Chapter5: The stochastic process followed by stock prices Flashcards
S_τ
model S_τ , price of an asset as a random variable
Brownian motion DEF
A Brownian motion is a family of random variables
{B_t | t ≥ 0}
on some probability space (Ω, F, P) such that:
(1) B_0 = 0, (starts at fixed position)
(2) for 0 ≤ s < t the increment B_t − B_s is normally distributed with mean 0 and variance t −s, (not only consecutive terms)
(3) for any 0 ≤ t1 < t2 < · · · < tn the increments
B_t_1 − B_0, B_t_2 − B_t_1
, . . . , B_t_n − B_t_{n−1}
are independent random variables, (indep means previous info doesn’t affect the future)
(4) For any ω ∈ Ω the function t → B_t (ω) is continuous.
brownian motion corres to ω1, ω2, ω3 ∈ Ω
graph B_t (ω_i) against t jagged motions above and below t axis
BROWNIAN MOTION PROPERTIES
(a) The function t → B_t (ω) is nowhere differentiable with probability 1, (although continuous)
(b) if B_t = x for some t then for any ε > 0 the set {τ : |τ − t| less than ε and Bτ = x} is infinite with probability one.
constructing an ito integral
- construct a stochastic process {X_t | t ≥ 0} on the same probability space (Ω, F,P)
on which Brownian motion {Bt |t ≥ 0} is defined with the property that the change of X over an
infinitesimal period of time dt is given by
dX = a(ω, t)dt + b(ω, t)dB
where a and b are themselves stochastic processes on (Ω, F,P) with continuous paths and where dB
is the change in the Brownian motion over the infinitesimal period of time dt.
…
* sum of normals is normal
- process defined
X_t(ω) = X_0(ω) + ∫ over {0,t} of
a(ω, s) ds +
∫ over {0,t} of b(ω, s) dB(ω)_s
for all t ≥ 0
an ITO integral
the following limit is an Ito integral
limit of ||P|| = max{s_1 - s_0,.., s_n - s_n-1}
denoted
∫ over {0,t} of b(ω, s) dB(ω)_s
integrate wrt B
ito processes
dX = a(X, t) dt + b(X, t) dB
to denote that fact that X is a stochastic process defined by
X_t(ω) = X_0(ω) + ∫ over {0,t} of
a(X_s(ω), s) ds +
∫ over {0,t} of b(X_s(ω), s) dB(ω)_s
We shall refer to stochastic processes of this form as Ito processes
first approximation:
first approximation: model the proportional increase in stock prices as a Brownian
motion. We could then derive the following discrete time version
dS/S =σdB
where dS is the change in the stock price over a short time from t to t + dt,
dB = B_{t+dt} − B_{dt} and
B is a Brownian motion.
increases in S are indep ie NO MEMORY
BUT unsatisfactory: it implies that the values of stocks vary without any long term
trend, i.e., E(dS/S) = 0.
DRIFT TERM
takes into account upward trend in stock prices exp growth + noise
dS/S = σdB+µdt.
for ito process
dS = μS.dt + σS.dB
process S is a GEOMETRIC BROWNIAN MOTION
GEOMETRIC BROWNIAN MOTION
for ito process
dS = μS.dt + σS.dB
process S is a GEOMETRIC BROWNIAN MOTION
E(dS/S ) = µdt.
expected value of the proportional change in S
volatility of the stock
µ
expected return of the stock
σ
higher implies higher noise
GEOMETRIC BROWNIAN MOTION approximation not accurate
(1) The model allows any real number to be a value for S, but in real life there is a smallest unit.
(2) We assume that prices are changing continuously but trades occur at discrete times.
(3) Shares are often traded through market makers. They buy at the bid price and sell at the ask price. So there are two prices! Sometimes it is crucial to model both.
(4) Sometimes, e.g., during market crashes, changes seem to have a “memory”.
ITOS LEMMA
Assume that G(x, t) is twice continuously differentiable with respect to x and continuously differentiable with respect to t.
The process Y = G(X, t) is also an Ito process.
In fact,
dY =
[(∂G/∂x)a + (∂G/∂t) + (1/2)(∂^2G/∂x^2)b^2] dt +
(∂G/∂x)b dB.
derived from stochastic process Y with chain rule for variable B despite it not being a variable
Consider a forward contract on stock paying no dividends maturing at time T ; let F(t) be its forward price at time t ≥ 0:
F(t) = S(t)e^r(T −t)
where S(t) is the spot price of the stock at time t. Regard F as a function of s and t, i.e.,
F = F(s, t) = se^{r(T −t)}
∂F/∂S = e^{r(T −t)}
∂^2F/∂S^2 = 0
∂F/∂t = -rse^{r(T −t)}
model assumes that dS = µSdt + σSdB
so Ito’s Lemma implies that
dF =
(e^{r(T −t)} µS − rSe^{r(T −t)})
dt + e^{r(T −t)} σSdB
= (µ − r)F dt + σF dB,
i.e., F follows a geometric Brownian motion with drift µ − r.
