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.


1 Answer 1


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(*).


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: $$ \frac{\partial V}{\partial t} + \frac{\partial V}{\partial X} \mu(t,X_t) + \frac{1}{2} \frac{\partial^2 V}{\partial X^2} \sigma^2(t,X_t) - r V = 0$$

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.

  • $\begingroup$ Ok, but say I am pricing a swaption, just in parallel with an equity option, it has a closed firm solution, but we can write the code for it, similar to BS Pde but the sde do swap rate is under annuity measure. So I i were to assume some sde dynamics for annuity under annuity measure(what could it be though, not sure since it is associated with its own measure, so I guess dA_t=0?) I would proceed as you described. In that case it would be hard for me to write all sdes under risk neutral measure. So it is natural to work with annuity measure here. $\endgroup$
    – Medan
    Aug 22, 2016 at 12:40
  • $\begingroup$ I don't understand. Are you talking about change of measures? $\endgroup$
    – Quantuple
    Aug 23, 2016 at 7:08
  • $\begingroup$ Assume I start with a swap rate and swaption as an instrument I would like to price. So, I can write an expectation under annuity measure and compute it. On the other hand, I can use FC to solve the corresponding PDE and get the same value. Now, I use the fact that I model the swap rate under annuity measure, so as the expectation defined in that measure. I will say $V_t/A_t$ is a martingale and apply Ito to set $dt$ term to 0 to get the PDE. I have not mentioned RN measure at all! So I used all the typical RN arguments but with annuity measure. Now I am to solve the PDE I derived. $\endgroup$
    – Medan
    Aug 23, 2016 at 13:54
  • $\begingroup$ So for a swaption, it was natural to write everything in the annuity measure. However, if I now do a change of measure and write everything under RN measure I am guessing I would end up with the same PDE as when modeled under RN measure but in much harder way as I don't know the dynamics of swap rate under annuity measure. So, what I am saying is that all examples are typically done under RN measure but they can be well applied for other instruments/measures! So, I wonder why there are no examples such as IR Swaption to show application of FK to obtain the solution in the books... $\endgroup$
    – Medan
    Aug 23, 2016 at 13:57

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