Building Price Paths from Copula-Sampled Log Returns

Summary

The discussion shows how to turn sampled, potentially non-normal log returns into asset price paths. For each asset, take its starting price and multiply it by the exponential of the cumulative sampled returns. A copula can supply joint return draws across assets, while the marginal return distributions represent each asset’s individual behavior. This provides an alternative to assuming normally distributed increments in a standard geometric Brownian motion setup.

The answer frames the approach under the physical probability measure for scenario simulation. For option valuation, it says the paths need risk-neutral drift: adjust the return distribution’s expected value and incorporate the risk-free rate through a compensator. It points to Lévy-process theory for non-Gaussian models and notes that Fourier methods are often used for option pricing in this setting. The guidance is brief and does not specify a particular Frank copula parameterization, return calibration procedure, or complete pricing implementation. Results therefore depend on the chosen marginal distributions, dependence model, and measure.

Key ideas

• Sampled log returns can be accumulated and exponentiated to construct future asset prices.
• A copula can model dependence across assets while separate distributions describe individual returns.
• Physical-measure paths for scenarios and risk-neutral paths for pricing require different drift treatment.
• Risk-neutral adjustment requires accounting for the return distribution’s expected value and the risk-free rate.
• Non-Gaussian price models connect to Lévy-process theory and may use Fourier methods for option pricing.

Tags

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


# 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.

## Answer by Richi Wa (score 2)

https://quant.stackexchange.com/a/18243

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).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.