Advances in Risk Management

GIUSEPPE DI GRAZIANO AND STEFANO GALLUCCIO 357

18.3 Analytical expression of the total hedging error

Consider the following situation: a trader sells at timeta European call
option struck atKexpiring atT>t, with payoffC(ST)=(ST−K)+. We will
first assume that the agent believes the BS model to be correct. In order to
hedge his exposure, he will form a dynamic portfolio consisting oft′shares,

wheret′=∂∂SC|t=t′is the option “Delta”, and invest an amountψt′=Ct′−βtt′St′

in the money market account fort≤t′≤T. If the “true” market dynamics
were lognormal with constant coefficient as postulated in the BS theory, the
agent would be able to perfectly replicate the option price and the hedg-
ing error would be identically zero (in a frictionless market). In general,
the replication error incurred by using the simple delta hedging strategy
does not vanish since, in particular, asset prices do not follow lognormal
processes.
In two separate works Carr and Madan (1997) and, independently, El
Karoui, Jeanblanc and Shreve (1997) proved that if the asset follows a contin-
uous diffusion process (with possibly time-dependent, adapted coefficients)
then the simple delta hedging strategy based on the BS model would result
in a total error:

Yt,T=

∫T

t

βT−s

S^2 s

2

∂^2 Cs

∂S^2

(σ^2 BS−σ(s)^2 )ds (18.3)

at timeT, whereσBSis the BS volatility corresponding to the price at which
the trader bought the option andσ(t) is the actual (or “realized”) volatility
process. The above fundamental formula indicates that the total hedging
error is a function of:

1 The difference of the squares of the volatilities. In particular the trader

realizes a gain if the realized volatility path is below the BS volatility and

to lose money in the opposite case.

2 The option’s Gamma∂

(^2) CS
∂S^2 .Ceteris paribus, the larger the Gamma the more
pronounced is the hedging error.
Although this result is strikingly simple and intuitive, the hedging strategy
is too simple to be realistic. For this reason we provide here a different
formula that postulates a “true” dynamics for the underlying asset as in
equation (18.1) and a hedging model at the trader disposal as in equation
(18.2). Moreover, we will also allow market agents to hedge their volatility
exposure. For the sake of simplicity we will limit ourselves to provide the
error formula and indicate a sketch of its proof. The interested reader will
find all details (in a more general setting) in Di Graziano and Galluccio
(2005).