An interesting observation was made recently by one of our trader friends. He is a commodities options trader in one of the banks in the City and he has recently witnessed extreme non-linearity in the volatility of oil options. In the last few months the volume in the exotic oil options have increased due to rising oil prices as well as corporate hedging activity. And apparently most of these exotics were barrier (knock-out and knock-in) trades. And our trader friend thinks that increased activity in oil call and put options near the barrier has given rise to non-linear volatility patterns.

What does it mean when we say that barrier makes the volatility non-linear?

Well, in the case of a vanilla option if we plot the implied volatility of a certain asset on the X axis and the corresponding Call option prices (for the asset) on the Y axis we more or less get a straight line. Of course, this is observation is trivial, because in a Black-Scholes world if you increase the volatility of an asset the value of a call option increases (assuming everything else remains constant). And this increase is quite linear. This means if we increase volatility by one unit the call option price will also change by a constant multiple of one unit and this multiple will remain constant.

However, for barrier options volatility becomes non-linear in a way such that barrier comes closer. As the volatility increases the price of a knock out increases as well, but if the increase in volatility becomes sufficiently large the probability of the barrier being hit (or the option being knocked-out) increases. This increase in probability then makes the knock out option price go down, i.e. after a point an increase in volatility will make the knock-out options cheaper. Thus the effect of rising volatility is to bring the barrier closer.

In fact an analysis of barrier options requires a good understanding of the theory of first passage time of a geometric Brownian motion. But that is beyond the scope of this article.

Let's look at how volatility actually behaves.

Say, an asset is trading at $100 and we buy a one month call with strike at $96 (in the money). For a certain interest rate (1.33%) the following is the plot of Volatility and the Call price as obtained from a Black-Scholes pricing engine.

Now consider a one month Knock-out call option on the same asset trading at $100, strike price $96 and a barrier (out-strike) at $80. By using a closed form pricing formula (with any rebate) the following is a plot of volatility and the corresponding Knock-out call.

Take the case of oil. If oil is trading at, say, $55 and the implied volatility is 15%. A trader who is short a knock-out call with a barrier at 53.00 will want to see that 53.00 is hit so that the option gets knocked out and he gets to keep the premium. Therefore, he would try to short oil futures as much as he can (of course he alone cannot move the market, but a couple of hundred very big traders like him can) so as to make it touch 53.00. Therefore, because of an increase in selling (shorting) activity in oil futures (or spot) in and around the barrier level of 53.00 will in turn cause the implied vols to rise further rapidly and this is where the market dynamics become extremely interesting. Keep in mind that our trader is short a knock-out which means he is actually long volatility and he will benefit from rising vols.

Rapid increase in implied vols actually brings down the price of the knock-outs due to higher probability of hitting the barrier. The sheer effect of volatility becoming higher near the barrier will shorten the expected life of the option. There will be a point of inflexion where the curvature of the volatility after become extreme starts to lose value (typically near the barrier). The vega of the knock-out will also become non-linear (as opposed to a vanilla where the vega remains linear).

Our trader friend told us of some very interesting structures in Oil that were traded in the last few months which gave rise such non-linear volatility patterns. Such options were apparently all customer driven (as opposed to prop trading) where by it appears that besides the usual hedging activity some synthetic notes were also issued with oil knock out options.

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