The Binomial Model

1 / 28

# The Binomial Model - PowerPoint PPT Presentation

The Binomial Model. \$120. \$20. \$100. C100 = ?. \$90. \$0. Strategy: Buy 1 stock sell 1.5 calls. The Binomial Model. CF today. CF at T (S = 90). CF at T (S=120). Buy Stock -\$100. \$90. \$120. Sell 1.5 calls \$1.5C. \$0. -\$30. ____________. _________. _______. 1.5C - 100. \$90. \$90.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## The Binomial Model

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
The Binomial Model

\$120

\$20

\$100

C100

= ?

\$90

\$0

Strategy: Buy 1 stock sell 1.5 calls

The Binomial Model

CF today

CF at T (S = 90)

CF at T (S=120)

\$90

\$120

Sell 1.5 calls \$1.5C

\$0

-\$30

____________

_________

_______

1.5C - 100

\$90

\$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

\$uS

Cu

\$S

1/Δ – hedge ratio

\$C

\$dS

Cd

uS - (1/Δ)*Cu

S – (1/Δ)*C

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

\$20

= 0

-

\$0

\$90

The Binomial Model

uS – {1/Δ}*Cu

S – {1/Δ}*C

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?

u > R > d

0 < P < 1

R=1.1

________________________________

d=0.9

u=1.2

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

Risk Aversion

P = q

Risk neutral

P > q

Risk seeking

P – the risk neutral probability

\$20

\$20

0.875

0.666

0.333

0.125

\$0

\$0

0.875*20=17.5

0.666*20=13.333

17.5/1.1=15.909

13.333/1.1=12.12

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

{0.666*44+0.333*8}/1.1

144

44

120

29.09

108

100

19.08

8

90

4.844

81

0

{0.666*29.09+0.333*4.844}/1.1

{0.666*8/1.1

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

P4

4

du3

P3 (1-P)

6

1

d2u2

P2(1-P)2

4

d3u

(1-P)3 P

1

d4

(1-P)4

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-jMax{0, ujdn-jS – E}]/Rn

Let a (number of ups) be the smallest

integer such that the option will mature in the

money

The Binomial Option Pricing Model

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]

The Binomial Option Pricing Model

S[Σj=a{n!/j!(n-j)!} [u/R]jPj (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]

The Binomial Option Pricing Model
• 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
The Binomial Option Pricing Model
• C = S*Φ{a, n, P’} - E*R-n*Φ*{a, n, P}
• P = [R-d]/[u-d]
• P’ = [u/R]P