A novel Monte Carlo approach to hybrid local volatility models

Abstract

We present in a Monte Carlo simulation framework, a novel approach for the evaluation of hybrid local volatility [Risk, 1994, 7, 18–20], [Int. J. Theor. Appl. Finance, 1998, 1, 61–110] models. In particular, we consider the stochastic local volatility model—see e.g. Lipton et al. [Quant. Finance, 2014, 14, 1899–1922], Piterbarg [Risk, 2007, April, 84–89], Tataru and Fisher [Quantitative Development Group, Bloomberg Version 1, 2010], Lipton [Risk, 2002, 15, 61–66]—and the local volatility model incorporating stochastic interest rates—see e.g. Atlan [ArXiV preprint math/0604316, 2006], Piterbarg [Risk, 2006, 19, 66–71], Deelstra and Rayée [Appl. Math. Finance, 2012, 1–23], Ren et al. [Risk, 2007, 20, 138–143]. For both model classes a particular (conditional) expectation needs to be evaluated which cannot be extracted from the market and is expensive to compute. We establish accurate and ‘cheap to evaluate’ approximations for the expectations by means of the stochastic collocation method [SIAM J. Numer. Anal., 2007, 45, 1005–1034], [SIAM J. Sci. Comput., 2005, 27, 1118–1139], [Math. Models Methods Appl. Sci., 2012, 22, 1–33], [SIAM J. Numer. Anal., 2008, 46, 2309–2345], [J. Biomech. Eng., 2011, 133, 031001], which was recently applied in the financial context [Available at SSRN 2529691, 2014], [J. Comput. Finance, 2016, 20, 1–19], combined with standard regression techniques. Monte Carlo pricing experiments confirm that our method is highly accurate and fast.

Keywords:

• Local volatility
• Monte Carlo
• Hybrid
• Stochastic volatility
• Stochastic local volatility
• Stochastic interest rates
• Stochastic collocation
• Regression
• SABR
• Heston
• Hull–White

AMS Subject Classifications:

• C15
• C63

Notes

No potential conflict of interest was reported by the authors.

1 Note that, in fact, the ‘pure’ SABR model is already a SLV model with a parametric local volatility component.

2 In Lipton and McGhee (2002) Lipton and McGhee present a more general form of SLV models including jumps.

3 In Piterbarg (2007) Piterbarg formalizes this procedure as the Markovian projection method.

4 Note that the general SLV model as described by equations (2) and (3) is an incomplete market model, which implies that a unique risk-neutral pricing measure does not exist, see e.g. Fouque et al. (2000).

5 A derivation of the local volatility component, consisting of Dupire’s local volatility and a conditional expectation, can be found in e.g. Gatheral (2006); Van der Stoep et al. (2014).

6 To prevent double use of the -notation we write the variance dynamics instead of the more common volatility dynamics. The traditional SABR model dynamics are given by the following two SDEs (Hagan et al. 2002, Rebonato et al. 2011):

with denoting the forward corresponding to expiry T and .

7 A well-known test for multi-variate normality is Mardia’s, see Mardia (1974), which is based on multivariate extensions of skewness and kurtosis measures.

8 We choose the collocation points in an optimal way, namely as the zeros of the Hermite polynomials (abscissas of the Gauss–Hermite quadrature) (Abramowitz and Stegun 1972, Grzelak et al. 2014).

9 Choosing the interpolation polynomial in the Lagrange form is well-accepted in the field of uncertainty quantification (when the stochastic collocation method is applied), see e.g. Sankaran and Marsden (2011).

10 Other, more complex types of basis functions we may use are the Laguerre, Hermite, Legendre, Chebyshev, Gegenbauer and Jacobi polynomials, see e.g. chapter 22 of Abramowitz and Stegun (1972). In this article we do not consider these basis functions, as the set of simple polynomials already yields highly satisfactory results, see the numerical experiments in section 2.3.

11 Numerical experiments demonstrate that merely applying the third correction, i.e. applying a vertical shift, typically yields worse pricing results compared to combining the second and third corrections mentioned in section 2.2.

12 In a stochastic volatility model these parameter values are representative choices for and , respectively.

13 Implied volatilities for the ‘pure’ Heston model are obtained based on Fourier techniques.

14 The Monte Carlo simulation consists of paths (20 seeds, each seed constitutes paths) and 200 time-steps per year, unless otherwise mentioned.

15 The boundaries of the 95%-confidence interval are , with and denoting the mean and standard deviation, respectively, and stands for the model implied volatility (obtained from Monte Carlo) corresponding to the ith seed.

16 E.g. when repeating the experiment for the Heston-SLV model (, ), given ‘Heston market’ Set 1, with 20 seeds, paths per seed, we obtain the errors 0.02, 0.00, 0.01, 0.01, 0.00, 0.03, 0.01 and corresponding standard deviations 0.02, 0.01, 0.01, 0.01, 0.01, 0.01, 0.02.

17 The expression for is obtained by decomposing the Hull–White model, see e.g. Pelsser (2000).

18 For notation purposes we suppress the t-superscript in the inverse of the CDF of S(t).

19 The Monte Carlo simulation consists of paths (20 seeds, each seed constitutes paths) and 200 time-steps per year, unless otherwise mentioned.

20 The boundaries of the 95%-confidence interval are , with and denoting the mean and standard deviation, respectively, and stands for the model implied volatility (obtained from Monte Carlo) corresponding to the ith seed.

21 We conclude this based on a simulation of the LV-HW model applying an ‘exact’, ‘brute-force’ approach to compute the expectation . The same bias was observed.

23 The joint distribution of two normal random variables does not need to be bivariate normal. Only the reverse holds in general.