Variance swaps under Lévy process with stochastic volatility and stochastic interest rate in incomplete market††thanks: This work was supported by the National Natural Science Foundation of China (11471230, 11671282).
Abstract. This study focuses on the pricing of the variance swap in the financial market where the stochastic interest rate and the volatility of the stock are driven by Cox-Ingersoll-Ross model and Heston model with simultaneous Lévy jumps, respectively. After transforming the physical probability measure to the forward measure, we obtain a closed-form solution of the related moment-generating function having the martingale property and the affine structure. Moreover, we get the fair delivery price of the variance swap via the derivation of the moment-generating function under some mild conditions. Finally, some numerical examples are given to show that the values of variance swaps not only depend on the stochastic interest rates but also are higher in the presence of jump risks.
Key Words and Phrases: Variance swap; Stochastic volatility model with jump; Stochastic interest rate; Volatility derivative; Pricing and hedging.
2010 AMS Subject Classification: 91G20, 91G80, 60H10.
In terms of finance volatility has always been considered a key measure. The development and growth of the financial market over last century have caused the role of volatility to change. Volatility derivatives in general is a special financial tool which give opportunity to display market fluctuation and provide methods to manage volatility risk for investors (see, e.g., [1, 2, 4, 5, 26, 30, 32, 33, 34]). Numbers of derivatives written on volatility/variance have been widely used in financial markets in the past twenty years. Volatility derivatives are traded for decision-making between long or short positions, trading spreads between realized and implied volatility, and hedging against volatility risks. The utmost advantage of volatility derivatives is their capability in providing direct exposure towards the assets volatility without being burdened with the hassles of continuous delta-hedging. Of all volatility derivatives, variance swaps are written on underlying assets’ historical volatility and are related to previous standard deviation of financial returns involving a specified time period. Variance swaps are often classified under historical-variance based volatility derivatives and due to their characteristics, they have become popular and are widely traded in financial markets all over the world (see, e.g. [5, 8, 29, 30, 31, 33]).
Along with the explosive growth of variance swaps, not only do investors in financial markets care for correct prices of variance swaps£¬ but researchers in financial mathematics also attempt to construct practical models and feasible methods for pricing variance swaps. In our contrast the important question is whether or not the market can determine a unique price for every given derivative, and this price explicitly computed. The ones to come up with a valid answer to this question was Nobel prize winners Robert Morton and Myron Scholes in the 1970s. Their model, the famous Black-Scholes (short for BS) model, is the market convention and provide an explicit formula to determine the price of European option. Actually, to relax BS’s assumptions and empirically support the market, some of the assumptions made have extended. In general, most of researchers focused on pricing variance swaps (or other volatility derivatives) by developing suitable market models for estimating values of variance swaps, in which three important market factors are considered: the stochastic volatility, the jump diffusion and the stochastic interest rate. The stochastic volatility of underlying assets is widely applied to avoid volatility skew or smile in financial markets, such as Heston model (see, e.g., [14, 21]). The jump diffusion is extensively applied to describe non-Gaussian characters of assets’ returns, such as Geometric Variance Gamma model, Merton model, and Geometric Stable Process Model (see, e.g., [23, 25]). The stochastic interest rate is often used to model the uncertainty of interest rate in the financial markets, such as Cox-Ingersoll-Ross (short for CIR) model, Hull-White (short for HW) model, and Heath-Jarrow-Morton (short for HJM) model (see, e.g., [3, 13, 19, 20]). These extensions are aimed to behave in a nice way and have been seen to give better empirical support.
Based on Heston stochastic volatility model, Grunbichler et al.  developed a pricing model for options on variance. Their important work showed the fundamental difference between volatility derivatives and usual equity options on traded assets. Carr and Madan  combined static replication using options with dynamic trading in the futures to price and hedge variance swaps without specifying the volatility process. This important study is very meaningful to later researches about variance swaps. It was later noted by Heston and Nandi  that specifying the mean reverting square root process has the disadvantage of unobservable underlyings. Thus, they proposed a user friendly model by working on the discrete-time GARCH volatility process with parametric specifications. This model had the advantage of real market practicability, as well as the capability to hedge various volatility derivatives using only a single asset. To distinguish the states of a business cycle, Elliott et al.  constructed a continuous-time Markovian-regulated version of the Heston stochastic volatility model to price variance swaps. Analytical formulas were obtained by using the regime-switching Esscher transform and then price comparisons were made between models with and without switching regimes. Their results showed that the properties and valuation of volatility swaps based on stochastic volatility model were significantly higher than those without stochastic volatility model.
