Robust pricing and hedging under trading restrictions and the emergence of local martingale models

We pursue the robust approach to pricing and hedging in which no probability measure is fixed, but call or put options with different maturities and strikes can be traded initially at their market prices. We allow the inclusion of robust modelling assumptions by specifying a set of feasible paths on which (super)hedging arguments are required to work. In a discrete-time setup with no short selling, we characterise absence of arbitrage and show that if call options are traded, then the usual pricing–hedging duality is preserved. In contrast, if only put options are traded, a duality gap may appear. Embedding the results into a continuous-time framework, we show that the duality gap may be interpreted as a financial bubble and link it to strict local martingales. This provides an intrinsic justification of strict local martingales as models for financial bubbles arising from a combination of trading restrictions and current market prices.     

Robustly Hedging Variable Annuities With Guarantees Under Jump and Volatility Risks

Recent variable annuities offer participation in the equity market and attractive protection against downside movements. Accurately quantifying this additional equity market risk and robustly hedging options embedded in the guarantees of variable annuities are new challenges for insurance companies. Due to sensitivities of the benefits to tails of the account value distribution, a simple Black–Scholes model is inadequate in preventing excessive liabilities. A model which realistically describes the real world price dynamics over a long time horizon is essential for the risk management of the variable annuities. In this article, both jump risk and volatility risk are considered for risk management of lookback options embedded in guarantees with a ratchet feature. We evaluate relative performances of delta hedging and dynamic discrete risk minimization hedging strategies. Using the underlying as the hedging instrument, we show that, under a Black–Scholes model, local risk minimization hedging can be significantly better than delta hedging. In addition, we compare risk minimization hedging using the underlying with that of using standard options. We demonstrate that, under a Merton's jump diffusion model, hedging using standard options is superior to hedging using the underlying in terms of the risk reduction. Finally, we consider a market model for volatility risks in which the at‐the‐money implied volatility is a state variable. We compute risk minimization hedging by modeling at‐the‐money Black–Scholes implied volatility explicitly; the hedging effectiveness is evaluated, however, under a joint model for the underlying price and implied volatility. Our computational results suggest that, when implied volatility risk is suitably modeled, risk minimization hedging using standard options, compared to hedging using the underlying, can potentially be more effective in risk reduction under both jump and volatility risks.     

Robust hedging with proportional transaction costs

A duality for robust hedging with proportional transaction costs of path-dependent European options is obtained in a discrete-time financial market with one risky asset. The investor’s portfolio consists of a dynamically traded stock and a static position in vanilla options, which can be exercised at maturity. Trading of both options and stock is subject to proportional transaction costs. The main theorem is a duality between hedging and a Monge–Kantorovich-type optimization problem. In this dual transport problem, the optimization is over all probability measures that satisfy an approximate martingale condition related to consistent price systems, in addition to an approximate marginal constraint.     

Risk management and market conditions

In this paper, we investigate the relation between hedging activity by commercial/merchant/producers to commodity prices and commodity market volatility using Commitments of Traders reports from commodity futures markets exchanges. Qualifying the body of literature which attributes hedging activity to departures from Modigliani-Miller theory, market imperfections and transactions cost, we address the paradoxes of hedging which is not value creating and the absence of hedging when firms might benefit, arguing that it may be related to the market conditions and risk appetite. We discover that prices and volatility are generally statistically significant contributors to hedging activity by commercial/merchant/producers’ users but with marked differences in their elasticities. For some commodities, price levels alone and not volatility are significant. We demonstrate that analysis of hedging in commodity markets should take cognisance of conditions and the degree of risk aversion, otherwise the implicit assumption is that hedging is invariant to such matters. Through considering both market conditions and the degree of risk aversion, understanding the motivation for hedging may be enhanced.     

Cross hedging under multiplicative basis risk

Cross hedging price risk in an incomplete financial market creates basis risk. We propose a new way of modeling basis risk where price risk and basis risk are combined in a multiplicative way. Under this specification, positive prudence is a necessary and sufficient condition for underhedging in an unbiased market. Using the example of cross hedging jet fuel price risk with crude oil futures, we show that the new specification is superior in describing the price series and that optimal cross hedges differ significantly from those derived under the traditional additive cross hedging model.     

Modelling and management of mortality risk: a review

In the first part of the paper, we consider the wide range of extrapolative stochastic mortality models that have been proposed over the last 15–20 years. A number of models that we consider are framed in discrete time and place emphasis on the statistical aspects of modelling and forecasting. We discuss how these models can be evaluated, compared and contrasted. We also discuss a discrete-time market model that facilitates valuation of mortality-linked contracts with embedded options. We then review several approaches to modelling mortality in continuous time. These models tend to be simpler in nature, but make it possible to examine the potential for dynamic hedging of mortality risk. Finally, we review a range of financial instruments (traded and over-the-counter) that could be used to hedge mortality risk. Some of these, such as mortality swaps, already exist, while others anticipate future developments in the market.     

