ñòð. 12 |

particular, dividends can be accounted. Also, r and s can be treated as

time-dependent parameters. BST has been expanded in several ways

(see [1â€“3, 7, 8] and references therein). One of the main directions

addresses so-called volatility smile. The problem is that if all charac-

teristics of the European option besides the strike price are fixed, its

implied volatility derived from the Black-Scholes expression is con-

stant. However, real market price volatilities do depend on the strike

price, which manifests in â€˜â€˜smile-likeâ€™â€™ graphs. Several approaches

have been developed to address this problem. One of them is introdu-

cing the time dependencies into the interest rates or/and volatilities

105

Option Pricing

(so-called term structure). In a different approach, the lognormal

stock price distribution is substituted with another statistical distri-

bution. Also, the jump-diffusion stochastic processes are sometimes

used instead of the geometric Brownian motion.

Other directions for expanding BST address the market imperfec-

tions, such as transaction costs and finite liquidity. Finally, the option

price in the current option pricing theory depends on time and price

of the underlying asset. This seemingly trivial assumption was ques-

tioned in [9]. Namely, it was shown that the option price might

depend also on the number of shares of the underlying asset in the

arbitrage-free portfolio. Discussion of this paradox is given in the

Appendix section of this chapter.

9.5 REFERENCES FOR FURTHER READING

Hullâ€™s book is the classical reference for the first reading on finan-

cial derivatives [1]. A good introduction to mathematics behind the

option theory can be found in [4]. Detailed presentation of the option

theory, including exotic options and extensions to BST, is given in

[2, 3].

9.6 APPENDIX: THE INVARIANT

OF THE ARBITRAGE-FREE PORTFOLIO

As we discussed in Section 9.4, the option price F(S, t) in BST is a

function of the stock price and time. The arbitrage-free portfolio in

BST consists of one share and of a number of options (M0 ) that hedge

this share [5]. BST can also be derived with the arbitrage-free port-

folio consisting of one option and of a number of shares MÃ€1 (see,

0

e.g., [1]). However, if the portfolio with an arbitrary number of shares

N is considered, and N is treated as an independent variable, that is,

F Â¼ F(S, t, N) (9:6:1)

then a non-zero derivative, @F=@N, can be recovered within the

arbitrage-free paradigm [9]. Since options are traded independently

from their underlying assets, the relation (9.6.1) may look senseless to

the practitioner. How could this dependence ever come to mind?

106 Option Pricing

Recall the notion of liquidity discussed in Section 2.1. If a market

order exceeds supply of an asset at current â€˜â€˜bestâ€™â€™ price, then the

order is executed within a price range rather than at a single price. In

this case within continuous presentation,

S Â¼ S(t, N) (9:6:2)

and the expense of buying N shares at time t equals

Ã°

N

S(t, x)dx (9:6:3)

0

The liquidity effect in pricing derivatives has been addressed in [10,

11] without proposing (9.6.1). Yet, simply for mathematical general-

ity, one could assume that (9.6.1) may hold if (9.6.2) is valid. Surpris-

ingly, the dependence (9.6.1) holds even for infinite liquidity. Indeed,

consider the arbitrage-free portfolio P with an arbitrary number of

shares N at price S and M options at price F:

P(S, t, N) Â¼ NS(t) Ã¾ MF(S, t, N) (9:6:4)

Let us assume that N is an independent variable and M is a parameter

to be defined from the arbitrage-free condition, similar to M0 in BST.

As in BST, the asset price S Â¼ S(t) is described with the geometric

Brownian process

dS Â¼ mSdt Ã¾ sSdW: (9:6:5)

In (9.6.5), m and s are the price drift and volatility, and W is the

standard Wiener process. According to the Itoâ€™s Lemma,

s2 2 @ 2 F

@F @F @F

dF Â¼ dt Ã¾ dS Ã¾ S dt Ã¾ dN (9:6:6)

@S2

2

@t @S @N

It follows from (9.6.4) that the portfolio dynamic is

dP Â¼ MdF Ã¾ NdS Ã¾ SdN (9:6:7)

Substituting equation (9.6.6) into equation (9.6.7) yields

@F s2 2 @ 2 F

@F @F

dP Â¼ [M Ã¾ N]dS Ã¾ [M Ã¾ S]dN Ã¾ M Ã¾S dt

@S2

2

@S @N @t

(9:6:8)

107

Option Pricing

As within BST, the arbitrage-free portfolio grows with the risk-free

interest rate, r

dP Â¼ rPdt (9:6:9)

Then the combination of equation (9.6.8) and equation (9.6.9)

yields

@F s2 2

2@ F

@F

Ã¾ N]dS Ã¾ [M Ã¾ MS Ã€ rMF Ã€ rNS]dtÃ¾

[M

@S2

2

@S @t (9:6:10)

@F

Ã¾ S]dN Â¼ 0

[M

@N

Since equation (9.6.10) must be valid for arbitrary values of dS, dt

and dN, it can be split into three equations

@F

Ã¾NÂ¼0

M (9:6:11)

@S

@F s2 2 @ 2 F

Ã¾S Ã€ rF Ã€ rNS Â¼ 0

M (9:6:12)

@S2

2

@t

@F

Ã¾SÂ¼0

M (9:6:13)

@N

Let us present F(S, t, N) in the form

F(S, t, N) Â¼ F0 (S, t)Z(N) (9:6:14)

