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,
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
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