Selective hedging strategies for crude oil futures based on market state expectations

This paper studies the ex-ante selective hedging strategies of crude oil futures contracts based on market state expectations and compares the hedging performances to the traditional minimum variance routine hedging strategies. The main advantage of the proposed method is that it achieves a trade-off between return and risk, rather than hedges risk at all costs. Specifically, we first use a multi-input Hidden Markov Model(HMM) to identify the market state, assess the market’s herding impact, and then integrate the findings of identification and measurement to forecast the price trend. We offer an adjustment criterion for the hedge ratios driven by GARCH²-type models based on the anticipated market state. We conducted an empirical analysis to examine the hedging effect of WTI and Brent crude oil futures, the results indicate that the proposed state-dependent hedging strategies are superior to the traditional model-driven hedging strategies concerning the hedged portfolio based on four criteria. The robustness check reveals that the proposed hedging strategies still outperform in different market situation. The findings can help traders in the crude oil markets, and the methodology can be applied to other energy markets.     

Pricing Weather Derivatives Using the Indifference Pricing Approach

Abstract

This paper adopts an incomplete market pricing model–the indifference pricing approach–to analyze valuation of weather derivatives and the viability of the weather derivatives market in a hedging context. It incorporates price risk, weather/quantity risk, and other risks in the financial market. In a mean-variance framework, the relationship between the actuarial price and the indifference price of weather derivatives is analyzed, and conditions are obtained concerning when the actuarial price does not provide an appropriate valuation for weather derivatives. Conditions for the viability of the weather derivatives market are examined. This paper also analyzes the effects of partial hedging, natural hedges, basis risk, quantity risk, and price risk on investors’ indifference prices by examining the distributional impacts of the stochastic variables involved.     

An Intertemporal International Asset Pricing Model: Theory and Empirical Evidence

We extend Campbell's (1993) model to develop an intertemporal international asset pricing model (IAPM). We show that the expected international asset return is determined by a weighted average of market risk, market hedging risk, exchange rate risk and exchange rate hedging risk. These weights sum up to one. Our model explicitly separates hedging against changes in the investment opportunity set from hedging against exchange rate changes as well as exchange rate risk from intertemporal hedging risk. A test of the conditional version of our intertemporal IAPM using a multivariate GARCH process supports the asset pricing model. We find that the exchange rate risk is important for pricing international equity returns and it is much more important than intertemporal hedging risk.     

Income hedging and portfolio decisions

We examine whether the decision to participate in the stock market and other related portfolio decisions are influenced by income hedging motives. Economic theory predicts that the market participation propensity should increase as the correlation between income growth and stock market returns decreases. Surprisingly, empirical studies find limited support for the income hedging motive. Using a rich, unique Dutch data set and the National Longitudinal Survey of the Youth (NLSY) from the United States, we show that when the income-return correlation is low, individuals exhibit a greater propensity to participate in the market and allocate a larger proportion of their wealth to risky assets. Even when the income risk is high, individuals exhibit a higher propensity to participate in the market when the hedging potential is high. These findings suggest that income hedging is an important determinant of stock market participation and asset allocation decisions.     

Hedging of Asian options under exponential Lévy models: computation and performance

In this paper we consider the problem of hedging an arithmetic Asian option with discrete monitoring in an exponential Lévy model by deriving backward recursive integrals for the price sensitivities of the option. The procedure is applied to the analysis of the performance of the delta and delta–gamma hedges in an incomplete market; particular attention is paid to the hedging error and the impact of model error on the quality of the chosen hedging strategy. The numerical analysis shows the impact of jump risk on the hedging error of the option position, and the importance of including traded options in the hedging portfolio for the reduction of this risk.     

A mixed C-vine copula model for hedging price and volumetric risk in wind power trading

When energy trading companies enter into long-term agreements with wind power producers, where a fixed price is paid for the fluctuating production, they are facing a joint price and volumetric risk. Since the pay-off of such agreements is non-linear, a hedging portfolio would ideally consist of not only forwards, but also a basket of e.g. call and put options. Illiquidity and an almost non-existent market for options challenge however the optimal hedging of joint price and volumetric risk in many market places. Here, we consider the case of the Danish power market, and exploit its strong positive correlation with the much more liquid German market to construct a proxy hedge. We propose a three-dimensional mixed vine copula to model the evolution of the Danish and German spot electricity prices and the Danish wind power production. We construct a realistic hedging portfolio by identifying various instruments available in the market, such as real options in the form of the right to transfer electricity across the border and the right to convert electricity to heat. Using the proposed vine copula to determine optimal hedging decisions, we show that significant benefits are to be drawn by extending the hedging portfolio with the proposed instruments.     

