How Fast Did Ito’s Lemma Kill The Market?
William C. Troy∗
May 5, 2009
Abstract
We investigate growth and extinction properties of solutions of the geometric brownian motion
equation dX = λXdt + σXdW (t), where W (t) is the Wiener process, λ > 0 is the drift term, and
σ > 0 determines the volatility of the process. In the Black-Scholes theory the random variable
X(t) denotes the price of an equity at time tge0. An application of Ito’s Lemma shows that there
√
is a critical value σcrit = 2λ such that (i) if 0 < σ < σcrit then solutions become unbounded
with probability 1 as t → ∞, and (ii) if σ > σcrit then solutions ‘go extinct’ and decay to zero with
probability 1 as t → ∞. However, the investor needs to know more, namely how fast will the the
stock price will rise or fall? To answer these questions we derive rigorous bounds to predict how
fast growth or extinction of solutions occurs in different parameter regimes. Numerical examples
illustrate our results.
keywords. Ito’s lemma, extinction.
AMS classifications. 34B15, 34C23, 34C11
1
Introduction
During the 2008-2009 global economic meltdown investors watched in dismay as their stock portfolios
plummeted in value. As the crises continued to deepen it became increasingly clear that leading experts
did not understand the underlying economic mechanisms responsible for this economic disaster [8].
The call went out for new analytic and modelling studies to understand the basic forces behind what
really went wrong [3]. In this paper we return to the fundamental Black-Scholes theory and use Ito’s
Lemma together with rigorous estimates to predict how fast a stock price will rise or fall. It is hoped
that our methods will shed new light on understanding the effects of market forces on equity prices.
The Black-Scholes theory assumes that the price of an equity is a stochastic process modelled by the
geoemtric brownian motion SDE
dX(t) = λX(t)dt + σX(t)dW (t), X(0) = X0 ,
∗ Department
of Mathematics, University of Pittsburgh, Pittsburgh PA 15260, U.S.A.
1
(1.1)
2
Did Ito’s Lemma Kill The Market?
where X(t) denotes the value of an equity, W (t) is the Wiener process, λ > 0 is the drift, and
σ > 0 deteremines the volatility of the process. An application of Ito’s Lemma [4] to the nonlinear
transformation Y (t) = ln (X(t)) gives
d ln (X(t)) =
λ−
σ2
2
dt + σdW (t), t ≥ 0.
(1.2)
Integrating (1.2) from 0 to t, we obtain
hence
σ2
ln (X(t)) = ln (X0 ) + λ −
t + σW (t), t ≥ 0,
2
(1.3)
σ2
t + σW (t) .
X(t) = X0 exp
λ−
2
(1.4)
It follows from (1.4) that the mean E (X(t)) variance Var(X(t)) are given by
2
E (X(t)) = X0 eλt and Var(X(t)) = X02 e2λt eσ t − 1 .
(1.5)
Because E(X(t)) becomes unbounded as t increases, one might expect that a typical realization of (1.1)
also become unbounded as t increases. However, this is not the case. Instead, it is well known (e.g.
√
see Ockesndal ([7],p. 61) that there is a critical value σcrit = 2λ such that
√
2λ then X(t) → 0 a.s. as t → ∞,
√
if σ < 2λ then X(t) → ∞ a.s. as t → ∞,
√
if σ = 2λ then X(t) fluctuates between large and small positive values as t → ∞.
if σ >
(1.6)
Of special interest is the counterintuitive property that solutions ‘go extinct’ with probability 1 when
σ 2 > 2λ, in spite of the fact that limt→∞ E (X(t)) = ∞. This extinction property is due to the presence
2
of the ’Ito term’ − σ2 in equation (1.2).
Related Results. In order to understand the ‘big picture,’ we mention two important studies in
which phenomena similar to (1.5)-(1.6) were observed.
(i) In 1969 Lewontin and Cohen [5] investigated the discrete population model
Xi+1 = γi+1 Xi , X0 > 0, i ≥ 1,
(1.7)
where the constants γi are independent, identically distributed positive random variables, with common pdf fl (l). Using a Central Limit Theorem based approach, they proved the following properties:
Growth: if E(l) > 1 then
E (Xn ) → ∞ as n → ∞.
