CFA Level IDerivativesHard
A stock currently trades at $50. Over one period, the stock price can move up by a factor of 1.20 or down by a factor of 0.90. The risk-free rate is 5% per period. A European call option has a strike price of $50. Using a one-period binomial model, what is the value of the call option today?
- A$4.76
- B$10.00
- C$5.00
- D$3.33
Show answer & explanationAnswer & explanation
Correct answer: A. $4.76
Up price = $60 (payoff $10), down price = $45 (payoff $0). Risk-neutral probability π = (1+r-d)/(u-d) = (1.05-0.90)/(1.20-0.90) = 0.50. Call value = [π×Cu + (1-π)×Cd]/(1+r) = [0.5×10 + 0.5×0]/1.05 = 5/1.05 = $4.76.
Why the other options are wrong
- B. This is simply the up-state payoff and ignores probability weighting and discounting.
- C. This is the undiscounted expected payoff and omits discounting by the risk-free rate.
- D. This results from using an incorrect risk-neutral probability calculation.
One-Period Binomial Option Pricing
The binomial model prices an option by computing a risk-neutral probability of an up move, then discounting the expected payoff at the risk-free rate.
- Risk-neutral probability π = (1+r-d)/(u-d)
- Call value = [π×Cu + (1-π)×Cd]/(1+r)
- No investor risk preferences are needed under risk-neutral valuation
Memory trick: Climb the tree with risk-neutral odds, then discount the branch payoffs home.