Estimation in Financial Models - RiskLab

wonder why people still use the Black-Scholes formula, since it is based on
such simple assumptions { unrealistically simple assumptions". Black [11]).
In real life volatility is not constant at all. It is non-uniform, that means on
days with major economic events volatility is usually higher than on other
days (especially on non-trading days). In addition, sometimes stock prices
jump which can be thought of as a sudden large increase in the stock's
volatility. Furthermore many economic time series have a mean-reversion
tendency: when the value of a random variable reaches a very high level then
it will rather go down than up and vice versa. Put dierently, it tends away
from extremely high or low values and reverts to some long-term mean.
All these observations lead to the assumption that volatility is a random variable
and a lot of stochastic volatility models have been recently introduced
in the literature.
Now the question arises in which way stochastic volatility 2 t = 2 (t; !) can
be modeled. The standard framework assumes the volatility specication
dv = m(v) dt + k(v) d ~W t ; (1.3)
with v = 2 t or v = ln 2 t and ~ W t a Wiener process. Out of the variety of
stochastic volatility models we will consider the following two:
I: dH t = ( t , 2 t
2 ) dt + t dW t
dv t = ( , v t ) dt + d~W t ; with v t =ln 2 t (1.4)
II: dH t = ( t , 2 t
2 ) dt + t dW t
dv t = b (a , v t ) dt + c p v t d ~W t ; with v t = 2 t (1.5)
where ; ; ; a; b and c are xed constants, W t and W ~ t are independent
Wiener processes.
In the model (1.4) volatility, ormore exactly ln 2 , is governed by an arithmetic
Ornstein-Uhlenbeck (or rst order autoregressive) process with a meanreverting
tendency. A large and growing literature treats this case as an
empirically relevant one (see Bollerslev, Engle and Nelson [14], Stein and
Stein [72], Wiggins [75]). In (1.5) a square root diusion model for stochastic
volatility is suggested. For this model see Ball [1] and literature given there.
One further reason why we have restricted our attention to the models (1.4)
and (1.5) above is to be able to show explicitly how certain statistical estimation
procedures are to be implemented and also compared and contrasted
in a specic nance context. Most of the techniques introduced do apply to
much more general set-ups.
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