4.1. Introduction

In the last chapter, we developed a model for stock price movements and used this to deduce a unique necessary arbitrage-free price. One might think this was the end of the story for vanilla call and put options - what else is there to say? Note that one consequence of the last chapter's arguments is that by investing the cost of an option, one can reproduce precisely the value of the option at pay-off. Given that this is true, why bother to buy options at all? Instead of purchasing the option, why not just carry out this dynamic replication strategy? In this chapter, we attempt to answer these questions and look at the practicalities of option hedging.

The first thing to observe is that even if our model holds for a bank with its easy access to the markets, it does not follow that it will hold for a general individual, and so one could view buying an option as outsourcing the hedging strategy to an institution better suited to carrying it out. There is also an economy of scale - hedging many options together is not much harder than hedging one as all the Delta hedges will add together and possibly even cancel each other,

4.2. Trading Vol

More fundamentally, it is important to realize that no model is perfect and it will never describe the market perfectly. Where are the imperfections in our model? Given certain parameters, our model produces a price. These parameters are strike, time to maturity, interest rates, current stock price and volatility. One of these parameters is vastly different from the others: all except volatility are either specified by the option contract or are observable in the market, A consequence of this is that if you call a trader and ask for a price on an option, he will not quote you a sum of money, instead he will quote a volatility or "vol" as traders typically say. Actually he will quote two vols, the price to buy and the price to sell, A "market maker" is expected to quote two prices, or vols, which are close together so the purchaser can be sure that he is not being cheated. An element of psychology comes in at this point, as the market maker tries to guess whether the purchaser wishes to buy or sell and slants his prices accordingly. Note that the difference between the two quoted prices is where the bank makes its profits. The true price is somewhere in between so whichever the customer chooses, the bank makes a small profit.

How does the trader estimate volatility? We can measure the volatility of an asset over any period in the past for which we have market data. We can therefore use the measured volatility over some past period. The problem is which past period? We could, for example, use a thirty-day average. Although the Blaek-Seholes model is based on an assumption of constant volatility it is really the average volatility that is important, or more precisely the root-mean-square volatility, A major news event will cause rapid movements in an asset's price and thus a spike in the volatility. We will use a higher value of volatility in the pricing formula if a major news event occurred in the last thirty days than if one did not. This is undesirable in that we will end up quoting a much lower vol thirty-one days after a major news event than thirty days after one, despite the fact that very little has changed. One solution is to use a weighted average ascribed to a given day decaying as it gets further in the past,

A more subtle issue is that it is not the past volatility that matters. It is the volatility that occurs during the life of the option which will cause hedging costs and the option should be priced thereby. The trader therefore has to estimate the future volatility. This could be based on market prices, past performance and anticipation of future news. It is important to realize that often announcements are expected in advance, and the information they contain will either push the asset up or down. The market knows that the asset price will move but cannot discount the information as it does not know whether it is good or bad. The options trader on the other hand does not care whether the information is good or bad, all he cares about is whether the asset price will move. Thus the anticipation of an announcement will drive estimated vols up.

The trading of options is therefore really about the trading of vol, and the options trader is taking views on the future behaviour of volatility rather than the movement of the underlying asset.


figure 4.3.1. Some possible smiles 4.3. Smiles

If we collect data on estimated vols from the prices of options and compare these to historical vols, two facts stand out. The first is that vols implied by the market prices of options are higher than historical vols. The second is that two different options on the same underlying with the same expiry date can imply different vols. Indeed, if one plots the implied volatility as a function of the strike of an option, one obtains a curve which is roughly smile shaped. The qualitative nature of the curve will vary according to the nature of the underlying but it is very rarely the flat horizontal line which the model would predict.

At this point, we would seem to have an arbitrage opportunity. Either the option with higher implied vol is overpriced or the option with lower implied vol is underpriced. We ought to be able to make some money from selling the one with high vol and buying the one with low vol. However, such smiles are a persistent feature of the markets and do not disappear with time, so the arbitrage opportunity is likely to be illusory.

How does the smile arise? Suppose a market maker spends his day buying and selling vanilla options. Each time a client calls him, he quotes a vol for buying and a vol for selling, with a little gap between them. If the first client buys from him then he wants the second client to sell. For if he sells to the first client for 11 dollars and buys from the second from 10 dollars, then he has made a riskless profit of one dollar; he will have to carry out no hedging, and he will be perfectly immune to market changes. He therefore slants his prices to encourage the second client to do the opposite of the first. If more and more clients buy then the price will get higher and higher. If more and more sell, then the price will get lower. Eventually the price will settle down when the number of buyers and sellers are similar.

The crucial point is that the buying and selling behaviour will be different for different values of strike which will drive the volatilities in different directions for the different strikes. Whilst one can hedge options for different strikes by each other, one is no longer perfectly hedged in a model-free sense, and one is taking on some risk on the basis of the imperfections in the model.

Thus the "smile" expresses the market's view of the imperfections of the Blaek-Seholes model. There are two obvious criticisms of the model; the first is that the model requires a certain hedging strategy to be carried out which is not practical, and the second is that stock and foreign exchange prices are simply not log-normally distributed.

