Risk Neutral Probability

Risk neutral probability:
π_U = ((1 + R_F) - (1/f_U))/((f_U) - (1/f_U))

π_D = 1 - π_U

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Underlying price: S₀ = $45
Strike price: K = $40
Up-move factor: f_U = 1.15
Risk-free rate: R_F = 4%

π_U = ((1 + 0.04) - (1/1.15)/(1.15 - (1/1.15)) = 0.608

π_D = 1 - 0.608 = 0.392

[1] One-period equity call option

Binominal model

t=0	t=1		Intrinsic value of the call
	45*(1.15)		Max(45*(1.15)-40,0)=11.75
$45
	45/(1.15)		Max(45/(1.15)-40,0)=0

The value of a one-period 40 call
= The probability weighted present value of the option payoff
= 0.608*11.75/1.04 + 0.392*0/1.04
= $6.87

[2] Two-period equity call option

Binominal model

t=0	t=1	t=2	Intrinsic value of the call
		45*(1.15)^2=59.51	Max(59.51-40,0)=19.51
	45*(1.15)
$45		45	Max(45-40,0)=5.00
	45/(1.15)
		45/(1.15)^2=34.03	Max(34.03-40,0)=0

The value of a two-period 40 call

= The probability weighted present value of the option payoff

= 0.608^2*19.51/1.04^2 + 2*0.608*0.392*5.00/1.04^2

= $8.87

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