Pricing Variance Swaps in a Hybrid Model of Stochastic Volatility and Interest Rate with Regime-Switching

Abstract

In this paper, we consider the problem of pricing discretely-sampled variance swaps based on a hybrid model of stochastic volatility and stochastic interest rate with regime-switching. Our modeling framework extends the Heston stochastic volatility model by including the Cox-Ingersoll-Ross (CIR) stochastic interest rate model. In addition, certain model parameters in our model switch according to a continuous-time observable Markov chain process. This enables our model to capture several macroeconomic issues such as alternating business cycles. A semi-closed form pricing formula for variance swaps is derived. The pricing formula is assessed through numerical implementation, where we validate our pricing formula against the Monte Carlo simulation. The impact of incorporating regime-switching for pricing variance swaps is also discussed, where variance swaps prices with and without regime-switching effects are examined in our model. We also explore the economic consequence for the prices of variance swaps by allowing the Heston-CIR model to switch across three different regimes.

Appendix: A Results on Variance Swaps Associated to the Simple-returns

In this appendix, we provide a detailed numerical calculation for pricing variance swaps via the simple-return realized variance given by formula (2), parallel to that in Section 4. By using the valuation of the fair delivery price for a variance swap, and summarizing the whole previous procedure, we can write the forward characteristic function for a variance swap as

$$\begin{array}{@{}rcl@{}} &&\mathbb{E}^{T}\left( \left( \frac{S(t_{j})}{S(t_{j-1})}-1 \right)^{2} \Big| \mathscr F_{1}(0) \vee \mathscr F_{2}(0) \vee \mathscr F_{3}(0) \vee \mathscr F_{X}(0) \right)\\ &=&\mathbb{E}^{T}\left( e^{2y(t_{j-1})}-2e^{y(t_{j-1})}+ 1 |\mathscr F_{1}(0) \vee \mathscr F_{2}(0) \vee \mathscr F_{3}(0) \vee \mathscr F_{X}(0)\right)\\ &=&f(2;0,t_{j-1},{\Delta} t,\nu(0),r(0))-2f(1;0,t_{j-1},{\Delta} t, \nu(0),r(0))+ 1, \end{array} $$

(48)

where $y(t_{j-1})=\ln S(t_{j})-\ln S(t_{j-1})$, Δt = t_j − t_j− 1, and the characteristic function f(ϕ; t, T, Δ, ν(t), r(t)) is given in (35). Hence, the fair strike price for a variance swap in terms of the spot variance ν(0) and the spot interest rate r(0) under T-forward measure is given as

$$\begin{array}{@{}rcl@{}} {}K\!&=&\mathbb{E}^{T}(RV_{d1})\\ {}&=&\frac{100^{2}}{T}\sum\limits_{j = 1}^{N} \left( f(2;0,t_{j-1},{\Delta} t, \nu(0), r(0)) -2f(1;0,t_{j-1},{\Delta} t, \nu(0), r(0))+ 1\right). \end{array} $$

(49)

Using the parameter values given in Table 1, we demonstrate the validation of formula (49) against Monte Carlo simulation in Fig. 5. Again, the sampling frequency varies from N = 1 up to N = 52, and the Monte Carlo simulation is conducted using the Euler-Maruyama discretization with 200,000 sample paths.

Fig. 5

Strike prices of variance swaps associated to the simple-returns for the Heston-CIR model with regime-switching using parameter values in Table 1 and the Monte Carlo simulation

By comparing results presented in Fig. 5 with those presented in Fig. 1, we can see that there is no significant difference between the results in using two formulae for measuring the realized variance of swaps.