LSV models with stochastic interest rates and correlated jumps

ABSTRACT

Pricing and hedging exotic options using local stochastic volatility models drew a serious attention within the last decade, and nowadays became almost a standard approach to this problem. In this paper we show how this framework could be extended by adding to the model stochastic interest rates and correlated jumps in all three components. We also propose a new fully implicit modification of the popular Hundsdorfer and Verwer and Modified Craig–Sneyd finite-difference schemes which provides second-order approximation in space and time, is unconditionally stable and preserves positivity of the solution, while still has a linear complexity in the number of grid nodes.

KEYWORDS:

• LSV model
• stochastic interest rates
• correlated Lévy processes
• PIDE
• splitting
• matrix exponential

2010 AMS SUBJECT CLASSIFICATIONS:

• 91G40
• 91G60
• 91G80
• 47G30
• 35R09
• 65L12

Acknowledgments

We thank Peter Carr, Alex Lipton and Alex Veygman for their useful comments and discussions. Also comments and suggestions of two anonymous referees are highly appreciated. We assume full responsibility for any remaining errors.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Here we do not discuss this conclusion. However, for the sake of reference note, that this could be dictated by some inflexibility of the Heston model where vol-of-vol is proportional to $v^{0.5}$. More flexible models which consider the vol-of-vol power to be parameter of calibration, [14,25], might not need jumps in v. See also [42] and the discussion therein.

2. If, however, somebody wants to determine these parameters by calibration, she has to be careful, because having both vol-of-vol and a power constant in the same diffusion term brings an ambiguity into the calibration procedure. Nevertheless, this ambiguity can be resolved if for calibration some additional financial instruments are used, for example, exotic option prices are combined with the variance swaps prices, see [25].

3. This is especially important at the first few steps in time because of a step-function nature of the payoff. So a smoothing scheme, for example, [36], is usually applied at the first steps, which, however, loses the second-order approximation at these steps.

4. The trick is motivated by the desire to build an ADI scheme which consists of two one-dimensional steps, because for the 1D equations we know how to make the rhs matrix to be an EM-matrix [26].

5. For the sake of clearness we formulate this Proposition for the uniform grid, but it should be pretty much transparent how to extend it for the non-uniform grid.

6. In our experiments 1–2 iterations were sufficient to provide the relative tolerance to be ${10}^{-6}$.

7. Using the same β as in the above. However, changing the first multiplier in the rhs of Equation (23(23) $\beta =10\underset{S,v}{max}\left[W\right(v)-{\rho }_{s,v}W(S\left)\right].$(23) ) can make the scheme working for the higher values of the time step as well.

8. Here EM is an abbreviation for an eventually M-matrix, see [27].

9. This can always be achieved by choosing a relatively small $\mathrm{\Delta }\tau$.

10. For instance, for the plain vanilla options the option price asymptotically is limited by the intrinsic value, therefore high-order derivatives rapidly vanish.

11. As mentioned by the referee, since the flop counts rarely predict accurately an elapsed time, this statement should be further verified.

12. Note, that, for example, for the HV scheme we need two sweeps per one step in time.

13. In this paper we do not analyse the convergence and order of approximation of the FD scheme, since the convergence in time is same as in the original HV scheme, and approximation was proven by the Theorem. For the jump FD schemes the convergence and approximation are considered in [26].

14. The values of parameters are taking just for testing, and could differ from those, for example, obtained by calibrating the model to the current market data.