Hedging Strategy and Electricity Contract Engineering - IFOR

dS t R SXQ – lnS tV S t dt F S t dW t G
70 Risk management
as the oil market in particular, despite the fact that electricity also differs substantially
from other commodities.
Black - 76 In the electricity market the concept of being able to replicate
options by continuously trading the underlying asset is unrealistic, since spot
electricity is non-storable. Instead it was argued that the option could be replicated
by trading futures on the spot. Futures naturally does not invoke any
storage problem, because they are simply financial papers. Hence the analytical
pricing formula by Black [Bla76] for options on forwards is frequently used
in the electricity market [Won01]. The spot price is assumed to follow a simple
GBM with a drift under the martingale measure Q equal to the risk free interest
rate minus the so-called convenience yield 13
dS t S r • V S t dt F S t dW t
P (3.7)
Schwarz The constant volatility assumption of the Black model is however
not consistent with empirical observations that longer dated forwards are less
volatile than short dated forwards. Schwartz [Sch90] introduced a more realistic
model, where mean reversion is introduced in the spot price process
– where is the market price of risk. The volatility of the forwards prices are then
given F by S F G t V F T e j H
T I tL , which better describe the empirical results of
commodity forward prices.
I
Gibson and Schwartz The Schwartz model (and the Black-76 model) is a
single factor models, since only one factor is stochastic, namely the spot price.
One of the deficiencies of that model is the simple volatility structure and its
convergence to zero with increasing maturity. This can however be resolved
13 Brennan [Bre91] defines the convenience yield as
. . . the flow of services which accrues to the owner of a physical inventory but
not to the owner of a contract for future delivery