On the pricing of forward-start variance swaps with stochastic volatility and stochastic interest rate

Variance swaps have gained an immense recognition in the financial market based on the tremendous spike in its trading volume since late 1990s. Being categorized under volatility derivatives, the substance of variance swaps can be related to the vital role of volatility in making investment decisions. In this paper, the price of discretely-sampled forward-start variance swaps is evaluated using an equity-interest rate hybrid model. The modeling framework involves an extension of the Heston stochastic volatility model, which is combined with the dynamics of the Cox-Ingersoll-Ross (CIR) stochastic interest rate model. Focus is given on the forward-start nature, identified by the starting time of the sampling period being a future date. Previous studies on variance swaps were mainly focusing on instantaneous-start variance swaps, whereas in reality, most of traded variance swap

The valuation of variance swaps under stochastic volatility, stochastic interest rate and full correlation structure

This paper considers the case of pricing discretely-sampled variance swaps under the class of equity-interest rate hybridization. Our modeling framework consists of the equity which follows the dynamics of the Heston stochastic volatility model, and the stochastic interest rate is driven by the Cox-Ingersoll-Ross (CIR) process with full correlation structure imposed among the state variables. This full correlation structure possesses the limitation to have fully analytical pricing formula for hybrid models of variance swaps, due to the non-affinity property embedded in the model itself. We address this issue by obtaining an efficient semi-closed form pricing formula of variance swaps for an approximation of the hybrid model via the derivation of characteristic functions. Subsequently, we implement numerical experiments to evaluate the accuracy of our pricing formula. Our findings confirm that the impact of the correlation between the underlying and the interest rate is significant for pricing discretely-sampled variance swaps

Valuation of discretely-sampled variance swaps under correlated stochastic volatility and stochastic interest rates

In this paper, we evaluate the price of discretely-sampled variance swaps using a equity-interest rate hybrid model. Our modeling framework extends the Heston stochastic volatility model by including the Cox-Ingersoll-Ross stochastic interest rates and imposes correlation between the stochastic interest rate and volatility. It is known that one limitation of the hybrid models is that the analytical pricing formula is often unavailable due to the non-affinity property of hybrid models. An efficient semi-closed form pricing formula is derived for an approximation of the fully correlated hybrid model. Our pricing formula which involves solving two phases of three-dimensional partial differential equations is evaluated through numerical implementations to confirm its accurac

Pricing variance swaps in a hybrid model of stochastic volatility and interest rate with regime-switching

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

Hybrid equity warrants pricing formulation under stochastic dynamics

—A warrant is a financial contract that confers the right but not the obligation, to buy or sell a security at a certain price before expiration. The standard procedure to value equity warrants using call option pricing models such as the Black–Scholes model had been proven to contain many flaws, such as the assumption of constant interest rate and constant volatility. In fact, existing alternative models were found focusing more on demonstrating techniques for pricing, rather than empirical testing. Therefore, a mathematical model for pricing and analyzing equity warrants which comprises stochastic interest rate and stochastic volatility is essential to incorporate the dynamic relationships between the identified variables and illustrate the real market. Here, the aim is to develop dynamic pricing formulations for hybrid equity warrants by incorporating stochastic interest rates from the Cox-Ingersoll-Ross (CIR) model, along with stochastic volatility from the Heston model. The development of the model involves the derivations of stochastic differential equations that govern the model dynamics. The resulting equations which involve Cauchy problem and heat equations are then solved using partial differential equation approaches. The analytical pricing formulas obtained in this study comply with the form of analytical expressions embedded in the Black-Scholes model and other existing pricing models for equity warrants. This facilitates the practicality of this proposed formula for comparison purposes and further empirical study

Modeling the price of hybrid equity warrants under stochastic volatility and interest rate

Previous studies revealed that most local researchers frequently used the Black Scholes model to price equity warrants. However, the Black Scholes model was perceived of possessing too many drawbacks, such as big errors of estimation and mispricing of equity warrants. In this work, we consider the problem of pricing hybrid equity warrants based on a hybrid model of stochastic volatility and stochastic interest rate. The integration of stochastic interest rate using the Cox-Ingersoll-Ross (CIR) model, along with stochastic volatility of the Heston model was first developed as a hybrid model. We solved the governing stochastic equations and come up with analytical pricing formulas for hybrid equity warrants. This provides an alternative method for valuation of equity warrants, compared to the usual practice of utilizing the Black Scholes pricing formula

A comprehensive literature review on pricing equity warrants using stochastic approaches

