Fugit

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.

In mathematical finance, fugit is the 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".

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 calculating "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.^[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, 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: "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", 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