MATH3090/7039: Financial mathematics Lecture 8
10 May 2018
Stochastic processes in continuous time
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Itˆo integrals
Itˆo’s rule
Incomplete market
In an incomplete market (Example: the one-period model with 3 possible outcomes and two assets B, S):
• From a replication standpoint: Some payoffs could not be replicated.
(In the Example, the only replicable payoffs are linear combinations of 1 and ST , or equivalently, affine functions of ST ).
For payoffs having no replicating portfolio, no-arbitrage alone may not be able to determine a unique price for the payoff.
• From the martingale / risk-neutral valuation standpoint:
There are many martingale measures consistent with the prices of the basic assets. The price for a payoff can depend on which martingale measure prevails in the market.
Completing a market
If we change the assumptions of the model, then we may be able to complete the market, and thus price all payoffs using no-arbitrage
• Could change the assumptions by adding more basic assets. In the example, we could complete the market by adding a third asset outside the span of B and S.
• Could change the assumptions by adding more trading opportunities.
In the trinomial example, we could complete the market by allowing trading of B and S at one intermediate time point
Now we will build models with infinitely many outcomes. We hope to replicate general payoffs by trading B and S continuously in time.
Stochastic processes in continuous time
Itˆo integrals
Itˆo’s rule
Filtrations
• Recall: we represent the arrival of information in a market by a filtration {Ft}t≥0. Each Ft represents what has been determined at or before time t.
• If we want to specify a filtration in continuous time, we cannot, say, draw a tree. Instead we could designate some process(es) – such as Brownian motion – that drive the risk in the market, and use the filtration generated by these “risk factor(s)”.
• Write {FtZ}t≥0 for the filtration generated by a process Z. This means FtZ contains all info about the history of Z through time t.
• Model asset prices as processes adapted to the filtration (so they are “functions of” the risk factors). Require trading strategies to be adapted to the filtration (so they don’t “look into the future”).
Martingales
Recall: we say Mt is a martingale with respect to a filtration {Ft} if: Mt isadaptedto{Ft},andforalls
Stochastic differential equations
Recall that in an Itˆo process
dXt = μtdt + σtdWt
the μt and σt can depend on the entire history of W up to time t.
Solutions of Itˆo stochastic differential equations (SDE) are a subclass of Itˆo processes. In an SDE, the μ and σ have the form μt = μ(Xt, t) and σt = σ(Xt, t) for some functions μ(x, t) and σ(x, t).
Typically, we specify the μ and σ functions, and define X to be the process that satisfies
dXt = μ(x, t)dt + σ(x, t)dWt, X0 = constant
Existence and uniqueness of a solution X can be guaranteed by Lipschitz-type technical conditions on μ and σ.
Example: dXt = aXtdt + bXtcdWt, X0 = 100 Plot some trajectories for a = −0.15, +0.15 and b = 0.20, 0.40
250 250 200 200 150 150 100 100
00 0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
250 250 200 200 150 150 100 100
00 0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Stochastic processes in continuous time
Itˆo integrals
Itˆo’s rule
Itoˆ’s rule for f : R → R
Given an Itˆo process Xt with dynamics dXt = μtdt + σtdWt, what are
dynamics of f(Xt), where f is a real-valued function.
• Example: Xt is some underlier, f(Xt,t) is value of a derivative
asset. Know dynamics of Xt. Want to learn dynamics of f(Xt,t). Itˆo’s rule: If f is sufficiently smooth, then f(Xt) is an Itoˆ process and
df(Xt) = ∂f dXt + 1 ∂2f (dXt)2, ∂x 2 ∂x2
where the partials of f are evaluated at Xt, and (dXt)2 is given by the “multiplication” rules
(dt)2 = 0, (dW )(dt) = 0, (dWt)2 = dt, which implies (dXt)2 = (μtdt + σtdWt)2 = (σtdWt)2 = σt2dt.
Itoˆ’s rule: idea of proof
It’s just a second order Taylor expansion
df(Xt) = ∂f dXt + 1 ∂2f (dXt)2,
If X were a differentiable function of t then dX = X′(t)dt, and (dX)2 = [X′(t)]2(dt)2, which is negligible relative to dt, so we could drop it. (Thus we obtained the chain rule of ordinary calculus.)
But if dXt = μtdt + σtdWt,
then (dXt)2 has terms involving dWt, which behaves like (dt)1/2.
(dt)2 ≪ dt (dt)(dWt) ∼ (dt)3/2 ≪ dt
(dWt)2 ∼ dt
so drop it
so drop it
cannot drop
Itoˆ’s rule: example
Geometric Brownian motion (GBM) S is defined by S0 > 0 and the dynamics
dSt = μStdt + σStdWt
where μ and σ are constant. (For now assume such S exists and is
positive.) Black-Scholes assumed GBM dynamics for stock prices. • What are the dynamics of log St?
Apply Itoˆ’s rule with f(s) = log s, f′(s) = 1/s, f′′(s) = −1/s2. Then
dlogSt= 1dSt+1(−1)(dSt)2 St 2 St2
= 1 (μStdt + σStdWt) − 1 1 (σ2St2dWt2) S t 2 S t2
= (μ − σ2/2)dt + σdWt
Itoˆ’s rule: example (cont)
Or in integrated form
logSt = logS0 + (μ−σ2/2)ds+ σdWs
=logS0 +(μ−σ2/2)t+σWt
• Distribution of St:
log St ∼ Normal(log S0 + (μ − σ2/2)t, σ2t)
so St is log-normal distribution (its log is normally distributed)
• Explicit expression for St in terms of Wt
St = elog St = S0e(μ−σ2/2)t+σWt .
Itoˆ’s rule for f : R2 → R
If Xt and Yt are Itˆo processes and f is sufficiently smooth, then
f(Xt, Yt) is an Itoˆ process and
df(Xt,Yt)= ∂fdXt+∂fdYt+1∂2f(dXt)2+1∂2f(dYt)2+ ∂2f (dXt)(dYt)
with the same “multiplication” rules as before. (For now, X and Y depend on the same W. Later, when we allow multiple Brownian motions, we will need one more multiplication rule.)
Important special case: Yt = t. Then dYt = dt, so
df(Xt,t)= ∂fdt+ ∂fdXt + 1∂2f(dXt)2. ∂t ∂x 2 ∂x2
∂x ∂y 2 ∂x2 2 ∂y2 ∂x∂y
Itoˆ’s rule: 2D example
Let Xt, Yt be Itˆo processes. Find the dynamics of Zt = XtYt. Solution:Letf(x,y)=xy,hence,∂f =y,∂f =x,∂2f =∂2f =0,
∂2f =1. ∂x∂y
∂x ∂y 2 ∂x2
+ 1 ∂2f (dYt)2 + ∂2f (dXt)(dYt)
Intuition:
d(XY)=(X+dX)(Y +dY)−XY =XdY +YdX+(dX)(dY)
Ordinary calculus says drop this if X, Y are differentiable in t. Itˆo calculus says keep this if X, Y are Itˆo processes.
The distinction is that (dt)(dt) ≪ dt but (dW )(dW ) = dt
Then by Itˆo’s rule:
df(Xt,Yt)= ∂fdXt + ∂fdYt + 1∂2f(dXt)2
∂x ∂y ∂x2 ∂y2
2 ∂y2 ∂x∂y ⇒ dZt = YtdXt + XtdYt + (dXt)(dYt).
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