A stock’s price today is S o = 100. It is known that at the end of 6 months it w

Question

A stock’s price today is So = 100. It is known that at the end of 6 months it will be either $120 or $80.
The risk-free interest rate is r = 5% per annum with continuous compounding. We are interested in
computing the today’s price of a T = 6M month European Put and Call options on this asset with the
same strike price K = 90 using a 1-step Binomial model.

• Give the Put option payoff at maturity PT as a function of the asset price at maturity ST
and use Risk- Neutral Pricing for the Put option to compute its price P0 at time 0.

• Give the Call option payoff at maturity CT as a function of the asset price at maturity ST
and use Risk- Neutral Pricing for the Call option to compute its price C0 at time 0.

I need step by step solution and formulas. Thank you.

Explanation / Answer

Answer: Stock price today is $100. It will either increase to $120 or decrease to $80 in one year. If the stock price rises to $120, he will exercise his call option for $110 and receive a payoff of $10 at expiration. If the stock price falls to $80, he will not exercise his call option, and he will receive no payoff at expiration.

First we need to solve for the risk neutral probability of the stock price going up.

100= (*120 + (1-)*80)/1.025 and thus = 0.5625

The risk-neutral probability of a rise in stock is 56.25%, and the risk-neutral probability of a fall instock is 43.75%.

Using these risk-neutral probabilities, determine the expected payoff to call option at expiration.

Expected Payoff at Expiration = (.5625)($10) + (.4375)($0) = $5.625

Since this payoff occurs 1 year from now, it must be discounted at the risk-free rate of 2.5% in order to find its present value:

PV(Expected Payoff at Expiration) = ($5.625 / 1.025) = $5.49

Therefore, given the information it has about stock price movements over the next year, a European call option with a strike price of $110 and one year until expiration is worth $5.49 today.

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