Spread Options Research Articles

The pricing of spread option using simulation

Monte Carlo simulation is one of the most common and popular method of options pricing. The advantages of this method are being easy to use, suitable for all kinds of standard and exotic options and also are suitable for higher dimensional problems. But on the other hand Monte Carlo variance convergence rate is which due to that it will have relatively slow convergence rate to answer the problems, as to achieve accuracy when it has been d-dimensions, complexity is . For this purpose, several methods are provided in quasi Monte Carlo simulation to increase variance convergence rate as variance reduction techniques, so far. One of the latest presented methods is multilevel Monte Carlo that is introduced by Giles in 2008. This method not only reduces the complexity of computing amount in use of Euler discretization scheme and the amount in use of Milstein discretization scheme, but also has the ability to combine with other variance reduction techniques. In this paper, using Multilevel Monte Carlo method by taking Milstein discretization scheme, pricing spread option and compared complexity of computing with standard Monte Carlo method. The results of Multilevel Monte Carlo method in pricing spread options are better than standard Monte Carlo simulation.

The multivariate mixture dynamics: Consistent no-arbitrage single-asset and index volatility smiles

ABSTRACTWe introduce a new arbitrage-free multivariate dynamic asset pricing model that allows us to reconcile single name and index/basket volatility smiles using a tractable and explicit dependence structure that goes beyond instantaneous correlation. Each asset volatility smile is modeled according to a density-mixture dynamical model while the same property holds for the multivariate process of all assets whose density is a mixture of multivariate basic densities. After introducing the model, we derive tractable index option smile formulas resulting from the model and related closed-form solutions for multivariate densities taking the form of multivariate mixtures. Using Markovian projection techniques, we relate our model to a multivariate uncertain volatility model and show a consistency result with geometric baskets with hints on possible uses in investigating triangular relationships between foreign exchange rates and the related smiles in practice. We also derive closed-form solutions for a number of terminal statistics of dependence and derive a precise relationship with a simpler, but less tractable, model based on a basic instantaneous correlation structure. Finally, closed-form solutions for volatility/asset correlations illuminating the relationship with the uncertain volatility model are introduced. The model tractability makes it particularly suited for calibration and risk management applications, where speed of calculations and tractability are essential. A few numerical examples on basket and spread options pricing conclude the article.

Algorithmic Approach to Delta-Hedging: Researching Realized FX Volatility at Micro Level

I present substantially extended back-testing results of the algorithmic approach to delta-hedging described in my earlier paper. Along with previously researched EURUSD straddles I research USDJPY and GBPUSD straddles as well as call and put spread option combinations. Back-tested period is extended from 5 to 7 years. The results re-confirm previously discovered phenomenon (weekly options sold at the end of the week are ”richer” than the options sold on Monday-Wednesday) and raise new interesting questions: ”cheap” GBPUSD options, substantial skew for USDJPY spread options that realizes at micro level, etc. The results also imply there is the same level of average ”break-even” volatility for all researched FX markets (weekly at-the-money volatility is around 8.8-9.0%), i.e. idiosyncratic element of volatility markets could be overestimated.

An improved basket of spread options heuristic for merchant energy storage

ABSTRACTPractitioners use the Basket of Spread Options (BSO) heuristic to model merchant energy storage as a portfolio of spread options and spot/forward sales. This method solves a linear program to obtain the composition of this portfolio and its associated BSO policy. Sequential reoptimization of this model yields the Rolling BSO (RBSO) policy. Although this policy performs well, typically dominating the BSO policy and often being near optimal, it can struggle when storage is fast. To attempt to obtain an improved RBSO policy, especially for fast storage, this article proposes a BSO heuristic that modifies the objective function of the BSO linear program based on exchange option prices and a tunable parameter. On a set of known natural gas storage instances, limited optimization of this adjustable quantity leads to modestly improved RBSO policies on average but substantially so when the original RBSO policies perform poorly, which occurs on some fast storage instances. Moreover, fixing this parameter to 0.6 gives RBSO policies that virtually match the performance of the best considered RBSO policies. The proposed BSO heuristic is thus as easy to use in practice as the original BSO heuristic.

