) dW_t^{\mathbb{Q}^B} $$ and consider the following contingent claims

• $V_t = V(t,X_t)$


• $N_t = N(t) = B_t$ with $ dB_t = B_t r dt$

Pick $N$ as a numéraire thereby introducing the pricing measure $\mathbb{Q}^B$ such that $$ \frac{V_t}{B_t}=\mathbb{E}_t^{\mathbb{Q}^B}\left[\frac{V_T}{B_T}\right] $$ we get, applying (bivariate) Itô's lemma:

\begin{align} d\left( \frac{V_t}{B_t} \right) &= \frac{1}{B_t} dV_t - \frac{V_t}{B_t^2} dB_t + \frac{1}{2}(0)d\langle V \rangle_t + \frac{1}{2}\frac{2V}{B_t^3}\underbrace{d\langle B \rangle_t}_{=0} - \frac{1}{B_t^2} \underbrace{d\langle V, B \rangle_t}_{=0} \\ &= \frac{1}{B_t} \left( \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial X} dX_t + \frac{1}{2} \frac{\partial^2 V}{\partial X^2} \underbrace{d\langle X \rangle_t}_{=\sigma^2(t,X_t)dt} - r V dt \right) \\ &= \underbrace{\frac{1}{B_t} \left( \frac{\partial V}{\partial t} + \frac{\partial V}{\partial X} \mu(t,X_t) + \frac{1}{2} \frac{\partial^2 V}{\partial X^2} \sigma^2(t,X_t) - r V \right) dt}_{=\text{Finite Variation Part}} + \frac{1}{B_t} \frac{\partial V}{\partial X} \sigma(t,X_t) dW_t^{\mathbb{Q}^B} \end{align} and setting the finite variation part to zero gives the well-known pricing PDE: $$ \frac{\partial V}{\partial t} + \frac{\partial V}{\partial X} \mu(t,X_t) + \frac{1}{2} \frac{\partial^2 V}{\partial X^2} \sigma^2(t,X_t) - r V = 0$$

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Usually, we change numéraires (for instance move from the traditional risk-neutral measure $\mathbb{Q}^B$ to an underlying asset related measure $\mathbb{Q}^S$) for mathematical convenience: it is sometimes easier to derive closed-form expressions under a different probability measure.

In your case, I do not directly see the benefits of moving to the measures you mention. So indeed it is possible but there is probably no point doing it, which would explain the lack of papers on the topic.

(*) If you instead assume jump-diffusion, just be careful as jump processes need to be compensated to emerge as martingales. You can have a look here, where the question is discussed with a very nice and thorough answer by Gordon.