Stochastic local volatility models and the Wei-Norman factorization method

Julio Guerrero; Giuseppe Orlando

doi:10.3934/dcdss.2022026

Article Contents

2022, Volume 15, Issue 12: 3699-3722. Doi: 10.3934/dcdss.2022026

This issue Previous Article A nonautonomous Cox-Ingersoll-Ross equation with growing initial conditions Next Article Global existence for equivalent nonlinear special scale invariant damped wave equations

Stochastic local volatility models and the Wei-Norman factorization method

Julio Guerrero^1, and
Giuseppe Orlando^2, ,

1.
University of Jaen - Department of Applied Mathematics, Campus Las Lagunillas, Jaen, 23071, Spain
2.
University of Bari - Department of Economics and Finance, Via C. Rosalba 53, Bari, 70124, Italy

^* Corresponding author: Giuseppe Orlando
^* Corresponding author: Giuseppe Orlando

Received: September 2021

Revised: December 2021

Early access: February 2022

Published: December 2022

The first author is supported by the Spanish MICINN grant PGC2018-097831-B-I00 and Junta de Andalucía grant FEDER/UJA-1381026

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Abstract

In this paper, we show that a time-dependent local stochastic volatility (SLV) model can be reduced to a system of autonomous PDEs that can be solved using the heat kernel, by means of the Wei-Norman factorization method and Lie algebraic techniques. Then, we compare the results of traditional Monte Carlo simulations with the explicit solutions obtained by said techniques. This approach is new in the literature and, in addition to reducing a non-autonomous problem into an autonomous one, allows for shorter time in numerical computations.

Keywords:

Wei-Norman,
Lie algebraic methods,
stochastic local volatility (SLV) model,
Monte Carlo simulations.

Mathematics Subject Classification: Primary: 60Gxx, 35C05, 35K05; Secondary: 17Bxx, 62P05.

Citation:

Full Text(HTML)

Figure 1. Setting: $ S_0 = 100 $, $ v_0 = 0.2 $, $ \beta = 1 $, $ \rho = 0.5 $, $ \alpha = 0.2 $. Notice that in absence of dividends and discount factors there is no drift (if we consider the drift coefficients $ k_\omega $ and $ k_\mu $ zero), therefore, the expected value should be equal to the initial value

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Figure 2. (a) Wei-Norman SLV prices. (b) Wei-Norman SLV/SABR implied volatility

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Figure 3.

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Figure 4.

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Figure 5. Scenarios: $ \alpha $ low = 0.1, $ \alpha $ high = 0.5, $ \alpha $ base = 0.2

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Figure 6. Scenarios: $ \alpha $ low = 0.1, $ \alpha $ high = 0.5, $ \alpha $ base = 0.2

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Figure 7. Scenarios: $ \rho $ low = -0.8, $ \rho $ high = 0.8, $ \rho $ base = 0.5

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Figure 8. Scenarios: $ \rho $ low = -0.8, $ \rho $ high = 0.8, $ \rho $ base = 0.5

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Table 1. Monte Carlo simulations: SABR vs Wei-Norman, case $ c = 0 $


Monte Carlo	CPU time (sec.)	$ \rho $
SABR	0.19040	1
Wei-Norman	0.10700	1
SABR	0.31250	0.5
Wei-Norman	0.18840	0.5
SABR	0.15460	0
Wei-Norman	0.10170	0
SABR	0.14600	-0.5
Wei-Norman	0.08960	-0.5
SABR	0.16660	-1
Wei-Norman	0.12870	-1

Simulations=10,000 for each model; Setting: T = 1, S₀ = 100, v₀ = 0.2, β = 1, α = 0.2

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