The Greeks and Hedging Explained

Peter Leoni

• Publisher: Palgrave Macmillan
• Year: 2014
• ISBN:
  9781137350749
• Topic: Business
• Subtopic: Financial Engineering
• Index: Business

A practical guide to basic and intermediate hedging techniques for traders, structerers and risk management quants. This book fills a gap for a technical but not impenetrable guide to hedging options, and the 'Greek' (Theta, Vega, Rho and Lambda) -parameters that represent the sensitivity of derivatives prices.

1	Hedging Contingent Claims

1.1 Introduction

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 factor 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.

As we progress through the different chapters, we will develop a deeper understanding of how a trader looks at various aspects of these instruments. There will be different steps of understanding. First of all, the fact that the price of such an instrument has a life of its own should never be neglected. Usually there is a model price that sets a range of prices, but as the instrument gets offered in the market, depending on the appetite of the other participants, the price might be pushed to its boundaries. Two observations need to be made here that are vitally important. First, the model starts from a set of assumptions. Usually, most of these assumptions can be challenged, as we will see throughout this book. We will start from a very strict set of assumptions, and then gradually relax them, leading the reader more and more in the direction of the real market. It can and has happened that a certain model [18] becomes the market standard although the basic assumptions are violated. If the market is well developed, this does not have to be a problem, because most participants know how to adjust their pricings to accommodate for this deficiency. This is the case for plain vanilla call and put options and the Black–Scholes model. We will discuss this in detail through the next chapters. The model can still provide a guideline to help the trader set his price. We will argue later that the real price of a derivative is determined by the cost of hedging and the residual risk. Both depend on the strategy the trader wishes to follow once the instrument becomes part of his book.

The second observation to bear in mind is that there can be situations where the model’s assumptions are wrong in the sense that they do not capture all of the essential risks. The situation can become even worse when this particular model becomes the market standard, leading to a misunderstanding of the price of the instrument by some participants [47]. This has been the case in the credit market with the valuation of products such as Collateralised Debt Obligations (CDO) [134]. It is important not to get a false sense of comfort because a model is available. When the boundaries of a model are well understood, it usually is a very useful tool [144].

This means that at all times the model should be questioned. However, how to model derivatives in more advanced ways will be beyond the scope of this book, as there are many books available that discuss in high mathematical detail how to set up different models [96, 22, 17, 36], depending on the particular asset class, instrument or market feature one is trying to capture [133, 122, 44, 124].

The focus of this book is to explain, once the risk has been taken by the trader and a transaction in this instrument is done, how to manage this risk properly. This brings us to the next step of understanding. We will start by setting up laboratory conditions such that it becomes possible to reduce the risk to almost zero. This is how a derivatives trader looks at his book, containing all different risks, at least as a first step. In principle his starting point is to manage and reduce, and if possible eliminate, the risks. For this, he will set up a trading strategy and typically the cost of this strategy should correspond to the price of the financial instrument.

Furthermore, the trader will take more active positions and challenge the current market levels of both the underlying asset and the model parameters. At this point, the trader starts taking on a risk. In later chapters we will show how the market adjusts every day, based on new information that becomes available. This flow of information leads to more insight and can lead to the understanding that the initial assessment of the price for the model parameters were not entirely correct and should be adjusted.

In this book, we will always look at the viewpoint of the trader who is willing to buy or sell any financial instrument to provide service to a client. The motivations for the client are endless and can go from reducing his risk through securing margins to speculation. Let us give an example of each one of these situations.

A pension fund may have built up huge positions in the stock market and wish to protect themselves from a short-term drop in the equity market. Rather than unwinding their entire portfolio, which would also mean giving up the upside potential, the fund manager can decide to buy an insurance against a sudden drop. For that, he will pay a premium that would pay back any drop below 10 per cent of the stock market. The financial instrument that the fund manager would buy from the trader is a put option, as we will define later.

A second example is a company that is located in Europe but exporting to the US. After signing the contract for delivery of the goods at a certain price, the company knows that he will receive a future cashflow in dollars, whereas his costs are in euros. In the negotiation, the current level of the FX rate has been taken into account, but if the dollar weakens the transaction might lose its profitability. Therefore, the company might wish to fix the current level of the FX rate and turn to a trader to step into a forward FX contract, ensuring him that at the future date when the client pays for his goods, he can exchange the dollar transaction into euros at the current levels.

A third example would be a speculant that enters into the market purely on a direction view. He might think that the stock market is currently at a historical low level and, rather than investing directly into the stocks, he chooses to invest into call options on the stocks. Typically the premium to such an option is relatively low compared to the stock price and the leveraged returns that are possible are enormous. However, there is a risk that the premium gets lost and the investor loses all of his investment.

However, the trader is in yet another position, as he offers this flexibility to any of the above participants, but he cannot afford to bet on the market like the speculant is doing, nor does he have the underlying portfolio to cover for the insurance in case the pension fund managers call upon his claim. This brings us nicely to the need for hedging. The trader will take over the risk and use his insights and large scale to hedge the risks, securing his own viability.

1.2 Setting Some Notation

Throughout the book, we will always refer to the price of a stock or index as S and, when relevant, we will explicitly mention the time dependence (or observation time) by using S (t). Typically we will omit the time dependence if we are focusing on the situations where the time is fixed as a moving component. The current time will be denoted as t₀ and hence the current price level is denoted as S (t₀). Very often we will want to use the short-cut notation S₀ = S (t₀) if we want to focus more on the initial level than on the time aspect. In many applications there will be future expiries or time horizons. We will denote these by a capital T.

The notation t and T are used to refer to specific times. Very often we will need the duration which we will denote as τ = τ (t, T) = T – t and it refers to the duration from t to T. In most of the theoretical considerations we will just use the short notation τ but in practical applications we will elaborate a bit on t...