Pricing and hedging contingent claims using variance and higher order moment swaps

This paper suggests perfect hedging strategies of contingent claims under stochastic volatility and random jumps of the underlying asset price. This is done by enlarging the market with appropriate swaps whose pay-offs depend on higher order sample moments of the asset price process. Using European options and variance swaps, as well as barrier options written on the S&P 500 index, the paper provides clear cut evidence that hedging strategies employing variance and higher order moment swaps considerably improves upon the performance of traditional delta hedging strategies. Inclusion of the third-order moment swap improves upon the performance of variance swap-based strategies to hedge against random jumps. This result is more profound for short-term out-of-the money put options.     

Insiders' hedging in a jump diffusion model

In this paper, we formulate the optimal hedging problem when the underlying stock price has jumps, especially for insiders who have more information than the general public. The jumps in the underlying price process depend on another diffusion process, which models a sequence of firm-specific information. This diffusion process is observed only by insiders. Nevertheless, the market is incomplete to insiders as well as to the general public. We use the local risk minimization method to find an optimal hedging strategy for insiders. We also numerically compare the value of the insider's hedging portfolio with the value of an honest trader's hedging portfolio for a simulated sample path of a stock price.     

Time and frequency dynamics of connectedness and hedging performance in global stock markets: Bitcoin versus conventional hedges

Some studies have revealed the hedging ability of Bitcoin against stock markets, but the knowledge of how it compares with other hedges is in its infancy. This paper presents the first study on time-frequency domain connectedness and hedging among five hedges (Bitcoin, crude oil, commodities, gold and the U.S. dollar (USD) index) and four stock indices (developed markets ex U.S., emerging markets ex China, U.S. and China). We find that the connectedness between hedges and stock markets varies by time across time horizons. Specifically, the connectedness between Bitcoin and stock indices is the smallest among all hedges, especially for the short horizon. Gold and USD are isolated from other markets at longer horizons. The hedging ratio, optimal portfolio weights and hedging effectiveness also vary across investment horizons. For short-term investment, gold has better hedging effectiveness, especially for emerging stock markets and the U.S. stock market. For median- and long-term investment, USD has better performance, especially for developed markets ex U.S. and emerging stock markets. Additionally, although Bitcoin has good hedging properties, it has high volatility compared with other hedging assets. In other words, if Bitcoin is included in a portfolio, investors should pay attention to its wide variation. These empirical findings highlight the important role that gold and USD play in hedging against global stock markets.     

Jump-diffusion volatility models for variance swaps: An empirical performance analysis

This paper studies a class of tractable jump-diffusion models, including stochastic volatility models with various specifications of jump intensity for stock returns and variance processes. We employ the Markov chain Monte Carlo (MCMC) method to implement model estimation, and investigate the performance of all models in capturing the term structure of variance swap rates and fitting the dynamics of stock returns. It is evident that the stochastic volatility models, equipped with self-exciting jumps in the spot variance and linearly-dependent jumps in the central-tendency variance, can produce consistent model estimates, aptly explain the stylized facts in variance swaps, and boost pricing performance. Moreover, our empirical results show that large self-exciting jumps in the spot variance, as an independent risk source, facilitate term structure modeling for variance swaps, whilst the central-tendency variance may jump with small sizes, but signaling substantial regime changes in the long run. Both types of jumps occur infrequently, and are more related to market turmoils over the period from 2008 to 2021.     

Model-independent hedging strategies for variance swaps

A variance swap is a derivative with a path-dependent payoff which allows investors to take positions on the future variability of an asset. In the idealised setting of a continuously monitored variance swap written on an asset with continuous paths, it is well known that the variance swap payoff can be replicated exactly using a portfolio of puts and calls and a dynamic position in the asset. This fact forms the basis of the VIX contract. But what if we are in the more realistic setting where the contract is based on discrete monitoring, and the underlying asset may have jumps? We show that it is possible to derive model-independent, no-arbitrage bounds on the price of the variance swap, and corresponding sub- and super-replicating strategies. Further, we characterise the optimal bounds. The form of the hedges depends crucially on the kernel used to define the variance swap.     

