Assume $(X_t)_{t\geq 0}$ follows an SDE of the form: $$dX_t = a(t, X_t) dt + b(t, X_t) dW_t$$ where $W$ is a standard $n$-dimensional Brownian motion, $a$ and $b$ are mappings from $\mathbb{R}_+\times\mathbb{R}^n$ to respectively $\mathbb{R}^n$ and $\mathcal{M}_{\mathbb{R}}(n,n)$ with all the usual regularity conditions, and $X_0=x_0$ for some deterministic $x_0\in\mathbb{R}^n$.
Now let $f$ be a mapping from $\mathbb{R}^n$ to $\mathbb{R}$. Assume we are interested in the sensitivity of $\mathbb{E}[f(X_t)]$ with respect to time ($f$ could be seen as a European payoff function and $X_t$ the value of the risk factors vector at maturity $t$), ie we want to compute $\partial_t \mathbb{E}[f(X_t)]$.
If we proceed via Monte-Carlo and a time-discretized scheme for our SDE (for example an Euler-Maruyama), how would one estimate the derivative with respect to time in a numerically stable way?
Assume $(\hat{X}_i)_{i\geq 0}$ is a time-discretized approximation of $X$ at discrete time-steps $0=t_0 < \dots < t_i = i h < \dots$ where $h>0$ is a constant step size. Then I see the following ways to approximate our differential (assuming $f$ is differentiable):
- estimate $\frac{1}{h}\mathbb{E}[f(\hat{X}_{\lfloor\frac{t}{h}\rfloor})-f(\hat{X}_{\lfloor\frac{t}{h}\rfloor-1})]$ (ie look at the whole pricing function as a black box);
- or estimate $\frac{1}{h}\mathbb{E}[(\hat{X}_{\lfloor\frac{t}{h}\rfloor}-\hat{X}_{\lfloor\frac{t}{h}\rfloor-1})^{\top} \nabla f(\hat{X}_{\lfloor\frac{t}{h}\rfloor})]$ (sort of a path-wise chain rule).
Unfortunately in both cases the resulting Monte-Carlo approximations have extremely high variance (I do use variance reduction techniques like common random numbers and antithetic variables, they only help marginally). Even in basic examples like a call price in a Black-Scholes with modest volatility ($30\%$) and even millions of paths, especially for small $h$ (I do take care to do everything in double precision to avoid round-off errors in the finite differences), when compared to sensitivities with respect to other diffusion parameters. For instance, the estimator of the sensitivity I get in both cases has a standard deviation that is of the order (or even bigger) of the ground truth value itself, meaning it's literally garbage.
Are there references about the calculation & numerical issues of the sensitivity with respect to time of value/price functions like the one introduced here using a time-discretization scheme for the SDE and Monte-Carlo simulations (no PDE based approaches)?
