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УДК 336
GEOMETRIC AVERAGE OPTIONS FOR SEVERAL FUTURES* Hagop Kechejian, V.K. Ohanyan and V.G. Bardakhchyan
Freepoint Commodities, Stanford, CT, USA Yerevan State University, Yerevan, Armenia
hkechejian(a)hotmail.com; victo(alaua.am; vardanbardakchyan(algmail.com
In this paper we concentrate on valuing OTC (Over-The-Counter) Average (Asian) options on commodity' futures where the averaging period involves consecutive futures contracts. In practice these options are valued on an ad hoc basis, where it is assumed some sort of average weighted price based on the futures contracts and also average implied volatility' taken from exchange traded vanilla options. Using Andersen's model for commodity futures we present a rigorous approach to pricing these options. At present we only consider the geometric average options and in later papers we will use it as a control variate to price Asian option using Monte Carlo or other approximate methods. Keywords: Asian option, optimal strategy, future price. Mathematical Subject Classification 2010: 97M30, 93E20, 91C20, 91B25.
1 Introduction
Asian options are options where the payoff depends on the average price of the underlying asset during at least some part of the life of the option. The payoff from an average price call is max ([Save — K, 0), where
1 T
Save = - JQ S(u)du or simply the arithmetic average of asset prices up to
the maturity of the contract (the time T). Average options are less expensive than regular options and are arguably more appropriate than regular options for meeting some needs of users. Another type of Asian option is an average strike option. An average strike call pays off max (ST — Save, 0) and an average strike put pays off max (Save — ST, 0). Average strike options can guarantee that the average price paid for an asset in frequent trading over a period of time is not greater than the final price. If the underlying assert price is assumed to be lognormally distributed
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and Saveis a geometric average of the ^"s, analytic formulas are available for valuing European average price options (see [11]). This is because the geometric average of a set of lognormally distributed variables is also lognormal.
In this paper we consider Asian options on future prices rather than spot. In that case one take several future contracts contemporary with the duration of the option, and the option price is calculated for those underlying future contracts. However the described options are not that easy to price. Generally it is done through geometric average options. We will start with the pricing of geometric average options, later to see them as control variable in Monte-Carlo simulation for Asian option (see [10]). However we will use future instead of spot prices. We continue investigations of [4] and [5].
Let F(t, 7") be the time t futures price for delivery at time T.
In continuous time, geometric average options have the following pay out formula:
0(F(t))= (exp(i/0T[ln (F(u,T')]du)-K)+ (1.1)
where T is the maturity of the option, and T' is the maturity of the underlying future contract. Here we take as an example only one future contract. Furthermore the future contract should have longer maturity time, for the option to work, for the case of one contract, i.e. T' > T.
The formula is an analog of the geometric average options for discrete time, because
71
AY
nZ—i
i=l
In (F(ti,T')
Now we want to replace spot prices by future prices. Here we use The Anderson's paper [1], where the future prices dynamics are given via
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stochastic differential equation. More precisely by the Ito's Lemma, in dynamics of future prices are given in the following SDE:
dln(F(u,T')) = —\(j(u,T')\2dt+ a(u,T')dW(u) (1.2)
where <r(ii, 7") is volatility process for all T' > u and W(u) is one-dimensional Brownian motion. Here we consider one-dimensional case, that is g is a function from RxR to M.
Using Assumption 1 from [1] we have <j(t,T') = a(t)/?CT'), where ¡3(T') , is a deterministic function. By integrating (1.2), solution for future price's logarithm has the following form:
ln[F(u,7")] = ln(F(0,7")) - \PO")y{u)(3{r) + /¡(T')x(u) (1.3) where
x(t) = f*a(z)dW(z) and y(0 = J0ta(z)a(z)dz In this case we can use this as an input for (1.1): <£(F(t))= (expg/or(ln(F(0,r))- -J(rfy(u) + p(t'Mu)) du] - k)+ (1.4)
To calculate this, we need to know the functional form of a(z). Moreover equation (1.4) is stated for only one future contract. We are particularly interested in the case for finitely many underlying futures.
*(F(0) = | (exP| [j^ In (F(u, TJdu + In (F(u, T2)du +
^ Tn~tn~t о
...Tn—lTn— trA nE »1 (F(u, Tn)du-K+ (1.5)
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where t0 is the beginning of the averaging period, while tn is a midpoint (not necessarily the exact middle) of the last covering future. And t0 is already lapsed time from the start of first future contract up to the issuing of the geometric average option. However we can take t0 to be 0, and just take the first future contract to be shorter. Here we have taken future contracts covering each other (Ts < Tm , when 5 < m). More precisely we take the contracts to start averaging at the time the previous contract expires. Let's state a more general case not considered in this paper, but easily derived from the simpler case.
