Mathematics of Financial Markets - Imperial College …mdavis/docs/math2001.pdfMathematics of Financial Markets ... Mathematical ﬁnance is a child of the 20th century. ... provided

Mathematics of Financial MarketsMark Davis∗

1. Background

Mathematical finance is a child of the 20th century. It was born on 29 March1900 with the presentation of Louis Bachelier’s doctoral dissertation Theoriede la speculation [1]. Now, one hundred years later, it is the basis of a hugeindustry, at the centre of modern global economic development, and the sourceof a great deal of interesting mathematics. Further, the theory and applicationshave proceeded in parallel in an unusually closely-linked way. This article aimsto give the flavour of the mathematics, to describe how the confluence of math-ematical ideas, economic theory and computer technology proved so effective,and to indicate how the theory relates to the practice of ‘financial engineering’.

Bachelier’s extraordinary thesis was years, and in some respects decades,ahead of its time. For example it introduces Brownian motion as a model forstock prices five years before Einstein’s classic paper [10] on that subject. Brow-nian motion is a continuous-path stochastic process (B(t), t ≥ 0) such that(a) B(0) = 0, (b) the increments (B(t4)−B(t3)), (B(t2)−B(t1)) are indepen-dent for t1 ≤ t2 ≤ t3 ≤ t4 and (c) the increment (B(t2) − B(t1)) is normallydistributed with mean zero and variance t2 − t1. It is simultaneously a Markovprocess and a martingale, though neither of those concepts had been named orclearly formulated in 1900. A martingale M(t) is the mathematical representa-tion of a player’s fortune in a fair game. The defining property is that, for t > s,E[M(t)|Fs] = M(s), where E[ . |Fs] represents the conditional expectationgiven Fs , the ‘information up to time s’: the expected fortune at some later timet is equal to the current fortuneM(s). Bachelier arrived at this by economic rea-soning. Arguing that stock markets have symmetry in that every trade involvesa buyer and a seller, and that there cannot be any consistent bias in favour ofone or the other, he formulated his famous dictum

‘L’esperance mathematique du speculateur est nulle’.

This is tantamount to the martingale property. Assuming that the price pro-cessB(t) is Markovian, Bachelier introduced the transition density p(x, t; y, s)defining the probabilities of moving from state y to state x:

P [B(t) ∈ [x, x + dx]|B(s) = y] = p(x, t; y, s) dx.

∗ Work supported by the Austrian Science Foundation (FWF) under grant Wittgenstein-Prize Z36-MAT.

page: 1 Engquist/Schmid (eds.): Mathematics Unlimited – 2001 and Beyond author: math2001_davis_final date: 3-Sep-2000 9:04

2 M. Davis

He assumed that this would be temporally and spatially homogeneous, i.e. thatp(x, t; y, s) = q(x − y, t − s) for some function q. By arguing that, startingat B(0) = 0, the transition probability should satisfy P [B(t + s) ∈ dx] =∫P [B(t + s) ∈ dx|B(s) = y]P [B(s) = y] dy he obtained what is now known

as the Chapman-Kolmogorov equation

q(x, s + t) =∫ ∞

−∞q(x − y, t)q(y, s) dy. (1.1)

He then showed that (1.1) is satisfied by the Brownian transition function

q(x, t) = 1√2πt

exp

(−x

2

2t

)(1.2)

(not worrying about uniqueness), and went on to show that this transition func-tion solves the heat equation

∂q

∂t= 1

2

∂2q

∂x2. (1.3)

Bachelier’s main objective was to study the valuation of options. A calloption on a stock with exercise time T and strike K is the right, but not theobligation, to buy a certain number of shares at time T for the fixed price K .If S(t) denotes the market price of this number of shares then the value of theoption at time T is S(T )−K if S(T ) > K , since the shares can be bought forK and immediately sold in the market for S(T ). On the other hand the optionis worthless, and will not be exercised, if S(T ) ≤ K . Briefly put, the value is[S(T )−K]+ = max(S(T )−K, 0), as shown in Figure 1 (above). A put optionis the right to sell, which has exercise value [K−S(T )]+. An option is Europeanif, as described above, it is exercised on a single date T , and American if it canbe exercised at any time at or before T . We do not discuss American options inthis article.

KS

KS0

(K– p)

–p

figure 1Above: Call option exercise value.Below: Call option exercise value,French style.

The valuation problem is to determine a ‘fair’ price or premium for acquiringthis right at an earlier time t < T . Nowadays option premia are paid up front,i.e. the right is acquired when the premium is paid, but in Bachelier’s day, atleast in France, the premium was paid at the exercise time of the option, andonly if the option is not exercised. This means that it is optimal to exercisewhen S(T ) > K −p, where p is the premium, giving the payoff profile shownin Figure 1 (below). Modelling the stock price as S(t) = σB(t) where σ is aconstant ‘volatility’ factor, Bachelier invoked the dictum quoted above to arguethat the expected exercise value at the time the contract is entered should bezero. A straightforward calculation using the transition function (1.2) showsthat this is so if

0 = E([S(T )− (K − p)]+ − p|S(t) = x

)= σ

√T − t

2πe− (K−p−x)2

2σ2(T−t) + (K − x − p)N

(K − p − x

σ√T − t

)+ x −K, (1.4)

which we now have to solve for p. Here N denotes the standard normal dis-tribution function. Bachelier noted that if K − p = x, meaning in today’s


Mathematics of Financial Markets 3

terminology that the option is at the money forward, then (1.4) is solved byp = σ

√(T − t)/2π , so that the option value is proportional to the square root

of the time to exercise. As we shall see, his approach to the pricing problem isvery close to the mark.

Not much happened for the next 65 years. The time wasn’t ripe for so-phisticated financial instruments, the technology could not have handled them,and there was the little matter of two world wars and the Great Depression todistract peoples’ attention. After 1945 the Bretton Woods agreements [20] onfixed exchange rates, and barriers to capital movements, provided little scopefor financial intermediation. Meanwhile, however, the mathematicians were farfrom idle. Einstein’s 1905 paper [10], deriving the transition function (1.2) byanalysing the ‘diffusion’ of particles in a perfect gas, put Brownian motion as amathematical model firmly on the map. But it was not until 1923 that NorbertWiener [35] gave a rigorous treatment, showing that it is possible to define aprobability measure on the space of continuous functions that corresponds to theBrownian transition function. In 1933 Kolmogorov [21] (who cites Bachelier)provided the axiomatic foundation for probability theory on which the subjecthas been based ever since. Martingales were introduced in the late 1930s and amagisterial treatment given by Doob [9]. In 1944 Kiyoshi Ito [19], attemptingto elucidate the connection between partial differential operators such as (1.3)above and Markov processes, introduced stochastic differential equations andthe famous ‘Ito stochastic calculus’.