On multi-asset spread option pricing in a Wick–Itô–Skorohod integral framework

We provide an elementary method for exploring pricing problems of one spread options within a fractional Wick–ItA´â€“Skorohod integral framework. Its underlying assets come from two different interactive markets that are modelled by two mixed fractional Black–Scholes models with Hurst parameters, \(H_1\neq H_2\), where \(1/2\leq H_i

Spread option pricing using ADI methods

Spread option contracts are becoming increasingly important, as they frequently arise in the energy derivative markets, e.g. exchange electricity for oil. In this paper, we study the pricing of European and American spread options. We consider the two-dimensional Black-Scholes Partial Differential Equation (PDE), use finite difference discretization in space and consider Crank-Nicolson (CN) and Modified Craig-Sneyd (MCS) Alternating Direction Implicit (ADI) methods for time stepping. In order to handle the early exercise feature arising in American options, we employ the discrete penalty iteration method, introduced and studied in Forsyth and Vetzal (2002), for one-dimensional PDEs discretized in time by the CN method. The main novelty of our work is the incorporation of the ADI method into the discrete penalty iteration method, in a highly efficient way, so that it can be used for two or higher-dimensional problems. The results from spread option pricing are compared with those obtained from the closed-form approximation formulae of Kirk (1995), Venkatramanan and Alexander (2011), Monte Carlo simulations, and the Brennan-Schwartz ADI Douglas-Rachford method, as implemented in MATLAB. In all spread option test cases we considered, including American ones, our ADI-MCS method, implemented on appropriate non-uniform grids, gives more accurate prices and Greeks than the MATLAB ADI method.

Pricing spread options by generalized bivariate edgeworth expansion

A spread option is a contingent claim whose underlying is the price difference between two assets. For a call, the holder of the option receives the difference, if positive, between the price difference and the strike price. Otherwise, the holder receives nothing. Spread options trade in large volume in financial, fixed-income, commodity, and energy industries. It is well known that pricing of spread options does not admit closed-form solutions even under a geometric Brownian motion paradigm. When price dynamics experience stochastic volatilities and/or jumps, the valuation process becomes more challenging. Following the seminal work of Jarrow and Judd, we propose the use of Edgeworth expansion to approximate the call price. In the spirit of Pearson, we reduce the cumbersome computation inherent in Edgeworth expansion to single numerical integrations. For an arbitrary bivariate price process, we show that once its product cumulants are available, either by virtue of the structural properties of the underlying processes or by empirical estimation using market data, the approach enables analysts to approximate the call price easily. Specifically, the call prices so estimated capture the correlation, skewness, and kurtosis of the two underlying price processes. As such, the approach is useful for approximate valuations based on Lévy-based models.

An Improved Basket of Spread Options Heuristic for Merchant Energy Storage

Practitioners use the basket of spread options (BSO) heuristic to model merchant energy storage as a portfolio of spread options and forward sales. This method solves a linear program to obtain the composition of this portfolio and its associated BSO policy. Sequential reoptimization of this model yields the rolling BSO (RBSO) policy. It is known that this policy can struggle when storage is fast but otherwise it performs well, often being near optimal. Further, the objective function of the BSO linear program is a lower bound on the value of the BSO policy associated with a feasible portfolio. To attempt to obtain an improved RBSO policy, especially for fast storage, this paper proposes a BSO heuristic that replaces this function with a weighted average of itself and a simple upper bound on the value of a given BSO policy. On a set of known natural gas storage instances, limited optimization of the value of the weight that defines this average yields improved RBSO policies. The observed improvement is substantial and modest, respectively, when the original RBSO policies perform poorly, which occurs on some fast storage instances, and on average. Moreover, setting the averaging weight equal to 0.6 leads to RBSO policies that virtually match the performance of the best considered RBSO policies. This BSO heuristic is thus as easy to use in practice as the original BSO heuristic.

