Asymptotics for the Discrete-Time Average of the Geometric Brownian Motion and Asian Options
.. ASYMPTOTICS FOR THE DISCRETE-TIME AVERAGE OF THEGEOMETRIC BROWNIAN MOTION AND ASIAN OPTIONS
DAN PIRJOL AND LINGJIONG ZHU
Abstract.
The time average of geometric Brownian motion plays a crucial role in thepricing of Asian options in mathematical finance. In this paper we consider the asymptoticsof the discrete-time average of a geometric Brownian motion sampled on uniformly spacedtimes in the limit of a very large number of averaging time steps. We derive almost surelimit, fluctuations, large deviations, and also the asymptotics of the moment generatingfunction of the average. Based on these results, we derive the asymptotics for the price ofAsian options with discrete-time averaging in the Black-Scholes model, with both fixed andfloating strike. Introduction
Asian (or average) options are widely traded instruments in the financial markets, whichinvolve the time average of the price of an asset S t . Most commonly S t is a stock price ora commodity futures contract price, for example oil or natural gas futures. An Asian calloption has payoff of the form(1) Payoff = max (cid:40) n n (cid:88) i =1 S t i − K, (cid:41) , where 0 ≤ t < t < · · · < t n is a sequence of strictly increasing times, called sampling oraveraging dates. Under risk-free neutral pricing, the price of such an option is given by theexpectation of the payoff in the risk-neutral measure. Assuming the Black-Scholes model oneis led to study the distributional properties of the discrete time average of the asset price(2) A n = 1 n n (cid:88) i =1 S t i under the assumption that S n follows a geometric Brownian motion(3) dS t = ( r − q ) S t dt + σS t dZ t , where Z t is a standard Brownian motion, r is the risk-free rate, q is the dividend yield and σ is the volatility.The main technical difficulty for pricing Asian options is that the probability distributionof the discrete time average (2) does not have a simple expression. If the averaging times areuniformly distributed, the time average can be well approximated, for sufficiently small timestep, by a continuous average(4) A n = 1 t n (cid:90) t n S t dt . Date : 15 September 2016.2010
Mathematics Subject Classification.
Key words and phrases.
Asian options, central limit theorems, Berry-Esseen bound, large deviations. a r X i v : . [ q -f i n . P R ] J un DAN PIRJOL AND LINGJIONG ZHU
When S t follows a geometric Brownian motion, the problem is reduced to the study of thedistributional properties of the time integral of the geometric Brownian motion, which hasbeen extensively studied in the literature. See [16] for a review of the main results and theirapplications to the Asian options pricing.A wide variety of methods have been proposed for pricing Asian options, and a brief surveyis given below.1. PDE methods [42, 49, 50, 35]. The pricing of an Asian option can be reduced to thesolution of a 1+1 partial differential equation, which is solved numerically. This method canbe applied both to continuous-time and discrete-time averaging Asian options [1]. See alsoAlziary et al. [2].2. The Laplace transform method [24, 7]: the Asian option price with random exponentiallydistributed maturity can be found in closed form for the case when the asset price S t followsa geometric Brownian motion. This reduces the problem of the Asian option pricing to theinversion of a Laplace transform.3. Spectral method [33]. The probability distribution of the time integral of the geometricBrownian motion can be related to that of a Bessel process [14, 13]. The transition density ofthis Bessel process can be expanded in an eigenfunction series [53], and Asian option pricescan be evaluated using the eigenfunction expansion, truncated to a sufficient high order [33].4. Bounds and control variates methods. There is a large literature on deriving bounds onAsian option prices. Both lower and upper bounds have been given, see [35] for an overview.They can be used also in conjunction with Monte Carlo methods as control variates. Oneprecise method of this type which is popular in practice was given by Curran [9]. Othermethods which take into account the discrete time averaging have been proposed in [22, 23,21].5. Monte Carlo simulation. See e.g. Kemna and Vorst [29], Fu, Madan, Wang [20], Lapeyreand Teman [30].6. Analytical approximations. Various numerical methods have been proposed whichapproximate the distribution of the arithmetic average A n using parametric forms, such aslog-normal [32] or inverse Gamma distributions [36].We note also the more general approach of [52] which can be applied for a wide class ofmodels.Most of the theoretical results in the literature concerning the distribution of the timeaverage of the geometric Brownian motion refer to the continuous time average. The discretesum of the geometric Brownian motion is a particular case of the sum of correlated log-normalswhich has been studied extensively in the literature, see [3] for an overview. Dufresne hasobtained in [15] limit distribution for the discrete time average in the limit of very smallvolatility σ →
