{{Short description|Mathematical finance term}} {{other uses}}
{| class="wikitable floatright" | width="300" |- style="font-size: 86% |- |'''Calculation of fugit:''' For Fugit — where {{mvar|n}} is the number of time-steps in the tree; {{mvar|t}} is the time to option expiry; and {{mvar|i}} is the current time-step — the calculation is as follows:<ref name = "Rubinstein"/><sup>; see also</sup> <ref name="Benhamou"/>
(1) set the fugit of all nodes at the end of the tree equal to {{math|1=''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 be 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 ({{math|1=''i'' = 0}}) is the current fugit.
Finally, to annualize the fugit, multiply the resulting value by {{math|''t'' / ''n''}}. |}
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 {{slink|Optimal stopping#Option trading}}. The term was first introduced by Mark Garman in an article "Semper tempus fugit" published in 1989.<ref name="RISK 1989">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</ref> The Latin term "{{lang|la|tempus fugit}}" means "time flies"<ref>{{cite web|title=Tempus it et tamquam mobilis aura volat|url=https://audiolatinproverbs.blogspot.com/2006/11/tempus-it-et-tamquam-mobilis-aura.html|publisher=Audio Latin Proverbs|accessdate=30 July 2012}}</ref> and Garman suggested the name because "time flies especially when you're having fun managing your book of American options".
==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.<ref name="Benhamou">Eric Benhamou: {{usurped|1=[https://web.archive.org/web/20110725103641/http://www.ericbenhamou.net/documents/Encyclo/Fugit%20_options_.pdf Fugit (options)]}}</ref> Fugit is thus used for the hedging of convertible bonds, equity linked convertible notes, and any putable or callable exotic coupon notes. Although see <ref name ="citigroup">Christopher Davenport, Citigroup, 2003. "Convertible Bonds A Guide".</ref> and <ref>Paul Wilmott's comment on [http://wilmott.com/messageview.cfm?catid=34&threadid=67833 a wilmott.com forum] {{Webarchive|url=https://web.archive.org/web/20150704172739/http://wilmott.com/messageview.cfm?catid=34&threadid=67833 |date=2015-07-04 }}: "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."</ref> for qualifications here. Fugit is also useful in estimating "the (risk-neutral) expected life of the option"<ref>Mark Rubinstein (1995). "[http://www.haas.berkeley.edu/groups/finance/WP/rpf241.pdf On the Accounting Valuation of Employee Stock Options] {{Webarchive|url=https://web.archive.org/web/20170811012829/http://www.haas.berkeley.edu/groups/finance/WP/rpf241.pdf |date=2017-08-11 }}", ''Journal of Derivatives'', Fall 1995</ref> for Employee stock options (note the brackets).
Fugit is calculated as "the expected time to exercise of American options",<ref name="RISK 1989"/> and is also described as the "risk-neutral expected life of the option"<ref name = "Rubinstein">Mark Rubinstein in an article "Guiding force"; the calculation is detailed on pages 43 and 44, as well as in [http://www.haas.berkeley.edu/groups/finance/WP/rpf220.pdf Exotic Options] {{Webarchive|url=https://web.archive.org/web/20150924024447/http://www.haas.berkeley.edu/groups/finance/WP/rpf220.pdf |date=2015-09-24 }}, a working paper by the same author.</ref> The computation requires a binomial tree — although a Finite difference approach would also apply<ref name="Benhamou"/> — where, a second quantity, additional to option price, is required at each node of the tree;<ref>[http://www.codeforge.com/read/151281/fugit_binomialtree.txt__html Example VBA code]</ref> see methodology aside. Note that fugit is not always a unique value.<ref name ="citigroup"/>
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”.<ref>[https://books.google.com/books?id=-5-OldaTjVQC&pg=PA178&dq Pg. 178] of Nassim Taleb (1997). ''Dynamic Hedging: Managing Vanilla and Exotic Options''. New York: John Wiley & Sons. {{ISBN|0-471-15280-3}}.</ref> Taleb terms this result “Omega” as opposed to fugit. The formula is <math display=block>\text{Omega} = \text{Nominal Duration} \times \frac{\rho_2 \text{ of an American option}}{\rho_2 \text{ of a European option}}</math> Here, {{math|''ρ''{{sub|2}}}} 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.<ref>See for example [http://www.nuclearphynance.com/Show%20Post.aspx?PostIDKey=125437 this discussion] on nuclearphynance.com.</ref> Taleb notes that this approach was widely applied, already in the 1980s, preceding Garman.<ref>Nassim Taleb: [http://www.fooledbyrandomness.com/rubinste.htm Review of ''Derivatives'' by Mark Rubinstein]</ref>
==References== {{reflist}}
Category:Mathematical finance Category:Options (finance)