The characteristic property of Brownian motion is that the quadratic varia-tion of each sample path is equal to the length of the time interval over whichit is calculated. More precisely, for s < t and tni = i/2n

limn→∞

∑{i:tni ∈[s,t]}

(B(tni+1)− B(tni )

)2 = t − s a.s. (1.5)

This means that in calculating Taylor expansions of functions of the Brownianpath, the second order term is of order dt and must be retained:

f (B(t + h)) = f (B(t))+ f ′(B(t)) dB + 1

2f ′′(B(t)) dB2 + . . . (1.6)

As McKean [24] nicely puts it, Ito calculus is the same as ordinary calculus,but using the multiplication table dB2 = dt , dB dt = 0, so in the limit (1.6)becomes

df (B(t)) = f ′(Bt ) dBt + 1

2f ′′(B(t)) dt. (1.7)

All of this came together in the 1960s in an extraordinary way. Paul-AndreMeyer’s supermartingale decomposition theorem [26, 27] opened the way todefining stochastic calculus for very general classes of semimartingales, notjust Brownian motion. (A semimartingale is a process that is the sum of amartingale and a process with sample paths of bounded variation; for example,f (B(t)) in (1.7) is a semimartingale.) Stroock and Varadhan [32] demonstratedin a definitive way the connection between martingales and Markov processes.


4 M. Davis

We can say that a second-order differential operator A is the ‘differentialgenerator’ of a continuous Markov processXt if for some class D of functionsf the process

Mf (t) = f (Xt )− f (X0)−∫ t

0Af (Xs) ds

is a martingale. For example when Xt is Brownian motion we see from (1.7)that for f ∈ C2

b , dMf = f ′dX and the generator of Brownian motion isAf = f ′′/2. Stroock and Varadhan turned this around and showed that themartingale property for quite a ‘small’ class D characterises the probabilitymeasure of the process Xt .

The net effect of these developments was to turn stochastic analysis from anarcane topic, of interest only to a few initiates, to a powerful body of techniqueaccessible to a wide range of applied scientists. The whole story can be foundin some excellent textbooks such as Rogers and Williams [30].

Bachelier’s Brownian motion model for stock prices is open to the objectionthat prices are by definition positive quantities while B(t), being normally dis-tributed, is negative with strictly positive probability. In 1965 Paul Samuelson[31] introduced what has now become the standard model, namely geometricBrownian motion in which the price S(t) satisfies the stochastic differentialequation

dS(t) = µS(t) dt + σS(t) dB(t). (1.8)

It is easily checked by Ito calculus that the solution to (1.8) is

S(t) = S(0) exp

((µ− 1

2σ 2

)t + σB(t)

)(1.9)

and hence that ES(t) = S(0)eµt (recall that B(t) ∼ N(0, t)). Expression (1.9)shows that S(t) > 0 with probability 1, and (1.8) has the nice interpretation thatthe log-return log(S(t+h)/S(t)) is normally distributed with mean (µ−σ 2/2)hand variance σ 2h. The ‘volatility’ parameter σ (though not the mean) is easilyestimated from financial time series. Measuring time in years, typical values arearound 10%–40%. Geometric Brownian motion provided a workable model forasset prices that led, eight years later, to the central result of modern finance,the Black-Scholes option-pricing formula [3].

The background to the discovery of the eponymous formula by FischerBlack, Myron Scholes and their collaborator Robert C. Merton [25] is recountedin detail by Bernstein [2]. The fundamental insight is the idea of perfect repli-cation. Selling an option, we receive the premium and acquire the obligation todeliver the exercise value at a later time T . The option being a derivative secu-rity, the exercise value depends only on the price movements of the underlyingasset, and this is tradable. Suppose we were able to trade in the underlyingasset and cash in such a way that the value of the resulting portfolio exactlymatches the option exercise value at time T . Then all the risk in writing theoption would have been ‘hedged away’, and the value of the option – i.e. thepremium we receive – must be exactly the amount required to establish the



replicating portfolio. Otherwise there is an arbitrage opportunity: the availabil-ity of riskless profit with no initial investment. It turns out that in a market modelwhere prices follow geometric Brownian motion, perfect replication is actuallypossible, giving a unique option price.

The ‘heroic period’ of finance was 1965–1980. Samuelson started it byintroducing geometric Brownian motion – and therefore Ito calculus – intofinance. The Black-Scholes formula was published in 1973, the same year thatoption trading started on the CBOE (Chicago Board Options Exchange). Alsoin 1973, the Bretton Woods system finally collapsed, leading to an immediaterequirement for management of exchange rate volatility. By 1980 arbitragepricing theory had become well understood, the close link with martingaletheory being established by Harrison, Kreps and Pliska [17, 16], and the interestrate swaps market was just about to take off. It is coincidental, but relevant,that 1979 was the date of the first IBM PC, ushering in the era of massivecomputational capacity and cheap memory without which the industry couldnot exist.

This article aims to explain what the mathematical questions behind thisnew industrial revolution are. The place to start is certainly the wonderful ‘bi-nomial model’, which contains virtually all the ideas in embryonic form. This iscovered in the next section, after which we return to the fundamental questionsof arbitrage pricing in Section 3, and to the classic Black-Scholes world in Sec-tion 4. In Section 5 consider the relation between theory and market practiceand the many interesting questions this brings to light. Concluding remarks aregiven in Section 6.

2. The Binomial Model

This model, introduced by Cox, Ross and Rubinstein [4] in 1979, has played adecisive role in the development of the derivatives industry. Its simple structureand easy implementation have given analysts the ability to price a huge rangeof financial derivatives in an almost routine way.

