Transcript Tools - Interest Rate and Currency Swaps

Prof. Avner Kalay - Options and Futures
The Binomial Model
$120
$20
C100
=?
$100
$90
Strategy: Buy 1 stock sell 1.5 calls
$0
Prof. Avner Kalay - Options and Futures
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
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
$uS
$S
Cu
1/Δ – hedge ratio
$C
$dS
Cd
uS - (1/Δ)*Cu
S – (1/Δ)*C
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
Delta
$120
$20
=0
-
$90
$0
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
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In our example: u=1.2, d=0.9, R=1.1, uS=120,
ds=90, E = 100, S=100
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P =[R-d]/[u-d] = [1.1-0.9]/[1.2-0.9]=2/3
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C= {(2/3)*20 + (1/3)*0}/1.1 = 12.12
Prof. Avner Kalay - Options and Futures
What is P?
u>R>d
0<P<1
R=1.1
________________________________
d=0.9
u=1.2
Prof. Avner Kalay - Options and Futures
What is P?
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P cannot be a probability since we do not know
the probability of a price increase – denoted q.
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Since the valuation of C is true for any q we can
assume (for our example) q = 0.5
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Do you feel comfortable with q = 0.5?
Prof. Avner Kalay - Options and Futures
What is P?
-
-
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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?
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Assume q=7/8=0.875.
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In our example P=[1.1-0.9]/[1.2-0.9] = 2/3
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E(Rs) = 0.875*20% + 0.125*(10%) = 16.25%
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Risk premium = 16.25 – 10 = 6.25%
Prof. Avner Kalay - Options and Futures
What is P?
-
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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
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P gets closer to q
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C=5/6*20/1.15 = 14.493
Prof. Avner Kalay - Options and Futures
What is P?
-
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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
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P is now equal to q
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C = {7/8}*20/1.1625 = 15.054
Prof. Avner Kalay - Options and Futures
P – the risk neutral probability
P < q
Risk Aversion
P = q
Risk neutral
P > q
Risk seeking
Prof. Avner Kalay - Options and Futures
P – the risk neutral probability
$20
$20
$0
$0
0.875*20=17.5
0.666*20=13.333
17.5/1.1=15.909
13.333/1.1=12.12
Prof. Avner Kalay - Options and Futures
Certainty equivalent
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The difference 17.5 – 13.333 = 4.167 is a
correction for risk in the numerator
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The option model is valuation by certainty
equivalents.
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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
{0.666*44+0.333*8}/1.1
144
44
120
29.09
108
100
19.08
90
81
8
4.844
0
{0.666*29.09+0.333*4.844}/1.1
{0.666*8/1.1
Prof. Avner Kalay - Options and Futures
Two Periods
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Cu = {P*Cuu + (1-P)*Cud}/R
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Cd = {P*Cud + (1-P)*Cdd}/R
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C = {P*Cu + (1-P)*Cd}/R
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C = {P2 Cuu + 2P(1-P)Cud + (1-P)2 Cdd}/R2
Prof. Avner Kalay - Options and Futures
Four periods
1
u4
P4
1
du3
P3 (1-P)
4
d2u2
P2(1-P)2
6
d3u
(1-P)3 P
4
d4
(1-P)4
1
Prof. Avner Kalay - Options and Futures
The Binomial Distribution
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The probability of a path with j ups and n-j downs
is
Pj(1 – P)n-j
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The number of paths leading to a node is
n!/{j!(n-j)!}
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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
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The probability to get to any one of the nodes is
Σj=0 [{n!/j!(n-j)!}Pj(1-P)n-j] = 1
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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
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C = S*Φ{a, n, P’} - E*R-n*Φ*{a, n, P}
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Σj=0{[n!/(j!(n-j)!]Pj(1-P)n-j}= 1
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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
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C = S*Φ{a, n, P’} - E*R-n*Φ*{a, n, P}
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P = [R-d]/[u-d]
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P’ = [u/R]P