Spread Option Pricing Research Articles

On the Use of the Moment-Matching Technique for Pricing and Hedging Multi-Asset Spread Options

The aim of this paper is to show the benefit of applying a moment matching technique to the short leg component in order to price and hedge multi-asset spread options: in particular, we approximate the real dynamics of the short leg component by taking a log-normal proxy, whose equivalent volatility can be computed by performing a two-moments matching approximation. The pricing of the option is then performed once the equivalent correlation parameter between the long leg underlying and the proxy short leg component has been calculated. The main advantage associated with the moment matching approach proposed in this paper is a reduction of the dimension of the pricing problem: we can, indeed, continue using all the option formulas available in the literature for two-legged spread options, i.e. spread options written on two underlyings. Besides it, the combined use of an option formula for two-legged spread options and the moment matching technique applied to the short leg component provides a good approximation to the Monte Carlo simulation. It is well-known that the Monte Carlo price and Greeks can be considered as the benchmark since no exact formula is available for the pricing and hedging of multi-asset spread options. The accuracy of our approach is even comparable to the one provided by using closed form approximation formulas based on three underlyings, where each underlying entering into the short leg component is treated separately.

Explicit approximations of multi-asset option prices including Greeks

In this article, we give a lower bound approximation to a stochastic programme that is typical for many complex financial derivatives, e.g., Bermudan options, Rainbow options and swing contracts. The approximation is based on an algorithm that does not require any simulation but uses a well-known spread option pricing formula. Since the method explicitly computes lower bounds for the claim’s value, sensitivities such as deltas and gammas are available as well as derivatives with respect to correlations and other inputs. Our resulting procedure is a powerful and practical tool for the pricing and risk analysis of exotic payoff structures. As such, it is useful in the day-to-day environment of financial institutions.

Pricing Spread Options with Stochastic Interest Rates

Although spread options have been extensively studied in the literature, few papers deal with the problem of pricing spread options with stochastic interest rates. This study presents three novel spread option pricing models that permit the interest rates to be random. The paper not only presents a good approach to formulate spread option pricing models with stochastic interest rates but also offers a new test bed to understand the dynamics of option pricing with interest rates in a variety of asset pricing models. We discuss the merits of the models and techniques presented by us in some asset pricing models. Finally, we use regular grid method to the calculation of the formula when underlying stock returns are continuous and a mixture of both the regular grid method and a Monte Carlo method to the one when underlying stock returns are discontinuous, and sensitivity analyses are presented.

Optimal portfolio allocations with tracking error volatility and stochastic hedging constraints

The performance of mutual fund or pension fund managers is often evaluated by comparing the returns of managed portfolios with those of a benchmark. As most portfolio managers use dynamic rules for rebalancing their portfolios, we use a dynamic framework to study the optimization of the tracking error–return trade-off. Following these observations, we assume that the manager minimizes the tracking error under an expected return goal (or, equivalently, maximizes the information ratio). Moreover, we assume that he/she complies with a stochastic hedging constraint whereby the terminal value of the portfolio is (almost surely) higher than a given stochastic payoff. This general setting includes the case of a minimum wealth level at the horizon date and the case of a performance constraint on terminal wealth as measured by the benchmark (i.e. terminal portfolio wealth should be at least equal to a given proportion of the index). When the manager cares about absolute returns and relative returns as well, the risk–return trade-off acquires an extra dimension since risk comprises two components. This extra risk dimension substantially modifies the characteristics of portfolio strategies. The optimal solutions involve pricing and duplication of spread options. Optimal terminal wealth profiles are derived in a general setting, and optimal strategies are determined when security prices follow geometric Brownian motions and interest rates remain constant. A numerical example illustrates the type of strategies generated by the model.

A general closed-form spread option pricing formula

We propose a new accurate method for pricing European spread options by extending the lower bound approximation of Bjerksund and Stensland (2011) beyond the classical Black–Scholes framework. This is possible via a procedure requiring a univariate Fourier inversion. In addition, we are also able to obtain a new tight upper bound. Our method provides also an exact closed form solution via Fourier inversion of the exchange option price, generalizing the Margrabe (1978) formula. The method is applicable to models in which the joint characteristic function of the underlying assets forming the spread is known analytically. We test the performance of these new pricing algorithms performing numerical experiments on different stochastic dynamic models.

