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Home arrow Business & Finance arrow Frequently Asked Questions in Quantitative Finance

Lesson 7: Feedback

Are derivatives a good thing or a bad thing? Their origins are in hedging risk, allowing producers to hedge away financial risk so they can get on with growing pork bellies or whatever. Now derivatives are used for speculation, and the purchase/sale of derivatives for speculation outweighs their use for hedging.

Does this matter? We know that speculation with linear forwards and futures can affect the spot prices of commodities, especially in those that cannot easily be stored. But what about all the new-fangled derivatives that are out there?

A simplistic analysis would suggest that derivatives are harmless, since for every long position there is an equal and opposite short position, and they cancel each other out.

But this misses the important point that usually one side or the other is involved in some form of dynamic hedging used to reduce their risk. Often one side buys a contract so as to speculate on direction of the underlying. The seller is probably not going to have exactly the opposite view on the market and so they must hedge away risk by dynamically hedging with the underlying. And that dynamic hedging using the underlying can move the market. This is the tail wagging the dog! This was quantified in Schonbucher & Wilmott (1995), important results from this work concern the role of gamma, and in particular its sign. For the following you will need to remember that hedging a short (long) gamma position is essentially the same as replicating a long (short) position.

There are two famous examples of this feedback effect:

• Convertible bonds - volatility decrease

• 1987 crash and (dynamic) portfolio insurance - volatility increase

Example When a company issues convertible bonds it does so with a discount to encourage people to buy. It would be very embarrassing if they remained unsold. This obviously presents a profit opportunity, buy the cheap CB and make some money. This is not quite that simple because if you own a CB then you are exposed to risk caused by movement of the underlying. To hedge this risk you must sell the underlying asset according to the dynamic-hedging theory. If all goes well you will then realize the profit, being on average the difference between the correct price, based on the actual volatility, and the market/issue price, based on implied volatility. So far so good. The problem arises because these days the vast majority of CBs are in the hands of banks and hedge funds. And that means if one firm is delta hedging, then so are they all.

If the stock rises, then because gamma is positive you have to sell some stock to maintain a delta-neutral position. If there are a lot of you simultaneously selling stock then this will put some downward pressure on the stock. This may or may not be enough to change its direction, that will depend on the relative sizes of the delta hedgers and other traders. But the larger the gamma the greater the pressure.

Now if the stock does fall then you will all have to buy back stock, and the pressure is now upwards. The result is that volatility can be suppressed. And that is a bit of a nuisance if you have just bought the CB for the potential profit; your actions aimed at realizing that profit cause the profit to decrease!

A simple simulation is shown in Figure 5.10. Here a long put with expiration five years is being hedged dynamically. The volatility should be 20%, and after expiration that is what you see, but before expiration the volatility is much lower. The level of the stock matters because the effect is more pronounced if the asset is around the strike, where gamma is largest.

The opposite side of the coin was seen in 1987. Dynamic portfolio insurance was partly blamed for the dramatic stock market crash. Portfolio insurance was invented in the late 1970s by Leland, O'Brien and Rubinstein and was the neat idea that if you were worried about losing money during a falling market then you could limit this by replicating a put option. As the market falls you sell, according to some Black-Scholes-like formula. This is replicating a long gamma position, or equivalently hedging a short gamma position. The opposite of the CB example. Now as markets fall you have to sell, putting downward pressure on the stock, increasing the rate of fall. Again, whether this matters depends on the ratio of insurers to other traders. In 1987 this ratio turns out to be big enough to cause or at least exacerbate the crash.[1] Again

Simulation when hedging long gamma.

Figure 5.10: Simulation when hedging long gamma.

the result was to cause precisely that event against which it was supposed to insure! Figure 5.11 shows a simulation. This uses the same method as in the previous figure, just with a sign change because here it is a positive gamma that is being replicated.

The simple lesson here is that derivatives do not make a zero-sum game, they can be beneficial or harmful, affecting the underlying via dynamic hedging or replication. Maybe you care about your impact on the world's economy, maybe you don't, but you should definitely look deeper into the impact on your P&L.

Simulation when hedging short gamma.

Figure 5.11: Simulation when hedging short gamma.

  • [1] This was not entirely Leland, O'Brien and Rubinstein's fault. Others in the market knew about this idea and were using it as well.
 
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