We address the hedging issues first. The model requires the option seller to hedge his exposure by holding units of the stock, S, at any time. This quantity is known as the "Delta" of the option and the hedging strategy is known as "Delta" hedging. Therefore as the stock moves up and down, the option seller has to continuously change his holding to remain Delta hedged. In the real world this is not practical, as it takes time execute a trade. Thus it is impossible to rehedge truly continuously. As a consequence, the seller will never be perfectly hedged. Another problem is that executing a trade costs money, whilst transaction costs may be low they will never be non-existent. The more trades that are made, the more transaction costs mount up and increase the costs of hedging the option.

One interesting feature of the log-normal model of asset prices is that the total amount of wobble is infinite. What is meant by wobble? If we do not let down moves cancel up moves but instead measure the total distance the asset has moved during the life-time of a contract, we obtain infinity, (The wobble is really the variation of the function.)

This means that the model requires an infinite amount of rehedging, and thus if we allow transaction costs, the cost of rehedging will be infinite and we are clearly worse off than if we had never hedged at all. The solution is, of course, to hedge discretely rather than continuously. We only rehedge when the hedge required by our model is more than a chosen distance from the hedge we currently hold. Whilst this is effective, we are no longer in the world of perfect hedging and a single necessary price.

One simplification to the trader's position arise from the fact that he will have bought and sold many different options' contracts on the same underlying. Each one of these has a Delta and as the model is linear, the trader can hedge them all simply by adding all these Deltas together. Note that the Deltas of long and short positions will have opposite signs and so if the portfolio is a mixture of short and long positions, the Deltas will at least partially cancel each other. It is only a short conceptual distance now to start thinking about using options to hedge options. Whilst there is not a great deal of point to this if our objective is to Delta hedge, the use of options allows us more sophisticated methods of hedging.

Recall that the purpose of the Delta hedge was to eliminate risk from the portfolio, by making the derivative of the portfolio with respect to the stock price zero, A portfolio with price which has zero derivative at a point will still change in value if the asset moves a short distance from that point. However, for small price changes will be proportional to the square of the distance from the point rather than being proportional to the distance from the point as in the case of a non-Delta hedged point. As the square of a small number is even smaller, this means the change for small movements is very small.

If we allow ourselves to use options to hedge, we can do better. The second derivative can also be matched and the portfolio's change in value for small changes in price of the underlying, will now be proportional to the cube of the change which is much smaller again. The second derivative with respect to the price is called the Gamma and the process we have discussed is called "Gamma hedging,"

4.4. The Greeks

Whilst in the Blaek-Seholes model, the price of the underlying is the only variable to change in a non-deterministic way, the real world is rather different and so traders often hedge their exposure to other


Figure 4.3.2. The changes in value of a call option Delta-hedged with spot and strike equal to 100.

quantities. Note that we can compute the derivative of the price of a portfolio with respect to any of the underlying parameters, and as with the stock price, we can buy options which match that derivative. In general, if we wish to hedge k of the parameters, we will need k — 1 different options to carry out the hedging, and we will need to construct the portfolio so as to match all the parameters at once by solving a system of linear equations. If it seems circular to hedge options with options, one should think in terms of hedging complicated options with less complicated ones. The complicated option could be an exotic or simply a far out-of-the money vanilla option.

The derivatives with respect to the various quantities are denoted by Greek letters with initials corresponding to the quantity differentiated, and the derivatives are collectively known as "the Greeks." The derivative with respect to the short rate, r, is therefore called "Rho," The derivative with respect to time, t, is called "Theta." The derivative with respect to volatility is known as "Vega" or, by purists, "Kappa," on the not unreasonable grounds that vega is not a letter in the Greek


Figure 4.3.3. The changes in value of a call option Delta-hedged with spot equal to 100 and strike equal to 110.

alphabet. Vega is a very important Greek as the Blaek-Seholes assumption of constant deterministic volatility is manifestly false, and the trader will wish to reduce his exposure to unexpected changes in

Typically, the options trader will monitor all his Greeks but not necessarily continually rehedge them all. Instead he will take a view on which ones are important, and on which risks he wishes to hedge. Equally importantly, he will have a view on which Greeks he does not wish to hedge. This expresses the trader's opinions about which direction market parameters will move in, and on which parameters the trader has a firm opinion that he wishes to place money on. For example, if the trader believes that the vol will increase in the near future, he will go ''long Vega" that is he will ensure that the derivative of his portfolio with respect to volatility is positive. If he believes that vol will decrease he will go "short Vega."


Figure 4.3.4. The changes in value of a Gamma-hedged call option with spot equal to 100 and strike 110. Hedging option struck at 100.

The Greeks are also important for the risk manager. The risk manager assesses the value of the bank's portfolio and tries to estimate the probability of the bank losing a lot of money. The Greeks describe to first order, via Taylor's theorem, the effect of the values of parameter changing. If F(S, t, r, a) is the value of an option, we have to first order

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