Interest Rate Risk Management in Major Finnish Firms

The interest rate policies of Finnish firms appear risk aversive, but hedging decisions are influenced by market view. Managers find they can forecast trends in interest rate development, and employ the forecasts in the choice of debt and hedging instruments. The use of risk assessment methods and hedging instruments are related to firm size but not to leverage. Most frequently employed hedging instruments are interest rate swaps and forward rate agreements. The respondents find their firms' interest rate risk management is successful, but performance is seldom measured against an explicitly defined benchmark.     

Analytics Underlying the Metallgesellschaft Hedge: Short Term Futures in a Multi-Period Environment

In a highly publicized example, a marketing and refining subsidiary of Metallgesellschaft controlled short term derivative positions reportedly equivalent to 160 million barrels of oil, 80 times the daily output of Kuwait. Presumably, the short term positions were taken to hedge oil contracts to customers over an extended period. This paper develops the analytics underlying the hedging of long term flow commitments with short term futures contracts. Its contributions includes the determination of minimum variance hedging paths for multiperiod flow portfolios and the evaluation of both period-by-period and end-of-horizon volatilities under various hedging schemes. The analysis allows for basis risk, non-zero cost-of-carry, and spot diffusion processes that have a non-zero market price of risk. The results are developed in the context of an efficient market and standard equilibrium pricing models.     

Corporate risk management and firm value: evidence from the UK market

The paper evaluates the effect of corporate risk management activities on firm value, using a sample of large UK non-financial firms. Following recent changes in financial reporting standards, we are able to collect detailed information on risk management activities from audited financial reports. This enables us to gain a better understanding of risk management practices and to investigate value implications of different types of hedging. Overall 86.88% of the firms in the sample use derivatives to manage at least one type of price risk. The hedging premium is statistically and economically significant for foreign currency derivative users, while we provide weak evidence that interest rate hedging increases firm value. The extent of hedging and the hedging horizon have an impact on the hedging premium, whereas operational risk management activities do not significantly influence the market value of the firm.     

Delta-hedging correlation risk?

While the Gaussian copula model is commonly used as a static quotation device for CDO tranches, its use for hedging is questionable. In particular, the spread delta computed from the Gaussian copula model assumes constant base correlations, whereas we show that the correlations are dynamic and correlated to the index spread. It might therefore be expected that a dynamic model of credit risk, which is able to capture the dependence between the base correlations and the index spread, will have better hedging performances. In this paper, we compare delta hedging of spread risk based on the Gaussian copula model, to the implementation of jump-to-default ratio computed from the dynamic local intensity model. Theoretical and empirical analysis are illustrated by using the market data in both before and after the subprime crisis. We observe that delta hedging of spread risk outperforms the implementation of jump-to-default ratio in the pre-crisis period associated with CDX.NA.IG series 5, and the two strategies have comparable performance for crisis period associated with CDX.NA.IG series 9 and 10. This shows that, although the local intensity model is a dynamic model, it is not sufficient to explain the joint dynamic of the index spread and the base correlations, and a richer dynamic model is required to obtain better hedging results. Moreover, although different specifications of the local intensity can be fitted to the market data equally well, their hedging results can be significant different. This reveals substantial model risk when hedging CDO tranches.     

Weather Derivatives and Weather Risk Management

Weather derivatives are a relatively recent kind of financial product developed to manage weather risks, and currently the weather derivatives market is the fastest-growing derivative market. The development of weather derivatives represents one of the recent trends toward the convergence of insurance and finance. This article presents an overview of weather risks, weather derivatives, and the weather derivatives market, and examines the valuation of weather derivatives in an incomplete market, the hedging effectiveness of standardized weather derivatives, as well as optimal weather hedging with the consideration of basis risk and credit risk.     

Discrete hedging of American-type options using local risk minimization

Local risk minimization and total risk minimization discrete hedging have been extensively studied for European options [e.g., Schweizer, M., 1995. Variance-optimal hedging in discrete time. Mathematics of Operation Research 20, 1–32; Schweizer, M., 2001. A guided tour through quadratic hedging approaches. In: Jouini, E., Cvitanic, J., Musiela, M., Option pricing, interest rates and risk management, Cambridge University Press, pp. 538–574]. In practice, hedging of options with American features is more relevant. For example, equity linked variable annuities provide surrender benefits which are essentially embedded American options. In this paper we generalize both quadratic and piecewise linear local risk minimization hedging frameworks to American options. We illustrate that local risk minimization methods outperform delta hedging when the market is highly incomplete. In addition, compared to European options, distributions of the hedging costs are typically more skewed and heavy-tailed. Moreover, in contrast to quadratic local risk minimization, piecewise linear risk minimization hedging strategies can be significantly different, resulting in larger probabilities of small costs but also larger extreme cost.     

Mean-variance hedging with oil futures

We analyze mean-variance-optimal dynamic hedging strategies in oil futures for oil producers and consumers. In a model for the oil spot and futures market with Gaussian convenience yield curves and a stochastic market price of risk, we find analytical solutions for the optimal trading strategies. An implementation of our strategies in an out-of-sample test on market data shows that the hedging strategies improve long-term return-risk profiles of both the producer and the consumer.