Spread Option Pricing Research Articles

Efficient Pricing of CMS Spread Options in a Stochastic Volatility LMM

In this paper we develop a fast yet accurate formula for pricing CMS spread options in a popular class of Libor market models with stochastic volatility. This formula makes it feasible to include quoted CMS spread option prices in the general calibration procedure and by this means to recover the correlation information contained in the market. Our approach is based on a new method for the fast evaluation of the density of an integrated Cox-Ingersoll-Ross (CIR) process. Combined with approximations for the swap-rate dynamics, this results in a semi-analytical formula for CMS spread option prices. Numerical experiments demonstrate the accuracy this formula.

Multi-asset spread option pricing and hedging

We provide two new closed-form approximation methods for pricing spread options on a basket of risky assets: the extended Kirk approximation and the second-order boundary approximation. The latter method generalizes the results in Li et al. [J. Deriv., 2008, 15, 58–80] to spread options on an arbitrary number of assets. Numerical analysis shows that while the latter method is more accurate than the former, both methods are extremely fast and accurate. Closed-form approximations for important Greeks are also derived. Our approximation methods enable the accurate pricing of a bulk volume of spread options on a large number of assets in real time, which offers traders a potential edge in a dynamic market environment.

Pricing of spread options on stochastically correlated underlyings

This paper proposes a method to price spread options on stochastically correlated underlying assets. Therefore, it provides a more realistic approach towards a dependence structure. We generalize a constant correlation tree algorithm developed by Hull (2002) and extend it using the notion of stochastic correlation. The resulting tree algorithm is recombining and easy to implement. Moreover, the average price error decreases approximately as n−1.6, where n is the number of time steps. We show that this level of convergence is similar to that of the algorithm for the constant correlation case. Our sensitivity analysis with respect to the stochastic correlation parameters shows that the constant correlation model systematically overprices spread options on two stochastically correlated underlying assets. Furthermore, we use our model to derive hedge ratios for the correlation of a spread option and show that the constant correlation model also overprices the hedge ratios.

Split bond ratings and rating migration

This paper examines the relationships between split ratings and ratings migration. We find that bonds with split ratings are more likely to have future rating changes. A one-notch (more-than-one-notch) split rating increases the probability of rating change within one year of initial issuance by about 3% (6%). Furthermore, we find that about 30% of split rated bonds have their two ratings converge after four years of initial issuance. The rating convergence tapers off after three years, and the rating agency with a higher (lower) initial rating generally maintains a higher (lower) rating in subsequent years if the two ratings do not converge. We also show that rating transition estimation can be improved by taking into consideration split ratings. We find that one-year rating transition matrices are significantly different between non-letter-split rated bonds and letter-split rated bonds, and we show that the difference has an economically significant impact on the pricing of credit spread options and VaR-based risk management models. Overall, our results suggest that split ratings contain important information about subsequent rating changes.

Multi-Asset Spread Option Pricing and Hedging

We provide two new closed-form approximation methods for pricing spread options on a basket of risky assets: the extended Kirk approximation and the second-order boundary approximation. Numerical analysis shows that while the latter method is more accurate than the former, both methods are extremely fast and accurate. Approximations for important Greeks are also derived in closed-form. Our approximation methods enable the accurate pricing of a bulk volume of spread options on a large number of assets in real time, which offers traders a potential edge in a dynamic market environment.

Modeling of certain problems in financial mathematics: Spread option pricing

The problem of valuating exotic options, namely, the option on the spread between two forward interest rates is considered. The price of the option is derived under the assumption that the dynamics of debt instruments and the interest rates are described by the Heath-Jarrow-Morton model. The parameters of the model are estimated, and the price of the option is numerically computed based on Russian bond market data.

On pricing of credit spread options

This paper describes and analyses different pricing models for credit spread options such as Longstaff–Schwartz, Black, Das–Sundaram and Duan (GARCH-based) models. The first two models, Longstaff–Schwartz and Black, assume respectively a mean-reverting dynamic and a lognormal distribution for the spread and are representative of the so-called “spread models”. Such models consider the spread as a unique variable and provide closed form solutions for option pricing. On the contrary Das–Sundaram propose a recursive backward induction procedure to price credit spread options on a bivariate tree, which describes the dynamic of the term structure of forward risk-neutral spread and risk-free rate. This model belongs to the class of structural models, which can be used to price a wider range of credit risk derivatives. Finally, we consider the pricing of credit spread options assuming a discrete time GARCH model for the spread.

Bivariate normal mixture spread option valuation

We discuss the pricing and hedging of European spread options on correlated assets when the marginal distribution of each asset return is assumed to be a mixture of normal distributions. Being a straightforward two-dimensional generalization of a normal mixture diffusion model, the prices and hedge ratios have a firm behavioural and theoretical foundation. In this ‘bivariate normal mixture’ (BNM) model no-arbitrage option values are just weighted sums of different ‘2GBM’ option values that are based on the assumption of two correlated lognormal diffusions, and likewise for their sensitivities. The main advantage of this approach is that BNM option values are consistent with both volatility smiles and with the implied correlation ‘frown’. No other ‘frown consistent’ spread option valuation model has such straightforward implementation. We apply analytic approximations to compare BNM valuations of European spread options with those based on the 2GBM assumption and explain the differences between the two as a weighted sum of six second-order 2GBM sensitivities. We also examine BNM option sensitivities, finding that these, like the option values, can sometimes differ substantially from those obtained under the 2GBM model. Finally, we show how the correlation frown that is implied by the BNM model is affected as we change (a) the correlation structure and (b) the tail probabilities in the joint density of the asset returns.

On the Pricing of Credit Spread Options: A Two Factor HW–BK Algorithm

In this article we describe what a credit spread option (CSO) is and show a tree algorithm to price it. The tree algorithm we have opted for is a two factor model composed by a Hull and White (HW) one factor for the interest rate process and a Black-Karazinsky (BK) one factor for the default intensity. As opposed to the tree model of Schonbucher 1999 the intensity process cannot become negative. Having as input the risk free yield curve and market implied default probability curve the model by construction will price correctly the associated defaultable bond. We then use Market data to calibrate the model to price an at the money (ATM) CSO call and then test it to price an out of the money (OTM) Bermudan CSO call on a CDS. Furthermore the discussions in this paper show in practice the difficulties and challenges faced by financial institutions in marking to market those instruments.

Pricing and Hedging Spread Options

We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present a general overview of the common features of all spread options by discussing in detail their roles as speculation devices and risk management tools. We describe the mathematical framework used to model them, and we review the numerical algorithms actually used to price and hedge them. There is already extensive literature on the pricing of spread options in the equity and fixed income markets, and our contribution is mostly to put together material scattered across a wide spectrum of recent textbooks and journal articles. On the other hand, information about the various numerical procedures that can be used to price and hedge spread options on physical commodities is more difficult to find. For this reason, we make a systematic effort to choose examples from the energy markets in order to illustrate the numerical challenges associated with these instruments. This gives us a chance to discuss an interesting application of spread options to an asset valuation problem after it is recast in the framework of real options. This approach is currently the object of intense mathematical research. In this spirit, we review the two major avenues to modeling energy price dynamics. We explain how the pricing and hedging algorithms can be implemented in the framework of models for both the spot price dynamics and the forward curve dynamics.