stochastic processes - Application of Vibrato Montecarlo methods

Ciao, I was studying Vibrato Montecarlo methods and I came up with a very simple question: what is an real application of this method? Let me explain.

In short the main idea of the method is the following: suppose you want to compute the sensitivities of a derivative with a discontinuous payoff then you should try the following:

1. Discretize time with pillars $t_1, \dots, t_N$.


2. Suppose you know the value of the stock at time $t_{N-1}$ and, by knowing it, analitically integrate the payoff function as $$V(S_{N-1}) = \mathbb{E}[f(S_N) | S_{N-1}]$$



3. $V(S_{N-1})$ is still a stochastic quantity depending on $S_{N-1}$ but know it is a differentiable function (essentially the integration works as molllification of the starting discontinuous payoff). At this point you can compute the sensitivity w.r.t. parameter $\lambda$ by using classical finite difference and Montecarlo methods applied to $V(S_{N-1})$ simulating trajectory $S_{N-1}$: $$\partial_\lambda V(\lambda) = \frac{V(\lambda + h, S_{N-1}) - V(\lambda, S_{N-1})}{h}$$

The procedure is quite smart and it works from a numerical point of view, but I do not understand how it could be usefull. In fact, usually, if you can integrate from $t_{N-1}$ to $t_N$, i.e. the step $2$, you can integrate starting from $t_0$ so that is does not make sense to use Montecarlo and finite difference.

This is evident for example for a Call option.

My question is then:

Can you present me a case in which the fully integrability is not possible (or too difficult) but the partial one (i.e. from $t_{N-1}$ to $t_N$ is easy to do (so that the Vibrato Montecarlo technique makes sense in such case)?

Thank you for help!!

Ciao ciao!

AM

Answer

To keep notations uncluttered assume zero funding costs and consider a European contingent claim priced as

$$V = \Bbb{E}_0 \left[ h(S_N) \right]$$ Suppose you would like to make the dependence of your pricing on a specific model parameter $\theta$ appear. You then usually have the following representation choices $$\begin{align} V &= \int_0^\infty h(S) q(S,\theta) dS \tag{1} \\ &= \int_{-\infty}^\infty h(S(z,\theta)) p(z) dz \tag{2} \end{align}$$ where $z$ is a random variable with known distribution. When expressing $\partial V/\partial \theta$ using the above expressions + Fubini, the first will give rise to the so-called Likelihood Ratio method the second to the Pathwise Sensitivities method $$\begin{align} \frac{\partial V}{\partial \theta} &= \int_0^\infty h(S) \frac{\partial q(S,\theta)}{\partial \theta} dS = \Bbb{E}_0 \left[ h(S_N) \frac{\partial \ln q}{\partial \theta} \right] \tag{1'} \\ &= \int_{-\infty}^\infty h(S(z,\theta)) p(z) dz = \Bbb{E}_0 \left[ h'(S_N) \frac{\partial S_N}{\partial \theta} \right] \tag{2'} \end{align}$$

The PS method requires the payout function to be differentiable, which may not always be the case. Think of digital option, differentiating an indicator function would yield a Dirac meaning that only simulated paths that end right on the strike price will matter, i.e. not practical.

The first is OK as long as the risk-neutral distribution of $S_N$ is tractable under the model at hand, which may not always be the case (think local volatility for instance). But when simulating the SDE the distribution of $S_N \vert S_{N-1}$ will be tractable so that the LR method is often rewritten as

$$\Bbb{E}_0 \left[ h(S_N) \sum_{n=1}^N \frac{\partial \ln q(S_n \vert S_{n-1})}{\partial \theta}\right]$$ One of the issues with the LR method is it does not exhibit extraordinary convergence property. The Vibrato is presented by its author as a smart hybrid between the 2.

It is easily illustrated in the case of a digital option, which does not have a differentiable payout function so tranditional PS method cannot be applied. The idea is that conditionally on $S_{N-1}$ the distribution of $S_N$ is tractable. So we shall use the LR method on the last time step. The parameters of the latter distribution will emerge as functions of $\theta$. To obtain the sensitivities of these parameters to $\theta$, we use the PS method.

So Vibrato in that case boils down to applying PS method for $t \in [0, t_{N-1}[$ to obtain the sensitivity of the parmaeters of the conditional distribution $q(S_N \vert S_{N-1})$ to $\theta$. Then, for each generated path, use the LR method over the last time step. Note that this gives rise to some kind of "nested" Monte Carlo step (which can be done analytically for vanilla and digital payouts) but not at all to using 'Finite Differences' as you suggest?