0(F(O) = (exp (jj In (F(u, T[)du + In (F(u, T'2)du + - +
Tn—ITn—tn\ n 0 fF(u, Tn ')du—K+.
Here we do not use the future contract up to their maturity (7/), but use them till Tt < T{ In the next section we compute the price for the case of (1.5).
2 Geometric average options on future prices
In this section we consider a particular case of equation (1.2), namely with the deterministic o function (see [1], pages 29-31).
ai(t,T) = e^h^'^-V + ea^hm ; a2(t,T) = eb^h2e-k^
So we can give the formula for solution for this particular case
([1]):
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In(F(t,T)) = ln(F(0,r)) + ea(T^ (zx(+ z2(t))
2 a(T)
--TT X [e2d(T)-2kT(e2kt - 1 )(hj + hi)
+ 4/i1/i00ed(T)-fcT(efct - 1) + 2hitk]
where
1) d(T) = b(T) — a(T)
2) Zl(t) = -kz^t)dt + M^CO + MW2(0; dziCf) =
= h^dW^ty. with Zi(0) = z2(0) = 0 is given (everything at time 0 is known).
As one can see the last term of the sum is also deterministic and will be integrated. However the second term has stochastic component. We have to compute the following:
0(F) = ( exp j \Tn-tJ In (F(u, Tt)dn + J rVг 1 1" (F(u,T2)du + - 7i
+ J fTn-tn 1 In (F(u, Tn)du J Tn-1 /, H + 1
where tn are the number of days before the expiration of the last future contract Tn . We take as starting point, the beginning of the geometric averaging period rather than the expiry of any future contract. Though the duration of the consequent futures here are the same (meaning that in general Tk — Tk_1 can be taken the same), the first period is shorter (i.e.
Ti < T2 ~ 7\).
Here we consider the discrete case as a sum, that is we take sum over {tj k}, where / stands for days, and k for the future contracts. For ex-
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ample, the third day of the second future contract (which is expiring at T2) we will have the price F(t3 2, T2).
In that case our expression takes the following form:
<P(F) =
( ( /1Щ m2
exP ( Tn Д„; (Zln (F(tu'Tl))+Zln + -
+ 2 In (F(ti>n, rj) J J-K
i=l
One can note that ln(F(t, T)) for any t 0, and T>t is normally distributed as being just Ito integral. Hence their sum is also normally distributed.
As we have concluded that the only randomness is in second component of (2.1), we should have the explicit form of zx(t) and z2(t). Here they will have the following form:
U h1eksdWl(s) + f h2eksdW2(s)
Zl(t) = e~kt 1 and
z2(0 = [ fcoodWiO) Jo
One can check that z:(t) and z2(t) are Gaussian processes. So we have that the sum of futures is normally distributed (see [10]):
mx 7i—l ms i<>
£ In (FfauTt)) + ^ Xln (F(ks'+ Zln Tn)) ~ ^ ' ct2)
( = 1 S = 1 ( = 1 ( = 1
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where the mean and variance of the distribution have the following forms:
71-1
ц = ^ ms In F(0, Ts) + i° In F(0, Fn)
S = 1
(ftf+fti)
\s=lt=l
;0
,2a(Tn)
n ms ^
, ^—, e2a(Ts)
e2d{Ts)-kTs^e2ktUs _ -q
Ah
| 6 е2й(Тп)-кТп^у^e2ktUn _ -Q t = l /
J 7^1 e2a(.Ts) e2a(Tn) ^
\s=1t=l t=l
and
/n-l / ms ms-l
2= hji ^ e2(a(Ts)-kTs+d(Ts)) + ^ (ms _ 0 _L
\s=l Vt=l t=l
/
4—i 1
+ e2(a(rn)-fcrn+d(Tn)) [ \ _e2ktin
\ Z_I 2/c
\i=i
i°-i t=i
n-l __ _
+ ^ ea(Ti)-kTi+d(Ti) у ea(Ts)-kTs+d(Ts)ms ^ ' lg2fctu
i=l s=i i = l
71-1 ™s
+ gd(T{)-kTi+d(T{)^ ea(Ts)-kTs+d(Ts)
t=l
71-1 71-1
LI.S
к "
S=i t=l
177
/п-1 ms-l
+ J ^ е2а№) ^ Ih,s(e-kTs+d(Ts)+kvhi + hoofdv
\ о — 1 .' — 1 о
\s = l t=l
71-1
+ e2a^y\ms
s=l i=1
-of ^(^e~kTs+d^+kvh1 + /I«,)
Jo
+ e2a(Tn) V ft,M(e-/cTn+d(Tn)+/c v^ + hmfdv
ЫJo
¿o ^