(1.8)
Extinction: If E(ln(l)) < 1 < E(l) then, for each K > 0,
lim Prob {Xn ≤ K} = 1,
n→∞
(1.9)
3
Did Ito’s Lemma Kill The Market?
(i) In 1990 Doering [2] investigated extinction phenomena in the stochastic Verhulst equation
dX(t) = λX(t) − X 2 (t) dt + σX(t)dW (t), X(0) = X0 .
He gave both numerical and theoretical evidence that extinction of solutions occurs when σ >
(1.10)
√
2λ.
Goals And Specific Aims. What is missing in the results described above is a description of exactly
how fast the growth or decay of solutions occurs in different parameter regimes. For example, when
the solution X(t) of (1.1) denotes the price of a stock, it is highly desirable to the investor to be able
√
√
to predict how fast the price will rise when σ < 2λ, or fall when σ > 2λ. Thus, our goal is to
address the following fundamental questions:
√
(I) If σ < 2λ what is practical lower bound for the probability that solutions of (1.1) become
unbounded as t → ∞?
√
(II) If σ > 2λ what is a practical upper bound for the probability that solutions of (1.1) ’go extinct”
as t → ∞?
In Section 2 we investigate the issues raised in (I) and (II). Theorem 2.1 gives precise estimates on
the growth or decay of solutions. We also give numerical examples to illustrate the prcticality of the
results of Theorem 2.1. Conclusions and suggestions for future research are given in Section 3.
2
Estimates on Growth and Extinction
In this section we prove our main theoretical result which addresses the questions raised in (I) and
(II) at the end of the previous section. Numerical examples are described following the proof of the
theorem.
Theorem 2.1 Let λ > 0, σ > 0, X0 > 0, and let X(t) denote a solution of (1.1).
(i) Growth. If M > 0 satisfies
σ2
2
− λ < −M < 0 then
Prob{X(t) ≥ X0 eMt } → 1 as t → ∞.
(ii) Extinction. If K > 0 satisfies
σ2
2
(2.1)
− λ > K > 0 then
Prob{X(t) ≤ X0 e−Kt } → 1 as t → ∞.
(2.2)
Proof. Part (i). Because ln(x) is an increasing function on (0, ∞), we conclude that
Prob X(t) > X0 eMt = Prob {ln (X(t)) > ln (X0 ) + M t} .
(2.3)
Substituting (1.3) into (2.3) gives
Prob X(t) > X0 e
Mt
σ2
= Prob
λ−
t + σW (t) > M t .
2
(2.4)
4
Did Ito’s Lemma Kill The Market?
√
tN (0, 1) into (2.4) and obtain, after a manipulation,
√ 2
t
σ
Mt
.
−λ+M
Prob X(t) > X0 e
= Prob N(0, 1) >
2
σ
As above, we set W (t) =
(2.5)
Thus,
2
x
1
dx.
(2.6)
√t √ exp −
σ2
2
2π
2 −λ+M
σ
2
√
− λ < −M it follows that limt→∞ σ2 − λ + M σt = −∞. This
Prob X(t) ≤ X0 eMt =
From the assumption that
σ2
2
and (2.6) imply that
Z
∞
Prob X(t) > X0 eMt → 1 as t → ∞.
(2.7)
Part (ii). Again, since ln(x) is an increasing function on (0, ∞), we conclude that
Prob X(t) > X0 e−Kt = Prob {ln (X(t)) > ln (X0 ) − Kt} .
(2.8)
Substituting (1.3) into (2.8) gives
Prob X(t) > X0 e−Kt = Prob
Setting W (t) =
σ2
t + σW (t) > −Kt .
