Use a three-time-step tree to value an American put option...

Use a three-time-step tree to value an American put option on the geometric average of the price of a non-dividend-paying stock when the stock price is $\$ 40$, the strike price is $\$ 40$, the risk-free interest rate is $10 \%$ per annum, the volatility is $35 \%$ per annum, and the time to maturity is three months. The geometric average is measured from today until the option matures.

Key Concepts

Binomial Tree Valuation

This concept refers to the use of a discrete-time lattice model to simulate the evolution of an asset's price over time. In the valuation of options, especially with shorter time horizons or when analytical solutions are difficult, a tree approach models the possible upward and downward movements at each time step. The valuation is then computed recursively by working backwards from the terminal nodes using risk-neutral probabilities.

American Options

American options are financial derivatives that allow the holder to exercise the option at any time up to and including the expiration date. This flexibility means that the valuation process must account for the possibility of early exercise. Therefore, when working with models like the binomial tree, it is necessary to compare the immediate exercise value with the continuation value at each node.

Geometric Averaging

In options where the payoff depends on the average price of the underlying asset, geometric averaging is one method used to summarize the asset's price path. The geometric average is particularly tractable mathematically and is used in certain types of Asian options. Its inclusion in the valuation problem introduces path-dependence, meaning that past price movements affect the payoff, which must be accounted for within the tree framework.

Risk-Neutral Pricing

Risk-neutral pricing is a fundamental concept in derivative pricing which posits that, under a risk-neutral measure, all assets are expected to grow at the risk-free rate. This allows for the pricing of options by discounting expected future payoffs at the risk-free rate, using probabilities that have been adjusted to reflect a world where investors are indifferent to risk.

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