Variance Swap Pricing under Markov-Modulated Jump-Diffusion Model

This paper investigates the pricing of discretely sampled variance swaps under a Markov regime-switching jump-diffusion model. The jump diffusion, as well as other parameters of the underlying stock’s dynamics, is modulated by a Markov chain representing different states of the market. A semi-closed-form pricing formula is derived by applying the generalized Fourier transform method. The counterpart pricing formula for a variance swap with continuous sampling times is also derived and compared with the discrete price to show the improvement of accuracy in our solution. Moreover, a semi-Monte-Carlo simulation is also presented in comparison with the two semi-closed-form pricing formulas. Finally, the effect of incorporating jump and regime switching on the strike price is investigated via numerical analysis.

Quadratic variation, models, applications and lessons

<p style="text-indent:20px;">Time changes of Brownian motion impose restrictive jump structures in the motion of asset prices. Quadratic variations also depart from time changes. Quadratic variation options are observed to have a nonlinear exposure to risk neutral skewness. Joint Laplace Fourier transforms for quadratic variation and the stock are developed. They are used to study the multiple of the cap strike over the variance swap quote attaining a given percentage price reduction for the capped variance swap. Market prices for out-of-the-money options on variance are observed to be above those delivered by the calibrated models. Bootstrapped data and simulated paths spaces are used to study the multiple of the dynamic hedge return desired by a quadratic variation contract. It is observed that the optimized hedge multiple in the bootstrapped data is near unity. Furthermore, more generally, it is exposures to negative cubic variations in path spaces that raise variance swap prices, lower hedge multiples, increase residual risk charges and gaps to the log contract hedge. A case can be made for both, the physical process being closer to a continuous time change of Brownian motion, while simultaneously risk neutrally this may not be the case. It is recognized that in the context of discrete time there are no issues related to equivalence of probabilities.</p>

Effect of Variance Swap in Hedging Volatility Risk

This paper studies the effect of variance swap in hedging volatility risk under the mean-variance criterion. We consider two mean-variance portfolio selection problems under Heston’s stochastic volatility model. In the first problem, the financial market is complete and contains three primitive assets: a bank account, a stock and a variance swap, where the variance swap can be used to hedge against the volatility risk. In the second problem, only the bank account and the stock can be traded in the market, which is incomplete since the idiosyncratic volatility risk is unhedgeable. Under an exponential integrability assumption, we use a linear-quadratic control approach in conjunction with backward stochastic differential equations to solve the two problems. Efficient portfolio strategies and efficient frontiers are derived in closed-form and represented in terms of the unique solutions to backward stochastic differential equations. Numerical examples are provided to compare the solutions to the two problems. It is found that adding the variance swap in the portfolio can remarkably reduce the portfolio risk.

VALUATION, HEDGING, AND BOUNDS OF SWAPS UNDER MULTI-FACTOR BNS-TYPE STOCHASTIC VOLATILITY MODELS

In this paper, we consider price weighted-volatility swap and price weighted-variance swap. The underlying asset considered in this paper is assumed to follow a general stochastic differential equation and exhibits stochastic volatility. We obtain analytical pricing formulas for the weighted-variance swap and approximate expression for the weighted-volatility swap. Nice bounds for the arbitrage-free variance swap price are also found. The proposed pricing formulas are easy to implement in real time and can be applied efficiently for practical applications. We consider the problem of hedging volatility swap with variance swap and obtain analytical formula for the hedge ratio. We also consider a problem of hedging an asset with variance swap and option. We determined the optimal amount of the underlying asset that has to be held for minimizing the hedging error by taking positions in options and weighted-variance swap. A numerical example is also provided.