# Fugit

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{{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](/source/mathematical_finance), '''fugit''' is the expected (or optimal) date to exercise an [American-](/source/Option_style) or [Bermudan option](/source/Bermudan_option). It is useful for [hedging](/source/Hedge_(finance)) purposes here; see [Greeks (finance)](/source/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](/source/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 option](/source/European_option)s.<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 bond](/source/convertible_bond)s, equity linked convertible notes, and any putable or callable [exotic](/source/exotic_derivatives) coupon notes.  Although see <ref name ="citigroup">Christopher Davenport, [Citigroup](/source/Citigroup), 2003. "Convertible Bonds A Guide".</ref> and <ref>[Paul Wilmott](/source/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](/source/Journal_of_Derivatives)'', Fall 1995</ref>  for [Employee stock options](/source/Employee_stock_option) (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](/source/risk-neutral) expected life of the option"<ref name = "Rubinstein">[Mark Rubinstein](/source/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](/source/Binomial_options_pricing_model) — although a [Finite difference approach](/source/Finite_difference_methods_for_option_pricing) 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](/source/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](/source/John_Wiley_%26_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|''&rho;''{{sub|2}}}} refers to sensitivity to dividends or the foreign interest rate, as opposed to [the more usual rho](/source/Greeks_(finance)) 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)

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