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Foreign currency options, particularly American options, are also useful when the hedger is uncertain about the date when the foreign currency will be needed.
The rest of paper is arranged as follows: In Section 2 we describe the American option pricing models with time-fractional derivatives; in Section 3 we introduce the Laplace transform methods for the problem; in Section 4 we give the boundary-searching finite difference methods for the problem; in Section 5 we provide numerical examples to compare the performance of the Laplace transform method with FDM; the conclusions are made in the final section.
We also compute the value of American option price under various values of volatility.
In the current article, when lapsation rationality is assumed, we use the American option framework--the dynamic approach, according to Milevsky and Salisbury (2006)--to price the contingent claim.
Bunch and Johnson [5] modified the Geske and Johnson [11] approach to approximate the American option values, using only the value of a European option ([P.sub.1]) permitting exercise at maturity date, T, and the value of a twice exerciseable option ([P.sub.2]) permitting exercise at time T/2 and T.
For example, Evnine and Rudd (1985) use intraday data for a two-month period in 1984 and find frequent violations of boundary conditions and put-call parity for S&P 100 and Major Market index options, both of which are American options.(5) Evnine and Rudd further conclude that these options are significantly mispriced relative to theoretical values based on the binomial option pricing model.
Before the investment threshold is reached, the follower makes a preparation to enter the market, the investment opportunity which is similar to an American option in a monopoly market.
This option to transfer would behave very much like an American option to lock in a GMDB at a time of the policyholder's own choosing.
Whaley, 1987, "Efficient Analytic Approximation of American Option Values", Journal of Finance, 42:301-320
The next section further discusses the American option element of the interest rate guarantee.
Due to the uncertainty of the optimal exercise time, the problem of American option pricing cannot be solved with analytical solution methods such as B-S formula but can be solved with numerical methods, such as binary tree method, finite difference method, and finite element method.

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