0. A recent work by the present authors [39] studied the properties of thediscrete time average at fixed σ in the limit n → ∞ , and its convergence to the continuoustime average as the time step τ → A n = n (cid:80) ni =1 S t i . We assume the Black-Scholes model, that is, the asset pricefollows a geometric Brownian motion(5) S t = S e σZ t +( r − q − σ ) t , where Z t is a standard Brownian motion. We would like to study the distributional propertiesof the average of the discretely sampled asset price (2) defined on the discrete times uniformlyspaced t i = iτ with time step τ . SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 3
We will derive in this paper asymptotic results about A n in the limit n → ∞ by keepingfixed the following combinations of model parameters β = 12 σ t n n = 12 σ τ n , (6) ( r − q ) τ n = ρ . (7)Note that β is always positive but ρ can be both positive and negative. We also note thatthe conditions (6) and (7) can be replaced by lim n →∞ σ τ n = β and lim n →∞ ( r − q ) τ n = ρ and all the results in this paper will still hold.The constraints (6), (7) include two interesting regimes: • When the maturity t n = τ n is constant, and so are the interest rates r and dividendyield q , then, (6) assumes that the volatility σ is of the order O ( √ n ). Therefore, theconditions (6) and (7) include the small volatility regime. • When the maturity t n = τ n is small, that is, t n → n → ∞ and in particularis of the order n , then by (7), the volatility σ is a constant. If the interest rates r and dividend yield q are constant, then (7) is replaced by lim n →∞ ( r − q ) τ n = 0, i.e., ρ = 0. Therefore, the conditions (6) and (7) include the short maturity regime.We emphasize that we do not make any assumptions about the values of ρ, β , and they canbe arbitrary. The validity of our asymptotic results require only that n (cid:29)
1, such that theseregimes cover most cases of practical interest, provided that the number of averaging timesis sufficiently large.We present in this paper three asymptotic results for the distributional properties of thediscrete time average of a geometric Brownian motion in the limit of a large number ofaveraging time steps n : i) almost sure limit and fluctuation results for A n , ii) an asymptoticresult for the moment generating function of the partial sums nA n for n → ∞ , and iii)large deviations results for P ( A n ∈ · ). Using these asymptotic results, we derive rigorouslyasymptotics for the prices of out-of-the-money, in-the-money and at-the-money Asian options.Section 2 presents the almost sure and fluctuations results for A n in the n → ∞ limit.Section 3 presents an asymptotic result for the Laplace transform of the finite sum of thegeometric Brownian motion sampled on n discrete times nA n , in the limit n → ∞ . InSection 4 we consider the asymptotics of fixed strike Asian options following from the largedeviations result iii), and in Section 5 we treat the case of the floating strike Asian options.These asymptotic results can be used to obtain approximative pricing formulas for Asianoptions, and in Section 6 we compare the numerical performance of the asymptotic resultagainst alternative methods for pricing Asian options under the BS model. Some of theproposed methods are known to be less efficient numerically in the small maturity and/orsmall volatility limit [24, 33]. The asymptotic results derived in this paper are of practicalinterest as they complement these approaches in a region where their numerical performanceis not very good. We demonstrate good agreement of our asymptotic results with alternativepricing methods for Asian options with realistic values of the model parameters.2. Asymptotics for the discrete time average of geometric Brownian motion
We have the almost sure limit:
Proposition 1.
We have (8) lim n →∞ A n = A ∞ ≡ S ρ ( e ρ − a.s. . DAN PIRJOL AND LINGJIONG ZHU
Proof.
Note that max ≤ i ≤ n σ | Z t i | = max ≤ i ≤ n (cid:113) βτ n | Z iτ | and from the property of Brownianmotion, n max ≤ i ≤ n | Z iτ | → n → ∞ . Moreover, σ t i = βin ≤ βn → n → ∞ uniformly in 1 ≤ i ≤ n . Therefore, S t i can be approximated by S e ( r − q ) t i uniformly in1 ≤ i ≤ n , that is, max ≤ i ≤ n | S t i − S e ( r − q ) t i | → n → ∞ . Finally, notice that(9) 1 n n (cid:88) i =1 e ( r − q ) t i = 1 n n (cid:88) i =1 e ρ in = 1 n e ρ − − e − ρn → ρ ( e ρ − , n → ∞ . Hence, we proved the desired result. (cid:3)
We have also the following fluctuation result:
Proposition 2.
