Lesson 2 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 2 of 6 · 12 min

The hedge ratio and the riskless portfolio

Combine an option with just the right number of units of the underlying, the hedge ratio, and the portfolio is worth the same whether the price goes up or down.

In short

  • Sell one call and buy h units of the underlying: V0=hS0−c0V_0 = hS_0 - c_0.
  • At the end, V1u=hS1u−c1uV_1^u = hS_1^u - c_1^u and V1d=hS1d−c1dV_1^d = hS_1^d - c_1^d. Choose h so the two are equal.
  • The hedge ratio is h∗=c1u−c1dS1u−S1dh^* = \dfrac{c_1^u - c_1^d}{S_1^u - S_1^d}: the change in option value divided by the change in the underlying.
  • For a sold call, h* is positive: buy h* units of the underlying per call (or sell 1/h* calls per unit).
  • For a put, h* is negative: the hedge is to buy the put and buy the underlying (or sell both).

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