Suppose we have an asset whose price is S today and whose price tomorrowcan only be one of two known values S0, S1 (we take S0 > S1); see Figure 2(left). This apparently highly artificial situation is the kernel of the binomialmodel. We also suppose there is a bank account paying a daily rate of interest α,so that $1 today is worth $R = $(1 + α) tomorrow. We assume that borrowingis possible from the bank at the same rate of interest α, and that the risky assetcan also be borrowed (sold short, in the usual financial terminology). The onlyother assumption is that S1 < RS < S0. If RS ≤ S1 we could borrow $B fromthe bank and buy B/S shares of the risky asset. Tomorrow these will be wortheither S0B/S or S1B/S, while only RB ≤ S1B/S has to repaid to the bank,so there is no possibility of loss and a positive profit in some instances. Thisis an arbitrage opportunity. There is also an arbitrage opportunity if RS ≥ S0,realised by short-selling the risky asset.

A derivative security, contingent claim or option is a contract that paystomorrow an amount that depends only on tomorrow’s asset price. Thus any such


6 M. Davis

S O1V00

uS

u2S O0

SV10

dS

V11

d2S O2

S

S0 O0

S1 O1

claim can only have two values, say O0 and O1 corresponding to ‘underlying’prices S0, S1, as shown in Figure 2 (left).

figure 2Left: 1-period binomial tree.Right: 2-period binomial tree.

Suppose we do the following today: form a portfolio consisting ofN sharesof the risky asset and $B in the bank (either or both ofN , B could be negative).The value today of this portfolio is p = B + NS and its value tomorrow willbe RB +NS0 or RB +NS1. Now choose B and N such that

RB +NS0 = O0

RB +NS1 = O1

i.e.N = (O0−O1)/(S0−S1) andB = (O0−NS0)/R. Then the portfolio valuetomorrow exactly coincides with the derivative security payoff, whichever waythe price moves. If the derivative security is offered today for any price otherthan p there is an arbitrage opportunity (realised by ‘borrowing the portfolio’and buying the option or conversely). It is easily checked that

p = 1

R(q0O0 + q1O1)

where q0 = (RS − S1)/(S0 − S1) and q1 = (S0 − RS)/(S0 − S1). Note thatq0, q1 ≥ 0, q0 + q1 = 1 and q0, q1 depend only on the underlying marketparameters, not on O0 or O1. We can therefore write the price of the derivativeas

p = EQ

(1

RO

),

the expected discounted payoff under the probability measureQ = (q0, q1) de-fined above. Note that this measure, the so-called risk-neutral measure, emergesfrom the ‘no-arbitrage’ argument. We said nothing in formulating the modelabout the probability of an upwards or downwards move and the above argu-ment does not imply that this probability has to be given byQ. A further featureof Q is this: if we compute the expected price tomorrow under Q we find that

S = 1

R(q0S0 + q1S1);

the discounted price process is a martingale. To summarize:

• There is a unique arbitrage-free price for the contingent claim.



• This price is obtained by computing the discounted expectation with respectto a certain probability measure Q.

• Q can be characterised as the unique measure such that the discounted un-derlying price process is a martingale.

Much of the classic theory of mathematical finance is concerned with identify-ing conditions under which these three statements hold for more general priceprocesses.

In the remainder of this section we show how more realistic models canbe obtained by generalising the binomial model to n periods. We consider adiscrete-time price process S(i), i = 0, . . . , n such that, at each time i, S(i)takes one of i + 1 values Si0 > Si1 > . . . > Sii . While we could considergeneral values for these constants, the most useful case is that in which the pricemoves ‘up’ by a factor u or ‘down’ by a factor d = 1/u, so that Sij = Sui−2j

where S = S(0); see Figure 2 (right) for the 2-period case. We can define aprobability measure Q by specifying that P [S(i + 1) = uS(i)|S(i)] = q0 andP [S(i + 1) = dS(i)|S(i)] = q1 where q0 and q1 are as before, i.e. in this caseq0 = (Ru − 1)/(u2 − 1), q1 = 1 − q0. Thus S(i) is a discrete time Markovprocess under Q with homogeneous transition probabilities.

Consider the 2-period case of Figure 2 (right) and a contingent claim withexercise valueO at time 2 whereO = O0,O1,O2 in the 3 states as shown. Bythe 1-period argument the no-arbitrage price for the claim at time 1 is V10 =(q0O0 +q1O1)/R if the price is uS and V11 = (q0O1 +q1O2)/R if the price isdS. But now our contingent claim is equivalent to a 1-period claim with payoffV10, V11, so its value at time 0 is just (q0V10 + q1V11)/R, which is equal to

V00 = EQ

[1

R2O

].

Generalizing to n periods and a claim that pays amounts O0, . . . , On at timen, the value at time 0 is

V00 = EQ

[1

RnO

]= 1

Rn

n∑j=0

Cnj qn−j0 q

j

1Oj

where Cnj is the binomial coefficient Cnj = n!/j !(n− j)!. By our original cal-culation the initial hedge ratio (the numberN of shares in the hedging portfolioat time 0) is

N = V10 − V11

uS − dS= 1

SRn−1(u− d)

n−1∑j=0

Cn−1j q

n−1−j0 q

j

1 (Oj −Oj+1).

For example, suppose S = 100, R = 1.001, u = 1.04 , n = 25 and O is a calloption with strike K = 100, so that Oj = [Sun−2j − K]+. The option valueis V00 = 9.086 and N = 0.588. The initial holding in the bank is thereforeV00 − NS = −49.72. This is the typical situation: hedging involves leverage(borrowing from the bank to invest in shares).

Now let us consider scaling the binomial model to a continuous limit. Take afixed time horizon T and think of the price S(i) above, now written Sn(i), as the


8 M. Davis

price at time iT /n = i*t . Suppose the continuously compounding rate of inter-est is r , so thatR = er*t . Finally, define h = log u andX(i) = log(S(i)/S(0));then X(i) is a random walk on the lattice {. . . − 2h,−h, 0, h, . . . } with rightand left probabilities q0, q1 as defined earlier and X(0) = 0. If we now takeh = σ

√*t for some constant σ , we find that

q0, q1 = 1

2± h

2σ 2

(r − 1

2σ 2

)+O(h2).

