Question

Consider an American put option with a maturity of one year and strike price 60 written on an underlying that pays no dividends. Suppose that S(0) = 60, the volatility σ = 35% per year and the risk-free rate r= 6% per year. Let's use a binomial tree of two periods.

i. Verify that with the usual choices we get u= 1.28 and d = .7803. Also check
that q=0.5007. (You can round up to .5.) Consider the option of working the
tree backwards (don't forget to check the possibility of exercising early!).
ii. Describe in each node the strategy that replicates. Verify that this strategy is
self-financing and that it replicates the payoff.

Answer 1

To address the student's question on valuing an American put option using a binomial tree, one needs to calculate the up-move, down-move, and risk-neutral probability, followed by working the tree backwards. The option's value at each node is determined by comparing it with the exercise value and ensuring that the replicating strategy of stock and risk-free bonds is self-financing.

Step-by-step explanation:

The subject matter of the student's question deals with the valuation of an American put option using a binomial tree model in a finance or business context. To verify the up (u) and down (d) factors, as well as the risk-neutral probability (q), one uses the following formulas where σ is the volatility, r is the risk-free rate, and Δt is the time step:

• u = e^{σ √Δt}
• d = 1/u
• q = (e^{rΔt} - d) / (u - d)

By plugging in the given values, one can calculate these factors. For example, with σ = 35% or 0.35, r = 6% or 0.06, and a Δt for two periods within a year, we get:

• u = e^{0.35 √(1/2)} = 1.28
• d = 1/1.28 = .7803
• q = (e^{0.06*(1/2)} - .7803) / (1.28 - .7803) = 0.5007

Using the calculated u, d, and q, and working backwards through the binomial tree, one can derive the value of the put option. At each node, the put option's value is compared with the exercise value (the strike price minus the stock price) to decide if exercising early is optimal. The option's value at each node is based on the discounted expected value of its possible future values, adjusted for risk neutrality. By replicating the option's payoff with a dynamic portfolio of the underlying stock and risk-free bonds, one ensures that the strategy is self-financing. This means that any adjustments to the portfolio do not require additional funds but are made by rebalancing between the stock and the bonds.