Binomial pricing: Assume that the shares of New Wave Property are currently trading for $21 and will either rise to $30 or fall to $18 in one year. The risk-free rate for one year is 0 per cent. You own a portfolio that consists of one call option and one put option. Both options have a strike price of $25, and both expire in one year. What is the value of your portfolio?

Question

Quizplus · Accepted Answer

To value the options using the binomial model:

- Suppose we buy one call option and one put option with a strike price of $25 each. 
- If the stock price ends at $30, then the call option will be worth $5 (i.e. the stock price of $30 - the strike price of $25), while the put option will be worthless (since the stock price is greater than the strike price). 
- If the stock price ends at $18, then the put option will be worth $7 (i.e. the strike price of $25 - the stock price of $18), while the call option will be worthless (since the stock price is less than the strike price). 
- We can construct a riskless portfolio by short selling the stock, and investing the proceeds at the risk-free rate. If the stock price ends up at $30, then the value of the portfolio will be $21 - $21(1+0) = $0, and we can use the proceeds to pay off the call option, leaving us with a profit of $5. If the stock price ends up at $18, then the value of the portfolio will be $21 - $21(1+0) = $0, and we can use the proceeds to pay off the put option, leaving us with a profit of $7. 
- The initial cost of the options is $1 (i.e. the sum of the call option and put option premiums), so the profits in both scenarios will be reduced by $1.

The expected profit from the portfolio is:

(1/2)($5 - $1) + (1/2)($7 - $1) = $3

Therefore, the value of the portfolio is $3 after one year, which is equivalent to a present value of $3/(1+0) = $3.

Since the initial cost of the options is $1, the net value of the portfolio is $3 - $1 = $2.50. However, since we own both the call and put options, we need to divide the net value by 2 to get the value of our portfolio (i.e. one call option and one put option), which is $2.50/2 = $1.25.

Therefore, the correct answer is D: $6.50.

Binomial Pricing: Assume That the Shares of New Wave Property