Spread option normal model
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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.
Spread options are financial instruments whose payoffs depend on the price difference of two underlying assets and are widely traded in a range of financial markets. For example, the price spread between heating oil and crude oil crack spread represents the value of production including profit for a refinery firm. In particular, to energy trading, the derivative contracts are a risk mitigation tool of crucial importance.
Carmona and Durrleman [ 1 ] presented a general overview of the common features of all spread options by discussing in detail their roles as speculation devices and risk management tools. Their review concentrated on the mathematics of the pricing and hedging of spread option.
Hurd and Zhou [ 2 ] showed a numerical integration method for computing spread options in two or higher dimensions using the FFT. Tt is based on square integrable integral formulas for the payoff function and, like those methods, it is applicable to a variety of spread option payoff in any model for which the characteristic function of the joint return process is given analytically.
Caldana and Fusai [ 3 ] proposed a new accurate method for pricing European spread options and obtained a lower bound, as in Bjerksund and Stensland [ 4 ], but for general processes. Lo [ 5 ] applied the idea of WKB method to provide a simple derivation of Kirks approximation and discussed its validity. Although spread options [ 1 — 9 ] have come into the limelight in recent years, little research has been published on the pricing of spread options with stochastic interest rates.
For options on bonds and other interest-rate-dependent instruments, movements in the interest rate are critical and the assumption of a constant interest rate is inappropriate. In this case, stochastic interest rate may significantly affect the prices. The pricing of options in the presence of stochastic interest rates can generally be difficult. Merton [ 12 ] used a bond maturing on the option expiration date to hedge European options.
However, sometimes it is inappropriate to assume that the volatility of the discounted bond price is a constant. There is no reason why interest rates should behave like stock prices.
Interest rates certainly do not exhibit the long-term exponential growth seen in the equity markets. On the other hand, mean-reverting models are used for modeling a random variable that is not going anywhere.
That is why they are often used for interest rates. The Vasicek model takes the following form: The model is mean-reverting to a constant level, which is a good property. It is easy to obtain the solution so that there are explicit formulae for many interest rate derivatives.
Thus, it is desirable to have a model that combines the strengths of the existing methods. Furthermore, the technique of measure changes is useful in a variety of asset pricing models. However, few papers focus on the discussion of the merits for it and its applications.
Credit risk arises from the possibility that borrowers and counterparties in derivatives transactions may default. Most financial institutes devote considerable resources to the measurement and management of credit risk. As was pointed out in Hull [ 15 ], the credit exposure on a derivatives transaction is more complicated than that on a loan.
This is because the claim that will be made in the event of a default is more uncertain. Klein [ 16 ] presented an improved method of pricing vulnerable Black-Scholes options under assumptions which were appropriate in many business situations. They used multiscale asymptotic analysis to derive an analytic approximation formula for the price of the vulnerable options and studied the stochastic volatility effect on the option price.
This study aims to fill the gaps and presents three novel spread options pricing models that permit the interest rates to be random. We assume that a discounted zero-coupon bond follows a stochastic process. The volatility of the discounted bond price is a function of time rather than a constant. The study presents the technique of measure changes to the computations of the spread options. We discuss the excellent feature for the models and techniques presented by us in some asset pricing.
The organization of the remaining part is as follows. Section 2 outlines a spread option pricing model with stochastic interest rates and presents a technique of measure changes to the price of the spread option.
Section 3 extends this analysis to jump diffusion model. Section 4 extends this analysis to a vulnerable spread option. Section 5 presents numerical analysis. The final section gives concluding remarks. This section constructs a spread option pricing model that permits the interest rate to be random.
A key point in the model is to assume that the volatility of the discounted bond price is a function of time rather than a constant. Using the technique of measure changes, we analyze the price of the spread option when underlying stock returns are continuous. We discuss the excellent features for the model and the technique in some asset pricing models. Let be a probability space on which is defined a three-dimensional Brownian motion, and let denote the risk neutral probability measure.
In particular, and are independent Brownian motions. Let, be the filtration generated by the Brownian motion. We assume that the interest rate process is adapted and define a discount process. Consider a zero-coupon bond that pays unit of currency at maturity. According to the risk-neutral pricing formula, the value of this bond at time is. Let and be two stock price processes.
The evolutions of the discounted prices will be modeled by the following stochastic differential equations: The value at time of a spread option, expiring at time with strike priceis. One defines three new measures. One defines the -forward measure by for all. Let be a probability space on which is defined a multidimensional Brownian motion. Lemma 2 Girsanov, multiple dimensions, [ 10 ]. Let be a d-dimensional adapted process. Define and assume that Then, under the probability measure given by the process is a d-dimensional Brownian motion.
Lemma 3 see [ 10 ]. Let be a Brownian motion and let be an adapted stochastic process that satisfies. Then, is a martingale. Under a new measure, the following price dynamics holds. Moreover, where is a three-dimensional Brownian motion under. The proof of Proposition 4 will be given in Appendix A. We need some notations and denote by the Euclidean norm in. For spread options, there is a long history of approximation methods for the computation of the prices.
The challenge in pricing spread options stems from the fact that there is no explicit solution. Furthermore, the pricing of options with stochastic interest rates can generally be difficult.
Therefore, pricing spread options with stochastic interest rates is a challenging task in finance and is important for both theory and applications. The price of the spread option, assuming the risk-neutral dynamics given in 2 and 3is with where the random variables and, have the bivariate standard normal distribution with correlation. For eventa number indicates the probability that will occur. The proof of Theorem 5 will be given in Appendix B.
The delta of an option or a portfolio of options is the sensitivity of the option or portfolio to the underlying. The sensitivities of the spread option price to initial stock prices, and spread are given by, and. The simplest way to calculate the delta, the gamma, and the theta of an option using Monte Carlo simulation is to estimate the option value twice.
According to our knowledge, the paper is the first one investigating spread option pricing with stochastic interest rates. For stochastic interest rates, the study presents a more generalized spread option pricing model.
Some models of stochastic interest rates are just a special case of the model presented by us. For Vasicek model 1interest rates can easily go negative, which is a very bad property. For a random variable that is not going anywhere, sometimes the assumption is inappropriate. We assume that the volatility of the discounted bond price is a function of time rather than a constant. The spread option pricing formula can be used in the standard call option.
We present also a generalized Black-Scholes-Merton option pricing formula with stochastic interest rates. In fact, the price of the European call, assuming the risk-neutral dynamics given in 2 and 3iswith where the random variables and have the standard normal distribution.
If the interest rate is a constantthen ; the result reduces to the usual Black-Shcholes-Merton formula. The price of an exchange option spread options withassuming the risk-neutral dynamics given in 2 and 3is with where the random variables and have the standard normal distribution and the random variables and have the bivariate standard normal distribution with correlation. The technique of measure changes is useful in a variety of asset pricing models.
The risk-neutral valuation of the spread option price involves a two-dimensional integration rather than a three-dimensional integration. Monte Carlo approximations always contain some randomness.