An improved method for pricing and hedging long dated American options

The majority of quasi-analytic pricing methods for American options are efficient near maturity but are prone to larger errors when time-to-maturity increases. We introduce a new methodology to increase the accuracy of almost any existing quasi-analytic approach in pricing long-maturity American options. The new methodology, called the “extension-method”, relies on an approximation of the optimal exercise price near the beginning of the contract combined with existing pricing approaches so that the maturity range for which small errors are attainable is extended. Our method retains the quasi-analytic nature of the methods it improves. Generic quasi-analytic formulae for the price of an American put as well as for its hedging parameter are derived. Our scenarios-based numerical study indicates that our method considerably improves both the pricing and the hedging performance of a number of established approaches for a wide range of maturities. The superiority of this approach is illustrated with real financial data by considering S&P 100TM LEAPS® options traded from January 2008 to May 2015

Dealing with commodity price uncertainty

Liberalization in commodity markets has brought profound changes in the way price risks are allocated and managed in commodity subsectors. Price risks are increasingly allocated to private traders and farmers rather than absorbed by the government. The success of market reform depends on the ability of the emerging private sector to make full use of the available range of modern commodity marketing, price risk management and financing instruments. Because farmers do not generally have access to these instruments, intermediaries must be developed. Larger private traders and banks are in the best position to become these intermediaries. Preconditions needed for accessing modern commodity marketing, price risk management, and financing instruments are: a) creating an appropriate legal, regulatory, and institutional framework; b) reducing government intervention; c) providing training and raising awareness; and d) improving creditworthiness and reducing performance risk. The use of commodity derivative instruments to hedge commodity price risk is not new. The private sectors in many Asian and Latin American countries have been using commodity futures and options for some time. More recently, commodity derivative instruments are being used increasingly in several African countries and many economies in transition. And several developing and transition economies have sought to establish commodity derivative exchanges.Markets and Market Access,Payment Systems&Infrastructure,Environmental Economics&Policies,Commodities,International Terrorism&Counterterrorism,Access to Markets,Crops&Crop Management Systems,Commodities,Environmental Economics&Policies,Markets and Market Access

Static Hedging of Standard Options

We consider the hedging of options when the price of the underlying asset is always exposed to the possibility of jumps of random size. Working in a single factor Markovian setting, we derive a new spanning relation between a given option and a continuum of shorter-term options written on the same asset. In this portfolio of shorter-term options, the portfolio weights do not vary with the underlying asset price or calendar time. We then implement this static relation using a finite set of shorter-term options and use Monte Carlo simulation to determine the hedging error thereby introduced. We compare this hedging error to that of a delta hedging strategy based on daily rebalancing in the underlying futures. The simulation results indicate that the two types of hedging strategies exhibit comparable performance in the classic Black-Scholes environment, but that our static hedge strongly outperforms delta hedging when the underlying asset price is governed by Merton (1976)'s jump-diffusion model. The conclusions are unchanged when we switch to ad hoc static and dynamic hedging practices necessitated by a lack of knowledge of the driving process. Further simulations indicate that the inferior performance of the delta hedge in the presence of jumps cannot be improved upon by increasing the rebalancing frequency. In contrast, the superior performance of the static hedging strategy can be further enhanced by using more strikes or by optimizing on the common maturity in the hedge portfolio. We also compare the hedging effectiveness of the two types of strategies using more than six years of data on S&P 500 index options. We find that in all cases considered, a static hedge using just five call options outperforms daily delta hedging with the underlying futures. The consistency of this result with our jump model simulations lends empirical support for the existence of jumps of random size in the movement of the S&P 500 index. We also find that the performance of our static hedge deteriorates moderately as we increase the gap between the maturity of the target call option and the common maturity of the call options in the hedge portfolio. We interpret this result as evidence of additional random factors such as stochastic volatility.Static hedging, jumps, option pricing, Monte Carlo, S&P 500 index options, stochastic volatility

Dynamic hedging strategies: an application to the crude oil market

International audienceThis article analyses long-term dynamic hedging strategies relying on term structure models of commodity prices and proposes a new way to calibrate the models which takes into account the error associated with the hedge ratios. Different strategies, with maturities up to seven years, are tested on the American crude oil futures market. The study considers three recent and efficient models respectively with one, two, and three factors. The continuity between the models makes it possible to compare their performances which are judged on the basis of the errors associated with a delta hedge. The strategies are also tested for their sensitivity to the maturities of the positions and to the frequency of the portfolio rollover. We found that our method gives the best of two seemingly incompatible worlds: the higher liquidity of short-term futures contracts for the hedge portfolios, together with markedly improved performances. Moreover, even if it is more complex, the three-factor model is by far, the best

Local time and the pricing of time-dependent barrier options

A time-dependent double-barrier option is a derivative security that delivers the terminal value $\phi(S_T)$ at expiry $T$ if neither of the continuous time-dependent barriers b_\pm:[0,T]\to \RR_+ have been hit during the time interval $[0,T]$. Using a probabilistic approach we obtain a decomposition of the barrier option price into the corresponding European option price minus the barrier premium for a wide class of payoff functions $\phi$, barrier functions $b_\pm$ and linear diffusions $(S_t)_{t\in[0,T]}$. We show that the barrier premium can be expressed as a sum of integrals along the barriers $b_\pm$ of the option's deltas \Delta_\pm:[0,T]\to\RR at the barriers and that the pair of functions $(\Delta_+,\Delta_-)$ solves a system of Volterra integral equations of the first kind. We find a semi-analytic solution for this system in the case of constant double barriers and briefly discus a numerical algorithm for the time-dependent case.Comment: 32 pages, to appear in Finance and Stochastic