Consider a model for security price evolution that superimposes random jumps according to a Poisson process on a geometric Brownian motion (GBM). Specifically, let S(t) denote the price of the

Consider a model for security price evolution that superimposes random jumps according to a Poisson
process on a geometric Brownian motion (GBM). Specifically, let S(t) denote the price of the security at
time t, and suppose that
N(t)
S(t) = S*(t) [ Ji,
i=1
where t 2 0, S*(t) is a GBM with volatility parameter o and drift parameter . Assume that jump sizes
Ji are independent and identically distributed random variables with mean zero. Also, assume that Ji’s,
the GBM, and the Poisson process N(t) with parameter > > 0 are independent from each other in the
sense explained in Section 8.4. (IT; J; is defined to equal 1 when N(t) = 0.) Show that risk-neutral
probabilities for the security’s price evolution will result if the intensity parameter of the Poisson process
A is given by A = p + o’/2 -r, where r is the risk-free rate and is assumed to be constant. nt: show
that under 1 = u + 02/2 -r, we will have E[S(t)] = S(0)ert.

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