Peter Leoni

Hedging strategies for energy derivatives

In this article, we define a hedging strategy in a setting typical for the commodity market. Firstly, we prove the existence of the locally risk-minimizing (LRM) hedging strategy for payment streams in this setting. Next, a three-step procedure is described to determine the LRM hedging strategy. Then the procedure is illustrated for stochastic volatility models, as these models are a special case of the non-traded situation which frequently occurs in the commodity markets. Finally, we introduce the (adjusted) LRM hedging strategy in the non-traded setting and for this specific setting we numerically show the outperformance of this strategy compared with current market practices.

The Balance between Gamma and Theta

We will dig into a small bit of theory because it will help us to understand the concept of hedging on a deeper level. For this we will use the Taylor expansion [138]. One can show that any function can be approximated by a polynomial function [103]. The coefficients for this polynomial are determined by the derivatives at a single point (the current market conditions). In general the approximation gets better as we add more terms to it. We will come back to this concept in Chapter 6. Recall that the option price π is a function of the underlying stock S and the time t. Of course the price of the stock depends on the time of observation but for the sake of simplicity of notation we will write S instead of S (t) as the time is already explicitly mentioned in the option price.

The Greek Approximation

This chapter will go in more depth into the Taylor expansion approach [103] we introduced earlier to explain some of the balances we encountered between the Greeks. Since the beginning of time, people have learned to copy from good practice, improve their observations and grow the complexity of their skillset to tackle yet more difficult problems. Some of the greatest pioneers in science have fully mastered the ability to reverse this process and to analyse complex problems and decompose them into simpler, more treatable problems that then got solved one by one. Although the hockey-stick payout of a derivative such as a call or put option is quite simple to write down or to interpret, mathematically speaking it has become non-linear, at least in the point where the option flips from being worthless to where it starts accumulating value. We already know that this transition point makes the difference between having a biased contract with only rights and no obligations, and the purchase of a stock itself.

Delta Hedging in the Perfect World

There are many ways to look at volatility, and people working in the industry are so used to this concept that it gets used in different contexts, which is sometimes confusing at first sight to new entrants. Throughout this book, we will keep coming back to this concept and, by breaking it down to its basics, the different points of view will all be clarified.

Volatility Term Structure

So far we have talked about the Black-Scholes model and we opened the hood to display all the machinery behind that remarkable engine. We discussed the relation between the price and the cost of hedging and we highlighted all the different Greeks to explain movements of prices. So far so good. But now we want to relate this to the actual market. How convinced are traders of this model? The Black-Scholes model is wide-spread in the industry, but not necessary any more in its original form. The best comparison is the following. Say you have to travel from London to Edinburgh and you can choose any car you want. Let’s make this super attractive and dream up a sunny day with blue skies, a fuel card, no time constraints to get there and nice company in the car. You caught the picture in your head? Good, tell us now which car you would choose?

Hedging Contingent Claims

In this chapter we will present a few simple examples that show what hedging is all about. In practice, understanding how to hedge risks is the single most important actor to become a good trader. In the general public traders, and in particular traders who are working with derivatives like options, are considered to be big risk takers [117, 31]. On some occasions, history has demonstrated that options can be tricky financial instruments that can introduce huge losses for the banks or institutions trading them [149, 117, 55]. However, the flexibility that these instruments bring to the investors or professional players in terms of reducing their risk cannot be denied [58, 52.]. Therefore, it is critical that both academics and practitioners develop a thorough understanding of both the theory, the market, the applications and the shortcomings or assumptions of the various models that are being deployed.

Trading Is the Answer to the Unknown

In the first part of this book, we focused on the hedging argument of a plain vanilla option. Within the Black-Scholes model, we explained that hedging away the risk can be done quite accurately up to finite-size effect. In real life there are many difficulties [35, 133] that prevent us from believing this model is a perfect representation of the world. However, as we will see later on, these flaws in the model do not take away the valuable lessons we can learn from it [79]. Moreover as it turns out, a solid understanding of the same basic flaws in the model allow a good trader to anticipate and take position in a smart way.

Vega as a Crucial Greek

Before we can start explaining how the trading activity around plain vanilla options came about [119], we first need to explain the effect of the volatility parameter on the price of an option. Up till now, we have always assumed this parameter to be known, at least up to certain accuracy. We already established at the end of Chapter 4 that this volatility seems to change over time. However, we were talking about the realised volatility. It might be true that there are periods of low and periods of high volatility, but once the option is sold, this will become apparent only through the hedging of the option, namely through the gamma and theta balance, which will be broken.

Skew and Smile

As we explained in Section 2.3.5, the implied volatility is basically set by two things: the market price of an option and the Black-Scholes formula. It is quite important to realise that this implied volatility is entangled with the Black-Scholes model. It is the mysterious parameter σ in the Black-Scholes model such that the model price equals the market price. If the model were to be perfect, we could follow the following recipe: pick one option arbitrarily, for example an ATM option with an expiry of approximately one year (or whichever maturity is closer to the one year one). We can then go into the market and find out how much we have to pay for this option. As the only reality is the cashflow associated with this purchase, we use this cash price to then determine what the implied volatility is in the Black-Scholes model such that the model price and market price match exactly. Once we found the volatility, if the model is perfect, we are pretty much set. After all, the σ parameter determines how much the underlying price will move up and down, at least statistically speaking. Theoretically, we should be able to price any other option, even if the strike is different. Admittedly, as we saw before in Chapter 7, one can understand that implied volatility can and in fact should depend on the hedging period, but at least for identical maturities one would expect this argument to hold.