At the end of this section let us point out a couple of difficulties that we still have to address, in order to convince you that there is still something important coming in the remaining part of this chapter.
First, there is the ItNo formula that we have used, but neither stated in generality, nor proved or at least motivated. Then there are a couple of concepts that we applied without really thinking too much about it. Using mathematics only led by intuition, but not by a correct foundation, can lead to wrong conclusions, and in finance wrong conclusions can be very very expensive, so maybe it is worth looking at the details.
There are other reasons: besides the Black-Scholes formula, we can get more results, for instance to price path-dependent options (at least numerically). Moreover, some of the fundamental assumptions in the Black-Scholes model are arguable at best:
why should stock prices follow a normal distribution? We have actually already seen that this is not really the case (as we have mentioned at first in Sect.3.4.2).
This will motivate us to study less handy, but more realistic models for stock prices.
We will also introduce methods to check the predictions of the Black-Scholes model empirically. This will again enable us to improve the model substantially.
8.2 Brownian Motion and It ¯o Processes
We consider a state space˝with a probability measurep.3We define aprocessas follows:
Definition 8.1 (Process4) Aprocess Xis a measurable functionXW˝Œ0;1/! R. We callX.t/WDX.;t/the value ofXat timet.
Processes will be used to describe the ups and downs of assets, in that they assign probabilities to states at any given time. One of the central problems in asset pricing is to find realistic and at the same time mathematically manageable classes of processes. Historically, this idea goes back to the year 1900 and Louis Bachelier’s seminal and unfortunately long forgotten work [Bac00]. He already applied Brownian motion as underlying process, which is even today still the most popular process among practitioners. We will see later, in Sect.8.8, that there are nowadays alternatives that model actual behavior of asset prices much better, but for now we will study Brownian motion:
3More precisely, we study a probability space.˝;F;p/, where F is a -algebra and p is a probability measure on˝with respect toF, see Sect.A.4for details.
4This and the following definitions can also be adapted to discrete-time problems.
Definition 8.2 (Brownian motion) A standard Brownian motionis a process B defined by the properties:
(a) B.0/D0a.s.
(b) For any times t0;t1 with t0 < t1, the difference B.t1/ B.t0/ is normally distributed with mean zero and variancet1t0.
(c) For any times 0 t0 < t1 < t2 < < tn < 1, the random variables B.t0/;B.t1/B.t0/; : : : ;B.tn/B.tn1/are independently distributed.
(d) For eachw2˝, thesample path t7!B.w;t/is continuous.
The idea of Brownian motion is that at any given time the change of a quantity is completely random and following a normal distribution. Brownian motion has first been described by the botanists Jan Ingenhousz in 1785 and Robert Brown in 1827 who noticed random movements in small particles under the microscope. The mathematical theory was first developed by Thorvald Thiele in 1880, but Bachelier independently re-invented it in his work.
When describing financial markets by processes, we are particularly interested in adapted processes, i.e., processes that “cannot see into the future”: in other words, only past events should be of interest to our model. To define this notion properly, we need the mathematical tool of the so-called “filtrations” which describe the information available at a given point in time:
Definition 8.3 (Filtration) A filtration of a measurable space˝ with-algebraF is a family of-algebrasfF.t/gt2.0;1/such that
(i) F.t/Ffor allt,
(ii) F.t1/F.t2/for allt1 t2.
The filtration will in a certain way reflect how much we know at a given timet:
condition (ii) ensures that the knowledge can only increase, never decrease. We can now define what we mean with an “adapted process”:
Definition 8.4 (adapted process) A processX is called adapted to the filtration F.t/of˝ifX.t/W˝ !Ris aF.t/-measurable function for eacht2Œ0;1/.
We have now a mathematically precise, although maybe slightly abstract model describing the price fluctuations on a financial market. In the next step we want to trade on this market.
Atrading strategyis described by a process that prescribes in every statewand at any timetthe assets a person should hold.5For simplicity, we will deal with only one asset, thus we a are looking for a functionW˝ Œ0;1/ ! R, thetrading strategy. As example take the fixed-mix strategy that keeps the value fraction of a
5We have seen such a trading strategy already in Sect.8.3: the delta hedge strategy.
