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Document Date: November 2, 2006

An Introduction To Derivatives And Risk Management, 7th

Edition

Don Chance and Robert Brooks

Technical Note: Derivation of the Dynamic Hedge Ratio for Portfolio Insurance,

Ch. 14, p. 485

This technical note supports the material in the Portfolio Insurance section of

Chapter 14 Advanced Derivatives and Strategies. The objective here is to derive the

stock-futures dynamic hedge and to derive the stock-T-bill dynamic hedge.

Stock-Futures Dynamic Hedge

The stock-put portfolio of N shares and N puts initially is worth

V = N(S + P).

By this definition, N must equal V/(S + P). The change in the portfolios value for a smallstock price change is given by the derivative of V with respect to S,

+

+=

+=

S

P1

PS

V

S

P1N

S

V.

This is the delta of the stock-put portfolio.

We assume the put is not available, so we shall match its delta to that of an actual

portfolio of NS shares of stock and Nffutures contracts. The value of the portfolio is

V = NSS + NfVf,

where Vfis the value of the futures contract. Remember that the initial value of a futures

contract is zero, so Vf

= 0. The number of shares then will be NS = V/S The delta of the

stock-futures portfolio is the derivative of V with respect to S,

+=

S

fNN

S

VfS .

Note that we must include Nf(f/S) because Vf/S = f/S. Assuming no dividends, the

futures price is

TrceSf=

Thus,

TrceS

f=

.

We can substitute V/S and NS and forf/S, givingTrce

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Trf

ceNS

V

S

V+

=

.

The objective is to equate the delta of the stock-futures portfolio to that of the

stock-put portfolio. Thus, we should set these two derivatives equal to each other:

Trf

ceNS

V

S

P1

PS

V+

=

+

+

.

Then we solve for Nf:

Tr-f

ceS

V

S

P1

PS

VN

+

+= .

This formula might look somewhat simpler if we recognize that V/(S + P) is simply

Vmin/X and that 1 + P/S =C/S Thus,

Tr-minfce

SV

SC

XVN

= .

Of course, C/S = N(d1) from the Black-Scholes-Merton model. Thus, if we sell Nf

futures, the stock-futures portfolio has the same delta as the stock-put portfolio.

StockRisk-Free Bond Dynamic Hedge

In the preceding section, we derived the delta of a portfolio of N shares of stock

and N puts. This value was shown to be

=

+

+=

S

C

X

V

S

P1

PS

V

S

V min .

A portfolio of stock and risk-free bond is worth

V = NSS + NBB.

Its delta is

SNS

V=

.

Note that the bond price does not change with a change in S. Setting this delta to the delta

of the stock-put portfolio and solving for NS gives

=

S

C

X

VN minS ,

IDRM7e, Don M. Chance and Robert-Brooks More on Interest Rate Parity2

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IDRM7e, Don M. Chance and Robert-Brooks More on Interest Rate Parity3

and C/S = N(d1). Thus, if we hold NS shares of stock and NB bond according to these

formulas, the delta of the stockbond portfolio will equal the delta of the stock-put

portfolio.