+ 2e2a(-Tn) - i) f 'i(e-^n+d(Tn)+fct;/liЛ(и)2
i — 1 0
dv
t=i
/ n-2 n-1 ms
'4<
+ | 2 ^ e2a<Ji) ^ e2a^ms ^ f '\e-kTi+d(-T^+kvh1
\ 1=1 s=l i=1
+ /i00)(e-fc7i+d^+ta,/i1 + /ioo)dv
n-l t
, ГЧ.П
+ /i00)(e"fc7i+d^+ta;/i1 + /ico)dv )
Here we have used the fact that covariance of any two Ito integrals
t S
of deterministic functions A"t = /0/(u)dM/„; Ks = f g(u)dWu is
0
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Also we have used the property of normal distributed variables i.e. if X~NQilr(rfi and Y~N(pl2 .of) then X + Y~N+ fi2.^x + +
2covX,Y.
And we have used the fact that cov(X + Y,Z) = cov(X.Z) 4-+cov(Y, Z) (see for example [6]).
Remark. What we have done is for the case where Ts denote exactly the number of days of future contracts (meaning that they are inte. TT . . _ number of days . . .
gers). However in general T = -1- can be any real number
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and to bring it to our case one just have to take tis = (i + —•
Ll=1ml
(for this technique see [10]).
For the further computations see [10] or [9].
*The present research of the second author was partially supported by the Mathematical Studies Center of Yerevan State University.
References
1. L. Andersen, "Markov Models for Commodity Futures: Theory and Practice". 1-45, 2008.
2. F. Jamshidian, "Commodity option valuation in the Gaussian futures term structure model", Review of Futures Markets, 10, 1992.
3. F.C. Klebaner, "Introduction to Stochastic Calculus With Applications", Imperial College Press, London, 2005.
4. H. Kechejian and V. K. Ohanyan, "Tolling contracts", Proceedings of the 6-th working conference "Reliability and optimization of structural systems" pp. 231_236, 2012.
5. H.Kechejian, V. K. Ohanyan and V. G. Bardakhchyan, "Tolling contracts with two driving prices", Russian Journal of Mathematical Research, Series A, vol. 1, pp. 14-19, 2015.
6. I. Karatzas and S.E. Shreve, "Brownian Motion and Stochastic calculus", 2nd ed., Springer -Verlag, New-York, 1998.
7. K.Miltersen, "Commodity price modelling that matches current Observables", Quantitative Finance, 3, 2003.
8. M. Ross and Erol A. Pekóz, "A Second Course of Probability", Boston, USA, 2007.
9. S. E. Shreve, "Stochastic calculus for finance II: Continous-time models", 1st ed., Springer 2008.
10. H. Zhang, "Pricing Asian Options using Monte Carlo Methods", 2009.
11. John C. Hull, "Options, Futures, and Other Derivatives" Sixth edition, New Jersey, 2006.
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АЗИАТСКИЕ ОПЦИОНЫ ИА ТОВАРНЫЕ ФЮЧЕРСЫ НА ФИНАНСОВЫХ ВНЕБИРЖЕВЫХ РЫНКАХ ЦЕННЫХ БУМАГ Кешишян А., Оганян В., Бардахчьян В.
Настоящая статья рассматривает азиатские опционы на товарные фючерсы на финансовых внебиржевых рынках ценных бумаг в случае, когда период усреднения вмещает несколько последовательных фьючерсных контрактов. На практике эти опционы оценивают, используя некоторую взвешено-усреднённую цену и некоторую усреднённую вола-тильность, которые в каждом случае берут на основе обычных опционов. Беря за основу модель Андерсона для товарных фючерсов, мы используем строгий подход для оценивания стоимости этих опционов. На данном этапе мы останавливаемся на оценках опционов с геометрическим средним, которые в дальнейшем будем использовать как контрольные переменные в методе Монте-Карло для приблизительной оценки азиатских опционов.
Дата поступления 18.06.2016.
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