Incorporating jump diffusions into models of pricing and hedging variance swaps, Carr et al.  and Huang et al.  studied the existence of many small jumps that cannot be adequately modelled by using finite-activity compound Poisson processes. In response, a new field of literature has been developed to consider more general jump structures, including infinite-activity Lévy processes. Recently, Ornthanalai  has confirmed Bate’s conclusion by using index options and returns from 1996 to 2009. Zheng and Kwok  presented a general analytic approach for pricing discretely sampled generalized variance swaps under the stochastic volatility models with simultaneous jumps in the asset price and variance processes. The resulting pricing formula of the gamma swap is in closed form while those of the corridor variance swaps and conditional variance swaps take the form of one-dimensional Fourier integrals. Through analytic calculations, they also verify the convergence of the asymptotic limit of the pricing formulas of the discretely sampled generalized variance swaps under vanishing sampling interval to the analytic pricing formulas of the continuously sampled counterparts. Very recently, Cui et al.  proposed a general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps. Their framework encompasses and extends the current literature on discretely sampled volatility derivatives, and provides highly efficient and accurate valuation methods. However, we note that all the mentioned work for the stochastic volatility models with jump diffusion have not incorporated stochastic interest rate into pricing variance swaps.
Considering the stochastic interest rate in financial markets, Kim et al.  proposed a model by combining the multi-scale stochastic volatility model with the Hull-White interest rate model and showed that the values of variance swaps depend on the variety of interest rate. Recently, by considering the effects of stochastic interest rate and the stochastic regime-switching volatility for pricing variance swaps, Shen et al.  investigated the pricing model involving the stochastic interest rate and the stochastic regime-switching volatility. By providing a numerical analysis in a two-state Markov chain case, they demonstrated the effect of both stochastic interest rate and regime-switching is significant in pricing variance swaps. Very recently, Cao et al.  studied the effects of imposing stochastic interest rate driven by the CIR process along with the Heston stochastic volatility model for pricing variance swaps under the discrete sampling times. Some numerical results were given to support the fact that the interest rate could impact and change the value of a variance swap and were reasonable with the real market phenomena. However, it is worth to point out that in the work mentioned above, the jump diffusion factor has not been considered to model the underlying assets’ price process.
Taken all together, we note that the values of variance swaps are dependent on the stochastic volatility, the jump diffusion and the stochastic interest rate. Thus, it would be important and interesting to consider the stated factors in pricing variance swaps. However, as far as we know, the work for pricing variance swaps based on the stochastic volatility with simultaneous jumps and the stochastic interest rate has not been reported. The main purpose of this paper is to make an attempt in this direction. We construct a hybridization model for pricing variance swaps in the financial market, in which the stochastic interest rate is driven by Cox-Ingersoll-Ross model and the volatility of the stock is driven by Heston model with simultaneous Lévy jumps. Then a solution to the fair delivery price of a discretely sampled variance swap is obtained via the analytical expression of the joint moment generating function of the underlying processes. Some numerical experiments are also given to support the main results of this paper.
The rest of this paper is organized as follows. Section 2 gives our analytical formulas for pricing of variance swaps under stochastic volatility model with simultaneous jumps and the stochastic interest rate in partial correlation case. In Section 3, we derive analytical formulas for pricing of variance swaps by solving the model given in full correlation case. Some numerical examples for pricing variance swaps are reported in Section 4. Finally, conclusions are given in Section 5.
2 Stochastic volatility model with jumps and CIR model: partial correlation case
Let be an underlying filtered complete probability space where is the physical probability measure. In this paper, we introduce a stochastic market interest rate with a form of the CIR model to a stochastic volatility model with simultaneous jumps. Precisely, the price of the underlying asset , its instantaneous volatility and the market interest rate are governed by the following system of stochastic differential equations:
where for and ; stands for the value of before a possible jump occurs; are three independent Brownian motions; is a compensated jump measure with respect to the jump measure and the Lévy kernel (density) satisfying ; the drift term is the expected return of the stock; the parameters , and are the mean-reverting speed, the long-term mean and the volatility of volatility (vol of vol) in the instantaneous volatility process , respectively; the parameters , and determine the speed of mean reversion, the long-term mean and the volatility of the interest rate in the stochastic instantaneous interest rate , respectively; is the correlation coefficient between the stock price and the volatility. Furthermore, in order to ensure that the square root processes are always positive, it is required that and (see ).
Some special cases of (1) are as follows.
2.1 Pricing kernel
We note that there are four uncertainties in the system driven by the Brownian motions and the jump process. In basic asset pricing theory, the pricing kernel is critical for determining the pricing and hedging of assets. In short, for any asset and its cash flow at time , the pricing kernel needs to satisfy
where is the conditional expectation at time in the physical probability measure . It is also called the martingale condition which requires that the multiply is a martingale. More precisely, the pricing kernel should follow the restriction
Due to the existence of the jump component, the market considered here is incomplete here. Hence, there are infinitely many equivalent martingale measures for pricing. Here we shall employ the idea of equilibrium pricing method to find an equivalent martingale measure and determine the corresponding risk-neural price dynamics of risk asset.