The time average A n converges in distribution to a normal distribution inthe n → ∞ limit (10) lim n →∞ √ n A n − A ∞ S = N (0 , βv ( ρ )) . with (11) v ( a ) := 1 a (cid:20) ae a − e a + 2 e a − (cid:21) . Proof.
We have √ n A n − A ∞ S (12) = 1 √ n n (cid:88) i =1 ( e σZ i +( r − q − σ ) t i − e ρ in ) + (cid:34) √ n n (cid:88) i =1 e ρ in − √ n e ρ − ρ (cid:35) (13) = 1 √ n n (cid:88) i =1 e ρ in ( e √ βn B i − β in −
1) + (cid:34) √ n n (cid:88) i =1 e ρ in − √ n e ρ − ρ (cid:35) , (14)where Z i = √ τ B i with B i a standard Brownian motion. We can rewrite the second term in(14) as 1 √ n n (cid:88) i =1 e ρ in − √ n e ρ − ρ = 1 √ n e ρ − − e − ρn − √ n e ρ − ρ (15) = ( e ρ −
1) 1 √ n (cid:34) ρn − ρ n + O ( n − ) − nρ (cid:35) = ( e ρ −
1) 1 √ n ρ n + O ( n − ) ρ ( ρn − ρ n + O ( n − )) → , as n → ∞ .The first term in (14) can be written further as1 √ n n (cid:88) i =1 e ρ in ( e √ βn B i − β in −
1) = 1 √ n n (cid:88) i =1 e ρ in √ βn B i + ξ n , (16)where we defined(17) ξ n ≡ √ n n (cid:88) i =1 e ρ in (cid:16) e √ βn B i − β in − √ βn B i − (cid:17) . SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 5
We claim that ξ n → n → ∞ .We have the following upper bound on ξ n .(18) ξ n ≤ √ n n (cid:88) i =1 e ρ in (cid:18) e √ βn B i − √ βn B i − (cid:19) ≡ ξ (up) n . The upper bound ξ (up) n is a non-negative random variable since e x − − x ≥ x .The expectation of ξ (up) n can be computed exactly E [ ξ (up) n ] = 1 √ n n (cid:88) i =1 e ρ in ( e βn i − √ n (cid:32) e ρ + βn − − e − ρn − βn − e ρ − − e − ρn (cid:33) = 1 √ n (cid:18) βρ + o (1 /n ) (cid:19) . This goes to zero as n → ∞ . The Markov inequality implies that ξ (up) n → n → ∞ .Next, let us estimate the lower bound on ξ n . We have ξ n ≥ √ n n (cid:88) i =1 e ρ in (cid:18) e √ βn B i − βn − (cid:18) √ βn B i − βn (cid:19) − − βn (cid:19) (20) ≥ √ n n (cid:88) i =1 e ρ in (cid:18) − βn (cid:19) = − β √ n e ρ − n (1 − e − ρn ) → , where we used again in the second step the inequality e x ≥ x .The first term in (16) is a normal random variable and converges in distribution to anormal distribution with mean zero and variance to be determined.(21) 1 √ n n (cid:88) i =1 e ρ in √ βn B i → N (0 , βv ( ρ )) . This can be computed by writing B i = (cid:80) i − j =0 V j with V j ∼ N (0 ,
1) i.i.d. normally dis-tributed random variables with mean zero and unit variance. The sum can be written as1 √ n n (cid:88) i =1 e ρ in √ βn B i = √ βn / n − (cid:88) j =0 V j n (cid:88) i = j +1 e ρ in (22) = √ βn / n − (cid:88) j =0 V j e ρ n − (cid:110) e ρ n +1 n − e ρ j +1 n (cid:111) . We can compute the variance of this random variable asVar (cid:32) √ n n (cid:88) i =1 e ρ in √ βn B i (cid:33) = 2 βn n − (cid:88) j =0 e ρ n − (cid:16) e ρ n +1 n − e ρ j +1 n (cid:17) (23) = 2 β n ( e ρ n − n − (cid:88) j =0 (cid:16) e ρ n +1 n − e ρ j +1 n (cid:17) n → βρ (cid:90) ( e ρ − e ρx ) dx, DAN PIRJOL AND LINGJIONG ZHU as n → ∞ , where we can compute that(24) 2 βρ (cid:90) ( e ρ − e ρx ) dx = 2 βρ (cid:20) ρe ρ − e ρ + 2 e ρ − (cid:21) . (cid:3) Moment generating function
