9. Derivative /Vicing Models

9.1.2 Stochastic Differential Equations

Despite this fact, the infinitesimal increment of Brownian motion, i.e., the limit of B(t+h)-B(t) as h approaches an infinitesimal of time (dt), has earnerl.the notation dB{l) with its own unique interpretation because it has become a fundamental building block for constructing other continuous-time processes.7 Heuristically, B(t+h)-B(t) can be viewed as Gaussian white noise (sec Chapter 2) and in the limit as h becomes infinitcsimally small, dB{l) is the continuous-lime version of white noise.

It is understood that dB(t) is a special kind of differential, a stochastic differential, not to be confused with the differentials dx and dy of the calculus. Nevertheless, dB(t) docs obey some of the mechanical relations lhat ordinary differentials satisfy. For example, (9.1.10) is often expressed in differential form as:

dp(t) = iidl + odB(t). (9.1.17)

However, (9.1.17) cannot be treated as an ordinary differential equation, and is called a stochastic differential equation to emphasize this facl. For example, the natural transformation dp(t)/dt = Д + dB(t)/dt does not make sense because dB(t)/dl is not a well-defined mathematical object (although <7й(0 is, by definition).

Indeed, thus far the symbols in (9.1.17) have little formal content beyond their relation lo (9.1.10)-one set of symbols has been defined lo be equivalent to another. To give (9.1.17) independent meaning, we must develop a more complete understanding of the properties of the stochastic differential dB(t). For example, since dB is a random variable, what are its moments? How do (dB)1 and (dB){dt) behave? To answer these questions, consider the definition of dB(i):

dll{t) = lint B(l 4- A)

h->dt

B(t)

(9.1.18)

and recall from (Bl) that increments of Brownian motion arc normally distributed with zero mean (since Ц - 0) and variance equal to ihe differencing interval A (since a - 1). Therefore, wc have

V.[dli] = Var[r/ J =

V.[(dll)(dll)} =

lim E[B(l + h) - B(l)} = 0 it-* tit

lim К[(й(/ + A) - B(l))2] = dt it-* iii

lim Y.\(H(t + A) - (/)) ]

и-*,11

(9.1.19) (9.1.20)

(9.1.21)

Л complete and rigorous exposition ol litownian motion and slocliaslir tlilleienlial cqua-jtions is beyond the scope ol this text, lloel. fori, and Stone (1972, Chapters t-(i). Merlon (1990, Chapter :i), and Schttss (19K0) provide excellent coverage ol this malerial.

9.1. lirownian Motion

V;\r\{dll)(dlS)\ = lim I F. ( (,-)-A)-/ /))I-A

= <>(.< > (9.1.22)

Y.[{dll)(dt)} = lim К[(й(, + Л) - Й(/))Л] = 0 (9.1.23)

Л- til

V;\r[(dll)(dl)] = VunV.[(ll(t + h) - IS(l))2lr] = ((/0.(9.1.24)

Л-* til

From (Bl) and (9.1.19)-(9.1.20) and we see that dlt(t) may be viewed as a normally distributed random variable with zero mean and infinitesimal variance dt. Although a variance of dt may seem like no variance at all, recall that we arc in a world of infinitesimals-after all, according to (9.1.17) the expected value оГ dpI) is full-so a variance of, is not negligible in a relative sense.

However, a variance of (fit)2 и negligible in a relative sense-relative to dt-since the square of an infinitesimal is much smaller than the infinitesimal itself. If we Heal terms of order o(dl) as essentially zero, then (9.1.21)-(9.1.24) shows that (dB)2 and (dB)(dl) are both non-stochastic (since the variances of the right-hand sides are olorder o(dl)) hence the relations (dB)~ = dt and (dB)(dt) = 0 are satisfied not just in expectation but exactly. This yields the well-known multiplication rules for stochastic differentials summarized in fable 9.1. To see why these rules are useful, observe that wc

Table 9.1. Multiplication rules /or stochastic ili rrriilinl.\.

can now calculate (dp)2:

, (dp)2 = (lull + odlS)2 (9.1.25)

= n2(dt)2 + a2(dl)) + 2iia(,IH)(dt) - °2dl. (9.1.26)

This simple calculation shows that although dp is a random variable, (dp)2 is not. It also shows that ( ; does behave like a random walk inclement in that the variance of dp is proportional lo ihe dilfcrcnring interval ( .

