Lesson 3 of 6 · All lessons in this module
  1. Why options need a price model: the one-period binomial setup
  2. The hedge ratio and the riskless portfolioVideo lesson
  3. From the riskless hedge to the option price
  4. Risk neutrality and risk-neutral probabilitiesVideo lesson
  5. What does and does not move a binomial option value
  6. Integrated cases: an underpriced put and a tree with a negative rate

Lesson 3 of 6 · 13 min

From the riskless hedge to the option price

A riskless portfolio must earn the risk-free rate, so discounting its certain end value tells you what it costs today, and the option's price is whatever makes that cost hold.

In short

  • If V1u=V1d=V1V_1^u = V_1^d = V_1, the portfolio is riskless and must earn the risk-free rate: V0=V1/(1+r)V_0 = V_1/(1+r).
  • For a sold call hedged with h* units: c0=h∗S0−V1(1+r)−1c_0 = h^*S_0 - V_1(1+r)^{-1}.
  • For a put hedged by buying the put and |h*| units: p0=V1(1+r)−1−∣h∗∣S0p_0 = V_1(1+r)^{-1} - |h^*|S_0.
  • If the option trades above its model value, sell it, buy h* units and borrow: you earn more than the risk-free rate. If it trades below, do the reverse.
  • Put-call parity holds for the binomial call and put values, a useful check.

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