Fugit

In mathematical finance, fugit is the expected (or optimal) date to exercise an American- or Bermudan option. It is useful for hedging purposes here; see Greeks (finance) and Optimal stopping#Option trading. The term was first introduced by Mark Garman in an article "Semper tempus fugit" published in 1989.[3] The Latin term "tempus fugit" means "time flies"[4] and Garman suggested the name because "time flies especially when you're having fun managing your book of American options".

For other uses, see Fugit (disambiguation).
Calculation of fugit:

For Fugit — where n is the number of time-steps in the tree; t is the time to option expiry; and i is the current time-step — the calculation is as follows:[1]; see also [2]

(1) set the fugit of all nodes at the end of the tree equal to i = n

(2) work backwards recursively:

  • if the option should be exercised at a node, set the fugit at that node equal to its period
  • if the option should not exercised at a node, set the fugit to the risk-neutral expected fugit over the next period.

(3) the number calculated in this fashion at the beginning of the first period (i=0) is the current fugit.

Finally, to annualize the fugit, multiply the resulting value by t / n.

Details

Fugit provides an estimate of when an option would be exercised, which is then a useful indication for the maturity to use when hedging American or Bermudan products with European options.[2] Fugit is thus used for the hedging of convertible bonds, equity linked convertible notes, and any putable or callable exotic coupon notes. Although see [5] and [6] for qualifications here. Fugit is also useful in estimating "the (risk-neutral) expected life of the option"[7] for Employee stock options (note the brackets).

Fugit is calculated as "the expected time to exercise of American options",[3] and is also described as the "risk-neutral expected life of the option"[1] The computation requires a binomial tree — although a Finite difference approach would also apply[2] — where, a second quantity, additional to option price, is required at each node of the tree;[8] see methodology aside. Note that fugit is not always a unique value.[5]

Nassim Taleb proposes a "rho fudge", as a “shortcut method... to find the right duration (i.e., expected time to termination) for an American option”.[9] Taleb terms this result “Omega” as opposed to fugit. The formula is

Omega = Nominal Duration x (Rho2 of an American option / Rho2 of a European option).

Here, Rho2 refers to sensitivity to dividends or the foreign interest rate, as opposed to the more usual rho which measures sensitivity to (local) interest rates; the latter is sometimes used, however.[10] Taleb notes that this approach was widely applied, already in the 1980s, preceding Garman.[11]

References
  1. Mark Rubinstein in an article "Guiding force"; the calculation is detailed on pages 43 and 44, as well as in Exotic Options Archived 2015-09-24 at the Wayback Machine, a working paper by the same author.
  2. Eric Benhamou: Fugit (options)
  3. Mark Garman in an article "Semper tempus fugit" published in 1989 by Risk Publications, and included in the book "From Black Scholes to Black Holes" pages 89-91
  4. "Tempus it et tamquam mobilis aura volat". Audio Latin Proverbs. Retrieved 30 July 2012.
  5. Christopher Davenport, Citigroup, 2003. "Convertible Bonds A Guide".
  6. Paul Wilmott's comment on a wilmott.com forum Archived 2015-07-04 at the Wayback Machine: "But, yes, remember that you need to put the real drift in there otherwise it's just the risk-neutral time and therefore not so relevant."
  7. Mark Rubinstein (1995). "On the Accounting Valuation of Employee Stock Options Archived 2017-08-11 at the Wayback Machine", Journal of Derivatives, Fall 1995
  8. Example VBA code
  9. Pg. 178 of Nassim Taleb (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. New York: John Wiley & Sons. ISBN 0-471-15280-3.
  10. See for example this discussion on nuclearphynance.com.
  11. Nassim Taleb: Review of Derivatives by Mark Rubinstein
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