VIX as a Variance Swap

In this chapter, we give an overview on applications of change of time methods considered in this book in Chapters 4–8 These applications include yet another (among many) derivation of the Black-Scholes formula; the derivation of option pricing formula for a mean-reverting asset in energy finance; pricing of variance, volatility, covariance, and correlation swaps for the classical Heston model; pricing of variance and volatility swaps in energy markets; pricing of financial and energy derivatives with multifactor Levy models; and pricing of variance and volatility swaps and hedging of volatility swaps for the delayed Heston model. This chapter not only describes the applications of the change of time method but also constitutes the ultimate difference between Barndorff-Nielsen-Shiryaev’s book (2010) and present book.

VIX and Volatility Derivatives: A Survey of Relevant Financial Instruments and Pricing Models.

In this paper, we consider volatility swap and variance swap when the underlying asset is described by a process with multiple stochastic volatility models. The model considered in this paper is the multi-factor Heston stochastic volatility model. We obtain pricing formulas for the weighted variance swap and approximate expressions for the weighted volatility swap. The bounds of the arbitrage-free variance swap price are also found. The proposed pricing formulas are easy to compute in real time and can be applied efficiently for practical applications. We study the problem of hedging volatility swap with variance swap. We also 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. From the numerical analysis, a couple of important features of the usefulness of the multi-factor Heston stochastic volatility model are discussed.

Volatility, Variance and Covariance Swaps for BN-S Type Levy Process Driven Financial Markets

We introduce a discrete Barndorff-Nielsen and Shephard (BNS) stochastic volatility model, where the non-Gaussian Ornstein-Uhlenbeck process governs the instantaneous variance of an underlying asset. A closed-form solution for pricing variance swaps in financial markets driven by a BN-S type Levy process is derived. Additionally, we propose a discrete-time framework for arbitrage-free pricing of variance swaps, supported by simulations that provide numerical validation of our results. The model fitting and parameter estimation processes are also discussed in detail.

Some of the newest and most recent financial products are variance and volatility swaps, which are useful for volatility hedging and speculation. The market for variance and volatility swaps has been growing, and many investment banks and other financial institutions are now actively quoting volatility swaps on various assets: stock indexes, currencies, as well as commodities. In this chapter, the authors model financial markets with semi-Markov volatilities and price variance, volatility, covariance and correlation swaps. Numerical evaluations of variance, volatility, covariance and correlations swaps with semi-Markov volatility are also presented. The novelty of the chapter lies in the pricing of volatility swaps in the closed form, and the pricing of covariance and correlation swaps in a market with two risky assets. Volatility swaps are forward contracts on future realized stock volatility, while variance swaps are similar contracts on variance, the square of the future volatility.

Discretely sampled variance and volatility swaps trade actively in OTC markets. To price these swaps, the continuously sampled approximation is often used to simplify the computations. The purpose of this paper is to study the conditions under which this approximation is valid. Our first set of theorems characterize the conditions under which the discretely sampled swap values are finite, given that the values of the continuous approximations exist. Surprisingly, for some otherwise reasonable price processes, the discretely sampled swap prices do not exist, thereby invalidating the approximation. Examples are provided. Assuming further that both swap values exist, we study sufficient conditions under which the discretely sampled values converge to their continuous counterparts. Because of its popularity in the literature, we apply our theorems to the 3/2 stochastic volatility model. Although we can show finiteness of all swap values, we can prove convergence of the approximation only for some parameter values.

This paper presents a robust methodology for the valuation of options on variance swaps and volatility swaps. While existing literature has often focused on the pricing of the swaps themselves under stochastic volatility models, the valuation of options on these second-order derivative products remains a challenge, particularly within a framework that captures empirical market properties such as volatility clustering and long memory. Our approach addresses this problem by relying on the precise calculation of the optimal exercise prices (or strike rates) for variance and volatility swaps. To achieve this, we introduce and utilize a multifractional extension of the classic SABR model. This enhanced version of the model incorporates a deterministic and time-varying Hurst function, h(t), which allows for the modeling of a richer and more flexible volatility dynamics, capable of replicating the varying persistence phenomena observed in real markets. The proposed methodology is detailed comprehensively. We derive the theoretical formulas for the swap prices, which form the basis for calculating the optimal exercise prices. The valuation of options (call and put) on these swaps is then performed by employing advanced numerical techniques, adapted to the complexity of the multifractional process.

The VIX and Future Information

We extract variance and skew risk premiums from volatility derivatives in a model‐free way and analyze their relationships along with volatility index and equity index returns. These risk premiums can be synthesized through option trading strategies. Using a time series of option prices on the VIX, we find that variance swap excess return can be partially explained by volatility index and equity index excess returns while these latter variables carry little information for the skew swap excess return. The results sharply contrast with those obtained for the equity index option market underlining very specific characteristics of the volatility derivative market.

Variance and Volatility Swaps in Energy Markets

Introducing the BITIX: The Bitcoin Fear Gauge

This paper shows the limitation of the cost-of-carry model which is used for pricing the theoretical value of the KTB futures, and proposes an alternative pricing model based on the term structure of interest rates. Under the assumption of 1-factor term structure, this paper treats the theoretical price of KTB futures price as a risk-neutral expectation of payoff function at maturity and derives the approximated formula for pricing the KTB futures. As compared with our price of KTB futures using the term structure of interest rates, the conventional KOFEX price based on the cost-of-carry model tends to be overvalued as the time to maturity increases. This result is due to the difference between the futures and forward prices which is caused by treating the futures contract as the forward contract in the conventional KOFEX pricing model. In particular, this discrepancy becomes more significant when the price of underlying asset and the interest rates are negatively correlated and the time to maturity is longer. The bond futures contract is a typical example of financial instrument whose price has a negative correlation with interest rates

Contractual arrangement plays an important role in mitigating market power in electricity markets. The issue of whether rational generators would voluntarily enter contract markets through a strategic incentive is examined, and the factors which could affect this strategic contracting behaviour. A two-stage game model is presented to formulate the competition of generators in bid-based pool spot markets and contract markets, as well as the interaction between these two markets. The affine supply function equilibrium (SFE) method is used to model competitive bidding for the spot market, while the contract market is modelled with the general conjectural variation method. The proposed methodology allows asymmetric, multiple strategic generators having capacity constraints and affine marginal costs with non-zero intercepts to be taken into account. It is shown that the presence of forward contract markets will complicate the solution to the affine SFE, and a new methodology is developed in this regard. Strategic contracting behaviours are analysed in the context of asymmetric, multiple strategic generators. A numerical example is used to verify theoretical results. It is shown that the observability of contract markets plays an important role in fostering generators' strategic contracting incentive, and that this contracting behaviour could also be affected by generators' cost parameters and demand elasticity.

The authors use a novel 32-year return series to study the risk, return, and predictability of a strategy that sells one-month S&P 500 variance swaps with fixed ex-ante tail risk. They find that unconditional short exposure in their sample is characterized primarily by two features: (1) a very high Sharpe ratio exceeding 1.2 and (2) a severe but infrequent crash risk. From a forecasting perspective, the authors find a generally lower premium following market sell-offs and crashes. However, they fail to find significant evidence linking returns to the level of either implied or realized volatility.