How does the market variance risk premium vary over time? Evidence from S&P 500 variance swap investment returns

We explore whether the market variance risk premium (VRP) can be predicted. We measure VRP by distinguishing the investment horizon from the variance swap’s maturity. We extract VRP from actual S&P 500 variance swap quotes and we test four classes of predictive models. We find that the best performing model is the one that conditions on trading activity. This relation is also economically significant. Volatility trading strategies which condition on trading activity outperform popular benchmark strategies, even once we consider transaction costs. Our finding implies that broker dealers command a greater VRP to continue holding short positions in index options in the case where trading conditions deteriorate.     

Hedging the smirk

This article presents a simple “model-free” method for inferring deltas and gammas from implicit volatility patterns. An illustration indicates that Black–Scholes deltas and gammas are substantially biased in the presence of the sort of smirks and smiles evident in stock index options.     

Is gold the best hedge and a safe haven under changing stock market volatility?

We evaluate the role of gold and other precious metals relative to volatility (Volatility Index (VIX)) as a hedge (negatively correlated with stocks) and safe haven (negatively correlated with stocks in extreme stock market declines) using data from the US stock market. Using daily data from November 1995 to November 2010, we find that gold, unlike other precious metals, serves as a hedge and a weak safe haven for US stock market. However, we find that VIX serves as a very strong hedge and a strong safe haven during our sample period. We also find that in periods of extremely low or high volatility, gold does not have a negative correlation with the US stock market. Our results show that VIX is a superior hedging tool and serves as a better safe haven than gold during our sample period. We highlight the practical significance of our results for financial market participants by conducting a portfolio analysis.     

Optimal investments in volatility

Volatility has evolved as an attractive new asset class of its own. The most common instruments for trading volatility are variance swaps. Mean returns of DAX and ESX variance swaps over the time period of 1995 to 2004 are strongly negative, and only part of the negative premium can be explained by the negative correlation of variance swap returns with stock market indices. We analyze the implications of this observation for optimal portfolio composition. Mean-variance efficient portfolios are characterized by sizable short positions in variance swaps. Typically, the stock index is also sold short to achieve a better portfolio diversification. To capture heterogeneous preferences for higher moments, we use a variant of the polynomial goal programming method. We assume that investors strive for a high Sharpe ratio, high skewness, and low kurtosis. Our analysis reveals that it is often not possible to achieve a balanced tradeoff between Sharpe ratio and skewness. Investors are advised to hold the extreme portfolios (Sharpe ratio driven, skewness driven, or kurtosis driven) and avoid the middle ground. This “all-or-nothing” characteristic is reflected in jumps of asset weights when certain thresholds of preference parameters are crossed. These empirical findings can explain why many investors are so reluctant to implement option-based short-selling strategies.

Pricing variance and volatility swaps in a stochastic volatility model with regime switching: discrete observations case

This study presents a set of closed-form exact solutions for pricing discretely sampled variance swaps and volatility swaps, based on the Heston stochastic volatility model with regime switching. In comparison with all the previous studies in the literature, this research, which obtains closed-form exact solutions for variance and volatility swaps with discrete sampling times, serves several purposes. (1) It verifies the degree of validity of Elliott et al.'s [Appl. Math. Finance, 2007, 14(1), 41–62] continuous-sampling-time approximation for variance and volatility swaps of relatively short sampling periods. (2) It examines the effect of ignoring regime switching on pricing variance and volatility swaps. (3) It contributes to bridging the gap between Zhu and Lian's [Math. Finance, 2011, 21(2), 233–256] approach and Elliott et al.'s framework. (4) Finally, it presents a semi-Monte-Carlo simulation for the pricing of other important realized variance based derivatives.     

Machine learning for quantitative finance: fast derivative pricing,hedging and fitting

In this paper, we show how we can deploy machine learning techniques in the context of traditional quant problems. We illustrate that for many classical problems, we can arrive at speed-ups of several orders of magnitude by deploying machine learning techniques based on Gaussian process regression. The price we have to pay for this extra speed is some loss of accuracy. However, we show that this reduced accuracy is often well within reasonable limits and hence very acceptable from a practical point of view. The concrete examples concern fitting and estimation. In the fitting context, we fit sophisticated Greek profiles and summarize implied volatility surfaces. In the estimation context, we reduce computation times for the calculation of vanilla option values under advanced models, the pricing of American options and the pricing of exotic options under models beyond the Black–Scholes setting.     

Willow tree algorithms for pricing Guaranteed Minimum Withdrawal Benefits under jump-diffusion and CEV models

This paper presents the willow tree algorithms for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB), where the underlying fund dynamics evolve under the Merton jump-diffusion process or constant-elasticity-of-variance (CEV) process. The GMWB rider gives the policyholder the right to make periodic withdrawals from his policy account throughout the life of the contract. The dynamic nature of the withdrawal policy allows the policyholder to decide how much to withdraw on each withdrawal date, or even to surrender the contract. For numerical valuation of the GMWB rider, we use willow tree algorithms that adopt more effective placement of the lattice nodes based on better fitting of the underlying fund price distribution. When compared with other numerical algorithms, like the finite difference method and fast Fourier transform method, the willow tree algorithms compute GMWB prices with significantly less computational time to achieve a similar level of numerical accuracy. The design of our pricing algorithm also includes an efficient search method for the optimal dynamic withdrawal policies. We perform sensitivity analysis of various model parameters on the prices and fair participating fees of the GMWB riders. We also examine the effectiveness of delta hedging when the fund dynamics exhibit various jump levels.     

