A Fourier Transform Method for Spread Option Pricing

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.

Valuing spread options with counterparty risk and jump risk

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's (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's (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.

A general closed-form spread option pricing formula

In this paper we propose a closed-form pricing formula for European basket and spread options. Our approach is based on approximating the risk-neutral probability density function of the terminal value of the basket using a Gauss-Hermite series expansion around the Gaussian density. The new method is quite general as it can be applied for a basket with a large number of assets and for all dynamics where the joint characteristic function of log-returns is known in closed form. We provide a simulation study to show the accuracy and the speed of our methodology.

Pricing of American lookback spread options

Using physical probabilistic measure of price process and the principle of fair premium , we deal with pricing formula of option on Foreign currency option under the assumption that foreign option price process driven by fractional Brownian motion process ,we obtain the pricing formula of foreign option.

Multi-Asset Spread Option Pricing and Hedging

The main objective of the present thesis is to provide efficient methods and tools for calibrating Libor Market Models (LMM) to market-prices of caps, swaptions and CMS spread options, and in this way to extract the volatility and correlation information implicitly contained in these products.

European Range Accrual Option pricing and deviation Formula has been deduced through observing the foundational asset probabilistic distribution characteristics with the help of Ito’s lemma, and through relaxing the boundary assumption to infinity and zero respectively, the classical Black-Scholes option formula has been worked out. This paper subsequently articulates the numerical value simulated computation algorithm using logic program language for the corresponding demonstration. From a statistical point of view, the American range option is not definitely more valuable than the corresponding European range option and the difference between their deviations is significant.

An analytical approximation formula for European option pricing under a new stochastic volatility model with regime-switching

Option markets, empirical price data, and theoretical arguments all indicate that asset prices in actively traded markets are driven by Lévy flights and/or tempered Lévy flights, not by Brownian motion. So here we model asset prices in the real world by (1) where dZL is a Lévy or tempered Lévy flight. Derivatives based on such assets cannot be made risk free by dynamic hedging, so these derivatives cannot be priced using the standard Black–Scholes–Merton (BSM) arbitrage-free pricing criterion. Therefore, we develop a pricing theory based on a more general pricing criterion: that the expected return of all diversifiable portfolios is at the risk-free rate. We show that even though derivatives based on Lévy or tempered Lévy-driven assets cannot be hedged to make risk-free portfolios, they can be hedged to make diversifiable portfolios. This allows us to conclude that these option prices are given by (2) where the expected value uses the “real-world” probability measure, but with the asset price X(t) replaced by the alternative “pricing process” (3) which grows at the risk-free rate. We analyze these models to obtain explicit asymptotic formulas for European option prices. This analysis shows that as the time to expiry decreases, eventually all these Lévy and tempered Lévy-based models reduce to the same canonical model, and that European option prices and implied volatilities are given by similarity solutions under this canonical model. These similarity solutions are then examined to assess the mishedging that arises from Brownian-based models in a Lévy world.

Pricing of interest rate derivatives, such as CMS spread or mid-curve options, depends on the modelling of the underlying single rates. For flexibility and realism, these rates are often described in the framework of stochastic volatility models. In this paper we allow rates to be modelled within a class of general stochastic volatility models, which includes the SABR, ZABR, free SABR and Heston models. We provide a versatile technique called Effective Markovian Projection, which allows a tractable model to be found that mimics the distribution of the more complex models used to price multi-rate derivatives. Three different numerical approaches are outlined and applied to relevant examples from practice. Finally, a new method that involves moment-matching of Johnson distributions is applied to facilitate closed-form pricing formulas.

Emphasis is placed on the valuation of plain vanilla option when the price process of underlying asset is described by the stochastic Verhulst-Gompertz Equation with network externality effects in a complete market. The method is based on the change of measure, Girsanov theorem and martingale valuation techinique. The application to an exchange option is made attempt and the valuation formula for this option like the Black-Scholes one is derived. A simple relationship similar to the put-call-parity between the exchange call option and the put option is provided. Our results will be useful to analyze and to hedge the price evolution at a sudden rise or crash of stock and commodity markets. 1 Introduction. The prices of some assets traded in the financial market have the ten- dency raised more when the prices begin to rise, it is because most traders expect that a price rises more and the purchase orders of other traders are induced as a result. Conversely, when the prices begin to fall, selling orders of other traders are induced. In the actual mar- ket, we often observe such a phenomenon called network externality price effects. These effects have been recognized as very significant concept in the field of marketing. Nerlove and Arrow(8) , Bass(2), Vidale and Wolfe(11) and Gould(4) made use of this deterministic version to study the dynamic behavior of new product and advertising policy. Their models were stimulated by a classical theory of logistic curve by Verhulst(10). Gompertz extended Verhulst's model to so-called a deterministic Verhulst-Gompertz model in order to investi- gate the growth of population. In the fields of finance, Schwartz(9) gave a stochastic version of Verhulst-Gompertz model and developed the random behavior of commodity price. His model was based on the biological problem studied by Goel and Richer-Dyn(3). The main purpose of this paper is devoted to describe the network externality effects by a generalized stochastic Verhulst-Gompertz equation and to derive pricing formulae for a plain vanilla and an exchange options when the price of underlying asset shows the network externality price effects. The organization of this paper is as follows. The next section briefly introduces the generalized stochastic Verhulst-Gompertz equation to express the network externality effects and in Section 3 we develope the martingale measure to evaluate the value of option. In Section 4 we derive a hedging strategy and a pricing formula of an exchange option that gives the holder the right which exchanges the underlying asset with the stochastic Verhulst-Gompertz equation for the asset with the geometric Brownian motion (Black-Scholes model) by means of martingale pricing method(7). If we regard the former as the spot price of energy commodity and the latter as the stock price of business firm that trades the energy commodity, such an exchang option will be very useful to hedge the sudden rise or heavy fall in price of energy and make a contribution to the stabilized price of energy. The paper concludes with a brief summary.

Pricing Barrier Options and Credit Default Swaps (CDS) in Spectrally One-Sided Levy Models: The Parabolic Laplace Inversion Method