Geometric limwnian Motion

If the arithmetic Brownian motion p(t) is taken to be the price of some asset, Property (111) shows that price changes over any interval will be normally distributed. But since the support of the normal distribution is the

у. Derivative Pricing Models

entire real line, normally distributed price changes imply that prices can be negative with positive probability. Because virtually all financial assets enjoy limited liability-the maximum loss is capped al -100% of the total investment-negative prices are empirically implausible.

As in Sections i .4.2 of Chapter I and 2.1.1 of Chapter 2, we may eliminate this problem by defining /;(/) lobe the natural logarithm of price P(l). Under this definition, j>(l) can be an arithmetic Brownian motion without violating limited liability, since the price /(/) = en is always non-negative. The price process /(/) = с1 is said lobe a geometric lirownian motion or lognormal diffusion. Wc shall examine the statistical properties of both arithmetic and geometric Brownian motion in considerably more detail in Section 0.3.

lids Lemma

Although the first complete mathematical theory of Brownian motion is due to Wiener (1923),* it is the seminal contribution oflto (19Г>1) that is largely responsible for the enormous number of applications of Brownian motion to problems in mathematics, statistics, physics, chemistry, biology, engineering, and of course, linancial economics. In particular, ho constructs a broad class olConiinuous-iime stochastic processes based on Brownian motion- now known as ltd processes or ltd stochastic differentia! equations-which is closed under general nonlinear transformations; that is, an arbitrary nonlinear function fip. I) of an Ilo process p and time / is itself an lto process.

More importantly, lto (1951) provides a formula-Itds Ijemma-for calculating explicitly ihe stochastic differential equation thai governs the dynamics of/(/>, t):

il/ i)f , a- / /</> I) = -/+ ~ + 5Г7<Ф> (9.1.27)

op ot - dp-

lhe modest term lemma hardly does justice lo the wide-ranging impact (9.1.27) has had; this powerful tool allows us lo quantify the evolution of complex stochastic .systems in a single step. For example, lei p denote the log-price process ol an asset and suppose ii satisfies (9.1.17); wbai are the dynamics of the price process /(/) = el,u)} llos Lemma provides us with an immediate answer:

HP , ;i-/>

= -rr/+,-7 (ДО (9.1.28)

op (l/i-

KScc Jctison. Singri. .nut Snoot к (II.Hi) lor an excellent liisiortY.il retrospective ol Wieners research which includes several articles about Wieners iiilliienceiiu nioclern financial economics.

У.2. Л Urief Review of Derivative Piicaig Methods

= еЫр+ \tf(dp)2 = P(p.dt + adU) Л- \P(aldl) dP = (pL-r\al)Pdt + aPdR.

In contrast to arithmetic Brownian motion (9.1.17), we see from (9.1.29) that the instantaneous mean and standard deviation of the geometric Brownian motion are proportional to P. Alternatively (9.1.29) implies thai the instantaneous percentage price change dP/P behaves like an arithmetic Brownian motion or random walk, which of course is precisely the case given the exponential transformation.

We provide a considerably less trivial example of the power of Itos Lemma in Section 9.2; Mertons derivation of the Black-Scholes formula.

9.2 A Brief Review of Derivative Pricing Methods

Although we assume that readers are already familiar with the theoretical aspects of pricing options and other derivative securities, we shall provide a very brief review here, primarily to develop terminology and define notation. Our derivation is deliberately terse and we urge readers unfamiliar with ] these models to spend some time with Mertons (1990, Chapter 8) definitive \ treatment of the subject. Also, for expositional economy we shall confine our attention to plain vanilla options in this chapter, i.e., simple call and put options with no special features, and the underlying asset is assumed to be common stock.9

Denote by G(P(t), I) the price at time / of a European call option with strike price X and expiration date Г > I on a stock with price P(t) at time Л1 Of course, G also depends on other quantities such as the maturity date 7, the strike price X, and other parameters but we shall usually suppress these arguments except when we wish to focus on them specifically.

However, expressing G as a function of the current stock price P(t), and not of past prices, is an important restriction that greatly simplifies the task of finding G (in Section 9.4 we shall consider options that do not satisfy

However, the techniques reviewed in this section have been applied in similar fashion to literally hundreds of other types of derivative securities, hence they are considerably more general than they may appear to be.

Recall thai a call option gives the holder the right to purchase the underlying asset for .Y and a put option gives the holder ilie right to sell the underlying asset for X, A European option is one that can be exercised only on the maturity date. An American option is one that can be exercised on or licfore the maturity dale. For simplicity we shall deal only with European options in this chapter. See Cox and Rubinstein (ШГ>), Hull (1993), and Merton (1990) for institutional details and differences between the pricing of American and European options.