Define the moment generating function of nA n as(25) F n ( θ ) := E [ e θnA n ] . For θ <
0, this is the Laplace transform of the distribution function of nA n .We are interested in the limit lim n →∞ n log F n ( θ ). We will compute this limit using thetheory of large deviations. Before we proceed, recall that a sequence ( P n ) n ∈ N of probabilitymeasures on a topological space X satisfies the large deviation principle with rate function I : X → R if I is non-negative, lower semicontinuous and for any measurable set A , we have(26) − inf x ∈ A o I ( x ) ≤ lim inf n →∞ n log P n ( A ) ≤ lim sup n →∞ n log P n ( A ) ≤ − inf x ∈ A I ( x ) . Here, A o is the interior of A and A is its closure. The rate function I is said to be good iffor any m , the level set { x : I ( x ) ≤ m } is compact. We refer to Dembo and Zeitouni [10] orVaradhan [48] for general background of large deviations and the applications.We have the following limit theorem for the generating function in the limit n → ∞ atfixed β . Theorem 3.
For any θ > , F n ( θ ) = ∞ and for any θ ≤ , lim n →∞ n log F n ( θ ) = sup g ∈AC [0 , (cid:40) θS (cid:90) e √ βg ( x ) dx − (cid:90) (cid:18) g (cid:48) ( x ) − ρ √ β (cid:19) dx (cid:41) . (27) Proof.
Since E [ e θX ] = ∞ for any θ > X , it is clear that E [ e θnA n ] = ∞ for any θ >
0. Next, for any θ ≤ E [ e θnA n ] = E (cid:20) e θ (cid:80) n − k =0 S e σZtk +( r − q − σ tk (cid:21) (28) = E (cid:20) e θS (cid:80) n − k =0 e σ √ τ (cid:80) kj =1 Vj +( r − q − σ kτ (cid:21) = E (cid:34) e θS (cid:80) n − k =0 e √ βn (cid:80) kj =1 Vj + ρkn − βn k (cid:35) = E (cid:34) e θS (cid:80) n − k =0 e √ βn (cid:80) kj =1( Vj + ρ √ β ) − βn k (cid:35) , where V j := √ τ ( Z j − Z j − ), 1 ≤ j ≤ k , are i.i.d. N (0 ,
1) random variables. Note that (cid:80) j =1 V j is defined as 0. By Mogulskii theorem, see e.g. [10], P ( n (cid:80) (cid:98)· n (cid:99) j =1 ( V j + ρ √ β ) ∈ · ) satisfies a largedeviation principle on L ∞ [0 ,
1] with the good rate function(29) I ( g ) = (cid:90) Λ( g (cid:48) ( x )) dx, SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 7 if g ∈ AC [0 , g (0) = 0 and I ( g ) = + ∞ otherwise, where(30) Λ( x ) := sup θ ∈ R (cid:110) θx − log E (cid:104) e θ ( V + ρ √ β ) (cid:105)(cid:111) = 12 (cid:18) x − ρ √ β (cid:19) . Let g ( x ) := n (cid:80) (cid:98) xn (cid:99) j =1 ( V j + ρ √ β ). Then,(31) (cid:90) e √ βg ( x ) dx = n − (cid:88) k =0 (cid:90) k +1 nkn e √ βg ( x ) dx = 1 n n − (cid:88) k =0 e √ βn (cid:80) kj =1 ( V j + ρ √ β ) . Moreover, we claim that(32) g (cid:55)→ (cid:90) e √ βg ( x ) dx is a continuous map. Let g n be any sequence in L ∞ [0 ,
1] so that g n → g in L ∞ [0 , | x | ≤ . | e x − | = (cid:12)(cid:12)(cid:12)(cid:12) x + x
2! + x