8.2 Brownian Motion and It ¯o Processes 293 risky asset constant: herewould be defined as.w;t/Dc=wwithcconstant ifw denotes the price of the risky asset.
If along a sample path is constant, it is relatively easy to compute the total return between timet0andt1asB..t1/B.t0//. This allows us to compute the gain also if is piecewise constant. For generalwe need to assume thatRT
0 .t/2dt<
1 a.s., then we can define the total gain,RT
0 .t/dB.t/, by approximating by sequencesnthat are piecewise constant along the sample path. This method called
“stochastic integration” needs in fact much more mathematical background then it seems and the interested reader is referred to [Kar88]. The general idea, however, is similar to the definition of the usual integral of a function via approximation with Riemann sums, as we know it from calculus. The stochastic integralRT
0 .t/dB.t/ also shares the same fundamental properties with the standard integral, in particular it is linear, i.e., for trading strategies; and real numbersa;bwe have
Z T
0 a.t/Cb.t/dB.t/Da Z T
0 .t/dB.t/Cb Z T
0 .t/dB.t/:
Brownian motion has a mean value of zero, real assets, however, have usually average returns larger than zero. Moreover, we want to study markets with more than one asset, and the returns of the assets will differ. Therefore, we need to go a step further and define a process that adds a “drift” to the Brownian motion, i.e., an additional directed price movement. Such processes are calledIt¯o processes.
Definition 8.5 (It¯o process) LetBbe a Brownian motion,x2R, 2 L2, i.e., is an adapted process withRT
0 .t/2dt < 1a.s. for allt, and2 L1, i.e.,is an adapted process withRT
0 j.t/jdt<1a.s. for allt. Then the It¯o processSis defined fort2Œ0;1/as
S.t/DS0C Z t
0 .s/dsC Z t
0 .s/dB.s/:
Informally and shorter, we denotedS.t/D.t/dtC.t/dB.t/,S.0/DS0. Under some additional technical conditions one can prove that and are nothing else than the rate of change of mean and variance ofS, precisely we have a.s.:
d
drEt.S.r//jrDtDt; d
drvart.S.r//jrDtDt2:
Consequently,is called thedrift processandthediffusion processofS.
Again, we can define a stochastic integral computing the total gain for a trading strategy, but this time has additionally to satisfy certain additional regularity conditions, see [Duf96, Chap. 5C] for details.
If we imagineS.t/to describe the price of an asset, then it would be interesting to study the process defined by a given function of this price (e.g., the payoff of a derivative based on this asset). That this can be done relatively easily is the merit of the result by Kiyoshi It¯o, the It¯o formula, that can be stated (in its easiest form) as follows:
Lemma 8.6 (It¯o formula) Let S be an It¯o process and fWR2 ! R twice continuously differentiable, then the process Y.t/ WD f.S.t/;t/is an It¯o process satisfying
dY.t/D
@f.S.t/;t/
@S .t/C@f.S.t/;t/
@t C1 2
@2f.S.t/;t/
@S2 .t/2
dt C@f.S.t/;t/
@S .t/dB.t/:
The proof of this result can be found, e.g., in [Duf96]. The intuition to it is as follows: if we expand the functionfWR2 !Rinto a Taylor series around the point .S.t/;t/, we obtain
f.a;b/Df.S.t/;t/C@f.S.t/;t/
@S .aS.t//C@f.S.t/;t/
@t .bt/
C1 2
@2f.S.t/;t/
@S2 .aS.t//2CO..aS.t//3; .bt/2/: (8.4) A Brownian motion has the property that dB.t/2is of order dt, in other words, the variance is the square root of the time increase. Therefore, the higher order terms in (8.4) vanish when we approximate the derivative offby taking the limita!S.t/, b !tand we obtain It¯o’s formula. (This is of course a mere intuition and should not fool us into assuming the proof would be nothing more than a straightforward expansion of this.)