Convergence analysis and optimal strike choice for static hedges of general path-independent pay-offs

In this paper, we propose a new algorithm to find the optimal static replicating portfolios for general path-independent nonlinear pay-off functions and give an estimate for the rate of convergence that is absent in the literature. We choose the static replication by designing an adaptation function arising in the error bound between the nonlinear pay-off function and the linear spline approximation and derive the equidistribution equation for selecting the optimal strikes. The numerical tests for variance swaps, swaptions, static quadratic hedges and also for a jump-diffusion process, allowing for the default of the underlying asset, show that the proposed iterative equidistribution equation algorithm is simple, fast and accurate. The paper generalizes and improves the results on static replication and approximation in the literature.     

Optimal Hedging in Futures Markets with Multiple Delivery Specifications

Nearly all futures contracts allow delivery of any of several qualities of the underlying asset. Consequently, the price of the futures contract is associated more with the price of the expected cheapest deliverable variety than with the price of the par-delivery variety. The delivery specifications introduce a delivery risk for every hedger in the market. We derive the optimal hedging strategies in these markets. Their hedging effectiveness is evaluated for wheat futures contracts in Chicago. Hedging optimally would have significantly reduced the variance of the rates of return on hedges while yielding similar mean returns.     

Foreign Currency Derivatives versus Foreign Currency Debt and the Hedging Premium

This paper compares the effect on firm value of different foreign currency (FC) financial hedging strategies identified by type of exposure (short‐ or long‐term) and type of instrument (forwards, options, swaps and foreign currency debt). We find that hedging instruments depend on the type of exposure. Short‐term instruments such as FC forwards and/or options are used to hedge short‐term exposure generated from export activity while FC debt and FC swaps into foreign currency (but not into domestic currency) are used to hedge long‐term exposure arising from assets located in foreign locations. Our results relating to the value effects of foreign currency hedging indicate that foreign currency derivatives use increases firm value but there is no hedging premium associated with foreign currency debt hedging, except when combined with foreign currency derivatives. Taken individually, FC swaps generate more value than short‐term derivatives.     

Consistent Variance Curve Models

We introduce a general approach to model a joint market of stock price and a term structure of variance swaps in an HJM-type framework. In such a model, strongly volatility-dependent contracts can be priced and risk-managed in terms of the observed stock and variance swap prices. To this end, we introduce equity forward variance term structure models and derive the respective HJM-type arbitrage conditions. We then discuss finite-dimensional Markovian representations of the fixed time-to-maturity forward variance swap curve and derive consistency results for both the standard case and for variance curves with values in a Hilbert space. For the latter, our representation also ensures non-negativity of the process. We then give a few examples of such variance curve functionals and briefly discuss completeness and hedging in such models. As a further application, we show that the speed of mean reversion in some standard stochastic volatility models should be kept constant when the model is recalibrated.     

Generic pricing of FX,inflation and stock options under stochastic interest rates and stochastic volatility

We consider the pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility, for which we use a generic multi-currency framework. We allow for a general correlation structure between the drivers of the volatility, the inflation index, the domestic (nominal) and the foreign (real) rates. Having the flexibility to correlate the underlying FX/inflation/stock index with both stochastic volatility and stochastic interest rates yields a realistic model that is of practical importance for the pricing and hedging of options with a long-term exposure. We derive explicit valuation formulas for various securities, such as vanilla call/put options, forward starting options, inflation-indexed swaps and inflation caps/floors. These vanilla derivatives can be valued in closed form under Schöbel and Zhu [Eur. Finance Rev., 1999, 4, 23–46] stochastic volatility, whereas we devise an (Monte Carlo) approximation in the form of a very effective control variate for the general Heston [Rev. Financial Stud., 1993, 6, 327–343] model. Finally, we investigate the quality of this approximation numerically and consider a calibration example to FX and inflation market data.     

