# Simulate (imaginary) asset prices using random numbers that follow a Frank Copula

I didn't understand how to simulate asset prices by using non normal random numbers.

I am assuming that it would be incorrect to use the standard Geometric Brownian Motion, since it is based solely on normally distributed random variables z~N(0,dt).

As discussed below it should be a good idea to simply assume that the random numbers are logreturns.

• For an asset, if you know the distribution function, you can simulate the price. The copula is employed to defined the joint distribution. – Gordon Jul 20 '15 at 16:10

For non-normal asset price models you could look at the theory of Lévy-processes.

If we assume that you work in the physical probability measure $P$ and that the random numbers that you have generated are daily log-returns, then you can do the following: Asset $i$ has starting price $S_0^i$ and for the future prices you can put $$S_t^i = S_0^i \exp(\sum_{k=1}^t r^i_k),$$ where the $(r_k^i)_{k=1}^N$ are the sampled returns from the copula that you have described.

EDIT after comment by the OP:

If you want to price an option then you can sample the paths but with a drift equal to the risk free rate. Thus you subtract the expected value (depending on your distribution) and add the risk free rate. If you look at the theory of Lévy processes this means that you calculate the compensator (for the theory of Lévy processes you can look here.

Usually in the context of non-Gaussian models options are priced using Fourier-transform techniques. To learn the standard theory of these processes you would work through "Financial Modelling with Jump Processes" by Cont and Tankov.

Note that using a different distribution for the log-returns than a Gaussian you already model jumps of the stock price (very small ones).

• I edited the question. for the risk neutral pricing you need to find the compensator of the Levy process. – Ric Jun 10 '15 at 12:49
• Let’s define D=Expected value minus risk free rate. 1. Question: Then do I need to subtract D from every single log return r_k that I simulated for the respective asset? And by doing so I follow the risk-neutral approach? 2. Question: For the expected value, should I compute the mean or use the parameter mu that I set when simulating the random returns. The values are close for a large number of random returns, but I think to make sure the expected value (after adjustment) equals the risk free rate, I should use the mean. – Tom Jun 11 '15 at 8:44
• (This is just to let you know: I also read about Lévy processes and that the compensator addresses somehow the number of jumps. But I wasn’t able to translate this to what I want to do. The hint regarding Fourier doesn’t work for me, because I want to just simulate the price paths, then take the average and discount it with the risk-free rate r.) – Tom Jun 11 '15 at 8:47
• To the comment witth $D$ -> yes .. then your expected return is the risk free rate. For the expected value you should use the theoretical one because the sample estimate bears the sampling error. Concerning the compensator: it addresses the jumps .. for the Brownian motion part (if it is there) you simply set the drift to zero ...Fourier is the thing to do if you want to price an option. – Ric Jun 11 '15 at 9:45
• No, no reference ... sorry – Ric Jul 8 '15 at 11:58