The Binomial Model

Prof. Avner Kalay - Options and Futures

The Binomial Model

$100

$120

$90

C100 = ?

$20

$0

Strategy: Buy 1 stock sell 1.5 calls


The Binomial Model

CF today

Buy Stock -$100

Sell 1.5 calls $1.5C

1.5C - 100

____________

CF at T (S = 90) CF at T (S=120)

$90

$0

_________

$90

$120

-$30

_______

$90


The Binomial Model

- Investment today of $100-1.5 C yields $90 for sure. Hence,

- [100-1.5C](1+r) = 90

- If r=10%

- C = (1/1.5)[100-90/1.1] = 12.12


The Binomial Model

$S

$uS

$dS

$C

Cu

Cd

1/Δ – hedge ratio

S – (1/Δ)*C

uS - (1/Δ)*Cu

dS - (1/Δ)*Cd


Delta

- Chose 1/Δ to hedge, thus;

uS - (1/Δ)*Cu = dS - (1/Δ)*Cd

1/Δ = {uS – dS}/{Cu – Cd}


Delta

$120

$90

$20

$0

- = 0


The Binomial Model

S – {1/Δ}*C uS – {1/Δ}*Cu

Investment Certain outcome

{S – [1/Δ}*C}*R = uS – {1/Δ}*Cu

R = 1 + rf and u > R > d

C = {S(R-u) + (1/Δ)Cu}/(1/Δ)R


The Binomial Model

- Substitute for 1/Δ to get

- C = {P*Cu + (1-P)*Cd}/R

- P = [R-d]/[u-d]


The Binomial Model

- In our example: u=1.2, d=0.9, R=1.1, uS=120, ds=90, E = 100, S=100

- P =[R-d]/[u-d] = [1.1-0.9]/[1.2-0.9]=2/3

- C= {(2/3)*20 + (1/3)*0}/1.1 = 12.12


What is P?

________________________________

d=0.9 u=1.2

R=1.1

u > R > d 0 < P < 1


What is P?

- P cannot be a probability since we do not know the probability of a price increase – denoted q.

- Since the valuation of C is true for any q we can assume (for our example) q = 0.5

- Do you feel comfortable with q = 0.5?


What is P?

- But if q=0.5 we can compute the expected return of the stock.

- E(Rs) = 0.5*20% + 0.5*(10%) = 5%

- Hence, E(Rs) < rf


What is P?

- Assume q=7/8=0.875.

- In our example P=[1.1-0.9]/[1.2-0.9] = 2/3

- E(Rs) = 0.875*20% + 0.125*(10%) = 16.25%

- Risk premium = 16.25 – 10 = 6.25%


What is P?

- Now reduce the risk aversion in the economy by reducing the risk premium to 1.25%. Increase the risk free rate to 15%.

- P = [1.15-0.9]/[1.2-0.9] = 5/6 = 0.833

- P gets closer to q

- C=5/6*20/1.15 = 14.493


What is P?

- Pushing it one step further, lets reduce the risk aversion in the economy to zero – R=1.1625

- P = [1.1625-0.9]/[1.2-0.9] = 7/8

- P is now equal to q

- C = {7/8}*20/1.1625 = 15.054


P – the risk neutral probability

P < q

P = q

P > q

Risk Aversion

Risk neutral

Risk seeking


P – the risk neutral probability

$20

$0

0.875

0.125

0.875*20=17.5

$20

$00.6

66

0.333

0.666*20=13.333

13.333/1.1=12.1217.5/1.1=15.909


Certainty equivalent

- The difference 17.5 – 13.333 = 4.167 is a correction for risk in the numerator

- The option model is valuation by certainty equivalents.

- Once we use P as if it is q we can take expectations and discount with the risk free rate


Two periods

100

120

90

144

108

81

19.08

29.09

4.844

44

8

0

{0.666*44+0.333*8}/1.1

{0.666*8/1.1{0.666*29.09+0.333*4.844}/1.1


Two Periods

- Cu = {P*Cuu + (1-P)*Cud}/R

- Cd = {P*Cud + (1-P)*Cdd}/R

- C = {P*Cu + (1-P)*Cd}/R

- C = {P2 Cuu + 2P(1-P)Cud + (1-P)2 Cdd}/R2


Four periods

1

u4

d4

du3

d2u2

d3u

P4

P3 (1-P)

P2(1-P)2

(1-P)3 P

(1-P)4

1

4

6

4

1


The Binomial Distribution

- The probability of a path with j ups and n-j downs is Pj(1 – P)n-j

- The number of paths leading to a node is

n!/{j!(n-j)!}

- The probability to get to a node is {n!/j!(n-j)!}Pj(1-P)n-j


The Binomial Distribution

- The probability to get to any one of the nodes is Σj=0 [{n!/j!(n-j)!}Pj(1-P)n-j] = 1

- The probability of at least a ups is Φ{a, n, P} = Σj=a{[n!/(j!(n-j)!]Pj(1-P)n-j} < 1


The Binomial Option Pricing Model

C = [Σj=0 {n!/j!(n-j)!}Pj(1-P)n-j Max{0, ujdn-jS – E}]/Rn

Let a (number of ups) be the smallest integer such that the option will mature in the

money



C = [Σj=a {n!/j!(n-j)!} Pj(1-P)n-j {ujdn-jS – E}]/Rn

S[Σj=a {n!/j!(n-j)!} Pj(1-P)n-j{ujdn-j/Rn}

=

ER-n[Σj=a {n!/j!(n-j)!} Pj(1-P)n-j]

-



S[Σj=a {n!/j!(n-j)!} [u/R]j Pj (1-P)n-j {d/R}n-j }

Let P’ = [u/R]P than 1 – P’ = [u/R]{(R-d)/(u-d)} = [d/R](1-P)

S[Σj=a {n!/j!(n-j)!} P’j (1-P’)n-j ]



- C = S*Φ{a, n, P’} - E*R-n*Φ*{a, n, P}

- Σj=0{[n!/(j!(n-j)!]Pj(1-P)n-j}= 1

- Φ{a, n, P} = Σj=a{[n!/(j!(n-j)!]Pj(1-P)n-j} < 1



- C = S*Φ{a, n, P’} - E*R-n*Φ*{a, n, P}

- P = [R-d]/[u-d]

- P’ = [u/R]P

The Binomial Model

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