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I would like to use Monte Carlo simulation to price some options. First I use standard approach where stock price is discribed by the following process: $$S_T = S_0\exp \left[(r - 0.5\sigma^2)T + \sigma \sqrt{T}\varepsilon\right],$$ where $S_0$ is an initial stock price at time $t = 0$, $S_T$ is a stock price at time $T$, $T$ is a time step, $r$ is the risk free interest rate, $\sigma$ is a standart deviation if stock returns and $\varepsilon$ is an independent normally distributed variable $\varepsilon \sim \phi(0, 1)$ There is an assumption under this model that log returns of a stock price have a normal distribution, such that $\log{\frac{S_T}{S_0}} \sim \phi(\mu, \sigma^2)$, where (from Ito's Lemma) $\mu = (r - 0.5\sigma^2)T$ and $var = \sigma^2T . $

Now I assume that returns of a stock price follow Normal Inverse Gaussian distribution (NIG). My question is am I able just to plug into my formula for $S_T$ NIG proccess with estimated parameters (using MLE) instead of $\varepsilon$ and than compare option prices? Proccess will look as following: $$S_T = S_0\exp \left[(r - 0.5\sigma^2)T + \sigma T\varepsilon^*\right],$$ where $\varepsilon^* \sim NIG(\hat{\mu}, \hat{\alpha}, \hat{\delta}, \hat{\beta}).$

I guess that properties of NIG don't allow me to do such manipulations but I am not sure where to find the solution to this problem. Any hints are welcome!

Many thanks @Quantuple for help:

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tosik
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1 Answers1

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Assuming you've used this definition for the NIG distribution and that you've managed to come up with estimates $(\hat{\alpha}, \hat{\beta}, \hat{\mu}, \hat{\delta} )$ of the individual NIG parameters, your question boils down to:

"How to simulate paths from the global log-return process $R_t = \ln(S_t/S_0)$ for all $t \in [0,T]$, assuming i.i.d. $NIG(\hat{\alpha}, \hat{\beta}, \hat{\mu}, \hat{\delta} )-$distributed periodic log-returns (in your case daily)?"

First of all, no, you cannot use the equation you mention.

Yet, because the NIG distribution is a special case of normal variance-mean mixture (see this document, page 14), if one lets \begin{align} \sigma^2 &\sim IG\left( \frac{\delta}{\gamma}, \delta^2 \right),\ \ \text{with } \gamma = \sqrt{ \alpha^2 - \beta^2 } \\ \varepsilon &\sim \mathcal{N}(0,1) \end{align} then the random variable $X$ defined as $$ X = \mu + \beta \sigma^2 + \sigma \varepsilon $$ follows a $NIG(\alpha,\beta,\mu,\delta)$ distribution.

Now, let $r_{\delta t, i}$ denote the $\delta t$-period log-return observed for a given $t_i \in [0,T]$ $$ r_{\delta t, i} := \ln\left( \frac{S_{t_i}}{S_{t_i-\delta t}} \right) $$

You can then proceed as follows to generate realisations of the global return process $(R_t)_{t\geq 0}$.

  1. Under the assumptions of i.i.d. NIG-distributed $\{r_{\delta t, i}\}$, estimate the NIG parameters $(\hat{\alpha}, \hat{\beta}, \hat{\mu}, \hat{\delta} )$ from historical data using your favourite method (Maximum Likelihood Estimation, Moment Matching etc.).

  2. To simulate the periodic log-returns $\{r_{\delta t,i}\}_{i=1,...,N}$ (in practice $N = T/\delta t$ where $T$ figures the horizon of your MC simulation and $\delta t$ the length of the period over which the individual periodic log-returns prevail) use the key result mentioned above by (i) generating $\sigma_i^2 \sim IG(\hat{\delta}/\hat{\gamma}, \hat{\delta}^2)\ \ \text{i.i.d.}$ (see here for instance), (ii) generating $\varepsilon_i \sim \mathcal{N}(0,1) \ \ \text{i.i.d.}$ (iii) computing $$ r_{\delta t,i} = \hat{\mu} + \hat{\beta} \sigma_i^2 + \sigma_i \varepsilon_i $$

  3. Once all $\{r_{\delta t, i}\}_{i=1,...,N}$ have been simulated, build the global return process $(R_t)_{t\geq 0}$ by aggregating a certain number $n$ of periodic returns. Indeed, for any fixed $t \in [0,T] $, the global log-return $R_t := \ln(S_t/S_0)$ computes as: \begin{align} R_t &= \ln\left(\frac{S_t}{S_0}\right) \\ &= \ln\left(\frac{S_t}{S_{t-\delta t}} \frac{S_{t-\delta t}}{S_{t- 2\delta t}} \dots \frac{S_{\delta t}}{S_{0}}\right) \\ &= \sum_{ t_i \in [\delta t, t] } \ln \left( \frac{S_{t_i}}{S_{t_i-\delta t}} \right) \\ &= \sum_{i \leq n} r_{\delta t,i} \end{align}

This method was first proposed by Rydberg in:

T. H. Rydberg. The normal inverse Gaussian Lévy process: simulation and approximation. Comm. Statist. Stochastic Models, 13(4):887–910, 1997.

Quantuple
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  • Thanks for your answer and sorry for mistakes. I made some corrections. I understand that I am not allowed to plug $\varepsilon^*$ instead of $\varepsilon$ and I would like to find a solution to this problem so I could run Monte Carlo assuming that $S_T/S_0 \sim NIG(\hat{\alpha}, \hat{\beta}, \hat{\mu}, \hat{\delta})$. So I need hints how to implement that. My code to simulate stock paths where $\varepsilon \sim N(0, 1)$ is correct as calculated option price converges to the BSM for a vanilla option. – tosik May 09 '16 at 10:33
  • @IvanovNikita, I've added details regarding how to simulate a NIG process. – Quantuple May 09 '16 at 12:08
  • Thanks again! So after doing this steps I will get set of variables $x_i$ which represent log returns, wright? To switch to the regular one I need to take exponent of them: $r_i = e^{x_i}$? – tosik May 09 '16 at 19:22
  • @IvanovNIkita, I've edited my answer because it seems like it wasn't clear enough. Following the steps described above will allow you to simulate paths of the global (log-)return process $R_t = \ln(S_t/S_0)$ (with iid NIG increments given by the periodic log-returns $r_{\delta t}$). To get back to classic stock price paths you need to write $S_t=S_0 \exp(R_t)$. From these price paths you can easily compute arithmetic returns if you want. – Quantuple May 09 '16 at 21:01
  • Yes, that's what I ment. If you have any references why to use the procces $x = \mu + \beta\sigma^2 +\sigma\varepsilon$ I would be thankful if you provide them – tosik May 10 '16 at 06:15
  • There's already an URL in my answer, see the part "Yet, because the NIG distribution is a special case of normal variance-mean mixture (... url here...). Or you can read the wikipedia definition of NIG. – Quantuple May 10 '16 at 06:55
  • Could you please see my edit – tosik May 10 '16 at 19:38
  • @IvanovNikita, Sorry for that: I had mixed up 2 different parametrisations of the Inverse Gaussian distribution. I have now fixed this (see my edits + reference for the method). Essentially you should draw $\sigma^2$ according to $IG(\delta/\gamma, \delta^2)$ and not what I mentioned before. Your plot of $r_i$ should now look exactly as your theoretical NIG pdf. I tested with a simple code in MATLAB and it works. – Quantuple May 11 '16 at 08:45