λ−
2
(2.9)
√
tN(0, 1), we find that (2.9) reduces to
Prob X(t) > X0 e
−Kt
= Prob N(0, 1) >
Combining (2.10) with the definition of N (0, 1) gives
Z ∞
−Kt
=
Prob X(t) > X0 e
2
σ
2
From the assumption that
(2.11) imply that
σ2
2
−λ−K
√t
σ
σ2
−λ−K
2
Z
∞
σ2
2
−λ−K
This completes the proof of the Theorem.
√t
σ
(2.10)
2
1
x
√ exp −
dx.
2
2π
− λ > K > 0 it follows that limt→∞
Prob X(t) ≤ X0 e−Kt = 1 −
√ t
.
σ
σ2
2
−λ−K
(2.11)
√
t
σ
= ∞. This and
2
x
1
√ exp −
dx → 1 as t → ∞.
2
2π
(2.12)
Numerical Examples. Figure 1 shows solutions of (1.1) which illustrate the the growth and extinction estimates derived in Theorem 2.1. Solutions were computed from the standard Euler scheme
√
X(tn+1 ) = X(tn ) + λX(tn )∆t + σ ∆t (randn) , X(t1 ) = X0 , n ≥ 1,
with step size ∆t = .005
(2.13)
5
Did Ito’s Lemma Kill The Market?
10
X
σ=.5
5
λ=1
M=.85
0
0
1
2
3
4
t
5
4
t
5
10
σ=2.25
X
λ=1
K=1
5
0
0
1
2
3
Figure 1: Upper panel: growth. Illustration of part (ii) of Theorem 2.1 showing a realization (solid
curve) of
(1.1) for parameters (σ, λ, M, X0 ) = (.5, 1, −.85, 1). Dashed curve is the lower bound
X0 exp(M t) = exp(.85t). Lower panel: extinction. Illustration of part (i) of Theorem 2.1 showing
a realization (solid curve) of equation (1.1) for parameters (σ, λ, K, X0 ) = (2.25, 1, 1, 5). Dashed curve
is the upper bound X0 exp(−Kt) = 5 exp(−t).
6
Did Ito’s Lemma Kill The Market?
3
Conclusions and Future Studies.
In Theorem 2.1 we have derived gives rigorous, practical bounds on the growth and decay of solutions
of (1.1). It would be interesting to compare our estimates with actual stock prices during the recent
dowwnturn of the economy when stocks lost large fractions of their values.
It is hoped that our techniques can be extended to a broader range of models. For example consider
the basic Malthusian [6] population model
dX
= µX, X(0) = X0 > 0,
dt
(3.1)
where X(t) denotes the population at time t ≥ 0 and µ > 0 is its growth rate. If we assume that µ
reflects environmental factors, then it is reasonable to expect that natural environmental fluctuations
will cause variations in µ. A simple way to express such variabliity is to assume that µ ha the form
µ = λ + σζ(t),
(3.2)
where λ > 0 is an average value, ζ(t) is white noise, and σ > 0 denotes the magnitude of the varaiation
of the fluctuations.
Ackmnowledgement. The author thanks Charles Doering for bringing the paper by Lewontin and
Cohen [5] to his attention.
References
[1] F. Black & M.Scholes The Pricing of Options and Corporate Liabilities. Journal of Political
Economy 81, No. 3 (1973), p. 637
[2] C. Doering Modeling Complex Systems: Stochastic Processes, Stochastic Differential Equations,
and Fokker-Planck Equations. The Proceedings Of The 1990 Complex Systems Summer School,
Santa Fe, New Mexico
[3] Maths and Markets: banks need quants and geeks to recover from the crisis. Financial Times,
March, 2009
[4] K. Ito On stochastic differential equations. Memoirs, American Mathematical Society (1951)
[5] R. C. Lewontin & D. CohenOn Population Growth in a Randomly Varying Environment.
PNAS vol. 62, no. 4 (1969), pp. 056-1060
[6] T. R. Malthus An Essay On The Principle Of Population. 1798 First Edition, London, JH.
Johnson Pub.
[7] B. Oksendal Stochastic Differential Equations. Springer-Verlag, (1995)
[8] C. Wald Crazy Money. Science vol. 322, 12 Dec. (2008), pp. 1624-1626