“How using derivative instruments and purposes affects performance of Islamic banks? Evidence from CAMELS approach”

The increase of the use of derivative instruments by Islamic banks for different purposes motivate us to conduct this study. This work has twice objective: firstly, to investigate the effect of each derivative instrument (forwards, futures, swaps or options) on the performance of Islamic banks, and secondly to examine the effect of each derivative purpose (hedging or trading) on the performance of Islamic banks.To reach this end, dynamic panel data econometrics with GMM system are conducted on 32 Islamic banks during the period from 2007 to 2017. The CAMELS approach is used to measure the performance of sample banks.Statistics on sample banks reveal that Islamic banks are substantial users of derivatives, prefer using derivatives for trading purpose than for hedging purpose, and have acceptable level of performance.The main results confirm that using options affects positively and moderately the performance of sample banks. In the same way, we find that swaps have positive and weak impact on the performance of sample banks. However, the results reveal that using forwards decrease the performance of sample banks. Finally, we find that futures have ambiguous and marginal effect on the performance of sample banks.As regards derivative purposes, results do not see which purpose mainly motivate the Islamic banks to invest in the derivatives market.As theoretical implication, we suggest for further studies to explore more the differences between using derivatives by Islamic banks for trading and hedging purpose.Finally, as practical implications, we recommend for managers of Islamic banks to enlarge their use of options and swaps, to supervise their use of forwards and to stop their use of futures.     

Information content and the contractual nature of hedging derivative incomes

In a sample of 87 banks representing 631 bank-years for the period 1996–2003, we examine whether information content of hedging derivative incomes is predicated on the contractual nature of the derivative. Of particular interest are the different abnormal trading volume reactions to incomes arising from executory contracts (i.e., cash flow and net investment hedges) and incomes arising from nonexecutory contracts (i.e., fair value hedges). We find a positive and significant relationship between two alternative measures of abnormal trading volume and incomes arising from cash flow and net investment hedges. The results are robust in an equity valuation framework. Our findings suggest that derivative incomes are informative, notably those incomes that are related to executory contracts. An implication for standard setters is that the complex rules for disaggregating incomes on hedging derivatives provide valuable information to the market.     

Pricing methods and hedging strategies for volatility derivatives

We explore the valuation and hedging of discretely observed volatility derivatives using three different models for the price of the underlying asset: Geometric Brownian motion with constant volatility, a local volatility surface, and jump-diffusion. We begin by comparing the effects on valuation of variations in contract design, such as the differences between specifying log returns or actual returns and incorporating caps on the level of realized volatility. We then focus on the difficulties associated with hedging these products. Delta hedging strategies are ineffective for hedging volatility derivatives since they require very frequent rebalancing. Moreover, they provide limited protection in the jump-diffusion context. We study the performance of a hedging strategy for volatility swaps that establishes small, fixed positions in vanilla options at each volatility observation.     

A forward started jump-diffusion model and pricing of cliquet style exotics

In this paper we present an alternative model for pricing exotic options and structured products with forward-starting components. As presented in the recent study by Eberlein and Madan (Quantitative Finance 9(1):27–42, 2009), the pricing of such exotic products (which consist primarily of different variations of locally/globally, capped/floored, arithmetic/geometric etc. cliquets) depends critically on the modeling of the forward–return distributions. Therefore, in our approach, we directly take up the modeling of forward variances corresponding to the tenor structure of the product to be priced. We propose a two factor forward variance market model with jumps in returns and volatility. It allows the model user to directly control the behavior of future smiles and hence properly price forward smile risk of cliquet-style exotic products. The key idea, in order to achieve consistency between the dynamics of forward variance swaps and the underlying stock, is to adopt a forward starting model for the stock dynamics over each reset period of the tenor structure. We also present in detail the calibration steps for our proposed model.     

