Spread Option Pricing on Single-Core and Parallel Computing Architectures

This paper introduces parallel computation for spread options using two-dimensional Fourier transform. Spread options are multi-asset options whose payoffs depend on the difference of two underlying financial securities. Pricing these securities, however, cannot be done using closed-form methods; as such, we propose an algorithm which employs the fast Fourier Transform (FFT) method to numerically solve spread option prices in a reasonable amount of short time while preserving the pricing accuracy. Our results indicate a significant increase in computational performance when the algorithm is performed on multiple CPU cores and GPU. Moreover, the literature on spread option pricing using FFT methods documents that the pricing accuracy increases with FFT grid size while the computational speed has opposite effect. By using the multi-core/GPU implementation, the trade-off between pricing accuracy and speed is taken into account effectively.

Pricing and hedging tools for spread option contracts

This thesis examines the problem of pricing and hedging spread options under market models with jumps driven by a Compound Poisson Process. Extending the work of Deng, Li and Zhou we derive the price approximation for Spread options in jump-diffusion framework. We find that the proposed model accurately approximates option prices and exhibits reasonable behavior when tested for sensitivity to the model parameters. Applying the method of Lamberton and Lepeyre, we minimize the squared error between the Spread option price and the hedge portfolio to arrive to an optimal hedging strategy for discontinuous underlying price modes. Additionally, we propose an alternative average Delta-hedging strategy that is derived by conditioning the underlying price processes on the number of jumps and summing over all the possible jump combinations; such an approach allows us to revert to a hedging problem in a Black-Scholes framework. Although the average Delta-hedging strategy offers a significantly simpler approach to hedge Spread options, we conclude that the former strategy performs better by examining the Profit and Loss Probability Density Function of the two competing strategies. Finally, we offer a model parameter calibration algorithm and test its performance using the transitional Probability Density Functions.

Pricing and hedging tools for spread option contracts

This thesis examines the problem of pricing and hedging spread options under market models with jumps driven by a Compound Poisson Process. Extending the work of Deng, Li and Zhou we derive the price approximation for Spread options in jump-diffusion framework. We find that the proposed model accurately approximates option prices and exhibits reasonable behavior when tested for sensitivity to the model parameters. Applying the method of Lamberton and Lepeyre, we minimize the squared error between the Spread option price and the hedge portfolio to arrive to an optimal hedging strategy for discontinuous underlying price modes. Additionally, we propose an alternative average Delta-hedging strategy that is derived by conditioning the underlying price processes on the number of jumps and summing over all the possible jump combinations; such an approach allows us to revert to a hedging problem in a Black-Scholes framework. Although the average Delta-hedging strategy offers a significantly simpler approach to hedge Spread options, we conclude that the former strategy performs better by examining the Profit and Loss Probability Density Function of the two competing strategies. Finally, we offer a model parameter calibration algorithm and test its performance using the transitional Probability Density Functions.

Pricing spread options under Levy jump-diffusion models

This thesis examines the problem of pricing spread options under market models with jumps driven by a Compound Poisson Process and stochastic volatility in the form of a CIR process. Extending the work of Dempster and Hong, and Bates, we derive the characteristic function for two market models featuring normally distributed jumps, stochastic volatility, and two different dependence structures. Applying the method of Hurd and Zhou we use the Fast Fourier Transform to compute accurate spread option prices across a variety of strikes and initial price vectors at a very low computational cost when compared to Monte-Carlo pricing methods. We also examine the sensitivities to the model parameters and find strong dependence on the selection of the jump and stochastic volatility parameters.

Pricing spread options under Levy jump-diffusion models

This thesis examines the problem of pricing spread options under market models with jumps driven by a Compound Poisson Process and stochastic volatility in the form of a CIR process. Extending the work of Dempster and Hong, and Bates, we derive the characteristic function for two market models featuring normally distributed jumps, stochastic volatility, and two different dependence structures. Applying the method of Hurd and Zhou we use the Fast Fourier Transform to compute accurate spread option prices across a variety of strikes and initial price vectors at a very low computational cost when compared to Monte-Carlo pricing methods. We also examine the sensitivities to the model parameters and find strong dependence on the selection of the jump and stochastic volatility parameters.

The valuation of a natural gas-fired power plant with multiple turbines using clean spark spread and weather options

Assessing the value of a power plant is an important issue for plant owners and prospective buyers. In a deregulated market, an owner has the option to operate the plant when the revenue from selling the electricity is higher than the cost of operating the plant. This option is known as the spark spread option. Under emission restrictions, when the carbon cost is deducted from the spark spread, the option is named as the clean spark spread option. This thesis presents an analysis on the spark spread and clean spark spread option based valuation methods for a power plant with multiple gas turbines having different input–output characteristics, emission rates, and capacities. Electricity, natural gas and carbon allowance prices are assumed to follow mean–reverting processes. Results demonstrate that CO2 allowance cost reduces the expected plant value, while the flexibility of switching among turbines adds value to the power plant. Weather also affects the power plant operation. This thesis also presents a valuation model for a power plant integrating spark spread and weather options. A cooler winter drawing more electricity could generate a higher payoff for the plant owner. A warmer winter, however, could lead to a lower payoff. An owner holding a long position in a temperature–based put option could exercise the option when the winter is milder. The exercise is triggered by the drop of heating degree days below a strike degree day. The number of weather contracts to buy is determined by minimizing the variance of the total payoff. Pricing of the weather option is calculated based on the mean–reverting behavior of temperature. Results demonstrate that the integrating weather option along with spark spread option adds value to the downward spark spread option based valuation of the plant in a warmer winter. A comparison of temperature modeling approaches with an aim to pricing weather option is also investigated. Regime–switching models generated from a combination of different underlying processes are utilized to determine the expected heating and cooling degree days. Weather option prices are then calculated based on a range of strike heating degree days.

