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Stochastic local volatility models and the Wei-Norman factorization method

  • * Corresponding author: Giuseppe Orlando

    * Corresponding author: Giuseppe Orlando

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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  • 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.

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

    Citation:

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  • 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

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

    Figure 3.   

    Figure 4.   

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

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

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

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

    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
    SABR 0.31250 0.5
    Wei-Norman 0.18840
    SABR 0.15460 0
    Wei-Norman 0.10170
    SABR 0.14600 -0.5
    Wei-Norman 0.08960
    SABR 0.16660 -1
    Wei-Norman 0.12870
    Simulations=10,000 for each model; Setting: T = 1, S0 = 100, v0 = 0.2, β = 1, α = 0.2
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