Variance Risk Premia

We propose a direct and robust method for quantifying the variance risk premium on financial assets. We theoretically and numerically show that the risk-neutral expected value of the return variance, also known as the variance swap rate, is well approximated by the value of a particular portfolio of options. Ignoring the small approximation error, the difference between the realized variance and this synthetic variance swap rate quantifies the variance risk premium. Using a large options data set, we synthesize variance swap rates and investigate the historical behavior of variance risk premia on five stock indexes and 35 individual stocks.Stochastic volatility, variance risk premia, variance swap, volatility swap, option pricing, expectation hypothesis

Variance Risk Premiums and Predictive Power of Alternative Forward Variances in the Corn Market

We propose a fear index for corn using the variance swap rate synthesized from out-of-the-money call and put options as a measure of implied variance. Previous studies estimate implied variance based on Black (1976) model or forecast variance using the GARCH models. Our implied variance approach, based on variance swap rate, is model independent. We compute the daily 60-day variance risk premiums based on the difference between the realized variance and implied variance for the period from 1987 to 2009. We find negative and time-varying variance risk premiums in the corn market. Our results contrast with Egelkraut, Garcia, and Sherrick (2007), but are in line with the findings of Simon (2002). We conclude that our synthesized implied variance contains superior information about future realized variance relative to the implied variance estimates based on the Black (1976) model and the variance forecasted using the GARCH(1,1) model.Variance Risk Premium, Variance Swap, Model-free Variance, Implied Variance, Realized Variance, Corn VIX

"Hedging European Derivatives with the Polynomial Variance Swap under Uncertain Volatility Environments"

This paper proposes a new hedging scheme of European derivatives under uncertain volatility environments, in which a weighted variance swap called the polynomial variance swap is added to the Black-Scholes delta hedging for managing exposure to volatility risk. In general, under these environments one cannot hedge the derivatives completely by using dynamic trading of only an underlying asset owing to volatility risk. Then, for hedging uncertain volatility risk, we design the polynomial variance, which can be dependent on the level of the underlying asset price. It is shown that the polynomial variance swap is not perfect, but more efficient as a hedging tool for the volatility exposure than the standard variance swap. In addition, our hedging scheme has a preferable property that any information on the volatility process of the underlying asset price is unnecessary. To demonstrate robustness of our scheme, we implement Monte Carlo simulation tests with three different settings, and compare the hedging performance of our scheme with that of standard dynamic hedging schemes such as the minimum-variance hedging. As a result, it is found that our scheme outperforms the others in all test cases. Moreover, it is noteworthy that the scheme proposed in this paper continues to be robust against model risks.

Hedging European Derivatives with the Polynomial Variance Swap under Uncertain Volatility Environments

This paper proposes a new hedging scheme of European derivatives under uncertain volatility environments, in which a weighted variance swap called the polynomial variance swap is added to the Black-Scholes delta hedging for managing exposure to volatility risk. In general, under these environments one cannot hedge the derivatives completely by using dynamic trading of only an underlying asset owing to volatility risk. Then, for hedging uncertain volatility risk, we design the polynomial variance, which can be dependent on the level of the underlying asset price. It is shown that the polynomial variance swap is not perfect, but more efficient as a hedging tool for the volatility exposure than the standard variance swap. In addition, our hedging scheme has a preferable property that any information on the volatility process of the underlying asset price is unnecessary. To demonstrate robustness of our scheme, we implement Monte Carlo simulation tests with three different settings, and compare the hedging performance of our scheme with that of standard dynamic hedging schemes such as the minimum-variance hedging. As a result, it is found that our scheme outperforms the others in all test cases. Moreover, it is noteworthy that the scheme proposed in this paper continues to be robust against model risks.

