A Simple Binomial Model
• A 3-month call option on the stock has a strike price of 21.
• A stock price is currently $20
• In three months it will be either $22 or $18

K=21

S_U = $22
Option Price = $1
S₀ = $20
S_D = $18
Option Price = $0

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Setting Up a Riskless Portfolio

Consider the Portfolio: 

a) long Δ shares; short 1 call option

b) long Δ shares; short 1 put option

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a) Portfolio is riskless when 22Δ – 1 = 18 Δ => Δ= 0.25
A riskless portfolio is therefore=>

Long : 0.25 shares
Short : 1 call option

b) -18Δ+1=-22Δ Δ => Δ= -0.25

A riskless portfolio is therefore=>

Long : 0.25 shares
Short : 1 call option

Valuing the Portfolio

The riskless portfolio is:
a) long 0.25 shares
short 1 call option

b) long 0.25 shares
short 1 put option

a) The value of the portfolio in 3 months is
S_uΔ-(S_u-K)=22 × 0.25 – (22-21) = 4.5

or

S_LΔ=18×0.25=4.5
• The value of the portfolio today is (if r_f =12%)
f=4.5e^{–0.12×0.25}=4.367

b) The value of the portfolio in 3 months is
S_uΔ+0 =-22 × (-0.25) = 5.5

or

S_LΔ+(S_u-K)=-18×(-0.25)+(22-21)=5.5
• The value of the portfolio today is (if r_f =12%)
f=5.5e^{–0.12×0.25}=5.337

Valuing the Option

 Stock price today = $20
• Suppose the option price = f

a) for call option

• The portfolio today is
ΔS₀-f=0.25 × 20 – f = 5 – f
It follows that
5 – f =4.367

So
f=5-4.367=0.633 ---- the current value
of option

(S₀Δ – f)e^rT = (20*0.25-f)e^0.12*3/12=4.5

(5.0-f)e^0.12*3/12=4.5

(5.0-f)=4.5/e^0.12*3/12

f=5.0-4.5/e^0.12*3/12

f=5-4.367=0.633

b) for put option

• The portfolio today is
ΔS₀-f=0.25 × 20 + f = 5 + f
It follows that
5 - f =5.337

So
f=-5.337+5=-0.337 ---- the current value
of option

(20 x 0.25 – f)e^0.12*3/12 = 5.5

(20 – f) = 5.5/e^0.12*3/12

f=5-5.5/e^0.12*3/12

f = -0.337

Generalization

S 0 = stock price
u= percentage increase in
the stock price
d= percentage decrease in
the stock price
ƒ = current option price whose stock price is S 0
ƒ_u =  payoff from the option when price moves up)
ƒ_d= payoff from the option when price moves down)
T= the duration of the option

Generalization (continued)

• Consider the portfolio that is long D shares and sh
ort 1 option

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The portfolio is riskless when S₀uΔ – ƒ_u  = S₀dΔ – ƒ_d
or
Δ=ƒ_u-ƒ_d/S₀u-S₀d

Generalization (continued)

• Value of the portfolio at time T is (S₀uΔ – ƒ_u )
• The cost of setting up the portfolio is S₀Δ – f

Hence S₀Δ – ƒ = (S₀uΔ – ƒ_u)e^–rT
ƒ = S₀Δ – (S₀uΔ – ƒ_u)e^–rT

Substituting for

Δ=ƒ_u-ƒ_d/S₀u-S₀d we obtain

f = [pf_u+(1-p)f_d]e^-rT (12.2)

where p=e^rt-d/u-d

Generalization (continued)

Ex. (see Figure11.1)
u=1.1, d=0.9,r=0.12,T=0.25, f_u=1, ƒ_d=0

p=e^rt-d / u-d=e^0.12×3/12-0.9 / 1.1-0.9=0.6523

f=[pf_u+(1-p)f_d]e-rT=[0.6523×1+(1-0.6523)×0]e^-0.12×3/12=0.633

Risk-Neutral Valuation

We assume p and 1-p as probabilities of up and down movements.
• Expected option payoff = p × ƒ_u  + (1 – p ) × ƒ_d

• The expected stock price at time T is
E(S_T) = pS₀u+(1-p)S₀d = pS₀(u-d) + S₀d

substituting p=e^rT-d / u-d =» E(S_T)=S₀e^rT

From this equation, we can see that the stock price grows, on average, at the risk-free rate. Because setting the probability of the up movement equal to p is therefore equivalent to assuming that the return on the stock equals the risk-free rate.

• In a risk-neutral world all individuals are
  indifferent to risk. In such a world, investors require no compensation for risk, and the expected return
  on all securities is the risk-free interest rate.
• Risk-neutral valuation states that we can assum
  e the world is risk neutral when pricing options.

Original Example Revisited

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• Since p is the probability that gives a return on the stock equal to the risk-free rate. We can find it from

E(S_T)=S₀e^rT => 22p + 18(1 – p ) = 20e^0.12×3/12 => p = 0.6523

• At the end of the three months, the call option has a 0.6523 probability of being worth 1 and
a (1-0.6523)=0.3477 probability of being worth zero. So the expect value is
Expected option payoff = p  × ƒ_u  + (1 – p )  × ƒ_d
0.6523 ×1 + 0.3477 ×0 = 0.6523
In a risk-neutral world this should be discounted at the
risk-free rate. The value of the option today is
0.6523e^{–0.12×0.25} = 0.633

• It is not easy to know the correct discount rate to apply to the expected payoff in the real world.
• Using risk-neutral valuation can solve this problem because we know that in a risk-neutral world the expected return on all assets is the risk-free rate.

Two-Step Binomial Model

• Stock price=$20 , u=10% , d=10%
• Each time step is 3 months
• r=12%, K=21