To determine the price of the non-standard option, we can use the binomial tree model, modified for the continuous compounding risk-free interest rate. The option's value at time ( t = 0 ) is given by the expected value of the option's payoffs at ( t = 3 ), discounted back to the present time ( t = 0 ).
Let's denote the probability of the stock price going up or down by 10% as ( p ). Given the equal probability of the stock price increasing or decreasing, ( p = 0.5 ).
At ( t = 1 ), the holder can choose between an American call or put option. Let's calculate the payoffs for both scenarios:
- If the holder exercises a call option, the payoff at ( t = 3 ) is:
- If the stock price goes up: ( 40 \times 1.1 - 40 = $4 )
-
If the stock price goes down: ( 40 \times 0.9 - 40 = $-4 )
-
If the holder exercises a put option, the payoff at ( t = 3 ) is:
- If the stock price goes up: ( 40 \times 1.1 - 40 = $0 )
- If the stock price goes down: ( 40 \times 0.9 - 40 = $-4 )
Now, we discount these payoffs back to the present using the continuously compounded risk-free rate of 5%:
Call option payoff: ( e^{-0.05 \times 2} \times (\$4 + (-\$4)) = ( e^{-0.1} \times (\$4 - \$4) )
Put option payoff: ( e^{-0.05 \times 2} \times (\$0 - \$4) = ( e^{-0.1} \times (-\$4) )
The expected value of the option is the average of the discounted call and put payoffs:
Expected value = ( 0.5 \times (e^{-0.1} \times (\$4 - \$4)) + 0.5 \times (e^{-0.1} \times (-\$4)) )
This gives us the price of the non-standard option at ( t = 0 ).
Please note that this is a simplified calculation, and in practice, the actual pricing might involve numerical methods or Monte Carlo simulations to account for the probabilistic nature of the option holder's choice at ( t = 1 ).