Preposterior analysis for option pricing

The second partial derivative of a European-style vanilla option with respect to the current price of the underlying asset—the option gamma—defines a probability density function for the current underlying price. By use of entropy maximization it is possible to obtain an option gamma, from which the associated option pricing formula can be recovered by integration. A number of pricing formulae are obtained in this manner, corresponding to different specifications of the constraints. When the available market information consists solely of a set of traded option prices, the entropic formulation leads to a model-independent calibration procedure. The result thus obtained also allows one to recover the relevant Greeks.     

Does country risks predict stock returns and volatility? Evidence from a nonparametric approach

We use the k-th order nonparametric causality test at monthly frequency over the period of 1984:1–2015:12 to analyze whether aggregate country risk, and its components (economic, financial and political) can predict movements in stock returns and volatility of eighty-three developed and developing economies. The nonparametric approach controls for the existing misspecification of a linear framework of causality, and hence, the weak evidence of causality obtained under the standard Granger tests cannot be relied upon. When we apply the nonparametric test, we find that, while there is no evidence of predictability of squared stock returns barring one case, at times, there are nearly 50 percent of the countries where the aggregate risks and its components tend to predict stock returns and realized volatility.     

Hedging jump risk,expected returns and risk premia in jump-diffusion economies

This article presents a pure exchange economy that extends Rubinstein [Bell J. Econ. Manage. Sci., 1976, 7, 407–425] to show how the jump-diffusion option pricing model of Black and Scholes [J. Political Econ., 1973, 81, 637–654] and Merton [J. Financ. Econ., 1976, 4, 125–144] evolves in gamma jumping economies. From empirical analysis and theoretical study, both the aggregate consumption and the stock price are unknown in determining jumping times. By using the pricing kernel, we determine both the aggregate consumption jump time and the stock price jump time from the equilibrium interest rate and CCAPM (Consumption Capital Asset Pricing Model). Our general jump-diffusion option pricing model gives an explicit formula for how the jump process and the jump times alter the pricing. This innovation with predictable jump times enhances our analysis of the expected stock return in equilibrium and of hedging jump risks for jump-diffusion economies.     

The impact of jump risks on nominal interest rates and foreign exchange rates

This article investigates the relationship between the nominal interest rate and inflation and also the forward exchange rate under a general specification of the underlying processes govering the foreign exchange rate. There are three distinct risks that affect the relation between the real rate of interest and the nominal rate namely, consumption risk, diffusion risk, and the existence of jump risks of inflation. Jump risks lower the nominal interest rate because of jump hedging of a nominal bond. The forward exchange rate depends on the expected depreciation of the domestic currency as well as these three risks. As the domestic jump risks increase, the domestic nominal interest rate decreases and the forward exchange rate decreases.     

Pricing mortgage insurance contracts under housing price cycles with jump risk: evidence from the U.K. housing market

Previous studies have investigated the determinants of housing price cycles in the housing market; however, we observed the phenomenon of housing price jumps in the 2007 subprime crisis. This paper presents a discussion on the housing price cycle and abnormal price jumps to describe the behavior of housing prices in the United Kingdom. The empirical results show that the impact factors of housing cycles are market risk and the switching factor. Furthermore, the impact factors of jump risks include the bursting of the housing bubble and financial crises. Therefore, in this paper, we employ the Markov switching model with jump risks to value the MI contracts and analyze the influences of housing price cycles, jump risks, risks of market interest rate, and the prepayment risks on MI premiums. The results of sensitivity analysis show that more volatile housing price index returns, as well as longer periods of higher volatility in housing prices, raise MI premiums. Moreover, the MI premium is positively related to the absolute value of the average jump amplitude and the shock frequency of abnormal events. There is the tradeoff between the market interest rate and the prepayment risk. The influences of market interest rate are different on MI premium with/without prepayment risks.     

Volatility and market structure

This study examines volatility within three related intra-day series – transaction returns, quote midpoint returns, and limit order book midpoint returns – for a set of NYSE-listed stocks. We document statistically significant GARCH effects both overall and surrounding earnings announcements in all three series for the majority of stocks in the sample. We then compare the extent of volatility clustering among the series. In addition, the relation between volatility and market structure is examined via a set of cross-sectional regressions, and relations among the series over time are studied in a vector autoregressive framework.     

Pricing and Hedging of Multi Type Contracts under Multidimensional Risks in Incomplete Markets Modeled by General Itô SDE Systems

The HARA and CARA theory of pricing, and the theory of partial, yet the most conservative hedging, of a single (liquid) tradable derivative contract under multidimensionality of risks in incomplete markets, including markets with non-hedgable interest rate risks, was developed by the author in a recent paper. In the present paper this theory is extended to the general case of simultaneous pricing and hedging of multiple (types of) such contracts. The results are based on the generalization of the “fundamental matrix of derivatives pricing and hedging” to include multiple contracts. Some applications are discussed as well.     

Smart expansion and fast calibration for jump diffusions

Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump processes. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black–Scholes volatilities for call options is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very rapid.