Recent high volatilities in the equity markets may have unsettled many a barrier option traders and investors holding structured products with knock-out and knock-in options. Volatility behaves in a non-linear way for barrier options. As the high actual (realized) volatilities start to impact the implied vols, the pressing problem for the seller of an barrier option becomes, what vols should be use to price the option that would capture the barrier better. Remember, the seller of certain type of barrier options, such as knock-outs, have a high incentive to see the barrier hit.?

As the vols increase, unlike a vanilla, the barrier gets flat vega. The non-linearity is due to the fact that as the spot comes closer to the barrier due to volatility the barrier starts to dominate. The vol will push the option nearer to the barrier and the probability of hitting the barrier will increase.

Below is the volatility profile for a one year down an out call, struck at the money (ATM) with barrier at 94% (T = 1, K = 100, KO = 94). A Monte Carlo simulator is used for pricing.

The non-linear profile of the barrier is not only an issue in trading but also an issue for pricing. Whenever a trader is pricing a barrier option, the million dollar question that confronts him is what vols to use? Most models - at least the standard ones - use a single volatility estimate to price a barrier. But since the first exit time of a barrier is uncertain and because if the market suddenly drops or rallies the spot moves closer to the barrier in a non-linear way an a priori estimate of barrier vols using both the term structure of the vol and smile/skew is important. One needs to know where the volatility is located on the term structure of vols. Also, because of smile/skew the volatility for the strike and the barrier will be different. How do we go about then?

Over the last decade a lot of math models, which are fairly computer intensive, have come up which try to fit Local Volatility Surfaces to option prices. One such vol surface is the Dupire-Derman-Kani Surface. Dupire presented the seminal notion of a two dimensional volatility surface. And then Derman and Kani developed a pricing technique in 1994 whereby a tree is built to capture the volatility smile and the term structure.

Fitting a local vol surface is basically constructing a binomial tree that captures the smile/skew as well as the term structure and then solving for the barrier value within this tree. This model is rigorous, mathematically savvy and certainly needs a lot of computer time (which may not be an issue these days at most banks).

However, ask any senior trader who has perhaps survived the onslaught of the markets for upwards of two decades as to what vols he would take to price a barrier and you'll possibly get an answer: " *simply use a single vol fudge". *And if you happen to say to him that a single volatility fudge does not work and is only partially correct, his answer would be: "*but it works*".

So, what is a single volatility fudge? For knock-out options it simply entails finding the expected stopping time of the barrier and estimating where it lies on the implied spot volatility curve. Say the spot vol term structure for a vanilla has the following profile.

If the expected stopping time of say, a 9 month KO (knock out) is close to 6 months then by interpolating on the curve above we will get a value of around 45%. The knock out option can then be priced using this interpolated value of 45%. But by only doing this the smile/skew dimension will be missed out. To add the skew dimension one can take a synthetic risk-reversal and price the knock out using that as well*.

For the corresponding KI (knock-in option) the trader needs to find the forward-forward volatility (or simply the forward vol) between the stopping time and the maturity. In the above example if the maturity is 9 months and the expected stopping time is approximately 6 months with corresponding vols of 40% and 45% respectively then the forward-forward vol, FFwd(6,9), will be 27.39%.

This is the volatility that we should use to price the corresponding Knock in option. In doing all this one has to remember that the expected stopping time is not a fixed point in time but rather it is a probability distribution. Knowing the distribution of first exit time is important for this whole exercise.

** See Nassim Taleb. His ***Dynamic Hedging** gives a good perspective on this problem.

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