144 Frequently Asked Questions In Quantitative Finance
the logarithm ofJis Normally distributed with standard
deviationσ′and if we write
k=E[J−1]
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).
In the above
λ′=λ(1+k), σn^2 =σ^2 +
nσ
′ 2
T−t
and
rn=r−λk+
nln(1+k)
T−t
,
andVBSis the Black–Scholes formula for the option
value in the absence of jumps. This formula can be inter-
preted as the sum of individual Black–Scholes values
each of which assumes that there have beennjumps,
and they are weighted according to the probability that
there will have beennjumps before expiry.
Jump-diffusion models can do a good job of capturing
steepness in volatility skews and smiles for short-dated
option, something that other models, such as stochastic
volatility, have difficulties in doing.
References and Further Reading
Cox, J & Ross, S 1976 Valuation of Options for Alternative
Stochastic Processes.Journal of Financial Econometrics 3
Kingman, JFC 1995Poisson Processes. Oxford Science Publica-
tions
