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Adding options to the portfolio can change its Gamma. In particular,

delta-neutral portfolio with Gamma G can be made gamma-neutral if

it is supplemented with n Â¼ Ã€G=GF options having Gamma GF .

Theta characterizes the time decay of the portfolio price. In add-

ition, two other Greeks, Vega and Rho, are used to measure the

portfolio sensitivity to its volatility and risk-free rate, respectively

@P @P

yÂ¼ ,rÂ¼ (9:4:13)

@s @r

Several assumptions that are made in BST can be easily relaxed. In

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

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