Prior literature's revealed that most researchers tend to employ the Black Scholes model to price equity warrants. However, the Black Scholes model was found deficient by contributing to large estimation errors and mispricing of equity warrants. Therefore( issues revolving equity warrants are discussed in this paper, by focusing on specific topics and respective stochastic models to provide a basis for improvements in future research. In recent years, stochastic approaches had been used to a great extent among researchers due to the expansive applications in both theoretical and practical sense. Subsequently, this paper provides the results of a comprehensive literature review on various stochastic modelling methods and its applications for pricing financial derivatives in terms of applications, modifications of methods, comparisons with other methods, and general related researches. Focus is given on two types of stochastic mod~ls namely stochastic volatility and stochastic interest rate models; along with the discussions associating these two types of models. This paper acts as a valuable source of information for academic researchers and practitioners not only for pricing financial instruments, but also in various other fields involving stochastic techniques

A Stochastic Hybrid Model for Pricing Forward-Start Variance Swaps

Recently, market players have been exposed to the astounding increase in the trading volume of variance swaps. In this paper, the forward-start nature of a variance swap is king inspected. where hybridizations or equity and interest rate models are used to evaluate the price of discretely-sampled forward-start variance swaps . The Hc~ton~ tmhasticv olatiliry model 1s being extended to incorporate the dynamic5 of the Cox-Tngersoll-Ross (CIR)s tochast~cln teresr rate model. This is essential since previous studies on variance swaps were mainly focusing on instantaneous-start variance swaps without considering the interest rate effects. This hybrid mcdc! pmduccp an cfficicnt scmi-closcd form pricing Formula through the dcvclopmen~ of tijnvnrd characteristic functions. The performance of this formula is investigated via simulations to demonstrate how the formula performs for different sampling times and again the real market scenario. Comparison done with the Monte Car1o simulation which was set as our main reference point reveals that our pricing formula gains almost the same precision in a shorter execution time

Pricing variance swaps under stochastic volatility and stochastic interest rate

In this thesis, we study the issue of pricing discretely-sampled variance swaps under stochastic volatility and stochastic interest rate. In particular, our modeling framework consists of the equity which follows the dynamics of the Heston stochastic volatility model, whereas the stochastic interest rate is driven by the Cox-Ingersoll-Ross (CIR) model. We first extend the framework of [119] by incorporating the CIR interest rate into their Heston model for pricing discretely-sampled variance swaps. We impose partial correlation between the asset price and the volatility, and derive a semi-closed form pricing formula for the fair delivery price of a variance swap. Several numerical examples and comparisons are provided to validate our pricing formula, as well as to show the effect of stochastic interest rate on the pricing of variance swaps. In addition, the pricing of discretely-sampled variance swaps with full correlation among the asset price, interest rate as well as the volatility is investigated. This offers a more realistic model with practical importance for pricing and hedging. Since this full correlation model is incompliant with the analytical tractability property, we determine the approximations for the non-affine terms by following the approach in [55] and present a semi-closed form approximation formula for the fair delivery price of a variance swap. Our results confirm that the impact of the correlation between the stock price and the interest rate on variance swaps prices is very crucial. Besides that, the impact of correlation coefficients becomes less apparent as the number of sampling frequencies increases for all cases. Finally, the issue of pricing discretely-sampled variance swaps under stochastic volatility and stochastic interest rate with regime switching is also discussed. This model is an extension of the corresponding one in [34] and is capable of capturing several macroeconomic issues such as alternating business cycles. Our semi-closed form pricing formula is proven to achieve almost the same accuracy in far less time compared with the Monte Carlo simulation. Through numerical examples, we discover that prices of variance swaps obtained from the regime switching Heston-CIR model are significantly lower than those of its non-regime switching counterparts. Furthermore, when allowing the Heston-CIR model to switch across three regimes, it is observable that the price of a variance swap is cheapest in the best economy, and most expensive in the worst economy among all

Analytical Pricing Formulas for Hybrid Variance Swaps with Regime-Switching

The problem of pricing discretely-sampled variance swaps under stochastic volatility, stochastic interest rate and regimeswitchin" e is beine considered in this oaoer. An extension of the Heston stochastic volatiliw model structure is done bv adding the - . . . Cox-lngersoll-Row (CIR) ctochawc intereel rare modcl In add~uon[,h e parameme of the mndel are perm~rtedlo have uan<luonc followane a Markov cham proccrs whtch I, conunuous and &>.uvcrable lh~shb bnd model cdn be uced lo tllusunre ccrrlrn macmec~nomicc onditions.-for example the changing phases of business stages. The outcome of our regime-switching hybrid model is presented in lerms of analytical pricing formulas for variance swaps