# Science in Finance IV: The feedback effect

First, many of those trading derivatives are hedging with the underlying and this can affect the behaviour of the underlying: hedging positive gamma can decrease volatility and hedging negative gamma can increase volatility. When hedging positive gamma (i.e. replicating negative gamma) as the price rises you have to sell more of the underlying, and when the price falls you buy back, thus reducing volatility if your trades are in sufficient size to impact on the market. But hedging negative gamma is not so nice, you buy when the price rises and sell when it falls, exacerbating the moves and increasing volatility. The behaviour of stocks on which there are convertible bonds is often cited as a benign example, with the rather more dramatic '87 crash, replicating a put i.e. hedging negative gamma, as the evil version. (See Wilmott, P. and Schonbucher, P 2000 The feedback effect of hedging in illiquid markets. SIAM J. Appl. Math. 61 232—272, also PS’s dissertation.) You will probably find some reluctance for people to sell certain derivatives if they are not permitted to dynamically hedge. (Not that it works particularly well anyway, but that is what people do, and that is what most pricing theory is based on. Static hedging with other derivatives is better, and does not cause such (in)stability problems.)

(We’ve had newspaper headlines about damage done by excessive risk taking, whether by single, roguish, individuals or by larger institutions such as hedge funds, or banks and corporates investing in products they don’t fully understand. I expect it won’t be long before the attempt to *reduce* risk is the cause of similar headlines!)

Second, with the leverage available with derivatives it is possible, and actually rather simple, for people to trade so much as to get themselves into a pickle when things go wrong. This has many consequences. For example a trader loses his bank so much money that the bank collapses or is taken over, job losses ensue and possibly the man in the street loses his savings. Is wealth conserved during this process, as would be the case in a zero-sum game? I think not.

Of course, we don’t know what proportion of derivatives trades are being used for hedging, speculation with leverage, etc. and how many are being dynamically hedged. But while derivatives trading is such a large business and while pricing theory is underpinned by dynamic hedging then we can say that the game of derivatives is not zero sum. Of course, this should spur on the implementation of mathematical models for feedback…which may in turn help banks and regulators to ensure that the press that derivatives are currently getting is not as bad as it could be.

P