The Feedback Effect: When Hedging Moves the Market

Every derivatives textbook makes the same quiet assumption: option trading does not affect the stock price. The stock does its random walk thing, the option value follows, and hedging is just a passive activity. But think about this. In many markets, the nominal value of options traded exceeds the value of trade in the underlying stock itself. When everyone is delta hedging, they are all buying and selling the stock in predictable amounts at predictable times. Can we really pretend this has no effect? Chapter 61 of Wilmott’s book says no, and the consequences are fascinating.

The Hidden Assumption in Black-Scholes

In the standard Black-Scholes derivation, the stock price follows a random walk driven by noise traders and random information flow. The replication strategy, buying and selling stock to hedge, is assumed to have zero market impact. For every buyer there is a seller, and the hedger’s trades are too small to move the price.

This works when the market has many participants and high liquidity. But portfolio insurance strategies are often implemented at enormous scale. And liquidity is sometimes limited. The October 1987 crash is the classic example. The Brady Commission report suggested that portfolio insurance trading (essentially, systematic put replication) contributed to making the crash worse.

Wilmott formalizes this intuition. What happens when hedging trades are large enough relative to the market’s liquidity to move the stock price itself? This creates a feedback loop: the stock price moves, which changes the hedge ratio, which forces more trades, which moves the stock price further.

The Setup: Excess Demand and Equilibrium

The model starts with basic economics. The excess demand function measures the gap between demand and supply, depending on price S, time t, and a random variable x (noise traders and new information). The equilibrium price is where excess demand equals zero.

Now add hedging demand. Someone replicating a put option holds delta shares at each moment, adding a deterministic component. There is a parameter alpha that measures market sensitivity to trades. Small alpha means a liquid market. Large alpha means an illiquid market where trades move prices significantly.

How Replication Distorts the Price Process

Using Ito’s lemma on this modified equilibrium condition, Wilmott derives the new stochastic process for the stock price. Both the drift and the volatility get modified. The key term in both is the denominator:

1 minus alpha times the gamma (second derivative) of the trading strategy.

When this denominator is close to 1, the feedback is negligible and we are back to the normal lognormal random walk. When it approaches zero, both the drift and volatility blow up. The stock price becomes infinitely sensitive to the random noise. This is where the interesting behavior happens.

The critical insight is about gamma. If you are replicating a derivative with positive gamma (like a long put or call), the extra demand has a positive slope: when the stock falls, you need to sell more stock (for put replication). This adds demand that pushes in the wrong direction. It destabilizes the market.

Specifically:

Positive gamma positions destabilize the market. The hedging trades amplify price movements. When the stock drops, the hedger sells, pushing it down further. When the stock rises, the hedger buys, pushing it up further.

Negative gamma positions stabilize the market. The hedging trades dampen price movements. This is less commonly discussed, but it is equally important.

The Tulip Curve: When Prices Jump

Here is where the analysis gets dramatic. Consider put replication as expiry approaches. The delta of a put becomes steeper and steeper near the strike price, approaching a step function at expiry. The hedging demand becomes a sharp kink.

When you add this kinked demand to the original smooth demand function, something breaks. The total demand function is no longer monotonic near the strike price. It develops a region where demand is positively sloped, creating an unstable equilibrium. In this region, there are three equilibrium prices instead of one. Two are stable and one (in the middle) is unstable.

The stock price cannot stay in the unstable region. If it enters from one side, it gets pushed to the nearest stable equilibrium. This creates deterministic jumps in the price. The stock hits a critical value and snaps to a distant price. No randomness involved. Just the mechanical consequence of hedging demand exceeding the market’s ability to absorb it.

Wilmott identifies four critical boundary points near the strike price close to expiry. He labels them A, B, C, and D. Points B and C are where the jumps originate. The stock can jump from B to D (jumping up) or from C to A (jumping down). Between B and C lies a barren zone that the stock price can never reach. It is pushed away from this region by the hedging activity.

The shape of these boundaries in the time-price plane looks like a tulip lying on its side. All four boundary curves are confined to a small region around the strike price and expiry date. The size of this region scales with the liquidity parameter alpha. In very liquid markets, the tulip is tiny and the effect is negligible. In illiquid markets, it can be significant.

Numerical Results: What It Looks Like

Wilmott presents numerical solutions for the probability density function under different scenarios. Without feedback, the stock spreads out in the usual lognormal fashion. With a time-independent trading strategy, a forbidden corridor develops near the strike. The stock cannot reach prices in that band and has to jump across.

The realistic case is put replication with time-dependent trading. The forbidden zone starts as a single point far from expiry, then fans out into what Wilmott calls the tulip shape. Close to expiry, the barren zone is widest. After expiry, the feedback vanishes and all prices become attainable again. The contour plot shows a dark tulip-shaped region of zero probability near the strike.

Attraction and Repulsion

The analysis above focuses on short gamma positions (hedging put or call replication). The end result is repulsion from the strike price near expiry. The stock is pushed away.

But what if people are, on balance, hedging long gamma positions? Then the signs flip and we get the opposite effect: attraction toward the strike price. The hedging demand now pulls the stock toward the strike as expiry approaches.

This is not just theory. It is commonly observed in practice. Stock prices near large option strikes tend to cluster around those strikes as expiry approaches. The phenomenon is sometimes called pinning. Wilmott connects this directly to the feedback model with obvious sign changes.

A related effect occurs with convertible bonds. When new convertible bonds are issued, the volatility of the underlying stock can be dampened for a while. This happens because convertible bond arbitragers delta hedge their bond positions, and this hedging activity (long gamma) stabilizes the stock price. It is a real-world example of the attraction effect.

What This Means in Practice

The feedback model reveals something that most pricing models completely ignore. The act of hedging affects the thing you are hedging against. This is a circular dependency that standard models break by assumption.

For most markets most of the time, the effect is small. Liquid markets have a small alpha parameter, and the feedback barely registers. But there are situations where it matters.

Large portfolios of options near expiry in less liquid markets can create significant feedback effects. Portfolio insurance programs that systematically sell into falling markets amplify crashes. Concentrated option positions at specific strikes create attraction or repulsion zones.

The practical implications are straightforward. If you know the hedging positions of other market participants, you have information about where the stock price is likely to go near expiry. Traders with knowledge of open interest at various strikes can exploit this.

The Bigger Picture

This chapter is a reminder that financial models are not closed systems. They interact with the markets they describe. When a model becomes popular and everyone uses it to hedge, the collective hedging activity changes the underlying dynamics. The model affects reality, which in turn should affect the model.

The October 1987 crash was partly driven by this feedback loop. Portfolio insurance strategies systematically sold into a falling market, which pushed it down further, which triggered more selling. Wilmott’s framework gives a mathematical foundation for understanding how that happens and why it is more likely in illiquid conditions.

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