Imagine you are given a choice between receiving a certain amount of money or taking a bet which has a 50 percent chance of winning $100 and a 50 percent chance of winning nothing. Clearly the bet has an expected value of $50. If you would rather receive a certain payoff of less than $50 you are risk averse. If you would take $50 you are risk-neutral. If you would only accept more than $50 you are risk-seeking.

The certain amount that you would settle for is the called the certainty equivalent. If we plot a graph of certainty equivalent as a function of the bet size we can visualize our utility function. For a risk-averse trader the utility curve will have downward curvature. This means that the utility of a given amount of money is greater than the amount of money itself. This can be seen in Figure 4.1. Conversely, the utility function for a risk-seeking

trader curves upwards. The utility of a given amount of money is less than the amount of money itself. Such a utility curve is shown in Figure 4.2.

The two most important aspects of (sensible) utility functions for a trader are that they slope up because we prefer more money to less, and they curve downward because we become more risk-averse when larger amounts are involved.

We can quantify our risk aversion by using the Arrow-Pratt absolute risk aversion, which is defined by r = UW (41)

A commonly used utility function is the exponential utility function. It has the functional form

It is unique in having constant absolute risk aversion as r = y is independent of the wealth, W.

Let's look at an example to see how we would find our risk tolerance. We do this by asking questions about the certainty equivalents of risky outcomes that are normally distributed.

Suppose the distribution of future wealth has a mean ¡i and a standard deviation a. For example, i = $10,000 and a = $2,000.

FIGURE 4.2 The Utility Function for a Risk-Seeking Trader

Money

FIGURE 4.2 The Utility Function for a Risk-Seeking Trader

(continued)

E [U] = E [- exp (-y W)] (4.3) ^-exp(-y(v - 2(4.4) So the certainty equivalent, W0, is given by

Wo = - 1y^2 (4.5) This can be inverted to give an expression for the risk aversion, X.

Now suppose that the trader is indifferent between this distribution of wealth and a certain outcome of $8,000; that is, W0 = $8,000. Here

We can go through this process for various wealth levels and various distributions. Quite a large amount of variability tends to be found. Real traders tend to exhibit little consistency in their risk appetites, either through time or as a function of the amount of the money at stake. Many behavioral finance studies directly address this point (Kahneman and Tversky 1979; Barberis et al. 2001).

For this and other reasons, the use of utility functions in economics has been subject to criticism (Mirowski 1989; McCauley 2004). This needn't overly concern us. As with our use of the BSM model, utility is just a framework for us to think within. While a trader may never specifically know his utility function and while his risk aversion may change, it isn't unreasonable to state that he prefers more to less and is risk-averse. This is all we really need.

Hodges and Neuberger (1989) recognized that as BSM really prices the replication strategy rather than the option directly, they could do the same thing but incorporating transaction costs. Actually, including transaction costs was first done by Leland (1985) but, while his results led to pricing the option with a modified volatility to allow for the costs of hedging, we still need to continuously adjust the hedge. So this doesn't address the practical problem of when best to hedge.

The most important idea in Hodges and Neuberger's paper is that there comes a point where the option trader is indifferent (in the utility sense)

to holding the risk associated with the mis-hedged option and the cost associated with hedging it. If we can specify our level of risk aversion, this strategy is optimal. They formulated the problem in terms of maximizing the exponential utility function, but it was subsequently proved (Davis et al. 1993; Andersen and Damgaard 1999) that the results are basically independent of the precise form of the utility function. As mentioned in the accompanying box on utility, we would be justified in being suspicious of any result that didn't have this property.

The mathematics required to formulate the problem are beyond the scope of this book. Sadly, the resulting valuation equation has no analytic solution and needs to be solved numerically. Even this is not easy. The required computations are prohibitively time consuming. There is no practical way to use the Hodges-Neuberger (HN) methodology as a real-time hedging guide.

