### Abstract

http://ssrn.com/abstract=914154

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

## Oleg Kovrizhkin

affiliation not provided to SSRN

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.

Number of Pages in PDF File: 36

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

JEL Classification: C00, C63, G13

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#### Suggested Citation

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

#### Contact Information

##### Oleg Kovrizhkin (Contact Author)
affiliation not provided to SSRN ( email )