Popular derivative pricing theory begins with assumptions about perfect, frictionless markets. All markets, however, are illiquid to a certain extent. Individual buy orders tend to make prices rise and sell orders tend to make prices fall. Even where the size of an order is too small to change the bid-offer spread in a security, any market (non-limit) trade reduces both the remaining order depth and, as a result, the minimum size needed for the subsequent order to cause a change in price.

Feedback is produced in markets by a change to price (or order depth) which:

* causes a modification to the way a market responds to

an existing order placement pattern or

* influences market participants' further order placement

behaviour

When a feedback mechanism accentuates a small change in the initial quantity, it is known as positive feedback. Conversely, when feedback counteracts a small change in the initial quantity, it is known as negative feedback.

**Positive Feedback**

Markets often demonstrate some level of short-term positive feedback between order placement, movement of prices and further order placement, even in the absence of non-linear derivatives such as options. For instance, where one large sell order significantly depletes the order book on the bid side of a market, further sell orders will have a relatively greater impact on price, causing a potential increase in short-term price volatility until new bids allow the market to recover equilibrium. Senior traders tend to reduce desk traders' limits through periods of high volatility thereby reducing overall market liquidity and increase limits during liquid, trending bull runs. This collective change in order placement behaviour is an amplifier of feedback.

Other interesting feedback effects can be predicted and observed with non-linear derivatives. Traders will be familiar with the gamma peaks and wells visible on 3D option risk maps. These represent potential 'trouble areas' through which their market exposure, delta, can change rapidly depending upon the path the underlying market takes. Where the option traders hedge their delta in a systematic fashion, these contingent delta trades can be viewed as an overlay of any pre-existing characteristic order placement function.

Although some consider derivative deals as a zero-sum game, the two parties involved in a trade usually vary in investment horizon or risk appetite. For those with small positions, long-term investment horizons or large trading limits, a large change in underlying price may not change their order placement behaviour. Those with large positions, small trading limits or short-term views are likely to modify their behaviour throughout the trading day.

Where trading horizons are mismatched it is common for delta hedging induced by price movement to have a feedback effect upon the underlying security.

**Following The Market**

To illustrate, when warrants on the CAC40 first became popular with retail investors, noticeable price acceleration in the direction in which the market had moved that day started to occur in advance of the close of the cash market. Local traders quickly spotted a pattern and started to trade in the direction of the market before the closing rush. Most of the retail investors that had purchased the warrants appeared to have a long term investment horizon. Most of the option traders re-hedged their delta at the end of the day. The investors were 'long gamma' but were seemingly unconcerned with day-to-day movements and so did not hedge themselves (if, indeed, they had the ability to do so) whereas the option traders were competing with each other to sell on 'down' days and buy on 'up' days. Due to the finite liquidity within futures markets, large warrant positions caused an increase in short-term volatility.

Where institutional investors with multiple-year investment horizons effectively become 'short gamma' through the use of derivatives, a reverse effect (negative feedback) should occur. The institutions' counterparties (usually investment banks) have to sell as the market rallies and have to buy as the market falls, damping the market's natural volatility. For example, convertible bonds usually end up being issued at low implied volatility because traders know a damping effect is likely to follow.

Many U.K. money managers hedge their traditionally 'long only' equity funds by entering into low or zero cost collars with investment banks, where the investment bank buys out-of-the-money calls and sells out-of-the-money puts. Given the size of the funds involved, the notional size of the derivative contracts traded is often extremely large relative to the daily volume in the underlying market. These positions can cause both positive and negative feedback.

How can this effect be modelled? A simple approximation is to

1. perturb P_{0} to P_{1} using the standard diffusion equation; and

2. calculate the change in total delta caused by this shift; and

3. increase (or decrease) the shift to compensate for the price

action caused by the hedgers.

If a security with pure liquidity is lognormally distributed with initial price P_{0}, price after time t equal to P_{1}, volatility * and a growth rate of r then, over a short time period t, its price evolution can be informally described as follows:

P_{1} = P_{0} exp [ (r**-** *^{2} /2)t + **(check)( t)]

(* is a random sample from a standard normal distribution).

If, in addition, *_{0} is the total initial delta held by active hedgers across the market (e.g. 0 in the delta neutral case), *_{1} is the total delta after time t and f is the logarithmic price impact function over the net delta (assuming hedgers wish to return to a delta neutral position), then we can say:

P_{1} ~ P_{0} exp [ (r **-** *^{2} /2)t + **(check)( t) + **f**( *_{1}- *_{0}) ]

Where the logarithmic price impact function is linear with a constant **k** then we can say:

Ln(P_{1}/P _{0}) ~ (**r -** *^{2}/2) t + **(check)( t) + **k*** _{1}-**k** *_{0}

If * is the gamma held by passive investors (here the opposite of the gamma held by active hedgers) and L is a liquidity constant greater than 0 (which is high for liquid securities), this can be approximated over certain ranges of the variables by:

Ln(P_{1}/P _{0}) ~ (**r -** *^{2} /2)t + **(check)( t) + *(P_{1}- P_{0})/L

Where the market for a security is illiquid, or has a high level of gamma, the market impact estimated above will have a noticeable effect on price evolution.

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**Example Trade**

Figure 1 shows a simulation of the daily price action within a market in which a large collar constructed from a 90% strike put and a 105% strike call has been traded between a dynamically hedging bank and a passive investing institution. Each random normal perturbation (diamond ordinates on the graph) of the market is overlain by the price impact caused by delta hedging. We assume that the price impact is linear although non-linear models can often be approximated by linear models within certain ranges of the variables. When the bank is short gamma (close to or below the put strike), price falls cause the bank to sell and price rises cause the bank to buy (positive feedback), exaggerating the movements as shown to the left of the graph (square ordinates).

When the bank is long gamma (near and above the call strike), price rises cause the bank to sell and price falls cause the bank to buy (negative feedback), damping the observed volatility as shown by the square ordinates to the right of the graph.

Sellers of credit derivatives sometimes hedge elements of credit risk by purchasing large numbers of out-of-the-money puts on the relevant stock. Again, the notional size of the OTM puts traded by various banks can be large relative to the daily volume of shares. Gamma induced crashes become feasible where a small downward move in the market can cause a rapid chain reaction to develop. In this scenario, the rate at which long delta accrues amongst dynamic hedgers (who then rapidly deplete the bid side of the order book by selling) far exceeds the rate at which 'bargain hunters' are drawn into the market. This positive feedback phenomenon is simulated in figure 2 for a 25% put. 'Pure' falls in the market (diamond series) trigger an avalanche of selling (square series). Sometimes calm can temporarily be restored to markets by suspending trading for a period which is long enough to allow the order book to be replenished. Stable equilibrium is only possible way below or above the put strike.

Any non-linear derivative hedged in different ways by the counterparties will modify the order placement behaviour for a given market. An analysis of the way large deals (such as typical 'corporate derivatives' and institutional 'low cost collars') will distort underlying markets should be conducted in advance of major transactions. Even in liquid markets, derivative-induced order placement can be observed to significantly change price dynamics.