Hedging European Derivatives with the Polynomial Variance Swap under Uncertain Volatility Environments

This paper proposes a new hedging scheme of European derivatives under uncertain volatility environments, in which a weighted variance swap called the polynomial variance swap is added to the Black-Scholes delta hedging for managing exposure to volatility risk. In general, under these environments one cannot hedge the derivatives completely by using dynamic trading of only an underlying asset owing to volatility risk. Then, for hedging uncertain volatility risk, we design the polynomial variance, which can be dependent on the level of the underlying asset price. It is shown that the polynomial variance swap is not perfect, but more efficient as a hedging tool for the volatility exposure than the standard variance swap. In addition, our hedging scheme has a preferable property that any information on the volatility process of the underlying asset price is unnecessary. To demonstrate robustness of our scheme, we implement Monte Carlo simulation tests with three different settings, and compare the hedging performance of our scheme with that of standard dynamic hedging schemes such as the minimum-variance hedging. As a result, it is found that our scheme outperforms the others in all test cases. Moreover, it is noteworthy that the scheme proposed in this paper continues to be robust against model risks.

Arbitrage Bounds for Prices of Weighted Variance Swaps

We develop robust pricing and hedging of a weighted variance swap when market prices for a finite number of co--maturing put options are given. We assume the given prices do not admit arbitrage and deduce no-arbitrage bounds on the weighted variance swap along with super- and sub- replicating strategies which enforce them. We find that market quotes for variance swaps are surprisingly close to the model-free lower bounds we determine. We solve the problem by transforming it into an analogous question for a European option with a convex payoff. The lower bound becomes a problem in semi-infinite linear programming which we solve in detail. The upper bound is explicit. We work in a model-independent and probability-free setup. In particular we use and extend F\"ollmer's pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are introduced. This allows us to establish the usual hedging relation between the variance swap and the 'log contract' and similar connections for weighted variance swaps. Our results take form of a FTAP: we show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which reproduces the observed prices via risk-neutral expectations of discounted payoffs.Comment: 25 pages, 4 figure

Arithmetic variance swaps

Biases in standard variance swap rates can induce substantial deviations below market rates. Defining realised variance as the sum of squared price (not log-price) changes yields an `arithmetic' variance swap with no such biases. Its fair value has advantages over the standard variance swap rate: no discrete-monitoring or jump biases; and the same value applies for any monitoring frequency, even irregular monitoring and to any underlying, including those taking zero or negative values. We derive the fair-value for the arithmetic variance swap and compare with the standard variance swap rate by: analysing errors introduced by interpolation and integration techniques; numerical experiments for approximation accuracy; and using 23 years of FTSE 100 options data to explore the empirical properties of arithmetic variance (and higher-moment) swaps. The FTSE 100 variance risk has a strong negative correlation with the implied third moment, which can be captured using a higher-moment arithmetic swap

The Term Structure of Variance Swap Rates and Optimal Variance Swap Investments

This paper performs specification analysis on the term structure of variance swap rates on the S&P 500 index and studies the optimal investment decision on the variance swaps and the stock index. The analysis identifies 2 stochastic variance risk factors, which govern the short and long end of the variance swap term structure variation, respectively. The highly negative estimate for the market price of variance risk makes it optimal for an investor to take short positions in a short-term variance swap contract, long positions in a long-term variance swap contract, and short positions in the stock inde

Forecasting the Term Structure of Variance Swaps

Recently, Diebold and Li (2003) obtained good forecasting results for yield curves in a reparametrized Nelson-Siegel framework. We analyze similar modeling approaches for price curves of variance swaps that serve nowadays as hedging instruments for options on realized variance. We consider the popular Heston model, reparametrize its variance swap price formula and model the entire variance swap curves by two exponential factors whose loadings evolve dynamically on a weekly basis. Generalizing this approach we consider a reparametrization of the three-dimensional Nelson-Siegel factor model. We show that these factors can be interpreted as level, slope and curvature and how they can be estimated directly from characteristic points of the curves. Moreover, we analyze a semiparametric factor model. Estimating autoregressive models for the factor loadings we get termstructure forecasts that we compare in addition to the random walk and the static Heston model that is often used in industry. In contrast to the results of Diebold and Li (2003) on yield curves, no model produces better forecasts of variance swap curves than the random walk but forecasting the Heston model improves the popular static Heston model. Moreover, the Heston model is better than the flexible semiparametric approach that outperforms the Nelson-Siegel model.Term structure, Variance swap curve, Heston model, Nelson-Siegel curve, Semiparametric factor model