1. Lec 20 Home Made Portfolio Insurance
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 D ? 120
100 A 100 E ? A E
80 ?
C 60 F C F t=
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.
Lec 20 Home Made Portfolio Insurance
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2. Lec 20 Home Made Portfolio Insurance
“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/ T= t=1/ T=1
140 D D ?
100 E E
Δ(140) + B(1.05) = 140
Δ(100) + B(1.05) = 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
Lec 20 Home Made Portfolio Insurance
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3. Lec 20 Home Made Portfolio Insurance
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/ T= t=1/ T=1
100 E E ?
C F 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
Lec 20 Home Made Portfolio Insurance
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4. Lec 20 Home Made Portfolio Insurance
at A
Portfolio Prices Desired Portfolio
t=1/ T= t=1/ T=1
120 B B
A ?
80 C C
Δ(120) + B(1.05) = 120
Δ( 80 ) + B(1.05) = Sol’n: Δ= shares, B = $
Therefore, at A we must have
/share = $ (amount in stock)
$ in Bonds
Total amount to invest = = $
Lec 20 Home Made Portfolio Insurance
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5. Lec 20 Home Made Portfolio Insurance
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 $
Portfolio Price Tree Desired Portfolio
B 140 D B D
100 A 100 E E
C 60 F C F t=
Lec 20 Home Made Portfolio Insurance
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6. Lec 20 Home Made Portfolio Insurance
“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
Lec 20 Home Made Portfolio Insurance
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7. Lec 20 Home Made Portfolio Insurance
Favara
Lec 20 Home Made Portfolio Insurance
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