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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Portfolio Management

and the portfolio standard deviation is

s2 Â¼ g2 s1 2 Ã¾ (1 Ã€ g)2 s2 2 Ã¾ 2g(1 Ã€ g)s12 (10:1:6)

In (10.1.6), s12 is the covariance between the returns of asset 1 and

asset 2. For simplicity, it is assumed further that the asset returns are

uncorrelated, that is, s12 Â¼ 0. The value of g that yields minimal risk

for this portfolio equals

gm Â¼ s2 2 =(s1 2 Ã¾ s2 2 ), (10:1:7)

This value yields the minimal portfolio risk sm

sm 2 Â¼ s1 2 s2 2 =(s1 2 Ã¾ s2 2 ) (10:1:8)

Consider an example with E[R1 ] Â¼ 0:1, E[R2 ] Â¼ 0:2, s1 Â¼ 0:15,

s2 Â¼ 0:3. If g Â¼ 0:8, then s % 0:134 < s1 and E[R] Â¼ 0:12 > E[R1 ].

Hence, adding the more risky asset 2 to asset 1 decreases the portfolio

risk and increases the portfolio return. This somewhat surprising

outcome demonstrates the advantage of portfolio diversification.

Finally, let us combine the risk-free asset with a portfolio that

contains two risky assets. The optimal combination of the risky

asset portfolio and the risk-free asset can be found at the tangency

point between the straight risk-return trade-off line with the intercept

E[R] Â¼ Rf and the risk-return trade-off curve for the risky asset

portfolio (see Figure 10.1). For the portfolio with two risky uncorrel-

ated assets, the proportion g at the tangency point T equals

gT Â¼ (E[R1 ] Ã€ Rf )s2 2 ={(E[R1 ] Ã€ Rf )s2 2 Ã¾ (E[R2 ] Ã€ Rf )s1 2 }

(10:1:9)

Substituting gT from (10.1.9) into (10.1.5) and (10.1.6) yields the

coordinates of the tangency point (i.e., E[RT ] and sT ). A similar

approach can be used in the general case with an arbitrary number

of risky assets. The return E[RT ] for a given portfolio with risk sT is

â€˜â€˜as good as it gets.â€™â€™ Is it possible to have returns higher than E[RT ]

while investing in the same portfolio? In other words, is it possible to

reach say point P on the risk-return trade-off line depicted in Figure.

10.1? Yes, if you borrow money at rate Rf and invest it in the portfolio

with g Â¼ gT . Obviously, the investment risk is then higher than that of

sT .

114 Portfolio Management

0.16

E[R]

P

T

0.12

0.08

RF

0.04

sigma

0

0 0.04 0.08 0.12 0.16

Figure 10.1 The return-risk trade-off lines: portfolio with the risk-free

asset and a risky asset (dashed line); portfolio with two risky assets (solid

line); Rf Â¼ 0:05, s1 Â¼ 0:12, s2 Â¼ 0:15, E[R1 ] Â¼ 0:08, E[R2 ] Â¼ 0:14.

10.2 CAPITAL ASSET PRICING MODEL (CAPM)

The Capital Asset Pricing Model (CAPM) is based on the portfolio

selection approach outlined in the previous section. Let us consider

the entire universe of risky assets with all possible returns and risks.

The set of optimal portfolios in this universe (i.e., portfolios with

maximal returns for given risks) forms what is called a efficient

frontier. The straight line that is tangent to the efficient frontier and

has intercept Rf is called the capital market line.2 The tangency point

between the capital market line and the efficient frontier corresponds

to the so-called super-efficient portfolio.

In CAPM, it is assumed that all investors have homogenous expect-

ations of returns, risks, and correlations among the risky assets. It is

also assumed that investors behave rationally, meaning they all hold

optimal mean-variance efficient portfolios. This implies that all invest-

ors have risky assets in their portfolio in the same proportions as the

entire market. Hence, CAPM promotes passive investing in the index

115

Portfolio Management

mutual funds. Within CAPM, the optimal investing strategy is simply

choosing a portfolio on the capital market line with acceptable risk

level. Therefore, the difference among rational investors is determined

only by their risk aversion, which is characterized with the proportion

of their wealth allocated to the risk-free assets. Within the CAPM

assumptions, it can be shown that the super-efficient portfolio consists

of all risky assets weighed with their market values. Such a portfolio is

called a market portfolio.3

CAPM defines the return of a risky asset i with the security market

line

E[Ri ] Â¼ Rf Ã¾ bi (E[RM ] Ã€ Rf ) (10:2:1)

where RM is the market portfolio return and parameter beta bi equals

bi Â¼ Cov[Ri , RM ]=Var[RM ] (10:2:2)

Beta defines sensitivity of the risky asset i to the market dynamics.

Namely, bi > 1 means that the asset is more volatile than the entire

market while bi < 1 implies that the asset has a lower sensitivity to the

market movements. The excess return of asset i per unit of risk (so-

called Sharpe ratio) is another criterion widely used for estimation of

investment performance

Si Â¼ (E[Ri ] Ã€ Rf )=si (10:2:3)

CAPM, being the equilibrium model, has no time dependence. How-

ever, econometric analysis based on this model can be conducted

providing that the statistical nature of returns is known [2]. It is

often assumed that returns are independently and identically distrib-

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