Pricing Structured Products in Gas Markets: A Tree Based Approximation

In this paper we propose a theoretical framework to price the most popular contracts traded in energy markets. We'll cope with three kind of products: swing options, virtual transportations and virtual storages. Swing options can be written both on fi xed or indexed price. To cope with these products we assume a classical vectorial brownian motion driven dynamic for the underlyings and we propose a moment matching approach, then we build a fast and efficient numerical scheme based on recombining trees' approximation. In addition we give analytical formula for the pricing of european spread options.

A General Closed-Form Spread Option Pricing Formula

We propose a new accurate method for pricing European spread options by extending the lower bound approximation of Bjerksund and Stensland (2011) beyond the classical Black-Scholes framework. This is possible via a procedure requiring a univariate Fourier inversion. In addition, we are also able to obtain a new tight upper bound. Our method provides also an exact closed form solution via Fourier inversion of the exchange option price, generalizing the Margrabe (1978) formula. The method is applicable to models in which the joint characteristic function of the underlying assets forming the spread is known analytically. We test the performance of these new pricing algorithms performing numerical experiments on different stochastic dynamic models.

American Basket and Spread Option Pricing by a Simple Binomial Tree

The return on a portfolio is the weighted average of the returns on the individual assets in the portfolio. But the dynamics of portfolio returns are not so simple. The standard assumption that the underlying asset for an option follows geometric Brownian motion is convenient for individual stocks, but it runs into trouble for combinations of stocks, because a linear combination of lognormal returns does not have a lognormal distribution. Luckily, the true portfolio return distribution can be closely approximated by a generalized lognormal using the technique of matching moments, even when some weights are negative, as in a spread trade. This makes it easy to price European options on baskets of stocks or spreads. In this article, the authors show how to extend the same basic idea to construct a binomial tree for the portfolio return, which allows for efficient pricing of contracts with American exercise. <b>TOPICS:</b>Options, VAR and use of alternative risk measures of trading risk

A Note to Enhance the BPW Model for the Pricing of Basket and Spread Options

The standard assumption for an underlying asset’s returns process is the lognormal diffusion. This works quite well for individual assets. Portfolios and indexes present a problem, however, because a weighted sum of lognormally distributed stock prices is not lognormal. But a lognormal approximation can typically still be used as long as all of the weights are positive. That condition fails for options on a spread, which generally involves approximately equally sized long and short positions. In the Summer 2007 issue of this journal, Borovkova, Permana, and Weide (BPW) presented a solution based on matching the moments of the returns distribution to a generalized lognormal. Here, Wu, Chang and Chen show how to speed up the BPW procedure considerably by using closed-form expressions for the relevant moments. <b>TOPICS:</b>Options, statistical methods

A Note to Enhance the BPW Model for the Pricing of Basket and Spread Options

A Note to Enhance the BPW Model for the Pricing of Basket and Spread Options

Explicit Approximations of Multi-Asset Option Prices Including Greeks

In this article we give a lower bound approximation to a stochastic program that is typical for many complex financial derivatives, e.g. Bermudan options, Rainbow options and swing contracts. The approximation is based on an algorithm that does not require any simulation but uses a well-known spread option pricing formula. Since the method explicitly computes lower bounds for the claim’s value, sensitivities such as deltas and gammas are available as well as derivatives with respect to correlations and other inputs. Our resulting procedure is a powerful and practical tool for the pricing and risk analysis of exotic payoff structures. As such, it is useful in the day-to-day environment of financial institutions.

Efficient Pricing of constant maturity swap spread options in a stochastic volatility LIBOR market model

Efficient Pricing of constant maturity swap spread options in a stochastic volatility LIBOR market model

On the Pricing and Sensitivity of Spread Options on Two Correlated Assets

We consider the exact analytical pricing of an European option written on the spread between two correlated underlying assets. General model-independent properties of the option price are observed and an exact analytical expression for the spread option value is derived in the case of two correlated geometric Brownian motions. The exact values are compared with existing approximations and with direct Monte-Carlo simulations. A graphical analysis of parameter sensitivity in both the spread option price and the option deltas is presented.