But because it is the optimal solution, its properties are important to understand. Figures 4.3 and 4.4 show the hedging bandwidth for a long call position and a short call position. When our position's delta moves outside the band, we hedge to bring our delta back to the edge of the band. (This is only true if we assume that the counterparty in an option trade has no ability to predict the direction of the underlying. There is some evidence that option customers are partially informed. In this case we should immediately hedge the delta of any new option trade to the BSM delta. However, subsequent delta rehedging would follow the HN scheme and we would only hedge if we moved outside the bands. It probably wouldn't be wise to overhedge the original trade to achieve the same directional risk as the counterparty, unless you had a very clear idea of the magnitude and duration of the customer's directional trading ability. This would require

Call Delta

FIGURE 4.3 Optimal Hedging Bands for the Long Call Option as Functions of the BSM Delta (Dashed Line)

Call Delta

FIGURE 4.3 Optimal Hedging Bands for the Long Call Option as Functions of the BSM Delta (Dashed Line)

FIGURE 4.4 Optimal Hedging Bands for the Short Call Option as Functions of the BSM Delta (Dashed Line)

FIGURE 4.4 Optimal Hedging Bands for the Short Call Option as Functions of the BSM Delta (Dashed Line)

extensive analysis.) The parameters chosen are not particularly realistic but have been chosen to exaggerate the properties of the solution. We chose the case of a one-year option with volatility of 0.3, transaction costs of 2 percent, zero interest and carry rates, and a risk aversion of one

Several points are immediately obvious. First of all, the short and long positions need to be hedged differently. The band for the short position is narrower. So we hedge our short positions more defensively. We have time decay on our side so we take less chance with delta, whereas when we are hedging long positions we need to let our deltas run. Interestingly, this is in accordance with trader folklore.

Another way to see why this should be so is to note that the hedger of a long position sees a different level of volatility to that seen by the hedger of a short position. As the underlying reaches a new high, a short gamma hedger will be inclined to buy. In doing so, he will make the high even higher by paying the offer. In contrast, a seller sells at the bid, slightly lower than the high. The cumulative effect of crossing bid/ask like this means that the long and short positions have different volatility levels to contend with.

This was the central result of Leland's. He showed that the adjusted volatility for a long option position was

where

X is the proportional transaction cost At is the time between rebalancing

For a short option position, the adjusted volatility is a = a a = a a V n At

While we do not explicitly use Leland's results to determine our hedging strategy, these are very important results. Before doing any option trade, we need to be aware approximately how much edge will be eroded by cumulative delta hedging. This effect can be considerable, particularly for low-volatility, illiquid stocks. For example, if fair volatility is 10 percent and the bid/ask spread is 1 percent, an option that is rehedged daily would need to be sold at an implied volatility of over 15.9 percent to account for this.

Additionally, the optimal delta band does not span the BSM delta. A perfectly hedged portfolio in the BSM world may need to be adjusted when transaction costs are present. This is also consistent with Leland's observation because if an out-of-the-money call sees a higher volatility, due to the dependency of the option's delta on volatility, it acts like it has a higher delta. Similarly, an in-the-money option acts like it has a lower delta. This causes the hedging band to be centered around an S-shaped modified delta, rather than the BSM delta.

The preceding analysis was done for European options but the general idea can be extended to cover American options. This is similar to the BSM model, where the partial differential equation is general but the methods used to solve it are dependent on the exact boundary conditions. This caveat also applies to the other models we look at. We can normally expect the results for American options to be similar to those obtained for European options (in most cases Americans can be thought of as Europeans).

Not evident from Figures 4.3 and 4.4, but the essential feature of the model, is that the width of the hedging band is dependent on the risk aversion. A large risk-aversion parameter means that the trader wants to accept little risk. So he wants tight hedging bands and will hedge often. Conversely, a trader with a small risk-aversion parameter will be prepared to hedge less frequently, accepting risk to reduce hedging costs. Neither of these choices is more correct than the other. As with all hedging methods, we need to decide how risk-averse we are. But given this choice, the HN formalism gives us the optimal balance between risk and reward.

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