fbpixel
101 Concepts for the Level I Exam

Essential Concept 79: Binomial Model: Expectations Approach


The expectations approach is given by the following equations:

c = PV[πc+ + (1 – π)c]  and

p = PV[πp+ + (1 – π)p]

π = the risk-neutral probability of an up move = (1 + r – d) / (u – d)

The expected terminal option payoffs can be expressed as:

E(c1) = πc+ + (1 – π)c and

E(p1) = πp+ + (1 – π)p

The option values can be written as:

c = PV[E(c1)] and

p = PV[E(p1)]

With the expectations approach,

  • The probability, π, is objectively determined and is called the risk-neutral (RN) probability. No assumption is made regarding the arbitrageur’s risk preferences.
  • The discount rate is not risk-adjusted. It is simply the risk-free interest rate.

Example: A non-dividend-paying stock is currently trading at €100. A call option has one year to mature, the periodically compounded risk-free interest rate is 5.15%, and the exercise price is €100. Assume a single-period binomial option valuation model, where u = 1.35 and d = 0.74. What is the call option value and put option using the expectation approach?

Solution: S+ = uS = 1.35(100) = 135

S = dS = 0.74(100) = 74

c+ = Max(0,uS – X) = Max(0,135 – 100) = 35

c = Max(0,dS – X) = Max(0,74 – 100) = 0

p+ = Max(0,100 – uS) = Max(0,100 – 135) = 0

p = Max(0,100 – dS) = Max(0,100 – 74) = 26

π = [1 + r – d]/(u – d) = (1 + 0.0515 – 0.74)/(1.35 – 0.74) = 0.51

c = PV[πc+ + (1 – π)c] = [(0.51)35 + (1 – 0.51)0] / 1.0515 = €17.00

p = PV[πp+ + (1 – π)p] = [(0.51)0 + (1 – 0.51)26] / 1.0515  = €12.11