Public User

# From Moving Average Local and Stochastic Volatility Models to 2-Factor Stochastic Volatility Models

36 Pages Posted: 15 Jul 2006 Last revised: 23 Aug 2008

## Oleg Kovrizhkin

affiliation not provided to SSRN

Date Written: October 6, 2006

### Abstract

We consider the following models:

1. Generalization of a local volatility model rolled with a moving average of the spot: dS = mu Sdt + sigma(S/A)SdW\$ where A(t) is a moving average of spot S.

2. Generalization of Heston pure stochastic volatility model rolled with a moving average of the stochastic volatility: dS = mu Sdt + sigma SdW, dsigma^2 = k(theta - sigma^2)dt + gamma sigma dZ where theta(t) is a moving average of variance sigma^2.

3. Generalization of a full stochastic volatility with the process for volatility depending on both sigma and S and rolled with a moving average of S: dS = mu Sdt + sigma SdW, dsigma = a(sigma, S/A)dt + b(sigma, S/A)dZ,
corr(dW, dZ) = rho(sigma, S/A)\$, where A(t) is a moving average of the spot S.

We will generalize these and other ideas further and show that they lead to a 2-factor pure stochastic volatility model: dS = mu Sdt + sigma SdW\$, sigma = sigma(v_1, v_2), dv_1 = a_1(v_1, v_2)dt + b_1(v_1, v_2)dZ_1,
dv_2 = a_2(v_1, v_2)dt + b_2(v_1, v_2)dZ_2, corr(dW, dZ_1) = rho_1(v_1, v_2), corr(dW, dZ_2) = rho_2(v_1, v_2), corr(dZ_1, dZ_2) = rho_3(v_1, v_2) and give examples of analytically solvable models, applicable for multicurrency models consistent with cross currency pairs dynamics in FX. We also consider jumps and stochastic interest rates.

Keywords: Local, Stochastic, moving average, jumps, Levy, multifactor

JEL Classification: C00, C63, G13

Suggested Citation

Kovrizhkin, Oleg, From Moving Average Local and Stochastic Volatility Models to 2-Factor Stochastic Volatility Models (October 6, 2006). Available at SSRN: https://ssrn.com/abstract=914154 or http://dx.doi.org/10.2139/ssrn.914154