ON MULTI-ASSET SPREAD OPTION PRICING IN A WICK–ITÔ–SKOROHOD INTEGRAL FRAMEWORK

We provide an elementary method for exploring pricing problems of one spread options within a fractional Wick–Itô–Skorohod integral framework. Its underlying assets come from two different interactive markets that are modelled by two mixed fractional Black–Scholes models with Hurst parameters, $H_{1}\neq H_{2}$, where $1/2\leq H_{i}<1$ for $i=1,2$. Pricing formulae of these options with respect to strike price $K=0$ or $K\neq 0$ are given, and their application to the real market is examined.

Vulnerable Exotic Derivatives

Understanding and managing counterparty credit risk exposure in derivatives contracts has become a crucial element of real-world trading as well as theoretical modeling. But existing models are limited in the number of securities involved and in the assumed dynamics of the underlying asset returns processes. In this article, the authors present a framework with two counterparties who enter into a derivatives contract in which either of them may default, and the derivative’s payoff depends on the joint distribution of n different assets. Three specific examples illustrate application of the approach: an option on a security in which both counterparties are subject to default risk; a vulnerable spread option, in which one risky counterparty issues an option tied to the price spread between two other assets; and a defaultable swap with one underlying and two risky counterparties who commit to a series of future cash flows. The resulting pricing formulas are mathematically complicated, but as closed-form solutions (with the caveat that integral expressions are approximated using finite step size and number of terms in an infinite series), they are much more efficient than Monte Carlo simulation in reaching a given level of accuracy.

Option Prices and Costly Short-Selling

We provide an analysis of option prices with costly short-selling. Short-sellers incur a shorting fee to borrow stock shares from investors who do not necessarily lend all their long positions (partial lending). We incorporate costly short-selling and options market-making into the classic Black-Scholes framework and obtain unique option bid and ask prices in closed-form. Option prices represent market-makers' expected cost of hedging, and are in terms of and preserve the well-known properties of the Black-Scholes prices. Consistently with empirical evidence, we show that bid-ask spreads of typical options and apparent put-call parity violations are increasing in the shorting fee. We also find that option bid-ask spreads are decreasing in the partial lending, and the effects of costly short-selling on option bid-ask spreads are more pronounced for relatively illiquid options with lower trading activity. We then apply our model to the recent 2008 short-selling ban period and obtain implications consistent with the documented behavior of option prices of banned stocks. Finally, our quantitative analysis reveals that the effects of costly short-selling on option prices are economically significant for expensive-to-short stocks and also sheds light on the behavior of option prices and apparent mispricings of the Palm stock in 2000.

The Model-Free Equivalence Condition for American Spread Options

A spread option involves the right to obtain the spread between two asset prices at a predefined strike price. This type of derivative security is frequently used in financial markets and academic finance. Furthermore, analysts use the spread option technique for real option modeling purposes. Some spread options are American-type in the sense that an option holder may exercise her option prior to the expiration. In this paper, we propose an equivalence condition for American spread options under which they are not exercised early, and are therefore equivalent to European options. Our theoretical results, developed within a model-free economic setting, suggest that the equivalence conditions documented by previous papers do not hold in a distribution-free environment. Traders, quantitative modelers, and financial programmers in various derivatives markets and the real option modeling area may use our results.

Forecasting Price Direction, Hedging and Spread Options in Oil Volatility

The energy market aims to manage risks associated with prices and volatility of the asset. It is a capital intensive market, rippled with a range of chaotic, complex and dynamic interaction among its supply and demand derivatives. Models help users forecast such interactions, to provide investors with empirical evidence of the price direction. Evolutionary modeling is an art, whose science seeks to analyze input data and yield an optimal, complete solution for which conventional methods yield a corresponding, non-cost effective solution. Its solutions are tractable, robust and low-cost with a tolerance of ambiguity, uncertainty and noise as applied to its input. Our study aims to predict the OPEC Oil market with data collected over the period.