(9.1.29) j

this restriction). In addition. Шаек and Scholes (1073) make the following assumptions:

{Al) There are no market imperfections, e.g., taxes, transactions costs, shorlsales constraints, and trading is continuous andfriclionless.

(A2) There is unlimited riskless bonowing and lending at the continuously compounded rale of return r; hence a $1 investment in such an asset over the time intervalx grows to%\ e . Alternatively, ifl){t) is the dale I price of a discount bond maturing at date T with face value $1, then for t e [0, 7] the bond price dynamics are given by

< )(() = rl)U)dl. (0.2.1)

\\(AJ) The stock price dynamics are given by a geometric llrownian motion, Ihe solution Vo the following Ho stochastic differential equation on /6 [0. 7] ;

dl(t) = fil(t)dl + alit)dB(t).

/40)

1\ > 0,

(9.2.2)

where li(t) is a standard Brownian motion, and at least one investor observes a without error.

(A4) There is no arbitrage.

9.2.1 The Black-Schbles and Merlon Approach

he goal is to narrow down the possible expressions for G, with the hope of obtaining a specific formula for it. Black and Scholes (1973) and Merlon

1973b) accomplish this by considering relations among the dynamics of the option price G, ihe slock price P, and die riskless bond D. To do this, we first derive the dynamics of the option price by assuming thai G is only a function of the current slock price / and (itself and applying llos l.emma (jScc Section 9.1.2 and Mcrton [ 1990, Chapter 3)) lo the function G{P(l), (), ydtich yields ihe dynamics of the option price:

where

dG = HgGdt + aGdlHl),

14 =

a. =

i)G dG a2!2 i)2G

W> + Tt+~W2

(9.2.3)

(9.2.4) (9.2.5)

Unfortunately, this expression does not seeinlO provide any obvious restrictions that miglu allow us to narrow down the choices for G. One possibility

is to set ;<A, equal lo some required rate ol return r, one that comes from equilibrium considerations of the corresponding risk of the option, much like the CAIM (see Chapter 5). II such an r caii be identified, the condition цк = r reduces to a 14)К which, under some regularity and boundary conditions, possesses a unique solution. This is the approach taken by Black and Scholes (1973).1 I lowever, this approach requires more .structure than we have imposed in (A 1 )-(A4)-a fully articulated dynamic model of economic equilibrium in which r can be determined explicitly must be specified.

Merlon (1973b) presents an alternative to the equilibrium approach ol Black and Scholes (1973) in which the same option-pricing formula is obtained but without any additional assumptions beyond (AI)-(A4). lie docs this by constructing a portfolio of stocks, options and riskless bonds that requires no initial net investment and no additional funds between 0 and T, a self/inanring portfolio where long positions are completely financed by short positions. Specifically, denote by ,(() the dollar amount invested in the stock al date /, /,/(() die dollar amount invested in riskless bonds at dale ( which mature at date T, and lK(l) the dollar amount invested in the call option at date (. Then the zero net investment condition may be expressed as:

/ ) +Ш 4- IgU) = 0. V/ e [0, 7j. (9.2.0)

Portfolios satisfying (9.2.0) are called arbitrage portfolios. Mcrton (1909) shows that the instantaneous dollar return dl lo this arbitrage portfolio is:

If Li ft

dl = -L dP + dl) + ~ dG. (9.2.7)

where the stochastic differentials dl and dl) are given in (9.2.2) and (9.2.1) respectively, and dG follows from llos l.emma:

dG = HuGdl + crgGdlt (9.2.8)

нРЯСГдР + ас/т + ii-G/;)/2

: / , < -;.-:- (9.2.9)

aPdG/dP G

(9.2.10)

Substituting the dynamics of /4), /*(), and (.(() into (9.2.7) and imposing (9.2.0) yields

dl = [in - >)!/, + (/< - г)/ ], + [a lt, + ctK ldlHl). (9.2.11)

11 In particular. IM.uk .ukI Scholes assume thai the (ЛГМ holds, and obtain ; by

appealing to the securiiy-markei-line relation winch links expected returns to beta. However, the CAIM is not a dynamic model of equilibrium returns and there are some subtle but significant inconsistencies between the (!Л1М and eoiuiiiuotis-iime opiion-pi it ing models (see. for example, Dybvij and Inersoll l)82j). Nevertheless, Rubinstein (!n7(i) provides л dynamic equilibrium model in which the lllack-SchoJcs formula holds.