Statistical & Financial Consulting by Stanford PhD

Fadeout is a derivative that pays a vanilla option at expiration but amortizes according to the following rule. If on any of the N observation dates the price of the underlying asset goes through a pre-specified knockout barrier, 1/N of the original notional is written off ("knocked out"). In particular, the fadeout is extinguished completely if the barrier breach happens on each observation date. Fadeouts have direct link to baseball options through the following equivalence of cashflows:

= baseball option which knocks out after 1 barrier breach and has notional X/N +

+ baseball option which knocks out after 2 barrier breaches and has notional X/N +

...

+ baseball option which knocks out after N barrier breaches and has notional X/N.

In the relationship above, all the derivatives are assumed to have the same contractual structure: same underlying vanilla option, same observation dates, same knockout barrier. Also, the baseball options are assumed to pay zero rebate in the event of knockout. It follows from the "no arbitrage" argument that the fadeout price equals the sum of the baseball option prices.

Compared to knockouts, fadeouts are a "softer" way of expressing the view that the knockout barrier will never be breached. As such, they are more expensive than knockouts but cheaper than vanilla options, other things being equal.

As is the case with many barrier options, fadeout has a twin, called fadein. Fadein does exactly the opposite: it does not promise to pay a vanilla option unless the knockout barrier has been touched. Whenever that happens the notional increases by a fixed increment until the maximum notional has been reached. Researchers never talk about fadein pricing in its own right because of the following cashflow equivalence:

In the relationship above the fadeout and fadein are assumed to have have the same maximum notional, observation dates, knockout barrier and underlying vanilla option. So fadeins are priced out of fadeouts and relatively liquid vanilla market.

Bouzoubaa, M. (2010). Exotic Options and Hybrids: A Guide to Structuring, Pricing and Trading. Wiley.

de Weert, F. (2008). Exotic Options Trading. Wiley.

Lipton, A. (2001). Mathematical Methods for Foreign Exchange: A Financial Engineer's Approach. World Scientific.

Nelken, I., ed (1996). The Handbook of Exotic Options: Instruments, Analysis and Applications. McGraw-Hill.

Clark, I. J. (2011). Foreign Exchange Option Pricing: A Practitioner's Guide. Wiley.

Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Application to Bond and Currency Options. The Review of Financial Studies 6, 327-343.

Carr, P., & Liuren W. (2007). Stochastic Skew in Currency Options. Journal of Financial Economics 86, 213-247.

Broadie, M., Glasserman, P. & Kou, S. G. (1999). Connecting discrete and continuous path-dependent options. Finance and Stochastics 3, 55-82.

Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. Springer-Verlag New York.

Gatheral, J. (2006). The Volatility Surface. John Wiley & Sons, New Jersey.

Taleb, N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. Wiley Finance, New York.