Chapter 9
Risk Management of Energy
Derivatives
Lu (Matthew) Zhao
Dept. of Math & Stats, Univ. of Calgary
March 7, 2007
“Lunch at the Lab” Seminar
Outline
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Introduction
Delta Hedging
Gamma Hedging
Vega Hedging
Factor Hedging
Summary
1 Introduction
• What is risk management?
Think of it as the immunization of risk. For
example, by setting up portfolios that contain
positions in the underlying energies and energy
derivatives, it can be achieved in such a way that
the portfolio is not affected by small changes in
the price of the underlying energy and other key
variables
• Sensitivities of components of the portfolio to
changes in their valuation parameters provide the
key information for risk management
2 Delta Hedging
• Delta hedging an option
It involves dynamically trading a position in the
underlying energy contract in a way that over each
small interval of time between trades, the change
in the option price is offset by an equal and
opposite change in the value of the position in the
underlying
• Hedged portfolio: option + position in the
underlying
• Example
Suppose we short an European call option. The delta of the
c

option is F and therefore to delta hedge this position
we should buy c of the underlying forward contract.
F
If P denotes the value of the hedged portfolio, then
The change in the hedged portfolio value is zero if the
forward price changes by a small amount.
• Theoretically, in order for a perfect hedge we must
consider the changes in F to become very small
leading to
• Example
Recall the price of an European call option, given
by
Thus the delta of the option is given by
• When the forward price is low to the strike price,
the delta of the option is close to zero, reflecting
the low probability of the option finishing in the
money
• For high forward price the delta is close to 1 as the
probability of finishing in the money is high
• The delta becomes steeper as the option maturity
decreases as the probability of the option finishing
in the money becomes more sensitive to small
changes in the forward price close to the strike
price
• The hedge can be seen to work well close to the
current future price but declines in effectiveness as
the forward moves away from the current forward
price
• Question: how often should the delta hedge be
rebalanced?
• Answer: it is not terms of a time interval but in
terms of how much the underlying price has
moved from the level at which the hedge was
established
• In practice, every time the hedge is
rebalanced, costs are incurred in trading in
the underlying asset. Efficient hedging
requires an appropriate trade-off between
risk reduction and trading costs. (Monte
Carlo simulation analysis)
3 Gamma Hedging
• One way to view the declining effectiveness
of the delta hedge is that the delta hedge is
sensitive to changes in the underlying asset.
• The closer the strike price is to the current
underlying price, the more severe the
problem is
• We can solve this problem by neutralising the
sensitivity of our delta hedge to changes in the
underlying price, known as gamma hedging
• The calculation of gamma is performed in a
similar way to delta:
• For standard European futures options:
• In order to neutralize the gamma of a portfolio we must
use another option since the gamma of a forward or
futures contract is zero. We require
which implies the position that has to be taken in the hedge
option to make the portfolio delta-gamma neutral is
• Since there might be a non-zero residual
delta left, we can take a position in the
underlying asset equal to the negative of the
residual delta. This delta hedge position will
not affect the portfolio’s gamma since the
underlying asset has a gamma of zero
• By comparison with figure 9.4, the delta-gamma
hedging error is significantly smaller than the
delta hedging error for a wide range of futures
prices and thus needs to be rebalanced much less
frequently
• However, trading costs in options markets are
typically much greater than in the futures markets,
therefore it’s still important to compare the
improvement in the hedge gained by gamma
hedging with the additional cost involved in
4 Vega Hedging
• The sensitivity of an option or portfolio to
changes in volatility is called vega and can be
calculated as follows:
• In many cases a trader may want to
neutralize delta, gamma and vega. This
requires trading in two different hedging
options, and we can neutralize both gamma
and vega at the same time by solving two
equations:
• With these solutions, the residual delta can
be calculated to obtain the position required
in the underlying energy asset
5 Factor Hedging
• A general approach to hedging a portfolio of
energy derivatives based on the multi-factor
model described in Chapter 8
• Step 1
Work out how the portfolio changes in value if the
forward curve were to be shocked by each of the
volatility functions separately
• Step 2
Compute the changes in the value of the portfolio
between the downward and upward shifts of the
forward curve for each factor
• Step 3
The three changes in the portfolio can be hedged
using three different forward contracts, choosing
appropriate positions in these contracts such that
the overall change in the hedged portfolio is zero
for each factor
• An alternative and more general solution method
is simply to minimize the sum of the squared
hedging errors
• This approach can be seen as a general form
of delta hedging, which suffers from the
same problem as the simple delta hedge
discussed before
• It can be improved in a similar way as for
the simple delta hedge – by using standard
European futures options to gamma hedge
the factors
6 Summary
• Basic concepts of delta, gamma and vega
hedging for a single option position
• Multi-factor forward curve model used to
generalize the delta and gamma hedging
• Effectiveness of delta, gamma and vega
hedging
THE END
THANK YOU