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In another section, we considered the pricing of rainbow options, which, practically speaking is an option which has 2 or more underlying assets. A spread option can be considered as a type of rainbow option in that it's payoff depends on 2 or 3 underlying assets.
Introduction || European Spread Approximation (Kirk) || 3-D Binomial || 3-D Trinomial || on 3 Assets || American Style Spread Options || Advanced Methods || Other Known Names & Variants || References || Advanced Readings In another section, we considered the pricing of rainbow options, which, practically speaking is an option which has 2 or more underlying assets. A spread option can be considered as a type of rainbow option in that it's payoff depends on 2 or 3 underlying assets. Spread options are commonly used within commodity markets as well as foreign exchange options to provide a payoff based on the difference between two or even 3 assets assets. For a 2 asset spread option, we can give the payoff for calls and puts respectively as:
Spread options are often called crack spreads, due to their use in the oil industry. Crack refers to crude oil contracts, and these spreads are usually a spread between crude oil and another commodity such as heating oil. Spark spreads refer to spreads related to the use of electricity. Kirk (1995) considered the pricing of European style spread option on futures contracts and formulated an approximation based on the original Black (1976) futures option model and gives the value of spread call and put options respectively as:
Where:
We generalised Kirk's futures price to being a general asset
price (S). X is the strike price, Since Cox, Ross & Rubinstein (1979) proposed a binomial method for the valuation of options, extensive research has been undertaken into the use of lattice methods. A more common model used is the Rubinstein (1994) three-dimensional lattice model which uses 4 branches in the tree to value based on up and down probabilities. These probabilities correspond to a discretisation with respect to two individual geometric Brownian motions which are correlated. For those who have not come across the concept of a 3-D binomial lattice, click here for an introduction. Hence, our probabilities can be given as:
Where:
The 4 branches of the binomial tree given by Rubinstein are then:
Comparing 3-Dimensional Binomial & Kirk's Approximation By implementing the models themselves, we find that the two models result in similar values given the cost of carry for the 3-dimensional model is 0 (that is r = D). Using the following inputs, we show that the absolute error of the two methods is minimal, even when only 100 time steps are used. S1 = 90, X = 5, r = D1 = D2 = 10%, V1 = V2 = 20%, T = 0.5 years, Correlation = 0.5
As with all binomial type lattices, we can extend this to incorporate the use of a trinomial lattice. The original trinomial lattice proposed for valuing options was given by Boyle (1986). He later extended this approach for use with options on 2 underlying assets - Boyle (1988) in a 3-D trinomial model, and in this model, Boyle considers 5 possible branches in the tree and uses the extension of his 1986 methodology to value spread options. We will graphically illustrate this in due course. The 3-D binomial method produces rapid convergence results already using a mere 100 time steps - and in the case with a 3-D trinomial lattice, one would expect to find even faster and more accurate convergence than the binomial style lattice. Much of what has been said so far concerns the pricing of European style spread option contracts. In the case of early exercise, we must first consider whether or not the underlying assets pay out any dividends during the life of the contract; in the case where no dividends are paid, the price of a European and American style contract are equal because of the fact that no exercises mean that Americans should not be exercised prior to maturity. In the case where there are dividends paid out, we highlight the main approach to price two asset American spreads. The first one would be to consider the 3-dimensional binomial tree given by Rubinstein (1994). The beauty of binomial models is that it is easy to structure an algorithm to incorporate early exercise. In the case of spread options, although the choice of early exercise is often more of a matter of convenience rather than matter, it is possible to model American spread options on 2 assets in a similar fashion to that of European style spread options. Although the aforementioned methods are highly popular in pricing spread options due to their simplicity of implementation and computation efficiency, there are a number of other methods available to price these types of options. Many methods which are used in pricing spread options rely on the assumption of constant volatility and correlation Carr & Madan (1999) proposed a Fast Fourier transform for standard option pricing, whic was extended by Dempster & Hong (2000). By using this method, one can incorporate stochastic volatility, jump diffusion or even a variance-gamma into the pricing of spread options. Dempster & Hong provide a framework in which European style spread options can be priced by considering an integral format. By reiterating the payoff of a 2 asset spread call option:
With this example, our payoff is given as:
Where E is the expected value, Q is the risk neutral martingale measure. Which can be shown to result to:
Where: m is the price of asset 1, and q(S1, S2) is the joint density function of the 2 assets. Dempster & Hong (D & H) elaborate on the pricing under a Fast Fourier transform and give a generalisation for use under stochastic volatility and also a two factor geometric Brownian motion. Solving the PDE has not been considered in other papers extensively, and D & H show examples of computational times of these various methods. D & H also show that using a Fast Fourier transform produces significantly better results than standard Monte Carlo simulation methods alone (without variance reducation techniques), with MCS methods requiring well in excess of 80,000 simulations to generate results which have comparable accuracy to Fourier transform methods. We point interested readers to the paper by D & H for further reading. You may have already thought about the question of using three underlying assets and the option value depending on two individual spreads. As with many options on a so-called 'basket' of options, the most important factor to be considered is the correlation between the assets. In our mountain range options section we highlighted the use of Monte Carlo simulation for use in finding a value for these more exotic types of options and although this has yet to be illustrated, Monte Carlo simulation does a relatively good job with pricing these options in terms of speed of computation albeit sacrificing accuracy. Much of the text which has been written on pricing options on numerous assets revolves around determining a suitable correlation matrix and then applying a so-called Cholesky decomposition onto it. The theory behind Cholesky decomposition is not the purpose of this text and we recommend readers to seek additional text on this topic. We can consider first, the payoff of the three asset (or dual spread) call and put options:
Due to the existence of 3 underlying assets, using a lattice method would be cumbersome and require perception of a 3 x 3 grid from which the assets could move in. i.e. An up movement for asset 1 could be accompanied by a down movement of 2 and an up movement of 3 and so forth. Effective valuation would most likely come in the form of the aforementioned simulation method using either standard MCS or a Quasi-MCS using a form of low-discrepancy sequence such as that of Halton, Sobol or Faure. We recommend QMCS as the most effective method as it is computationally efficient, robust and the accuracy of the simulation is good. Furthermore, one can implement the Moro (1995) Inversion instead of a standard normal Gaussian inversion in order to generate even more accurate results.
Converging to a true value of 22.333 for a 3-asset spread call option. Asian Spread Options See a table of computational time for Asian Spread Options under Monte Carlo simulation here. Basket Options Additional/Useful List of resources Papers: Bermin, H., "Exotic Lookback options: The case
of Extreme Spread Options", Working Paper, Department of Economics,
Lund University, Sweden (1996) Andreas, A., Engelmann, B., Schwendner, P., Wystup,
U., "Fast Fourier Method for the Valuation of Options on
Several Correlated Currencies", Working Paper, (2001) |
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is
the correlation between the 2 assets, 
is
the volatility of the respective assets and r is the risk free rate.
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