The current price of the common stock of Internet Enterprises is $100. Over the course of a year, the stock's price will either increase by 100% or decrease by 50%. The stock pays no dividends. The current prices of one-period and two period zero coupon risk free bonds are $909.09 and $826.45 respectively ($1000 face value and the period here is 6 months).

A special European option has recently been created on the common stock of Internet with the following terms: On the expiration date the holder of the option has the right to sell the underlying asset for the highest stock price that occurred during the life of the option inclusive of the date on which it was issued.

What is the current value of this newly issued option on Internet Enterprises.?

Current value of this newly issued option on Internet Enterprises = $25

Explanation:

Risk free rate for 6 month or period 1 = (1000-909.09) / 909.09=10%

Risk free rate for 1 year = (1000-826.45) / 826.45=21%

Hence, risk free rate for period 2 = (1+21%) / (1+10%) - 1=10%

Now, Risk free rate factor for period 1 (R1) = 1+10%=1.1

Risk Free rate factor for period 2 (R2) = 1+10%=1.1

Upward price factor for a period (u) = (1+100%) ^ (1/2) = 1.414

Downward price factor for a period (d) = (1-50%) ^ (1/2) = 0.707

Probability of upward price = (R-d) / (u-d) = (1.1-0.707) / (1.414-0.707) = 0.55

Probability of downward price = 1-0.55=0.45

After period 1: Upward price=100*1.414=141.4 with probability 55%

Downward price = 100*0.707=70.7 with probability 45%

After period 2:

Upward Price will be = 141.4*1.414=200 with probability = 55%*55%=30.25%

Downward price will be=70.7*0.707=50 with probability=45%*45%=20.25%

Mid price will be = 141.4*0.707 or 70.7*1.414=100 with probability = 2*45%*55%=49.5%

Now, the highest price the stock can go is $200 with probability 30.25% and it was issued at $100

Hence, expected payoff of the option=30.25% * (200-100) = $30.25

So, current value of the newly issued option = 30.25 / (1+21%) = $25