Integrated EUA and CER Price Modeling and Application for Spread Option Pricing

In this paper we propose a market consistent futures price dynamics model for cap-and trade schemes, designed in the spirit of the European Union's Emissions Trading Scheme (EU ETS). Historical price dynamics for the EU ETS suggest that, both, European emission Allowances (EUAs) and Certified Emission Reductions (CERs), certificates which are generated through the Clean Development Mechanism (CDM) - a non-domestic offset mechanism - are significantly related. We use an equilibrium framework to demonstrate that compliance regulation singles out special joint futures price dynamics. Based on this result we propose an arbitrage free futures price model and apply it to the pricing of spread options between EUAs and CERs.

Pricing two dimensional derivatives under stochastic correlation

In this paper, we develop a framework for pricing two dimensional derivatives under stochastic correlation. Closed form approximations for the price of these derivatives are provided based on Taylor’s expansions of known price function under constant correlation. Two families of stochastic dynamics for the correlation are considered. The framework is applied in the pricing of spread options and compo options.

Extracting Correlations from the Market: New Correlation Parameterizations and the Calibration of a Stochastic Volatility LMM to CMS Spread Options

We present a new generic method for constructing correlation parameterizations that are always positive definite, and derive new flexible parametric forms.Furthermore, we use the CMS spread option pricing formula from Kiesel & Lutz to calibrate a stochastic volatility LMM to caplets, swaptions and CMS spread options, and in this way extract the implied correlation information available from the market.We investigate the performance of several correlation parameterizations and compare the implied correlation matrices with the corresponding empirical correlation matrices.

A numerical method for pricing spread options on LIBOR rates with a PDE model

In this paper we present a new numerical method for solving a Black–Scholes type of model for pricing a class of interest rate derivatives: spread options on LIBOR rates. The interest rates are assumed to follow the recently introduced LIBOR Market Model. The Feynman–Kac theorem provides a PDE model for the spread option pricing problem which is initially posed in an unbounded domain. After a localization procedure and the consideration of appropriate boundary conditions in a bounded domain, we propose a Crank–Nicholson characteristic time discretization scheme combined with a Lagrange piecewise quadratic finite element for the spatial discretization. In order to illustrate the performance of the PDE model and the numerical methods, we present a real example of spread option pricing.

PRICING CMS SPREAD OPTIONS IN A LIBOR MARKET MODEL

We present two approximation methods for the pricing of CMS spread options in Libor market models. Both approaches are based on approximating the underlying swap rates with lognormal processes under suitable measures. The first method is derived straightforwardly from the Libor market model. The second one uses a convexity adjustment technique under a linear swap model assumption. A numerical study demonstrates that both methods provide satisfactory approximations of spread option prices and can be used for calibration of a Libor market model to the CMS spread option market.

A Fourier Transform Method for Spread Option Pricing

Spread options are a fundamental class of derivative contracts written on multiple assets and are widely traded in a range of financial markets. There is a long history of approximation methods for computing such products, but as yet there is no preferred approach that is accurate, efficient, and flexible enough to apply in general asset models. The present paper introduces a new formula for general spread option pricing based on Fourier analysis of the payoff function. Our detailed investigation, including a flexible and general error analysis, proves the effectiveness of a fast Fourier transform implementation of this formula for the computation of spread option prices. It is found to be easy to implement, stable, efficient, and applicable in a wide variety of asset pricing models.

Fast and Accurate Pricing and Hedging of Long-Dated CMS Spread Options

We present a fast method to price and hedge CMS spread options in the displaced-diffusion co-initial swap market model. Numerical tests demonstrate that we are able to obtain sufficiently accurate prices and Greeks with computational times measured in milliseconds. Further, we find that CMS spread options are weakly dependent on the at-the-money Black implied volatility skews.