Topic 9
Portfolio Insurance
Basic Premise
•  We know that
C(S,X,t) = S – B(X,t) + P(S,X,t)
•  We assume that for a very short time
C(S,X,t) = ∂1S – ∂2B(X,t)
1
Here’s a Picture of ∂1
Call
Call
∆C/∆S
C2
C1
Stock
B
B(X,t)
(X,t)
S1
Stock
S2
Here’s a Picture of ∂2
Call
Call
∆C/∆X
C1
C2
BB(X,t)
(X1,t)
S
B (X2,t) Stock
Stock
2
Important Relationships
•  ∂1 and ∂2 are always less than 1
•  ∂2 is always less than ∂1
•  Exact values can be estimated using the
Black-Scholes OPM
•  Proportions must be continuously adjusted
Example
•  Suppose
∂1 = 0.7
∂2 = 0.6
S = $50
B(X,t) = $46
•  Then C(S,X,t) =
(.7*50) – (.6*46) =
$7.40
•  Then, sell 100-option
contract (receive $740)
  Sell 60 bonds (receive
another $2760, making total
$3500)
  Buy 70 shares stock (pay
$3500)
  Zero net investment
•  After a moment
  S = $50.125
  B(X,t) = $46.10
  C(S,X,t)=7.4275
•  Close position, net
zero
3
Example
•  Now, suppose calls selling on CBOE for
$7.75 per share
 Sell 100-share call contract for $775
 Create synthetic for $740
 Pocket profit of $35
•  How will adjustment process work?
Basic Premise for Portfolio
Insurance
•  From put-call parity, we know that
C(S,X,t) + B(X,t) = S + P(S,X,t)
•  Then let’s make a simple rearrangement
S + P(S,X,t) = C(S,X,t) + B(X,t)
•  We assume that for a very short time
C(S,X,t) = ∂1S – ∂2B(X,t)
•  Then for a short time
S + P(S,X,t) = B(X,t) + ∂C(S,X,t)
1S – ∂2B(X,t)
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
4
How does insurance work?
Call
Call
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
∂ close to 1
∂ declines
Less in stock
More in bonds
B (X,t)
B(X,t)
Stock
S2
Stock
S1
How does insurance work?
Call
Call
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
∂ closer to 1
Sell bonds
Buy more stock
∂ less than 1
B (X,t)
B(X,t)
Stock
S1
Stock
S2
5
How might this process lead to
runup?
Call
Call
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
∂ closer to 1
Sell bonds
Buy more stock
∂ less than 1
B (X,t)
B(X,t)
Stock
S1
Stock
S2
How might this process lead to
meltdown?
Call
Call
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
∂ close to 1
∂ declines
Less in stock
More in bonds
B (X,t)
B(X,t)
Stock
S2
Stock
S1
6