3! + · · · (cid:12)(cid:12)(cid:12)(cid:12) ≤ | x | (1 + | x | + | x | + · · · ) ≤ | x | . (33)Let n be sufficiently large so that √ β (cid:107) g n − g (cid:107) L ∞ [0 , ≤ . Therefore, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e √ βg n ( x ) dx − (cid:90) e √ βg ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e √ βg ( x ) (cid:16) e √ β ( g n ( x ) − g ( x )) − (cid:17) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ e √ β (cid:107) g (cid:107) L ∞ [0 , (cid:90) (cid:12)(cid:12)(cid:12) e √ β ( g n ( x ) − g ( x )) − (cid:12)(cid:12)(cid:12) dx ≤ (cid:112) βe √ β (cid:107) g (cid:107) L ∞ [0 , (cid:107) g n − g (cid:107) L ∞ [0 , which converges to 0 as n → ∞ . Hence the map is continuous. Let us recall the celebratedVaradhan’s lemma from large deviations theory, see e.g. [10]. if P ( Z n ∈ · ) satisfies a largedeviation principle with good rate function I : X → [0 , + ∞ ], and if φ is a continuous mapand(34) lim M → + ∞ lim sup n →∞ n log E (cid:104) e nφ ( Z n ) φ ( Z n ) ≥ M (cid:105) = −∞ , then lim n →∞ n log E [ e nφ ( Z n ) ] = sup x ∈X { φ ( x ) − I ( x ) } . In our case, φ ( g ) = θS (cid:90) e √ βg ( x ) dx is a continuous map. Moreover, for θ ≤ φ ( g ) ≤ n →∞ n log E (cid:34) e θS (cid:80) n − k =0 e √ βn (cid:80) kj =1( Vj + ρ √ β ) (cid:35) (35) = sup g ∈AC [0 , (cid:40) θS (cid:90) e √ βg ( x ) dx − (cid:90) (cid:18) g (cid:48) ( x ) − ρ √ β (cid:19) dx (cid:41) . DAN PIRJOL AND LINGJIONG ZHU
Finally, notice that E (cid:34) e θS (cid:80) n − k =0 e √ βn (cid:80) kj =1( Vj + ρ √ β ) (cid:35) ≤ E [ e θnA n ](36) ≤ E (cid:34) e θS e − βn (cid:80) n − k =0 e √ βn (cid:80) kj =1( Vj + ρ √ β ) (cid:35) . Hence, for any θ ≤ n →∞ n log E [ e θnA n ](37) = sup g ∈AC [0 , (cid:40) θS (cid:90) e √ βg ( x ) dx − (cid:90) (cid:18) g (cid:48) ( x ) − ρ √ β (cid:19) dx (cid:41) . (cid:3) Solution of the Variational Problem.
The variational problem in Theorem 3 canbe re-stated as(38) lim n →∞ n log F n ( θ ) = λ ( − θS , (cid:112) β ; ρ )where λ ( a, b ; ρ ) is the solution of the variational problem(39) λ ( a, b ; ρ ) = sup g ∈AC [0 , (cid:26) − a (cid:90) e bg ( x ) dx − (cid:90) (cid:16) g (cid:48) ( x ) − ρb (cid:17) dx (cid:27) . Here we have a, b >
Proposition 4.
The function λ ( a, b ; ρ ) is given by one of the two expressions λ ( a, b ; ρ ) = a (cid:26) (cid:18) δ (cid:19) (cid:18) − ρδ + ρ δ (cid:19) − − ρδ sinh δ (cid:27) (40) + 2 b ρ log (cid:20) cosh (cid:18) δ (cid:19) + ρδ sinh (cid:18) δ (cid:19)(cid:21) − ρ b , or λ ( a, b ; ρ ) = a (cid:26) − sin ξ (cid:18) ρξ − ρ ξ (cid:19) + ρ − ξ sin(2 ξ ) (cid:27) (41) + 2 ρb log (cid:20) cos ξ + ρ ξ sin ξ (cid:21) − ρ b . In (40) δ is the solution of the equation (42) ρ − δ = 2 ab (cid:18) cosh (cid:18) δ (cid:19) + ρδ sinh (cid:18) δ (cid:19)(cid:19) , and in (41) ξ is the unique solution ξ ∈ (0 , ξ max ) of the equation (43) 2 ξ (4 ξ + ρ ) = ab (2 ξ cos ξ + ρ sin ξ ) .ξ max is the smallest solution of the equation tan ξ max = − ξ max /ρ .For given ( a > , b, ρ ) only one of the two equations (42) and (43) has a solution, suchthat the solution of the variational problem is unique. SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 9
Proof.