The valuation of a natural gas-fired power plant with multiple turbines using clean spark spread and weather options

Assessing the value of a power plant is an important issue for plant owners and prospective buyers. In a deregulated market, an owner has the option to operate the plant when the revenue from selling the electricity is higher than the cost of operating the plant. This option is known as the spark spread option. Under emission restrictions, when the carbon cost is deducted from the spark spread, the option is named as the clean spark spread option. This thesis presents an analysis on the spark spread and clean spark spread option based valuation methods for a power plant with multiple gas turbines having different input–output characteristics, emission rates, and capacities. Electricity, natural gas and carbon allowance prices are assumed to follow mean–reverting processes. Results demonstrate that CO2 allowance cost reduces the expected plant value, while the flexibility of switching among turbines adds value to the power plant. Weather also affects the power plant operation. This thesis also presents a valuation model for a power plant integrating spark spread and weather options. A cooler winter drawing more electricity could generate a higher payoff for the plant owner. A warmer winter, however, could lead to a lower payoff. An owner holding a long position in a temperature–based put option could exercise the option when the winter is milder. The exercise is triggered by the drop of heating degree days below a strike degree day. The number of weather contracts to buy is determined by minimizing the variance of the total payoff. Pricing of the weather option is calculated based on the mean–reverting behavior of temperature. Results demonstrate that the integrating weather option along with spark spread option adds value to the downward spark spread option based valuation of the plant in a warmer winter. A comparison of temperature modeling approaches with an aim to pricing weather option is also investigated. Regime–switching models generated from a combination of different underlying processes are utilized to determine the expected heating and cooling degree days. Weather option prices are then calculated based on a range of strike heating degree days.

The Impact of Implied Volatility Fluctuations on Vertical Spread Option Strategies: The Case of WTI Crude Oil Market

This paper aims to analyze the impact of implied volatility on the costs, break-even points (BEPs), and the final results of the vertical spread option strategies (vertical spreads). We considered two main groups of vertical spreads: with limited and unlimited profits. The strategy with limited profits was divided into net credit spread and net debit spread. The analysis takes into account West Texas Intermediate (WTI) crude oil options listed on New York Mercantile Exchange (NYMEX) from 17 November 2008 to 15 April 2020. Our findings suggest that the unlimited vertical spreads were executed with profits less frequently than the limited vertical spreads in each of the considered categories of implied volatility. Nonetheless, the advantage of unlimited strategies was observed for substantial oil price movements (above 10%) when the rates of return on these strategies were higher than for limited strategies. With small price movements (lower than 5%), the net credit spread strategies were by far the best choice and generated profits in the widest price ranges in each category of implied volatility. This study bridges the gap between option strategies trading, implied volatility and WTI crude oil market. The obtained results may be a source of information in hedging against oil price fluctuations.

European Spread Option Pricing with the Floating Interest Rate for Uncertain Financial Market

In this paper, we investigate the pricing problems of European spread options with the floating interest rate. In this model, uncertain differential equation and stochastic differential equation are used to describe the fluctuation of stock price and the floating interest rate, respectively. We derive the pricing formulas for spread options including the European spread call option and the European spread put option. Finally, numerical algorithms are provided to illustrate our results.

ON SPREAD OPTION PRICING USING TWO-DIMENSIONAL FOURIER TRANSFORM

Spread options are multi-asset options with payoffs dependent on the difference of two underlying financial variables. In most cases, analytically closed form solutions for pricing such payoffs are not available, and the application of numerical pricing methods turns out to be nontrivial. We consider several such nontrivial cases and explore the performance of the highly efficient numerical technique of Hurd & Zhou[(2010) A Fourier transform method for spread option pricing, SIAM J. Financial Math. 1(1), 142–157], comparing this with Monte Carlo simulation and the lower bound approximation formula of Caldana & Fusai[(2013) A general closed-form spread option pricing formula, Journal of Banking & Finance 37, 4893–4906]. We show that the former is in essence an application of the two-dimensional Parseval’s Identity. As application examples, we price spread options in a model where asset prices are driven by a multivariate normal inverse Gaussian (NIG) process, in a three-factor stochastic volatility model, as well as in examples of models driven by other popular multivariate Lévy processes such as the variance Gamma process, and discuss the price sensitivity with respect to volatility. We also consider examples in the fixed-income market, specifically, on cross-currency interest rate spreads and on LIBOR/OIS spreads.