Euler Discretization to use with Monte Carlo simulation and Local Volatility Model

Question

Like in the title, I am working on running Monte Carlo simulations to price options with the Local Volatility model as a project. I just want to make sure that I am understanding the process, especially the discretization correctly.

The risk neutral dynamics under the Local Volatility model is:

$$ \frac{d S_t }{S_t } = \mu_t dt + \sigma(t,S_t) dW_t $$

Applying Itô's lemma gives:

$$ d \ln(S_t) = (\mu_t-\frac{1}{2}\sigma^2(t,S_t)) dt + \sigma(t,S_t) dW_t $$

Using Euler-Maruyama discretization scheme for simplicity:

\begin{align} \ln(S_{t+\delta t}) &= \ln(S_{t}) + \int_t^{t+\delta t}(\mu_t-\frac{1}{2} \sigma^2(u,S_u)) du + \int_t^{t+\delta t} \sigma(u, S_u) dW_u \\ &\approx \ln(S_{t}) + (\mu_t - \frac{1}{2} \sigma^2(t,S_t)) \delta t + z \sqrt{\sigma^2(t, S_t)\delta t} \tag{1} \end{align}

Then I can incorporate the local volatility model (and the skew/smile) into my simulations by splitting the time interval between 0 and T into smaller intervals and use the volatility given by the local volatility surface and time step, plug these two into (1) (assuming that I can build a smooth LV surface).

I have two questions.

1/ Would it be correct to use the drift rate equal to the risk free rate for pricing options ?

2/ If I want to use Monte Carlo simulations to get an idea on the probability of the underlying asset ending up between an interval after a defined time period, then I would have to use the "expected return" of the underlying asset instead of the risk free rate ?

Thanks!

Answer 1

Use the risk free rate for pricing

You use the risk free rate (using the risk neutral measure $\mathbb{Q}$) so that you can use the formula $$ V(t) = \underbrace{\exp(-r(T-t))}_{\text{because we used $\mathbb{Q}$}} \mathbb{E}^{\mathbb{Q}}(P(S_T)), $$ where because we used $\mathbb{Q}$ we were able to discount the expectation after doing all the MC simulations. If you want to use the physical measure $\mathbb{P}$ then you need to move a discounting factor inside the expectation, and things just all get a bit more awkward.

Use the real-world/physical rate for computing probabilities

For getting the probability of some event $A$ happening at time $T$ use the physical measure $\mathbb{P}$ and make use of $$ \mathbb{P}(A_T) = \mathbb{E}^{\mathbb{P}}(\mathbb{1}_{\{S_T\in A_t\}}), $$ and then make use of normal Monte Carlo to compute the expectation.

A comment on your Euler-Maruyama scheme

If you wish to simulate $\log(S_t)$ rather than $S_t$ then make sure your local volatility is modified appropriately to use $\log(S_t)$. On a more important note, for monotonic transformations such as taking $\exp(\cdot)$ then the confidence interval you had for $\log(S_t)$ will directly give you a correct interval for $S_t$. In general though this is not true, and can be easily seen, such as if you took $\sin(\cdot)$. (In fairness I can't think of any common place example(s) of this, but It is nonetheless something to keep in mind).