Thus Z(i) := X(i)−X(i − 1) are independent random variables with

EZ(i) = h2

σ 2

(r − 1

2σ 2

)+O(h3)

=(r − 1

2σ 2

)*t +O(n−3/2)

andvar(Z(i)) = σ 2*t +O(n−2).

Hence Xn(T ) := X(n) = ∑ni=1 Z(i) has mean µn and variance Vn such that

µn → (r − σ 2/2)T and Vn → σ 2T as n → ∞. By the central limit theorem,the distribution of Xn(T ) converges weakly to the normal distribution withthe limiting mean and variance. If the contingent claim payoff is a continuousfunctionO = g(Sn(n)) then the option value converges to a normal expectationthat can be written as

V0(S) = e−rT√2π

∫ ∞

−∞g

(Se(r−σ 2/2)T+σ√

T x)e−

12 x

2dx. (2.1)

This is in fact one expression of the Black-Scholes formula. It can be given inmore explicit terms when, for example, g(S) = [S − K]+, the standard calloption.

3. The Fundamental Theorem of Asset Pricing

In the binomial model we discovered that there is a unique martingale measureand that the unique no-arbitrage option value is the discounted expectation ofits payoff under this measure. To what extent do these properties generalize toother market models? This question turned out to be surprisingly delicate anddefinitive answers were not given until the 1990s.

Let us first consider a general discrete-time, finite-time model, followingthe treatment given by Elliott and Kopp [11]. We start with a probability space(.,F , P ) where, as will be seen, the only role of P is to determine the nullsets of F . We take a filtration {F t , t = 0, 1, . . . , T } and a (d + 1)-vectorprocess S(t) = (Si(t), i = 0, . . . , d) adapted to F t . Components i = 1, . . . , drepresent asset prices while S0(t) represents the value of $1 invested in theriskless bank account. Without essential loss of generality we can assume thatS0(t) ≡ 1. A trading strategy is a predictable process θ(t), meaning that θ(t) is



F t−1-measurable for each t . θi(t) is the number of units of asset i held betweentimes t − 1 and t , and the initial value of the portfolio corresponding to θ is justI0(θ) = θ(1)S(0) := ∑d

i=0 θi(1)Si(0). The value at time t isVt (θ) = θ(t)S(t).

The gain from trade is given by

Gt(θ) =t∑s=1

θ(s) (S(s)− S(s − 1)) ,

and the portfolio is self-financing if

Vt (θ) = I0(θ)+Gt(θ), t = 1, . . . , T . (3.1)

An arbitrage opportunity means the existence of a self-financing strategy θ suchthat I0(θ) = 0, VT (θ) ≥ 0 a.s. and EVT (θ) > 0. The market model is viable ifthere are no arbitrage opportunities.

A contingent claim is a non-negative F T -measurable random variable H ,and H is attainable if H = VT (θ) a.s. for some self-financing strategy θ . It iseasily seen that in a viable market if we also have H = VT (θ

′) for some otherstrategy θ ′ then Vt (θ) = Vt (θ

′) for all t (otherwise there would be an arbitrageopportunity).

If Q is an equivalent martingale measure (EMM), i.e. Q is a measure onF with the same null sets as P and the price processes Si(t) are martingalesunder Q, then the gain from trade process Gt(θ) is a martingale. From (3.1) itfollows that the price at time 0 of an attainable claim is justEQH , a number thatis independent of the particular EMM chosen. The simplest form of the Funda-mental Theorem is the following. A complete sigma-field is finitely generatedif it is the completion of a sigma-field containing a finite number of sets.

� theorem 3.1 (Harrison-Kreps [16]) Suppose F is finitely generated. Thenthe market model is viable if and only if there exists an EMM.

That existence of an EMM implies viability follows readily from simple mar-tingale arguments. The key point is proving the converse. When F is finitelygenerated this can be done by a finite-dimensional separating hyperplane argu-ment; see [11].

The next idea is completeness. A viable market model is complete if everycontingent claim is attainable. By the remarks about attainability above thismeans that if Q, Q′ are EMMs then EQH = EQ′H for every non-negativeF -measurable random variable H . But this means that Q = Q′, so the EMMis unique.

Theorem 3.1 remains true without the condition that F be finitely gen-erated, as long as the time horizon remains finite; this was shown by Dalang,Morton and Willinger [5]. However, if either T = ∞ or trading takes place incontinuous time then things are more complicated. Both cases can be handledsimultaneously by taking the time set asR+ and the asset prices as a vector semi-martingale on a stochastic basis (.,F , (F t )t∈R+ , P ). Trading strategies arepredictable processes (θt ) and the gains from trade process isGt(θ) = (θ · S)t ,the semimartingale integral; but some restrictions have to be imposed on θt toavoid obvious arbitrage. For example, consider betting on successive tosses of


10 M. Davis

a coin with heads probability p. The classic ‘doubling strategy’ is as follows.Bet $K on the first toss. If heads comes up, stop. Otherwise, bet $2K on thenext toss and double your stake on successive bets until heads comes up. Thenstop. Whenever you stop, your net winnings are $K . It is easy to prove that (a)with probability one, heads will occur at some finite time whatever the valueof p, and (b) if you must stop after n plays or if there is a house limit L (youmust stop if your fortune ever reaches −$L) then this strategy fails to generatearbitrage. In the general theory we therefore restrict admissibility to those trad-ing strategies such that Gt(θ) ≥ −aθ for all t almost surely, for some positivenumber aθ .

Let L0 (L0+, L∞) denote the set of finite-valued (non-negative valued, es-sentially bounded) F -measurable functions, and define

K0 = {G∞(θ) : θt is admissible and limt→∞(θ · S)t exists a.s.}.

We denote by C0 the cone of functions dominated by elements of K0, i.e.C0 = K0−L0+, and defineK = K0∩L∞,C = C0∩L∞. The no-arbitrage (NA)condition can now be stated as C ∩ L∞+ = {0}. The existence of an equivalentmartingale measure implies (NA) but the converse is generally false, meaningthat some stronger condition is required to obtain equivalence. A major step wastaken by David Kreps [23], who realised that the purely algebraic condition ofno-arbitrage has to be supplemented by a topological condition. He introducedsuch a condition under the name of ‘no free lunch’. In the general semimartingalesetting, Delbaen and Schachermayer [7] gave an apparently weaker, but in factequivalent, condition called ‘no free lunch with vanishing risk’ (NFLVR). Itstates that C ∩ L∞+ = {0}, where C denotes the closure of C in the normtopology of L∞.

