STOCHASTIC METHODS IN FINANCE 2021–22 STAT0013
Exercises 5 – Brownian motion
1. a) Before observing any of the path of a standard Brownian motion, Bt, (i.e., considering only the fact that B0 = 0), what is the mean and variance of the Brownian motion at time T ?
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b) Given that we now observe that B1 = k, for a constant k, then what is the mean and variance of B3?
2. Consider a process {St} that follows the SDE: dSt =μdt+σdBt
i.e. follows generalised Brownian motion. For the first three years, we have that μ = 2 and σ = 3; for the next three years, μ = 3 and σ = 4. If the initial value is S0 = 5, what is the distribution of S6?
3. A company’s cash position, measured in millions of pounds, follows a generalised Brownian motion, with a drift rate of 0.5 per quarter and a variance rate of 4 per quarter. How high does the company’s initial cash position have to be for the company to have less than a 5% chance of a negative cash position by the end of one year?
4. Determine whether Zt := −Wt is a Brownian motion, if Wt is itself a Brownian motion.
Hint: Are the increments independent? How are they distributed? Are the paths continuous? Is it true that Z0 = 0?
5. The partial differential equation (PDE)
∂f = 1 ∂2f
is called the (forward) diffusion equation – or sometimes the heat equa- tion – and is a model of the flow of heat (or diffusion of gas) in a con- tinuous medium, in one dimension. Here t is time, x is distance and f(x,t) is the temperature at time t and at a position given by x.
You are reminded that the density of a normal distribution with mean μ and variance σ2 is
1 (x−μ)2 f(x) = σ√2π exp − 2σ2 ,
for−∞
6. Show that for a Brownian motion {Wt} we have that E [ WsWt ] = min(s, t)
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