Chapter 9 Risk Management of Energy Derivatives

1
Chapter 9Risk Management of Energy Derivatives

• Lu (Matthew) Zhao
• Dept. of Math Stats, Univ. of Calgary
• March 7, 2007
• Lunch at the Lab Seminar

2
Outline

• Introduction
• Delta Hedging
• Gamma Hedging
• Vega Hedging
• Factor Hedging
• Summary

3
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

4
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

5

• Example
• Suppose we short an European call option. The
  delta of the option is and therefore
  to delta hedge this position we should buy
  of the underlying forward contract.
• 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.

6
(No Transcript)
7

• Theoretically, in order for a perfect hedge we
  must consider the changes in F to become very
  small leading to

8

• Example
• Recall the price of an European call option,
  given by
• Thus the delta of the option is given by

9
(No Transcript)
10

• 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

11
(No Transcript)
12
(No Transcript)
13

• 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

14

• 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)

15
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

16

• 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

17
(No Transcript)
18

• For standard European futures options

19

• 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

20

• 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 portfolios gamma since the underlying asset
  has a gamma of zero

21
(No Transcript)
22
(No Transcript)
23

• 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 its still important to
  compare the improvement in the hedge gained by
  gamma hedging with the additional cost involved in

24
4 Vega Hedging

• The sensitivity of an option or portfolio to
  changes in volatility is called vega and can be
  calculated as follows

25

• 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

26
(No Transcript)
27

• With these solutions, the residual delta can be
  calculated to obtain the position required in the
  underlying energy asset

28
5 Factor Hedging

• A general approach to hedging a portfolio of
  energy derivatives based on the multi-factor
  model described in Chapter 8

29
(No Transcript)
30

• 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

31
(No Transcript)
32

• Step 2
• Compute the changes in the value of the
  portfolio between the downward and upward shifts
  of the forward curve for each factor

33

• 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

34

• An alternative and more general solution method
  is simply to minimize the sum of the squared
  hedging errors

35
(No Transcript)
36

• 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

37
(No Transcript)
38
(No Transcript)
39
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

40
THE ENDTHANK YOU