� theorem 3.2 (Delbaen-Schachermayer [7]) Let S be an Rd -valued semi-martingale.

(a) If S is bounded then there is an equivalent martingale measure if and onlyif S satisfies NFLVR.

(b) If S is locally bounded then there is an equivalent local martingale measureif and only if S satisfies NFLVR.

Part (b) covers in particular the case of continuous-path semimartingalesS. The reader is referred to [7] for discussion of NFLVR and the various otherconditions that have been proposed, and a guide to related literature. See alsothe companion paper [8] in which results are given for processes that are notlocally bounded.

4. The Classic Black-Scholes Model

The probability space (.,F , (F t )0≤t≤T , P ) for this model is the canonicalWiener space:. is the space of continuous functions C[0, T ], F 0

t is the filtra-tion generated by the coordinate process wt(ω) = ω(t), P is Wiener measureon F 0

T , F is the P -completion of F 0T and, for each t , Ft is F 0

t completed



with all null sets of F . Then (wt ,F t ) is Brownian motion (we only considerthe scalar case here; everything extends to vector Brownian motion withoutdifficulty).

The set of measures equivalent to Wiener measure P was identified byI.V. Girsanov.

� theorem 4.1 ([15])

(a) Suppose Q ≈ P . Then there exits an adapted process Φt such that∫ T0 Φ2

s ds < ∞ a.s. and

dQ

dP= exp

(∫ T

0Φs dws − 1

2

∫ T

0Φ2s ds

). (4.1)

(b) Let Φt be any adapted process with∫ T

0 Φ2s ds < ∞ a.s., and let L denote

the right-hand side of (4.1). If EL = 1 then dQ = LdP is equivalent toP .

(c) Under measure Q given by (4.1), the process

wt = wt −∫ t

0Φs ds (4.2)

is an F t -Brownian motion.

The content of Girsanov’s theorem is that an absolutely continuous changeof measure just adds an absolutely continuous ‘drift’ to Brownian motion. Thebest general condition ensuring that EL = 1, due to Novikov, is that

E exp

(1

2

∫ T

0Φ2s ds

)< ∞.

As we have already seen, the (Samuelson-) Black-Scholes price model is

dS(t) = µS(t) dt + σS(t) dwt (4.3)

where µ and σ are the ‘drift’ and ‘volatility’. We also assume a riskless rate ofinterest r .

Define Q by (4.1) with Φs = (r − µ)/σ . Certainly E(dQ/dP ) = 1, soQ ≈ P and from (4.2) we can write (4.3) as

dS(t) = µS(t) dt + σS(t) (dwt +Φt dt)

= rS(t) dt + σS(t) dwt . (4.4)

By the Ito formula

d(e−rtS(t)) = σe−rtS(t) dwt .

Thus the discounted price is a martingale and Q is the EMM. The market iscomplete due to the martingale representation theorem for Brownian motion.

This gives us all we need to price derivative securities. Suppose for examplethat our claim is H = g (S(T )). The solution of (4.4) is


12 M. Davis

S(T ) = S(0) exp

((r − 1

2σ 2

)T + σwT

)(4.5)

and the option value at time 0 is

EQ(e−rT g(S(T ))

). (4.6)

In view of (4.5), the expectation is a 1-dimensional gaussian integral that, asthe reader can check, coincides with (2.1), the limiting result obtained from thebinomial model.

Let C(t, s) be the solution of the parabolic partial differential equation

∂C

∂t+ rs

∂C

∂s+ 1

2σ 2s2 ∂

2C

ds2− rC = 0 (4.7)

with boundary dataC(T , s) = g(s). (4.8)

If one defines the portfolio strategy θ1(t) = Cs(t, S(t)), θ0(t) = C(t, S(t))−

S(t)θ1(t), whereCs = ∂C/∂s, then it is easily checked using the Ito formula thatthe portfolio value is exactly Vt (θ) = C(t, S(t)) and hence VT (θ) = g(S(T ))

in view of (4.8). In fact, this is how Black and Scholes originally obtained thepricing formula: hypothesizing that the value is C(t, S(t)) for some smoothfunction C, they showed that perfect replication is obtained if and only if Csatisfies (4.7), (4.8) and θ is given as above. Note that the replicating strategyis delta-hedging: the number of units of S(t) in the hedging portfolio at timet is * = Cs(t, S(t)). For this argument – which can be found in Hull [18] –no measure change is required. One can work directly in the original measureP and one finds that the drift parameter µ in (4.3) simply cancels out. In therisk neutral measure Q the connection between (4.6) and (4.7) is through theFeynman-Kac formula: (4.7) can be written

∂C

∂t+ AC − rC = 0

where A is the differential generator of S(t) satisfying (4.4). The Feynman-Kac formula gives a probabilistic representation for the solution of this equation,which is exactly (4.6).

The Black-Scholes methodology extends to pricing a whole range of com-plex option products: barrier options, basket options, look-back options, Amer-ican options and many others. Rather than describe these, we will in the nextsection concentrate on a more basic question: what is the relation between theoryand practice in Black-Scholes?

5. Black-Scholes and Market Practice

The most obvious question to start with is whether the process (4.3) is actu-ally a good model for financial asset prices. Recall that under (4.3) log-returnslog(S(t + h)/S(t)) are normally distributed. Figure 3 shows the empirical dis-tribution of daily log-returns for the S&P500 index over the period 1988–2000,



-3.20

Rel

ativ

e fr

eque

ncy

Standard deviations

0.02

0.04

0.06

0.08

0.10

0.12

0.14

-2.8 -2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2

May 216½

EURO STYLE FTSE 100 INDEX OPTION (LIFFE) £10 per full index point

C¼P

116½C

¼P

16½C

¼P

¼C

83½P

¼C

183½P

¼C

283½P

¼C

383½P

¼C

483½P

5825 5925 6025 6125 6225 6325 6425 6525

19 May

Jun 310½ 76½ 241½ 107 179 144 127 119½ 84 248½ 52 316 30½ 393½ 15 477½Jul 410 144½ 347 181 288 221 224 256 175½ 306 134 363½ 98 426½ 69 496½Sep 506½ 216 441½ 249 380 286 323½ 327 271 373 224 424 181½ 479½ 145½ 541½Dec 663½ 301½ 597 331 533½ 364½ 474 401½ 418½ 442½ 366½ 487½ 320 537 273½ 587½Calls 15,531; Puts 32,579. * Underlying Index value. Premiums shown are based on settlement prices.dated expiry months.

