p is the instantaneous correlation between the asset price and its volatility.
Hence, in the usual notation (we make use of dWf = ʌ/l — pdWtr+pt dWtv)
the forward price F is given by
dFt = Ftσf (√l — p2λ<1 >(t) + λ<2>(t)ft) dt + Ftσζ dW,r , 0 ≤ t ≤ T,
(16)
where Wf and Wv have correlation p.
It is important to notice that even though the information process and its
volatility process are uncorrelated, the asset price process and its volatility
process are correlated. This is in contrast to an assumption in many sto-
chastic volatility models (see for example Hull, White [18] or Stein, Stein
[31] ) and has already been criticized by Pham and Touzi [27]. Corollary 1
establishes conditions for p ≡ 0.
Corollary 1 The correlation p is zero if and only if
(T) σy = 0
or
(TT) ʌɑ = = 0 a.s.
Condition (T) is trivial since it implies that the volatility of the information
process is not governed by a Brownian motion. Condition (TT) implies that
the correlation p is zero, if ʌɑ∖ the risk premium relative to the source of
uncertainty Wʃ, is zero. Thus Corollary 1 gives an economic foundation
for stochastic volatility models where the asset price and its volatility are
correlated.9
Corollary 2 Assume that the market price of risk ʌɑi is positive (resp. λ^
negative) and σv > 0. Then the correlation p between the forward price and
its volatility is negative (resp. positive).
This result is obvious from equation (15). Ruling out the implausible case
that the market price of risk is not positive, we can conclude that the corre-
lation is negative. Thus, our analysis supports the usual result in empirical
studies that volatility and price of an asset are negatively correlated. But
our argument is inverse to the argument which bases on the leverage effect.
The intuition of Corollary 2 is as follows: For a higher volatility σ1t, hence
a higher risk, at time t a risk averse investor requires a higher reward. And
9See for example Schobel, Zliu [30] and Heston, Nandi [15].
18