the binomial model n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
The Binomial Model PowerPoint Presentation
Download Presentation
The Binomial Model

Loading in 2 Seconds...

play fullscreen
1 / 28

The Binomial Model - PowerPoint PPT Presentation


  • 74 Views
  • Uploaded on

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.

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

PowerPoint Slideshow about 'The Binomial Model' - nansen


An Image/Link below is provided (as is) to download presentation

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
The Binomial Model

$120

$20

$100

C100

= ?

$90

$0

Strategy: Buy 1 stock sell 1.5 calls

the binomial model1
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

the binomial model2
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 model3
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
Delta
  • Chose 1/Δ to hedge, thus;

uS - (1/Δ)*Cu = dS - (1/Δ)*Cd

1/Δ = {uS – dS}/{Cu – Cd}

delta1
Delta

$120

$20

= 0

-

$0

$90

the binomial model4
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 model5
The Binomial Model
  • Substitute for 1/Δ to get
  • C = {P*Cu + (1-P)*Cd}/R
  • P = [R-d]/[u-d]
the binomial model6
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
What is P?

u > R > d

0 < P < 1

R=1.1

________________________________

d=0.9

u=1.2

what is p1
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 p2
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 p3
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 p4
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 p5
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 – the risk neutral probability

P < q

Risk Aversion

P = q

Risk neutral

P > q

Risk seeking

p the risk neutral probability1
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
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
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 periods1
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
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 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 distribution1
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
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 model1
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 model2
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 model3
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 model4
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