# Modelling Geometric Browian Motion price model with stochastic volatility

I'd like to generate scenarios (simulate several paths of the process) for several stocks using multinomial Geometric Brownian Motion under Stochastic volatility assumption. I'm going to use it in my portfolio optimization task. Firstly, I tried to model stochastic volatility using Copula-GARCH model (because it is essential for the portfolio to model volatility(dispersion) of each stock and dependency(covariance)). I tried to find some articles, which uses a similar approach but haven't found it.

So I have two questions: why are these models like this unpopular? And what are the alternatives, that I could model dependencies between financial assets?

I found that researches added to GBM another process that modelling volatility, like this:

$$dS_t = \mu S_{t}dt + \sigma(Y_t)S_tdW_{1t},$$

$$dY_t = \theta(w-Y_t)dt + \epsilon \sqrt(Y_t)dW_{2t}$$

But I don't understand how to model dependencies in this case.

Thank you.

• You could correlate the Brownian motions in the volatilities/returns processes. – Kermittfrog Jul 20 '20 at 20:06

Let me try to answer, this topic is much deeper than my answer

1. Why are these models like this unpopular?

• First, these models produce marginal distributions that does not fit the market, which means they cannot reproduce vanilla option prices traded in the market
• SV models, e.g. Heston model, may fit to a few vanilla prices, they cannot fit the entire surface, acc to Gyongy's lemma $$E[v_t|S_1]=\sigma_{Dupire}(S_1,t)^2$$
• $$v_t =$$ stoch variance of the asset
• This condition must be satisfied if your model wants to fit the iv surface
• If you are trading exotics like basket options / autocalls, you typically hedge it with vanillas. Using a model that cannot fit the implied vol surface, means the model value of your hedge instruments are wrong

2. What are the alternatives, that I could model dependencies between financial assets?

• You can start with multi-asset Local Volatility (LV) model $$\frac{dS_i}{S_i}=\sigma_{Dupire_i}(S_i,t)dW_i$$ $$dW_idW_j=\rho_{ij}dt$$
• Multi asset LV models can fit implied vol surface of each underlying, i.e. correct marginal distributions implied by market
• But they have constant instantaneous spot/spot correlation, while markets typically exhibit correlation skew
• And they assume 100% spot/vol correlation, which is unrealistic
• Multi asset Local-Stochastic Volatility (LSV) model would have a SV component and LV component $$\frac{dS_i}{S_i}=A_i(S_i,t)\sqrt{v_i}dW_i$$

$$dv_i = \alpha(v_i,t)dt + \beta(v_i,t)dW_{v_i}$$

$$\sigma_{Dupire_i}(S_i,t)^2 = A_i(S_i,t)^2E[v_i|S_i]$$

$$dW_idW_j=\rho_{ij}dt,\ dW_idW_{v_i}=\rho_{S_iv_i}dt,\ dW_idW_{v_j}=\rho_{S_iv_j}dt$$

• It perfectly fits the implied vol surface for each underlying while keeping SV dynamics that you desire $$E[A_i(S_i,t)^2v_i|S_i]=E[\frac{\sigma_{Dupire_i}(S_i,t)^2}{E[v_i|S_i]}v_i|S_i]=\sigma_{Dupire_i}(S_i,t)^2$$

• LSV typically exhibit a correlation skew

• The choice of a good SV is also paramount, even if you have LV component to adjust for vanilla prices, if your SV dynamics far from the vol dynamics in reality, the model would give ridiculous prices for multi asset payouts