Question

The stock price of Comet Inc. is currently $30. The stock price a year from now will be either $50 or $10 with equal probabilities. The interest rate at which investors can borrow is 5%. Using the binomial option pricing model (OPM), the value of a call option with an exercise price of $40 and an expiration date one year from now should be worth what?

Answer 1

To find the call option price using the binomial model, we follow the following steps -

1) Compute hedge ratio

Hedge ratio = (Value of option if price goes UP - Value of option if price goes Down) / (Upper price - Lower Price)

or, Hedge ratio = ($10 - $0) / ($50 - $10) = 1 / 4

You will buy the shares equal to the numerator of Hedge ratio and write/ sell options equal to denominator of the hedge ratio. So, you will buy 1 share and sell 4 call option (having 1 share each)

2) Compute loss in both situations

Situation 1 - Price goes UP to $50

If price goes UP, you will sell your share and option buyer (to whom you sold the options) will also exercise his options and to honor the options contract, you buy @50 from the market and sell @40 to him.

Loss = ($50 - $30) x 1 - [($40 - $50) x 4] = (-)$20

Situation 2 - Price goes down to $10

If price goes down, you will sell share@10, but the option buyer will not exercise his option as he would not want a loss.

Loss = $10 - $30 = (-)$20

Loss will always be equal in both situations because it is a hedge. (If it is not, you're probably doing something wrong)

3) Use the quation to find value of call option

(Vs - n x Vo) (1 + rt) = Net Investment

Where, Vs = Value of shares bought = $30, n = no. of options sold = 4, Vo = Value of option, r = interest rate = 5%, t = time to expiry = 1 year, Net investment = Initial investment - Loss = $30 - $20) = $10

($30 - 4 x Vo) (1 + 0.05 x 1) = $10

or, Vo = $5.12