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Hawaiian options are a relatively new breed of exotic options, a combination of Asian and American style options. To recap, Asian options are a form of path-dependent options which look back in time an takes the average of the underlying price to determine its payoff. These options are generally more suitable for investors who want to take up an option position but do not want to be exposed to the variations in asset price, particularly near expiry date. American style options simply mean that early exercise is possible and allows for more flexibility for the investor. Combining the two effectively gives an Asian option which allows for early exercise.
Introduction || Pricing || Binomial Method || PDE || MCS || QMCS || Finite Differences || Other Known Names / Variants || Readings || Advanced Readings Hawaiian options are a relatively new breed of exotic options, a combination of Asian and American style options. To recap, Asian options are a form of path-dependent options which look back in time an takes the average of the underlying price to determine its payoff. These options are generally more suitable for investors who want to take up an option position but do not want to be exposed to the variations in asset price, particularly near expiry date. American style options simply mean that early exercise is possible and allows for more flexibility for the investor. Combining the two effectively gives an Asian option which allows for early exercise. As we can see, these two options on its own are already complex to price , and more than often, closed form solutions for the respective options are hard to come by. In the case of the so-called Hawaiian options, or Asian-American options, authors have tried to blend the pricing methodologies from Asian and American models in order to price Hawaiian options. Taking Asian options on its own, we can find several dozen variants which make up the Asian family (floating, fixed, arithmetic, geometric, forward start, lookback, call, put, etc.), making the choice of pricing a Hawaiian option, equally if not more expansive. Effective computation of Hawaiian options requires solving a two-dimensional pricing problem (both the path of the asset price, as well as the moving average price which governs the Asian aspect of the option). Because of this inherent quality, analytical approaches are not possible and we must look towards numerical methods used to price these options. Although pricing methologies for American-Asian options are not abundant compared to that of their European style siblings - which have numerous numerical and analytical pricing methods, some attempts have been made to price these highly path-dependent options. We know from pricing standard American contracts that early exercise can often create difficulties in calculating an option price, and one might argue that a lattice method extended from that of Cox, Ross & Rubinstein (1979) and Boyle (1986) and determining the boundary conditions and early exercise regions can be used to deliver pricing methods. American style contracts have long been considered as a "no-go" in using Monte Carlo simulation because it is difficult to determine an optimal exercise date - Hull & White (1993), but we see in an eventual section that such simulation methodology can be implemented. Andreasen (1998) attempts to reduce the dimensionality problem associated with Hawaiian options by introducing a reduction space variable - given as the ratio between the moving average price of the underlying and the current underlying price for floating strike Hawaiian options. This reduces the problem to a one-dimension problem and allows for better evaluation of the option price. The Partial Differential Equation (PDE): A number of authors have attempted to utilise the Partial Differential Equation governing the pricing of a Hawaiian option. Miltersen, Jorgensen & Jorgensen (1997), (hereon referred to as MJJ) show that Hawaiian options can be valued through the use of an inhomogenous PDE covering the entire state space:
Where
Further details can be found in their 1999 paper. Earlier, we noted that much of the past literature on American style contracts has suggested that simulation methods cannot be applied because of the inability to determine optimal exercise for such contracts. However, Grant, Vora & Weeks (1997) suggest otherwise, by implementing an iterative search process to determine every possible exercise date throughout the life of the contract. They relate it to the special case of pricing an American put options under a binomial tree and reference a critical price method given by Merton (1973) which attempts to find the critical stock price which maximises the expected value of the American put. With standard American style contracts on a non-dividend paying stock, Merton (1973) shows that it is never optimal to undertake early exercise - see also Hull (2002). In the case of Hawaiian options, this is not the case. Grant, Vora & Weeks (GVW) (1997) illustrates this with the case of a Hawaiian call option. Because of its Asian style features, a fall in the underlying asset price, which in turn brings down the averaging price reduces the cost of early exercise. In cases where the benefits of early exercise exceed the cost of early exercise, Hawaiians will be exercised. By determining the critical value at each possible exercise date and comparing this to the cost of early exercise, one can determine when it is optimal to exercise this type of option through a backwards recursive technique, which in some ways can be compared to a standard binomial lattice. The authors, on page 10 of their informative paper, suggest the reasons why MCS can be applied to the pricing of these options: "We can do this because it is feasible to initiate a Monte Carlo simulation at any date and for any set of initial conditions; to interrupt a simulation at any point or points, to investigate the consequences of decisions at these points, and to continue the simulation." Although the methodology presented seems straightforward, the matter of computation efficiency comes into play. We have not implemented the pricing via this method ourselves, but the impression presented to us by the paper is that it is not particularly quick to price Hawaiian options via simulation. Furthermore, the aspect of simulation alongside calculation of the critical price at each possible exercise point, appears to be open for debate in using MCS as an effective way of valuing these options in terms of the payoff between time and accuracy. With said in mind, Fu & Wu (2000) extend the GVW approach and detail an optimal exercise policy used to determine the boundary condition of an American-Asian. Their method converges rapidly to a viable Hawaiian price and taking data from the authors' paper, we can see convergence to values near to that of GVW within milliseconds for 2,000,000 simulations. Without testing the GVW, we have been unable to make direct comparisons at this time. Results will hopefully be posted shortly. Ameriasian Options Additional/Useful List of resources Papers: Ameur, H., Breton, M., & L'Ecuyer, P., "A Dynamic Programming Procedure for Pricing American-style Asian Options", GERAD Report, G-99-39, (1999) Wu, R., & Fu, M., "Optimal Exercise Policies & Simulation-based Valuation fo American-Asian Options", Working Paper, Apr 2000 |
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is defined based on what type of Hawaiian is given. MJJ give 4 Hawaiian types as follows:
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