Question

# You want to price a European call option on stock X, which currently trades at \$40...

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