Simulating Local Volatility Paths with Time-Stepped Monte Carlo

# Local Volatility with Monte Carlo Simulation


# Local Volatility with Monte Carlo Simulation












I am trying to implement a Monte Carlo Simulation using Local Volatility Model (Dupire’s Equation).

I’m pretty sure I can build a very good LV surface, however, I do not know how to use it in the MC Simulation.

The point is since I always start the simulation from the current spot price, in the first step I will always end up with the ar-the-money volatility at first place. Therefore, in case I want to price a plain Vanilla option in or out-of-the money, with a maturity equal to the first time step (first tenor of my surface) all the paths will use the at-the-money vol and price will not be consistent with market.

For sure I’m doing something wrong, can someone help me?

## Answer by Quantuple (score 6)

https://quant.stackexchange.com/a/46738

Let the risk-neutral dynamics under your LV model be given by $$ \frac{d S_t }{S_t } = \mu_t dt + \sigma(t,S_t) dW_t $$ Let's drop the drift contribution (not relevant here) and apply Itô's lemma to obtain: $$ d \ln(S_t) = -\frac{1}{2}\sigma^2(t,S_t) dt + \sigma(t,S_t) dW_t $$ In order to simulate from this SDE, you need to choose a particular discretisation scheme.

The most simple choice would be to opt for the Euler-Maruyama scheme yielding, conditional on the information available at $t$:

\begin{align} \ln(S_{t+\delta t}) &= \ln(S_{t}) - \frac{1}{2}\int_t^{t+\delta t} \sigma^2(u,S_u) du + \int_t^{t+\delta t} \sigma(u, S_u) dW_u \\ &\approx \ln(S_{t}) - \frac{1}{2} \sigma^2(t,S_t) \delta t + z \sqrt{\sigma^2(t, S_t)\delta t} \tag{1} \end{align} where we have also assumed a Euler discretisation of the time integrals + used Itô isometry. At the end of the day you see that indeed $S_{t+\delta t} \vert S_t$ will be lognormal.

So if you want to simulate from $t=0$ to $t=T$ ($T$ being your date of interest), using only one time step $\delta t = T$, you'll have a lognormal distribution as in Black-Scholes (or equivalently, no IV skew).

However, if you break the interval $[0,T]$ into $i=1,\dots,N$ sub-intervals $$ 0 =: t_0 < t_1 = \delta t < \dots < t_N := T $$ and apply the previous method to successively generate $S_{1} \vert S_0$ then $S_2 \vert S_1$ until $S_{T} \vert S_{T-\delta t}$, $S_{T}$ will not be lognormal anymore.

Intuitively this is because you will gradually start using a path-dependent volatility to simulate future price evolutions, contrary to the single volatility figure framework (i.e. Black-Scholes).

My other comment was related to other discretisation schemes where at each time step the conditional distribution won't be lognormal. This is because the path-dependence of the diffusion coefficient will be accounted for within the discretisation of the SDE itself (see e.g. Milstein). Whether with those you can use a single time step and hope to directly to match the theoretical distribution is then a convergence + discretisation bias question.

## Answer by Salah (score 0)

https://quant.stackexchange.com/a/63469

True, but you use LV at the money very "locally". If your K>S0 or K<S0, for your first step you will use LV(new K = S0,0) which considered at the money localy but not in global since S0!=K(at the end). It s very tricky and it s how the process of LV is defined.