Question

Consider a call option with an exercise price of $110 and one year to expiration. The underlying stock pays no dividends, its current price is $110, and you believe it has a 50% chance of increasing to $120 and a 50% chance of decreasing to $100. The risk-free rate of interest is 10%

What is the hedge ratio?

What is the value of the riskless (perfectly hedged) portfolio one year from now?

What is the value of the call option ?

Answer 1

Solution:

We assume that this is Euorpean Call Option and use binomial model to solve the sum.

Lets see what details are given.

- Current stock price is $110 i.e. S0 is $110

- Expiration is one year i.e. T is 1

- Price will go up to 120 ie. u is 120/110 = 1.0909

- Price will do down to 100 i.e. d is 100/110 = 0.909

- Risk Free rate is 10% i.e. r is 0.1

- Stike Price is $110 ie K is $110

So the binomial model looks this way

At Node B: Option value fu = Max(120-110,0) = 10

At Node C: Option value fd = Max(100-110,0) = 0

A] Hedge Ratio:

Hedge ratio can be calculated using formula

= (10-0)/(120-100) = 0.5

Hedge ratio is 0.5

B] What is the value of the riskless (perfectly hedged) portfolio one year from now?

The value of the perfectly hedged portfolio will be S0u*Hedge Ratio - fu = (120*0.5 - 10) = 50

C] What is the value of the call option ?

Option value is calculated as under

= (e^(0.1*1) - 0.909)/(1.0909 - 0.909) = 1.078455

= e^(-1*0.1*1) * (1.078455 * 10 + (1-1.078455) * 0) = 9.758

The value of the call option is 9.758.

I hope you understand the solution. Thanks.

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