Long

and the normal distribution with the same mean and variance. The fit is notparticularly good, the empirical distribution being negatively skewed and sig-nificantly leptokurtic (the skewness and kurtosis are −0.5 and 7.1 respectively).The general appearance is representative of many financial time series. Does itmatter?

figure 3Empirical distribution of daily log-returns and best Normal fit

figure 4FTSE100 Index option prices, FinancialTimes, 19 May 2000. Spot=6045

In the markets there are exchange-traded options and OTC (‘over thecounter’) options. The former are standardised call and put contracts on, forexample, the major stock indices, available typically with a range of strikelevels and with maturity times less than 1 year. OTC options, on the otherhand, are negotiated on a case-by-case basis between banks and may involvelonger maturities and/or ‘exotic’ features. Their prices are not publicly quoted.Figure 4 shows the prices of put and call options on the FTSE-100 stock indextraded at the London option exchange LIFFE on 19 May 2000. The level of theindex was 6045. (The exercise date for these options is the third Friday of themonth.)

Traders do not need a model to tell them the prices of exchange-tradedoptions: they can read the prices on their screens. These prices can be used toobtain ‘implied volatilities’; thinking of the Black-Scholes formula as a mapfrom volatility σ to price p, the implied volatility is the inverse map. In an ideal


14 M. Davis

5825

15%

59256025

6125

63256225

64256525

DecSept

JulyJune

Impl

ied

vola

tili

ty

17%

19%

21%

23%

25%

27%

29%

Strike

Black-Scholes world the implied volatility would be constant across all strikesand maturities, but in reality this is far from being the case. Figure 5 showsthe implied volatilities for the FTSE-100 index European call options whoseprices are given in Figure 4. There is a significant dependence on strike level,with out-of-the-money options having lower implied volatilities, and generallyan increase of implied volatility with maturity. Further evidence is given inFigure 6, which shows that the evolution of implied volatility over time is quite‘random’.

figure 5Implied volatility for the European calloptions in Figure 4.

A model is required to price an OTC derivative. The normal procedure isto build a Black-Scholes style model and then ‘calibrate it to the market’ byusing volatility parameters equal to the implied volatility from ‘comparable’exchange-traded options. In this process, the Black-Scholes formula is just aninterpolation formula: given the prices of exchange-traded options, produce aconsistent price for something that is not exchange traded. For this purpose,any reasonably smooth map from σ to p will give essentially the same answer:model error is not an important factor.

Much more serious is the question of hedging. Even for exchange-tradedoptions a model is required to determine the hedge ratio* = ∂C/∂s and otherrisk management parameters. Here a remarkable ‘robustness’ property of theBlack-Scholes hedging procedure comes in. Suppose the price process is inreality accurately modelled by a stochastic differential equation

dS(t) = α(t, ω)S(t) dt + β(t, ω)S(t)dwt (5.1)

where the coefficients α, β are general Ft -adapted processes. (The multiplica-tive dependence αS, βS in (5.1) is just for notational convenience, since α, β



Jan 885%

10%

15%

20%

25%

30%

35%

Jan 89 Jan 90 Jan 91 Jan 92 Jan 93 Jan 94 Jan 95 Jan 96 Jan 97

could depend on S(t).) As traders, however, we believe that S(t) satisfies (4.3)with some specific volatility σ , and we price and hedge accordingly. Thus if wewrite an option with exercise value g(S(T )), our estimate of its value at t < T

is C(t, S(t)), where C is the solution of (4.7), (4.8). We form a self-financinghedge portfolio with valueX(t) at time t by holding θ1

t = ∂C/∂s(t, S(t)) unitsof S(t) and placing the residual value X(t) − θ1

t S(t) in the riskless account.The increment in value in time dt is then

dX(t) = θ1t dS(t)+ (X(t)− θ1

t S(t))r dt, (5.2)

figure 6Implied volatility for ATM options onthe S&P500 index, 1988–1997.

with X(0) = C(0, S(0)) if we write the option at the Black-Scholes price.Define Y (t) = X(t) − C(t, S(t)). Using (5.1), (5.2), the Ito formula and theBlack-Scholes PDE (4.7) we find that

dY (t) = rY (t)dt + 1

2S2(t)

∂2C

∂s2(σ 2 − β2)dt

with Y (0) = 0, so that

Y (T ) = 1

2

∫ T

0er(T−s)S2(t)

∂2C

∂s2(σ 2 − β2)dt. (5.3)

Since ∂2C/∂s2 > 0 for put and call options, this shows that Y (T ) ≥ 0, i.e. ourhedging strategy makes a profit, with probability one as long asσ 2 ≥ β2. (Recallthat at time T ,C(T , S(T )) = g(S(T ))which is model-independent.) What thisshows is that successful hedging is entirely a matter of good volatility estimation:we consistently make a profit if the Black-Scholes volatility σ dominates the‘true’ diffusion coefficient β, regardless of other details of the price dynamics.If the true price process has jumps then almost sure profits cannot be obtained,but Y (T ) has positive expectation if σ is sufficiently large. Formula (5.3) is akey result of the whole theory: without something like it, attempts at practical


16 M. Davis

hedging would be wrecked by the effects of model error. This result can befound in the recent book [12] and in a number of earlier research papers but not,for some reason, in most of the standard textbooks.

Traders are very aware of the need to hedge against volatility risk. Thisis known as ‘vega hedging’, the vega of an option C being v = ∂C/∂σ , thesensitivity of the Black-Scholes value to changes in the volatility σ . If we holdoption C (say an OTC option) we could in principle hedge the volatility risk byselling v/v′ units of an exchange traded option C′ whose vega is v′, giving a‘vega neutral’ portfolio C − (v/v′)C′. Effectively we are — quite correctly —treating the exchange-traded option as an independent financial asset.