Housing Price Differences Under the Development of New Towns: Using Taiwan as an Example

The purpose of this study is to examine the housing price differences between new towns and old urban areas. In addition, related issues such as spread options, the anchoring effect, and the ripple effect are discussed. By investigating the static spread, dynamic spread, and new town theory, we test seven hypotheses and verify most of these hypotheses. Furthermore, the key causes of the spread are identified: For example, spread options for different residence types depend on risk attitudes; and the ripple effect and anchoring effect of neighboring areas are determined by regional compactness and market transparency. The findings and relevant factors addressed in study can serve as a reference for investment decision making as well as for governments to implement projects of new town development or old urban area revitalization.

Gold Medal Spreads

Spread options themselves are more likely to be found in energy markets but I've taken the liberty of linking them with the Rio Olympics, in view of Team GB's success.

A Stochastic Correlation Model with Time Change for Pricing Credit Spread Options

In this paper, we introduce the stochastic correlation processes for modeling the credit spread. We first model the components of spread process as correlated Ornstein-Uhlenbeck processes and correlation as Jacobi process. Using the properties of Jacobi process, we are able to obtain the analytical solutions for the credit spread option prices. To further enhance the model’s ability to capture the abrupt changes in the observed correlation time series, we construct a new model where the correlation is modeled by a Jacobi process time change by Levy subordinators. We employ the eigenfunction expansion methods to obtain the closed-form solutions for the option prices. Our empirical study indicates the time changed Jacobi process fits the correlation series significantly better than the Jacobi process.

From the Samuelson volatility effect to a Samuelson correlation effect: An analysis of crude oil calendar spread options

Abstract Our first aim in this paper is to introduce a futures-based model able of capturing the main features displayed by Crude Oil futures and options contracts, such as the Samuelson volatility effect and the volatility smile. We calculate the joint characteristic function of two futures contracts in the model in analytic form and use it to price calendar spread options. In an empirical application we show that the model, in contrast to simpler nested models, can be successfully calibrated to market prices of vanilla and calendar spread options. Our second aim is to use this model to analyze the dependence structure of Crude Oil futures contracts. To this end, we propose analytical expressions giving the copula and copula density directly in terms of the joint characteristic function. These tools allow us to perform an in-depth analysis for pairs of futures, and we observe a phenomenon we call the Samuelson correlation effect.

Integrating weather and spark spread options for valuing a natural gas-fired power plant with multiple turbines

ABSTRACTA model that integrates spark spread and weather options in valuing a power plant with multiple turbines is presented. The spark spread option entails the right to operate the plant when it is profitable to run. Weather also affects a plant’s operation. A mild winter inducing a lower heating requirement could decrease the payoff. An owner holding a long position in a temperature-based put option could exercise the option when accumulated heating degree days drop below a strike level of heating degree days. Results demonstrate that integrating spark spread with weather options adds value during a mild winter.

A General Closed Form Approximation Formula for Pricing and Hedging Multi-Asset Spread Options

The aims of this paper are twofold. Firstly, we present an approximating formula for pricing basket and multi-asset spread options, which genuinely extends Caldana and Fusai (2013) two-asset spread options formula. Secondly, under the lognormal setting, we show that our formula becomes a Black and Scholes type formula, extending Bjerksund and Stensland (2011). Numerical experiments and comparison with Monte Carlo simulations and other methods available in the literature are discussed. The main contribution of this paper is to provide practitioners with a pricing formula, which can be used for pricing basket and multi-asset spread options, even under a non-Gaussian framework.

Option spread trades: Returns on directional and volatility trades

Option spread trades can be set up in a variety of different ways and offer flexibility for an investor to target specific goals in terms of risk-return profile. I study different setups of Bull spread trades with Calls to illustrate which setups are largely directional trades and which setups are largely volatility trades. Analytical derivations reveal that setups involving short positions in out-the-money Calls offer the strongest directional plays, whereas other setups are mostly long or short volatility. An analysis of spread returns using S&P 500 Index options data reveals high average returns, strong negative skewness in short volatility setups, and positive skewness in long volatility setups. Returns are heavily affected by costs of trading, but setups including short positions in out-the-money Calls return strong average returns even after costs. Factor analysis reveals that positive alpha is more readily obtained for trades held until maturity rather than for trades closed out before maturity.