You want to price a European call option on stock X, which currently trades at $40 per share (this stock does not currently pay dividends). Suppose there are two possible outcomes for share prices of stock X next period: It can go up by 15%, or it can drop by 10%.
The option expires in one period, and has a strike price of $41. The risk-free rate over the next period is 5% (you can lend and borrow at the riskless rate). Markets are efficient and there are no arbitrage opportunities.
(a) What is the value of this call option today – i.e., what is the call premium?
Price of the stock at time 1 if the stock goes up by 15% = 40 * 1.15 = 46
Price of the stock at time 1 if the stock goes down by 10% = 40 * 0.9 = 36
value of call option at time 1 if the stock goes up = 46 - 41 = 5
Value of call option at time 1 if the stock goes down = 41 - 46 = 0
(Please not: since -5 will be less than 0, value of call option will be 0 if the stock goes down)
Risk neutral probability of 'up' move = [( 1 + R) - down move) / ( up move - down move)
Risk neutral probability of 'up' move = [( 1 + 0.05) - 0.9) / ( 1.1 - 0.9)
Risk neutral probability of 'up' move = 0.15 / 0.2
Risk neutral probability of 'up' move = 0.75
Risk neutral probability of 'down' move = 1 - 0.75 = 0.25
Call option value today = [0.75 ( 5) + 0.25 ( 0)] / ( 1 + 0.05)
Call option value today = 3.75 / 1.05
Call option value today = $3.5714
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