An immense amount of effort has gone into establishing a firm mathematicalbasis for these trading ideas. One major line of enquiry is stochastic volatility, inwhich the volatility parameter is treated as a stochastic process, not a constant.Thus our market model, in the physical measure P , takes the form

dS(t) = µS(t) dt + σ(t)S(t)dwt

dσ (t) = a(S(t), σ (t)) dt + b(S(t), σ (t))dwσt

where a, b define the volatility model and wσt is a Brownian motion withEdwtdw

σt = ρ dt , i.e. wσt is possibly correlated with the asset price Brow-

nian motion wt . We can write wσt = ρwt + ρ′w′t where w′

t is a Brownianmotion independent of wt and ρ′ = √

1 − ρ2. Measures Q equivalent to Pthen have densities of the form

dQ

dP= exp

(∫ T

0Φsdws − 1

2

∫ T

0Φ2s ds +

∫ T

0Ψsdw

′s − 1

2

∫ T

0Ψ 2s ds

)(5.4)

for some integrands Φ, Ψ . Taking Φ = (r − µ)/σ and Ψ = Ψ (S, σ ) we findthat the equations for S, σ under measure Q are

dS(t) = rS(t) dt + σS(t)dwt

dσ (t) = a(S(t), σ (t)) dt + b(S(t), σ (t))dwσt

where w, wσ are Q-Brownian motions with Edwdwσ = ρ dt and a(S, σ ) =a+ bρΦ + bρ′Ψ . Then S(t) has the riskless growth rate r , but σ is not a tradedasset so arbitrage considerations do not determine the drift of σ , leavingΨ as anarbitrary choice. Suppose we now have an option written on S(t) with exercisevalue g(S(T )) at time T . We define its value at t < T to be

C(t, S(t), σ (t)) = EQ[e−r(T−t)g(S(T ))

∣∣S(t), σ (t)].C then satisfies the PDE

∂C

∂t+ rs

∂C

∂s+ a

∂C

∂σ+ 1

2σ 2s2 ∂

2C

∂s2+ 1

2b2 ∂

2C

∂σ 2+ ρσsb

∂2C

∂s∂σ− rC = 0



and we find that the process Y (t) := C(t, S(t), σ (t)) satisfies

dY (t) = rY (t) dt + ∂C

∂sσSdw + ∂C

∂σbdwσ . (5.5)

If the map σ �→ y = C(t, s, σ ) is invertible, so that σ = D(t, s, y) for somesmooth function D, then the diffusion coefficients in (5.5) can be expressed asfunctions of t, S(t), Y (t) and we obtain an equation of the form

dY (t) = rY (t) dt + F(t, S(t), Y (t))dwt , (5.6)

where wt is another Brownian motion, again correlated with wt . S(t) and Y (t)are linked by the fact that, at time T , Y (T ) = g(S(T )). We have now created acomplete market model with traded assets S(t), Y (t) for whichQ is the uniqueEMM. By trading these assets we can perfectly replicate any other contingentclaim in the market. We have however created a whole range of such models,one for each choice of the integrand Ψ in (5.4). The choice of Ψ ultimatelydetermines the ‘volatility structure’ F of Y (t) in (5.6), which is all that isrelevant for hedging. This choice is an empirical question. The relationshipwith implied volatility is clear: if BS(t, S, σ ) denotes the Black-Scholes priceat time t with volatility parameter σ , then the implied volatility σ (t) mustsatisfy Y (t) = BS(t, S(t), σ (t)), so each stochastic volatility model implicitlyspecifies a model for implied volatility.

A comprehensive survey of the statistics and financial economics of stochas-tic volatility will be found in Ghysels, Harvey and Renault [14], while empiricalevidence is given by Tompkins [34] and an interesting new angle by Fouque,Papanicolaou and Sircar [12]. Other explanations for non-constant impliedvolatility and smile behaviour have been advanced, for example introducingjumps into the asset price process or making volatility a function of asset price.Based on the evidence of [34] however, it seems clear that extra random factorsare needed to get the whole picture. These investigations are of vital importancein improving trading practice and risk management in the derivatives industry.

6. Concluding Remarks

Far more has been left out of this article than has been put in. The most glaringomission concerns the interest-rate markets. Figure 7 shows the yield curvesfor three currencies as they existed on May 22, 2000. These are the interbankinterest rates available over 6-month periods stretching 20 years out into thefuture, implied by contracts traded today. The interest rate swap market, onwhich these curves are largely based, is one of the biggest financial marketsthere is, with billions of dollars notional value being traded every day. In 1980,this market didn’t exist. Along with ‘plain vanilla’ swaps there is a range ofderivative contracts: options on short-term interest-rate futures, swaptions, capsand floors, together with cross-currency swaps and various other multi-currencyproducts. The textbooks by Musiela and Rutkowski [28] and Rebonato [29] givea comprehensive picture of the theoretical and trading aspects.


18 M. Davis

00

For

war

d ra

te (

%)

Time (years)

1

2

3

4

5

6

7

8

9

5 10 15 20

JPY

EUR

USD

From the mathematician’s perspective interest-rate modelling is a wonderfulplayground. Yield curves are (or appear to be) infinite-dimensional objects andare therefore most naturally modelled by stochastic partial differential equations– an extra layer of sophistication beyond the simple models described in thisarticle. In general, finance is perhaps unique among the application areas ofmathematics both in the level of the mathematics involved and in the shortgap between pure mathematical research and its application in a commercialenvironment. Anyone trading exotic options really does need to understandmartingales, the Girsanov theorem, how to price American options, do quantoadjustments, and many other things; those who don’t will just lose money tothose who do.

figure 7Yield curves: forward 6-month interestrates on 22 May, 2000, for US Dollar,Yen and Euro.

In recent years the traditional range of derivative contracts – equities, interestrates and commodities – has been extended in three main directions: credit,energy and non-traditional underlying assets. ‘Credit’ refers to the risk thatcounterparties may default. There is now a big market in credit derivatives,which involve payments contingent on default or downgrading; see Tavakoli[33]. Privatisation of energy supply in Europe and regulatory developmentsin the USA have led to huge markets in which electricity is traded and tothe introduction of options. Finally, among the other ‘underlyings’ on whichderivative contracts have been written are insurance loss indices and the weather.See Krapeli [22] and Geman [13] for surveys of these areas.