Performance of utility-based strategies for hedging basis risk

The performance of optimal strategies for hedging a claim on a non-traded asset is analysed. The claim is valued and hedged in a utility maximization framework, using exponential utility. A traded asset, correlated with that underlying the claim, is used for hedging, with the correlation ρ typically close to 1. Using a distortion method (Zariphopoulou 2001 Finance Stochastics 5 61–82) we derive a nonlinear expectation representation for the claim’s ask price and a formula for the optimal hedging strategy. We generate a perturbation expansion for the price and hedging strategy in powers of ε²?=1?ρ². The terms in the price expansion are proportional to the central moments of the claim payoff under the minimal martingale measure. The resulting fast computation capability is used to carry out a simulation-based test of the optimal hedging program, computing the terminal hedging error over many asset price paths. These errors are compared with those from a naive strategy which uses the traded asset as a proxy for the non-traded one. The distribution of the hedging error acts as a suitable metric to analyse hedging performance. We find that the optimal policy improves hedging performance, in that the hedging error distribution is more sharply peaked around a non-negative profit. The frequency of profits over losses is increased, and this is measured by the median of the distribution, which is always increased by the optimal strategies. An empirical example illustrates the application of the method to the hedging of a stock basket using index futures.     

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.

Hedging with Two Futures Contracts: Simplicity Pays

We propose to use two futures contracts in hedging an agricultural commodity commitment to solve either the standard delta hedge or the roll‐over issue. Most current literature on dual‐hedge strategies is based on a structured model to reduce roll‐over risk and is somehow difficult to apply for agricultural futures contracts. Instead, we propose to apply a regression based model and a naive rules of thumb for dual‐hedges which are applicable for agricultural commodities. The naive dual strategy stems from the fact that in a large sample of agricultural commodities, De Ville, Dhaene and Sercu (2008) find that GARCH‐based hedges do not perform as well as OLS‐based ones and that we can avoid estimation error with such a simple rule. Our semi‐naive hedge ratios are driven from two conditions: omitting exposure to spot price and minimising the variance of the unexpected basis effects on the portfolio values. We find that, generally, (i) rebalancing helps; (ii) the two‐contract hedging rules do better than the one‐contract counterparts, even for standard delta hedges without rolling‐over; (iii) simplicity pays: the naive rules are the best one–for corn and wheat within the two‐contract group, the semi‐naive rule systematically beats the others and GARCH performs worse than OLS for either one‐contract or two‐contract hedges and for soybeans the traditional naive rule performs nearly as well as OLS. These conclusions are based on the tests on unconditional variance ( Diebold and Mariano, 1995 ) and those on conditional risk ( Giacomini and White, 2006 ).     

Cross hedging single stock with American Depositary Receipt and stock index futures

This paper investigates the cross hedging effectiveness of individual stock in a market that does not have single stock futures traded using American Depositary Receipt (ADR) and stock index futures. We apply Caporin and Billio’s Multivariate regime switching GARCH to capture the state-dependent covariance structure of underlying stock, ADR and stock index futures. Empirical results indicate that in general simultaneous hedging with both ADR and index futures creates hedging gains and incorporating regime switching effects further increases the hedging performances.     

Volatility as an Asset Class: European Evidence

Abstract

Volatility movements are known to be negatively correlated with stock index returns. Hence, investing in volatility appears to be attractive for investors seeking risk diversification. The most common instruments for investing in pure volatility are variance swaps, which now enjoy an active over-the-counter (OTC) market. This paper investigates the risk-return tradeoff of variance swaps on the Deutscher Aktienindex and Euro STOXX 50 index over the time period from 1995 to 2004. We synthetically derive variance swap rates from the smile in option prices. Using quotes from two large investment banks over two months, we validate that the synthetic values are close to OTC market prices. We find that variance swap returns exhibit an option-like profile compared to returns of the underlying index. Given this pattern, it is crucial to account for the non-normality of returns in measuring the performance of variance swap investments. As in the US, the average returns of selling variance swaps are found to be strongly positive and too large to be compatible with standard equilibrium models. The magnitude of the estimated risk premium is related to variance uncertainty and past index returns. This indicates that the variance swap rate does not seem to incorporate all past information relevant for forecasting future realized variance.     

Moment swaps

In this paper we discuss moment swaps. These derivatives depend on the realized higher moments of the underlying. A special case is the nowadays popular variance swaps. After introducing moment swaps we discuss how to hedge these derivatives. Moreover, we show how the classical hedge of the variance swap in terms of a position in log-contracts and a dynamic trading strategy can be significantly enhanced by using third moment swaps.