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# The Valuation of American Options

The same process used for valuing a bond applies for an American option. In order to obtain an option value, we start from its terminal values given the interest rate values simulated at this date. Once all possible terminal values are determined from the terminal values of rates, the recursive process serves for deriving the values at other dates. For American options, whose exercise is feasible at any time between now and maturity, there is a choice at any date when the option is in-the-money. Exercising the option depends upon whether the borrower is willing to wait for opportunities that are more profitable or not. An American option has two values at each node (date i) of the tree: a value under no exercise and the exercise value. If exercise is deferred to later periods, the value is the average of the two possible values at the next date t + 1. If exercised, the value is the difference between the strike price and the gain. The calculation rule is to use the maximum of those two values. With this additional rule, the value of the options results from the backward process along the tree, starting from terminal values which are known.

# The Current Value of the Prepayment Option

This methodology applies to a loan amortized with constant annuities, of which maturity is 5 years and fixed rate is 10%. The current rate is also 10%. The yearly volatility of rates is 20% (in percentage of interest rate). The borrower considers a renegotiation when the decline in interest rates generates future savings whose present value exceeds the penalty of 3%. However, being in-the-money does not necessarily trigger exercise, since immediate renegotiation may be less profitable than a deferred exercise. The borrower makes the optimal decision by comparing the immediate gain with the expected gains of later periods. The expected payoff is the expected value of all discounted gains one period later, while the immediate payoff is the exercise value.

The first step of the process consists of simulating the interest rate values at all dates. Equal annual periods divide the horizon. The values of u and d correspond to a yearly volatility of 20%. The binomial tree is the same as above, and extends over five periods. Dates are as of end of year (Eoy), so that date 1 is the end of period 1. The binomial tree simulates short-term rates, which are valid from one date to the next. The interest rate is not a market rate since it is a client's rate. If the bank's spread remains constant, the fixed rate follows the same process as a market rate. The strike price is 1.03 times the outstanding principal of the original loan. The payoff under immediate exercise is the difference between the value of the original loan and the strike price. Finally, the optimum payoff is determined at each node. The present value of these optimum payoffs is the option value.

We need some assumptions for making the calculations simple. Consider that the original and the new loan are fully amortized at the same date. The relevant rate for calculating the payoff for the client is the fixed rate of the loan for its remaining maturity, which declines over time.

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