The proof will be given in the Appendix. (cid:3)
Let us recall that lim n →∞ ( r − q ) τ n = ρ , and in the short maturity limit t n → r, q , we have ρ = 0. Therefore, the special case ρ = 0 is of practical interest when consideringthe short maturity limit. For this case it is clear that only (43) has a solution for a > Corollary 5.
The function λ ( a, b ; 0) in the ρ = 0 limit is given by (44) λ ( a, b ; 0) = a (cid:18) cos ξ − ξ sin(2 ξ ) (cid:19) , where ξ is the solution of the equation ξ = ab cos ξ . (45)In conclusion, the result of Theorem 3 and Proposition 4 gives an asymptotic expressionfor the Laplace transform of the discrete sum of the geometric Brownian motion nA n in thelimit n → ∞ , of the form E [ e − θnA n ] = exp( nλ ( θS , √ β ; ρ ) + o ( n )). This result could beused for numerical simulations of nA n , similar to the approach presented in [31] using anasymptotic result for the Laplace transform of the sum of correlated log-normals. Anotherpossible application would be to obtain a first-order approximation of Asian options pricesin the asymptotic n (cid:29) Asymptotics for Asian options prices
Asymptotics for the option pricing is a well studied subject in mathematical finance. Thereis a vast literature on the asymptotics for option pricing, especially the asymptotics for thevanilla option pricing and the corresponding implied volatility for various continuous-timemodels, see e.g. [4, 25, 17, 18, 46]. We are interested in the asymptotics for the pricing ofthe Asian options in the discrete time setting, under the assumptions (6) and (7).Let us consider an Asian option with strike price K , in the Black-Scholes model withvolatility σ , risk free rate r and dividend yield q . The prices of the put and call options attime zero are given by P ( n ) := e − rt n E [( K − A n ) + ] , (46) C ( n ) := e − rt n E [( A n − K ) + ] , (47)respectively, where A n = n (cid:80) ni =1 S t i and the expectation are taken under the risk-neutralprobability measure under which the asset price satisfies the SDE dS t = ( r − q ) S t dt + σS t dW t .Also notice that e − rt n = e − rr − q ( r − q ) τn = e − rr − q ρ . Recall that we have proved that A n → A ∞ = S ρ ( e ρ −
1) a.s. as n → ∞ . Since ( K − A n ) + ≤ K , by the bounded convergence theorem fromreal analysis, we have(48) lim n →∞ P ( n ) = e − rr − q ρ lim n →∞ E [( K − A n ) + ] = e − rr − q ρ (cid:18) K − S ρ ( e ρ − (cid:19) + . From put-call parity, C ( n ) − P ( n ) = e − rt n E [ A n − K ] = e − rr − q ρ (cid:34) n n (cid:88) i =1 E [ S t i ] − K (cid:35) = e − rr − q ρ (cid:34) n S n (cid:88) i =1 e ρ in − K (cid:35) → e − rr − q ρ (cid:18) S ρ ( e ρ − − K (cid:19) , as n → ∞ . Therefore,(49) lim n →∞ C ( n ) = e − rr − q ρ (cid:18) S ρ ( e ρ − − K (cid:19) + . Out-of-the-Money Case.
