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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!

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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).

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  • $\begingroup$ Thanks for your answer. Is there an intuitive way to explain why in the second case (calculating probability) we still use the risk free rate? Since there is no discounting factor to be included. Also, I can understand the need to use the risk free rate in pricing since the BS model is built upon creating a riskless hedging portfolio. But when it comes to real probability wouldn't it make more sense to use an expected return for the drift rate? For example if risk free rate is negative or 0 even then every stock in the universe would drift down over time? $\endgroup$ – Bach Pham Jul 22 '20 at 15:38
  • $\begingroup$ For the second case this is the natural/intuitive way to simulate. The non-intuitive bit is the risk neutral measure for pricing. Intuition only stems from Girsanov's theorem. This is required for taking the price (dimensional) and mapping this to a corresponding value today. As probability does not have dimensions, there is no discounting, and hence the simpler expression. For real probabilities it makes sense to use the expected return, which I'm advocating, as stocks move according to $\mu$, (not $r$). Being able to make this change is primarily just a neat mathematical trick for pricing. $\endgroup$ – oliversm Jul 23 '20 at 8:55

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