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

Quantitative Finance Asked on November 5, 2021

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!

One Answer

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 $$ mathbb{P}(A_T) = mathbb{E}^{mathbb{P}}(mathbb{1}_{{S_Tin A_t}}), $$ 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).

Answered by oliversm on November 5, 2021

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