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Figure 2: The daily log returns of the USS price.

3 The models

The behavior of the USS price will be considered as two folds. One is that it has a

continuous sample path so that it is assumed to behave as similar as the price of the underlying

asset of Black-Scholes model (Black and Scholes, 1973):
dS   S dt   S dB . (2)
t t t t
The other is that it has jumps so that it is assumed to behave as similar as the price of the
underlying asset of Merton Jump Diffusion (MJD) model:

dS   S dt   S dB   y   1 S dN . (3)
t t t t t t t
In model (2) the price is driven by Brownian motion B where the constants  and  are drift
t
and volatility respectively. While the model (3) can be seen as if it is an improvement of model

(2) proposed by Merton (Merton, 1976) by adding an independently and identically jump part

which has Poisson distribution. Suppose that in a small interval t the asset price jumps from

S to y S . We call y an absolute price jump size. Then the relative price jump size is
t t t t
dS y S  S
t  t t t  y  1
S S t
t t