Lecture 20: “Home Made” Portfolio Insurance
Risk Management: Trade a stock/bond portfolio
as if it were a {+S, +P} hedged portfolio.
Binomial Methodology
Suppose we manage a well diversified stock portfolio and we expect
the price to evolve as follows (1-yr Time horizon):
Portfolio Price Tree
Desired Portfolio
B
140 D
B
140D
120
? 120
100 A
100 E
?A
100 E
80
? 95.238
C
60 F
C
100 F
t=0
1
2
0
1
2
There is a chance the price will fall to 60 (in one year).
Portfolio sponsor does not want to take that chance.
We have two choices:
a) Buy a put (over the counter) with K = 100.
Or, b) Create our own {Stock+Put} combination.
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“Home Made Portfolio Insurance”. Assume riskless rate 5%/6months
at B set up a Portfolio {Δ shares, $B in bonds}
to achieve the desired cash flows at D and E.
Portfolio Prices
Desired Portfolio
t=1/2
T=1
t=1/2
T=1
140 D
140 D
120
?
100 E
100 E
Δ(140) + B(1.05) = 140
Δ(100) + B(1.05) = 100 Sol’n:
Δ= 1, and B= $0
Therefore, at B we must be 100% in stock. No bonds,
Desired Portfolio Value = 1(120) - 0 = $120
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at C set up a Portfolio {Δ shares, $B in bonds}
to achieve the desired cash flows at D and E.
Portfolio Prices
Desired Portfolio
t=1/2
T=1
t=1/2
T=1
100 E
100 E
80
?
C
60 F
100 F
at C we want:
Δ(100) + B(1.05) = 100
Δ( 60 ) + B(1.05) = 100 Sol’n: Δ = 0, B = 100/1.05 = $95.238
(no stock, all bonds)
Therefore, at C we must be 100% in Bonds, No stock.
Desired Portfolio Value = $95.238
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at A
Portfolio Prices
Desired Portfolio
t=1/2
T=1
t=1/2
T=1
120 B
120 B
A 100
?
80 C
95.238 C
at A
Δ(120) + B(1.05) = 120
Δ( 80 ) + B(1.05) = 95.238 Sol’n: Δ= 0.61905 shares, B = $43.537.
Therefore, at A we must have
0.61905 shares @ 100/share = $61.905 (amount in stock)
$43.537 in Bonds
Total amount to invest = 61.905 + 43.537 = $105.442
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In sum
▸ Invest $100 in stock, then there is a chance of losing $40.
▸ Invest $105, then the portfolio value will not drop below 100.
▸ If the floor of $90, then you only need to invest $104.082
Portfolio Price Tree
B
140 D
120
100 A
100 E
80
C
60 F
t=0
1
2
Desired Portfolio
B
140D
120
105.44
100 E
95.238
C
100 F
0
1
2
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“Home Made Portfolio Insurance”, Black-Scholes Methodology
At any point in time t, the Put price is:
Pt = -St N(-d1 ) + K e-rT N(-d2 )
d1 = [ln(St /K)+(r+σ2/2)T ]/(σ√T) , and d2 = [ ln(St /K)+(r-σ2/2)T ]/(σ√T)
a {+S, +P} combination will be:
St + Pt = St [1 - N(-d1 )] + K e-rT N(-d2 )
= St Delta +Bonds
~=~S_t`~~~Delta_t ~+~~~~~ Bond_t
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a
Favara
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