A numerical approach to pricing exchange options under stochastic volatility and jump-diffusion dynamics

Abstract

We consider a method of lines (MOL) approach to determine prices of European and American exchange options when underlying asset prices are modeled with stochastic volatility and jump-diffusion dynamics. As with any other numerical scheme for partial differential equations (PDEs), the MOL becomes increasingly complex when higher dimensions are involved, so we first simplify the problem by transforming the exchange option into a call option written on the ratio of the yield processes of the two assets. This is achieved by taking the second asset yield process as the numéraire. Under the equivalent martingale measure induced by this change of numéraire, we derive the exchange option pricing integro-partial differential equations (IPDEs) and investigate the early exercise boundary of the American exchange option. We then discuss a numerical solution of the IPDEs using the MOL, its implementation using computing software and possible alternative boundary conditions at the far limits of the computational domain. Our analytical and numerical investigation shows that the near-maturity behavior of the early exercise boundary of the American exchange option is significantly influenced by the dividend yields and the presence of jumps in the underlying asset prices. Furthermore, with the numerical results generated by the MOL, we are able to show that key jump and stochastic volatility parameters significantly affect the early exercise boundary and exchange option prices. Our numerical analysis also verifies that the MOL performs more efficiently, than other finite difference methods or simulation approaches for American options, since the MOL integrates the computation of option prices, greeks and the early exercise boundary, and does so with the least error.

Keywords:

• Exchange options
• Jump-diffusion processes
• Method of lines
• Put-call transformation
• Stochastic volatility

JEL Classification:

• C60
• G12

Acknowledgments

The authors thank the managing editor and the anonymous reviewers for their valuable suggestions to improve the paper. The first author is supported by a Research Training Program International (RPTi) scholarship awarded by the Australian Commonwealth Government and by a Faculty Development Grant from the Loyola Schools of Ateneo de Manila University.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The time-discrete MOL involves discretizing the PDE in all but one spatial variable, as opposed to most applications where time is left as the continuous variable (see Schiesser and Griffiths 2009). This approach is also known as Rothe's method or the horizontal MOL.

2 Cheang and Chiarella (2011) assumed that only one asset price process had jumps, while the other was modeled as a pure-diffusion process. Quittard-Pinon and Randrianarivony (2010) discuss in greater detail the European exchange option pricing problem under a similar model specification.

3 See Runggaldier (2003) for more details.

4 In contrast, Cheang and Garces (2020) assume one variance process for each asset price. However, their analytical representations require that the asset price processes are uncorrelated with each other and with the variance processes. The current model specification allows such dependence structure.

5 The Radon-Nikodým derivative $L\left(T\right)=\frac{\mathrm{d}\hat{\mathbb{Q}}}{\mathrm{d}\mathbb{P}}$ can be used to characterize any probability measure $\hat{\mathbb{Q}}$ equivalent to $\mathbb{P}$ as parameterized by the vector process $\left\{\mathit{\theta }\right(t\left)\right\}$ and the constants ${\gamma }_1,{\gamma }_2,{\nu }_1,{\nu }_2$. We assume that ${\gamma }_1,{\gamma }_2,{\nu }_1,{\nu }_2$ are constant to preserve the time-homogeneity of the intensity and the jump-size distribution. As the market under the SVJD is generally incomplete, one can construct multiple equivalent martingale measures consistent with the no-arbitrage assumption.

6 This assertion can be proved using Itô's Lemma on $\tilde{S}\left(t\right)$ and $\tilde{M}\left(t\right)$ and eliminating the resulting drift term as required by the martingale representation for jump-diffusion processes (see Runggaldier 2003, Theorem 2.3).

7 In view of Proposition 2.1, we note that $\mathrm{d}{\overline{W}}_j\left(t\right)={\psi }_j\left(t\right)\mathrm{d}t+\mathrm{d}{W}_j\left(t\right)$.

8 See Cheang and Garces (2020) for a similar derivation of the joint conditional characteristic function for a pair of correlated Bates (1996) SVJD processes.

9 Mishura and Shevchenko (2009) analyzed, in further detail, the properties of the exercise region of the finite-maturity American exchange option in a pure diffusion setting. In the same setting, Villeneuve (1999) established the nonemptiness of exercise regions of American rainbow options, which include spread and exchange options as special cases.

10 In this situation, the investor is unable to adjust the decision to exercise in response to the instantaneous jump in asset prices and is therefore vulnerable to the rebalancing cost described earlier. A similar phenomenon in the context of consumption-investment problems with transaction costs in a Lévy-driven market is explored in greater technical detail by De Vallière et al. (2016).

11 This is in contrast to the proposition of Carr and Hirsa (2003), in their analysis of the one-asset American put option where the log-price is driven by a Lévy process, that the limit of the early exercise boundary is only dependent on the dividend yield and the risk-free rate.

12 The LSMC algorithm only simulates the optimal exercise strategy and is unable to estimate the early exercise boundary. For a simulation-based method for approximating the early exercise frontier, see Ibáñez and Zapatero (2004). More recently, Bayer et al. (2020) proposed a new simulation-based method that estimates exercise rates of randomized exercise strategies, an optimization problem which they show is equivalent to the original optimal stopping formulation.

13 Provided that the Feller condition is satisfied, the quadratic extrapolation combined with the MOL approach produces a consistent approximation of the pricing equation as $v\to 0$ (Chiarella et al. 2009, Appendix).

14 The convergence criterion can also include the early exercise boundary.

15 Such an analysis was also done by Kang and Meyer (2014) for American calls under stochastic volatility and stochastic interest rates. Meyer (2015) provides a discussion of the Fichera theory applied to common financial problems.

16 We do not seek to formally define Venttsel boundary conditions in this paper. The reader is referred to Meyer (2015, Section 1.2.3) for a discussion of Venttsel boundary conditions in the context of financial pricing problems.

17 Model parameter values used in this paper are similar to those used by Chiarella et al. (2009), with the values of the additional parameters for the second asset chosen appropriately.

18 Less emphasis on the jump-size distribution concentrates the analysis on the effect of the jump intensities. Alternative jump-size distributions may be considered as well, although the form of the density function dictates the quadrature formula for approximating the integral terms.

19 To ensure the convergence of the PSOR algorithm, a larger value for N (N = 150) was used in the time-discretization of the LCP.

20 This is not to be confused with the correlation parameters ${\rho }_1$ and ${\rho }_2$. Our implementations assume that these correlations are negative, reflecting the well-known leverage effect that is observed empirically between the diffusion components in the asset prices and in the underlying (stochastic) volatility (Cont 2001). The parameters ${\sigma }_1$ and ${\sigma }_2$ govern the direct contemporaneous relationship between the asset price and the instantaneous volatility level.

21 Chiarella and Ziogas (2009) proposed this method as an alternative to the local analysis of the option PDE for small time-to-maturity options as was done by Wilmott et al. (1993) in the pure diffusion case.

22 By assuming that $S_2\left(0\right)=1$, the MOL prices are then expressed in monetary units rather than in units of the second asset yield process.