From the mathematical standpoint, most of these new areas are incompletemarkets in which Black-Scholes style replication will be impossible. Attitudesto risk are no longer irrelevant and any pricing formula must involve some bal-ance of the risks involved. Davis [6] describes one possible approach, based on‘marginal substitution’ ideas of long standing in economics. Indeed, the moveto incomplete markets means that mathematical finance, hitherto narrowly fo-cussed around arbitrage pricing, will inevitably engage more with mathematical



economics, econometrics, statistics and insurance mathematics – see ClaudiaKluppelberg’s article in this volume – as analysts continue to grapple with pric-ing and risk-management issues in a fast-moving world.

Acknowledgements. This article was written at the Technical University of Vienna, sup-ported by funds from the Wittgenstein Prize awarded to Professor Walter Schachermayer.I thank Professor Schachermayer for providing this opportunity and for many conver-sations on the topics covered. Thanks go to Dr Robert Tompkins for discussions ofstochastic volatility and for providing data, and to Dr Josef Teichmann and Dr MarkOwen for helpful comments and eagle-eyed error detection. Discussions with ProfessorTerry Lyons have also been influential.

References

1. L. Bachelier: Theorie de la Speculation. Ann. Sci. Ecole Norm. Sup. 17 (1900) 21–86 [English translation in P. Cootner (ed.) The Random Character of Stock Prices.MIT Press, 1964, reprinted Risk Books, London 2000]

2. P. L. Bernstein: Capital Ideas. The Free Press, New York 19923. F. Black and M. Scholes: The pricing of options and corporate liabilities. J. Political

Econom. 81 (1973) 637–6544. J. Cox, S. Ross and M. Rubinstein: Option pricing, a simplified approach. J. Financial

Economics 7 (1979) 229–2635. R. C. Dalang, A. Morton and W. Willinger: Equivalent martingale measures and no-

arbitrage in stochastic securities market models, Stochastics and Stochastics Reports29 (1989) 185–202

6. M. H. A. Davis: Option pricing in incomplete markets, in M. A. H. Dempster andS. R. Pliska (eds.) Mathematics of Derivative Securities. Cambridge University Press1997

7. F. Delbaen and W. Schachermayer: A general version of the fundamental theoremof asset pricing. Math. Ann. 300 (1994) 463–520

8. F. Delbaen and W. Schachermayer: The fundamental theorem of asset pricing forunbounded stochastic processes. Math. Ann. 312 (1998) 215–250

9. J. L. Doob: Stochastic Processes. Wiley, New York 195310. A. Einstein: Uber die von der molekularkinetischen Theorie der Warme geforderte

Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen. Annalen derPhysik 17 (1905) 549–560

11. R. J. Elliott and E. Kopp: Mathematics of Financial Markets. Springer, Berlin Hei-delberg 1999

12. J.-P. Fouque, G. Papanicolaou and K. R. Sircar: Derivatives in Financial Marketswith Stochastic Volatility. Cambridge University Press 2000

13. H. Geman: Insurance and Weather Derivatives: From Exotic Options to ExoticUnderlyings. Risk Books, London 1999

14. E. Ghysels, A. C. Harvey and E. Renault: Stochastic volatility. In: G. S. Maddalaand C. R. Rao (eds.) Handbook of Statistics, vol. 14. Elsevier, Amsterdam 1996

15. I. V. Girsanov: On transforming a certain class of stochastic processes by absolutelycontinuous substitution of measures. Theory Prob. Appl. 5 (1960) 285–301

16. J. M. Harrison and D. M. Kreps: Martingales and arbitrage in multiperiod securitiesmarkets. J. Econ. Theory 20 (1979) 381–408

17. J. M. Harrison and S. R. Pliska: Martingales and stochastic integrals in the theoryof continuous trading. Stoch. Proc. & Appl. 110 (1981) 215–260

18. J. C. Hull: Options, Futures and Other Derivatives, 4th edn. Prentice-Hall 199919. K. Ito: Stochastic integral. Proc. Imperial Acad. Tokyo 20 (1944) 519–52420. P. B. Kenen: Bretton Woods system, in The New Palgrave Dictionary of Money and

Finance. Macmillan, London 1992


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21. A. N. Kolmogorov: Grundbegriffe der Wahrscheinlichkeitsrechnung. Ergebnisse derMathematik, vol. 2, no. 3, 1933

22. E. N. Krapeli: Electricity Trading and Hedging. Risk Books, London 200023. D. Kreps: Arbitrage and equilibrium in economies with infinitely many commodi-

ties. J. Math. Econom. 8 (1981) 15–3524. H. P. McKean: Stochastic Integrals. Wiley, New York 196925. R. C. Merton: The rational theory of option pricing. Bell J. Econ. Manag. Sci. 4

(1973) 141–18326. P.-A. Meyer: A decomposition theorem for supermartingales. Illinois J. Math. 6

(1962) 193–20527. P.-A. Meyer: Decomposition of supermartingales: the uniqueness theorem. Illinois

J. Math. 7 (1963) 1–1728. M. Musiela and M. Rutkowski: Martingale Methods in Financial Modelling.

Springer, Berlin Heidelberg 199729. R. Rebonato: Interest-Rate Option Models, 2nd edn. Wiley, Chichester 199830. L. C. G. Rogers and D. Williams: Diffusions, Markov Processes and Martingales,

vols. I & II. Cambridge University Press 200031. P. A. Samuelson: Proof that properly anticipated prices fluctuate randomly. Industrial

Management Review 6 (1965) 41–5032. D. W. Stroock and S. R. S. Varadhan: Diffusion processes with continuous coeffi-

cients I & II. Comm. Pure & Appl. Math. 22 (1969) 345–400 and 479–53033. J. M. Tavakoli: Credit Derivatives. Wiley, Chichester 199834. R. Tompkins: Stock index futures markets: stochastic volatility models and smiles.

To appear in: Journal of Futures Markets, 200035. N. Wiener: Differential space. J. Math. Phys. 2 (1923) 131–174


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