prof. avner kalay - options and futures the binomial model $100 $120 $90 c100 = ? $20 $0 strategy:...
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Prof. Avner Kalay - Options and Futures
The Binomial Model
$100
$120
$90
C100 = ?
$20
$0
Strategy: Buy 1 stock sell 1.5 calls
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
Delta
- Chose 1/Δ to hedge, thus;
uS - (1/Δ)*Cu = dS - (1/Δ)*Cd
1/Δ = {uS – dS}/{Cu – Cd}
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
The Binomial Model
- Substitute for 1/Δ to get
- C = {P*Cu + (1-P)*Cd}/R
- P = [R-d]/[u-d]
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
What is P?
________________________________
d=0.9 u=1.2
R=1.1
u > R > d 0 < P < 1
Prof. Avner Kalay - Options and Futures
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?
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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%
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
P – the risk neutral probability
P < q
P = q
P > q
Risk Aversion
Risk neutral
Risk seeking
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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
Prof. Avner Kalay - Options and Futures
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]
-
Prof. Avner Kalay - Options and Futures
The Binomial Option Pricing Model
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 ]
Prof. Avner Kalay - Options and Futures
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
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