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 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 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.

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

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