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where Z(N) satisfies the condition

Z(1) Â¼ 1 (9:6:15)

Then it follows from equation (9.6.11) that

@F0

M Â¼ Ã€N= Z (9:6:16)

:

@S

This transforms equation (9.6.15) and equation (9.6.16), respectively,

to

@F0 s2 2 @ 2 F0

@F0

Ã¾ rS Ã¾S Ã€ rF0 Â¼ 0 (9:6:17)

@S 2

2

@t @S

dZ @F0

Â¼ (S=F0 ) (Z=N), (9:6:18)

dN @S

108 Option Pricing

Equation (9.6.17) is the classical Black-Scholes equation (cf. with

(9.4.6)) while equations (9.6.16) and (9.6.18) define the values of M

and Z(N). Solution to equation (9.6.18) that satisfies the condition

(9.6.15) is

Z(N) Â¼ Na (9:6:19)

@F0

Â¼ Ã€MÃ€1 . Equation (9.6.13) and equa-

where a Â¼ (S=F0 )D, D Â¼ 0

@S

tion (9.6.16) yield

M Â¼ Ã€N1Ã€a =D Â¼ N1Ã€a M0 (9:6:20)

Hence, the option price in the arbitrage-free portfolio with N shares

equals

F(S, t, N) Â¼ F0 (S, t)Na (9:6:21)

It coincides with the BST solution F0 (S, t) only if N Â¼ 1, that is when

the portfolio has one share. However, the total expense of hedging N

shares in the arbitrage-free portfolio

Q Â¼ MF Â¼ Ã€(N=D)F0 Â¼ NM0 F0 (9:6:22)

is the same as within BST. Therefore, Q is the true invariant of the

arbitrage-free portfolio.

Invariance of the hedging expense is easy to understand using the

dimensionality analysis. Indeed, the arbitrage-free condition (9.6.9) is

given in units of the portfolio and therefore can only be used for

defining part of the portfolio. Namely, the arbitrage-free condition

can be used for defining the hedging expense Q Â¼ MF but not for

defining both factors M and F. Similarly, the law of energy conser-

vation can be used for defining the kinetic energy of a body,

K Â¼ 0:5mV2 . Yet, this law alone cannot be used for calculating the

bodyâ€™s mass, m, and velocity, V. Note, however, that if a body has

unit mass (m Â¼ 1), then the energy conservation law effectively yields

the bodyâ€™s velocity. Similarly, the arbitrage-free portfolio with one

share does not reveal dependence of the option price on the number of

shares in the portfolio.

109

Option Pricing

9.7 EXERCISES

1. (a) Calculate the Black-Scholes prices of the European call and

put options with six-month maturity if the current stock

price is $20 and grows with average rate of m Â¼ 10%, vola-

tility is 20%, and risk-free interest rate is 5%. The strike price

is: (1) $18; (2) $22.

(b) How will the results above change if m Â¼ 5%?

2. Is there an arbitrage opportunity with the following assets: the

price of the XYZ stock with no dividends is $100; the European

put options at $98 with six-month maturity are sold for $3.50;

the European call options at $98 with the same maturity are sold

for $8; T-bills with the same maturity are sold for $98. Hint:

Check the put-call parity.

**3. Compare the Itoâ€™s and Stratonovichâ€™s approaches for derivation

of the Black-Scholes equation (consult [12]).

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Chapter 10

Portfolio Management

This chapter begins with the basic ideas of portfolio selection.

Namely, in Section 10.1, the combination of two risky assets and

the combination of a risky asset and a risk-free asset are considered.

Then two major portfolio management theories are discussed: the

capital asset pricing model (Section 10.2) and the arbitrage pricing

theory (Section 10.3). Finally, several investment strategies based on

exploring market arbitrage opportunities are introduced in Section

10.4.

10.1 PORTFOLIO SELECTION

Optimal investing is an important real-life problem that has been

translated into elegant mathematical theories. In general, opportun-

ities for investing include different assets: equities (stocks), bonds,

foreign currency, real estate, antique, and others. Here portfolios

that contain only financial assets are considered.

There is no single strategy for portfolio selection, because there is

always a trade-off between expected return on portfolio and risk of

portfolio losses. Risk-free assets such as the U.S. Treasury bills guar-

antee some return, but it is generally believed that stocks provide

higher returns in the long run. The trouble is that the notion of â€˜â€˜long

runâ€™â€™ is doomed to bear an element of uncertainty. Alas, a decade of

111

112 Portfolio Management

market growth may end up with a market crash that evaporates a

significant part of the equity wealth of an entire generation. Hence,

risk aversion (that is often well correlated with investor age) is an

important factor in investment strategy.

Portfolio selection has two major steps [1]. First, it is the selection

of a combination of risky and risk-free assets and, secondly, it is the

selection of risky assets. Let us start with the first step.

For simplicity, consider a combination of one risky asset and one

risk-free asset. If the portion of the risky asset in the portfolio is

a(a 1), then the expected rate of return equals

E[R] Â¼ aE[Rr ] Ã¾ (1 Ã€ a)Rf Â¼ Rf Ã¾ a(E[Rr ] Ã€ Rf ) (10:1:1)

where Rf and Rr are rates of returns of the risk-free and risky assets,

respectively. In the classical portfolio management theory, risk is

characterized with the portfolio standard deviation, s.1 Since no

risk is associated with the risk-free asset, the portfolio risk in our

case equals

s Â¼ asr (10:1:2)

Substituting a from (10.1.2) into (10.1.1) yields

E[R] Â¼ Rf Ã¾ s(E[Rr ] Ã€ Rf )=sr (10:1:3)

The dependence of the expected return on the standard deviation is

called the risk-return trade-off line. The slope of the straight line

(10.1.3)

s Â¼ (E[Rr ] Ã€ Rf )=sr (10:1:4)

is the measure of return in excess of the risk-free return per unit of

risk. Obviously, investing in a risky asset makes sense only if s > 0,

that is, E[Rr ] > Rf . The risk-return trade-off line defines the mean-

variance efficient portfolio, that is, the portfolio with the highest

expected return at a given risk level.

On the second step of portfolio selection, let us consider the port-

folio consisting of two risky assets with returns R1 and R2 and with

standard deviations s1 and s2 , respectively. If the proportion of the

risky asset 1 in the portfolio is g(g 1), then the portfolio rate of

return equals

E[R] Â¼ gE[R1 ] Ã¾ (1 Ã€ g)E[R2 ] (10:1:5)

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