Now we consider a money market account whose price M(t) with interest rate as follows
If we define the expected excess return of the stock (equity premium) as , then the expected return of the stock will be decomposed into two parts: the interest rate and the equity premium. Assume that there is a representative investor whose portfolio is which means the fraction of wealth invested in the stock and money market, respectively, and the consumption rate of the investor is . Then the investor’s wealth process satisfies the following stochastic differential equations:
Moreover, we assume the representative investor has a CRRA utility
where the relative risk aversion coefficient and . Choosing the portfolio and the consumption rate , the representative investor maximizes naturally his/her expected objective function (3) in an infinite horizon, that is,
Equilibrium in our production economy is defined in a standard way: equilibrium consumption portfolio pairs are such that the representative investor maximizes his/her expected objective function (4) with .
After solving the optimal consumption and portfolio problem by using the Hamilton-Jacobi-Bellman (HJB) method, we get the following lemma.
In the production economy with a representative investor who has CRRA utility and with a production process governed by equations (1), the equilibrium equity premium is given by
where , and are determined by the following equations
Then satisfies the following HJB equation
Following the equilibrium condition in Definition 2.1 and taking , we have
Suppose the value function
Based on the fact that the pricing kernel is related to four risk sources, we define the pricing kernel as follows:
where is the market price of the -th diffusion risk (risk premium) from respectively and the new compensated jump measure
of is modelled by another jump measure and Lévy kernel for . Making use of the martingale condition of pricing kernel , we can get the following lemma in the view of four diffusion risks.
Under the physical measure , the pricing kernel of model (1) satisfies the following equation
where the Lévy kernel satisfies the following martingale condition restriction
Similar to the proof of Proposition 5 in , we can obtain this result easily considering the risks cased by and in the pricing kernel. ∎
We would like to mention that we cannot obtain the risk premium with respect to the volatility simply by applying the martingale condition to our Brownian motion . In fact, the risk premium is not only related to the volatility itself but also to the price of stock . The same holds true for the risk premium with respect to the interest rate . Moreover, the distribution of the Lévy kernel in (14) can be arbitrary, as long as it follows the constraint (15) (see, for example, [5, 6, 25]).
Suppose the stock price and the interest rate are modelled by stochastic processes (1) with discrete or continuous time parameter . The evolution of the system is governed by the underlying probability measure . In general, to price derivatives one usually looks for a so-called equivalent martingale measure which is a probability measure . Recognizing the fact that determining , and the distribution of jump process is not easy in the physical measure , we need to get the particular forms of the stochastic processes in a risk-neutral measure . We define a Radon Nikodým derivative as follows to produce a risk-neutral probability measure:
Then the asset pricing formula can be rewritten under the risk-neutral measure .
Under the probability measure , the asset pricing formula is given by
By the definition of pricing kernel, we have
which just proves our result since is -measurable. ∎
From Lemmas 2.2 and 2.1, using Girsanov Theorem (see [26, 27, 30, 31]), the stochastic system (1) can be changed from the physical measure to the risk-neutral measure. More precisely, Brown motions, the compensated jump measure and the parameters in (1) can be transformed from to . In order to ensure the mean-reverting speed of the volatility and interest rate in measure are constants, we employ Heston’s assumptions on the parameters of mean-reverting process.
As illustrated in , Heston applied Breeden’s consumption-based model to yield a volatility risk premium of the form for the CIR square-root process. Inspired by Heston’s idea, we also adopt risk premiums of the volatility and the interest rate as follows
where are two constants. Then it follows that parameters in the kernel are chosen to be and , respectively. Consequently, the stock price, the volatility process and the interest rate process at time in the risk-neutral probability measure can be rewritten as
where is a Brownian motion under risk-neutral measure for ,
are the risk-neutral parameters.
2.2 Pricing formula for variance swaps
We recall that a variance swap is a forward contract on the future realized variance (RV) of the returns of a specified asset with a maturity and a constant strike level . At the maturity time , the payoff of a variance swap can be evaluated as where is the notional amount of the swap in dollars and RV is the sum of squared returns of asset. In the risk-neutral world, the value of a variance swap is the conditional expectation of its future payoff with respect to defined by
Since it is defined in the class of forward contracts, we know that . The calculation of above expectation is difficult due to it involves the joint distribution of the interest rate and the future payoff. Noticing that the price of a -maturity zero-coupon bond at is given by
we can change the risk-neutral measure to the -forward measure and so
where denotes the expectation under with respect to at . Thus, the fair delivery strike price of variance swap is given by
under the measure and
under the measure , where
Using Itô Lemma, we have
where and are the partial derivatives with respect to .
From the above discussion, we have the following result.
Equations (17) under the risk-neutral measure can be transformed into the following system under the forward measure :
Let be the log-price process and . Then the joint moment-generating function of joint processes , and in (18) can be defined as follows
where , and are constant parameters. Moreover, if
then the value of at is given by
where and satisfy
with initial conditions
It follows from (18) that
Next we prove that is an -martingale. In fact, by the Markov property of , one has
Thus, it suffices to prove that
for each . Denote
Since and , we know that and so Corollary 5 (for the one-dimensional case) in  shows that
which shows that is an -martingale.
Now by applying Itô Lemma to , we can obtain a partial integral-differential equation (PIDE) for as follows
Denoting , we get
If all the conditions of Theorem 2.3 holds, then can be expressed as