Quantitative Finance Asked on October 27, 2021
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.
Let me try to answer, this topic is much deeper than my answer
1. Why are these models like this unpopular?
2. What are the alternatives, that I could model dependencies between financial assets?
$$dv_i = alpha(v_i,t)dt + beta(v_i,t)dW_{v_i}$$
$$sigma_{Dupire_i}(S_i,t)^2 = A_i(S_i,t)^2E[v_i|S_i]$$
$$dW_idW_j=rho_{ij}dt, dW_idW_{v_i}=rho_{S_iv_i}dt, dW_idW_{v_j}=rho_{S_iv_j}dt$$
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
Answered by ryc on October 27, 2021
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