280 Frequently Asked Questions In Quantitative Finance
Asymptotic analysis If the volatility of volatility is large
and the speed of mean reversion is fast in a stochastic
volatility model,
dS=rS dt+σSdX 1 and dσ=
p−λq
dt+
q
√
dX 2
with a correlationρ, then closed-form approximate
solutions (asymptotic solutions) of the pricing equation
can be found for simple options for arbitrary functions
p−λqandq. In the above model the represents a
small parameter. The asymptotic solution is then a
power series in
1 /^2.
Sch ̈onbucher’s stochastic implied volatility Schonbucher begins ̈
with a stochastic model for implied volatility and
then finds the actual volatility consistent, in a no-
arbitrage sense, with these implied volatilities. This
model calibrates to market prices by definition.
Jump diffusion
Given the jump-diffusion model
dS=μSdt+σSdX+(J−1)Sdq,
the equation for an option is
∂V
∂t
+^12 σ^2 S^2
∂^2 V
∂S^2
+rS
∂V
∂S
−rV
+λE[V(JS,t)−V(S,t)]−λ
∂V
∂S
SE[J−1]= 0.
E[·] is the expectation taken over the jump size.
If the logarithm ofJis Normally distributed with
standard deviationσ′then the price of a European
non-path-dependent option can be written as
∑∞
n= 0
1
n!
e−λ
′(T−t)
(λ′(T−t))nVBS(S,t;σn,rn),