When
K < S ρ ( e ρ − n →∞ P ( n ) = 0 and the put optionis out-of-the-money and the decaying rate of P ( n ) to zero as n → ∞ is governed by the lefttail of the large deviations of A n . When K > S ρ ( e ρ − n →∞ C ( n ) = 0 and the calloption is out-of-the-money and the decaying rate of C ( n ) to zero as n → ∞ is governed bythe right tail of the large deviations of A n . Before we proceed, let us first derive the largedeviation principle for P ( A n ∈ · ). Proposition 6. P ( A n ∈ · ) satisfies a large deviation principle with rate function (50) I ( x ) = inf g ∈AC [0 , , (cid:82) e √ βg ( y ) dy = xS (cid:90) (cid:18) g (cid:48) ( x ) − ρ √ β (cid:19) dx, for x ≥ and I ( x ) = + ∞ otherwise.Proof. We proved already that n (cid:80) n − k =0 e √ βn (cid:80) kj =1 ( V j + ρ √ β ) = (cid:82) e √ βg ( x ) dx , where g ( x ) = n (cid:80) (cid:98) xn (cid:99) j =1 ( V j + ρ √ β ) and the map g (cid:55)→ (cid:82) e √ βg ( x ) dx is continuous in the supremum norm.Since P ( n (cid:80) (cid:98)· n (cid:99) j =1 ( V j + ρ √ β ) ∈ · ) satisfies a large deviation principle on L ∞ [0 ,
1] with ratefunction (cid:82) (cid:16) g (cid:48) ( x ) − ρ √ β (cid:17) dx if g ∈ AC [0 ,
1] and + ∞ otherwise. From the contractionprinciple, and the fact that e − βn ≤ e − βn k ≤ ≤ k ≤ n −
1, we conclude that P ( A n ∈ · ) satisfies a large deviation principle with rate function defined in (50). Finally,notice that A n is positive and thus I ( x ) = + ∞ for any x < (cid:3) Remark 7. I ( x ) = 0 in (50) if and only if the optimal g satisfies g (cid:48) ( x ) = ρ √ β which isequivalent to g ( x ) = ρ √ β x since g (0) = 0 . This gives us (cid:82) e √ βg ( y ) dy = (cid:82) e ρy dy = e ρ − ρ .Thus I ( x ) = 0 if and only if x = S e ρ − ρ = A ∞ which is consistent with the a.s. limit of A n as n → ∞ . Remark 8.
We have proved that Γ( θ ) := lim n →∞ n log E [ e θnA n ] exists for any θ ≤ and isdifferentiable and Γ( θ ) = + ∞ for any θ > . Since Γ( θ ) = + ∞ for any θ > , we cannot useG¨artner-Ellis theorem to obtain large deviations for P ( A n ∈ · ) . One may speculate that wemight have subexponential tails. But the intriguing fact is that we still have large deviationsas stated in Proposition 6. We can further analyze and solve the variational problem (50). For ρ (cid:54) = 0, the solution isgiven by the following result. SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 11
Proposition 9.
The rate function of the discrete time average of the geometric Brownianmotion is given by (51) I ( x ) = 12 β J ( x/S , ρ ) , with (52) J ( x/S , ρ ) = (cid:40) J ( x/S , ρ ) xS ≥ ρ J ( x/S , ρ ) xS ≤ ρ , where J (cid:18) xS , ρ (cid:19) = 12 ( δ − ρ ) (cid:32) − δ ) δ + ρ tanh( δ ) (cid:33) (53) − ρ log (cid:20) cosh (cid:18) δ (cid:19) + ρδ sinh (cid:18) δ (cid:19)(cid:21) + ρ , J (cid:18) xS , ρ (cid:19) = 2 (cid:18) ξ + 14 ρ (cid:19) (cid:26) tan ξξ + ρ tan ξ − (cid:27) − ρ log (cid:18) cos ξ + ρ ξ sin ξ (cid:19) + ρ , (54) and δ, ξ are the solutions of the equations (55) 1 δ sinh δ + 2 ρδ sinh (cid:18) δ (cid:19) = xS . and (56) 12 ξ sin(2 ξ ) (cid:18) ρ ξξ (cid:19) = xS . Proof.
The proof is given in the Appendix. (cid:3)
Remark 10.
We note that the equations for J , ( K/S , ρ ) can be put into a unique form bydenoting z = 2 ξ = iδ . Expressed in terms of this variable we have J (cid:18) xS , ρ (cid:19) = 12 ( z + ρ ) (cid:32) (cid:0) z (cid:1) z + ρ tan (cid:0) z (cid:1) − (cid:33) (57) − ρ log (cid:104) cos (cid:16) z (cid:17) + ρz sin (cid:16) z (cid:17)(cid:105) + ρ , where z is the solution of the equation z sin z + 2 ρz sin (cid:16) z (cid:17) = xS . (58) Remark 11.
The rate function J ( K/S , ρ ) vanishes for K = A ∞ = S ρ ( e ρ − , as expectedfrom the general properties of the rate function. Since we have ρ ( e ρ − ≥ ρ for any ρ ∈ R , this zero occurs for J ( K/S , ρ ) . We note that the rate function J ( K/S , ρ ) vanishesat δ = ± ρ . Both these values of δ satisfy (55) with K/S = ρ ( e ρ − , which corresponds to K = A ∞ . However, the true solution of the variational problem (50) corresponds to δ = − ρ which gives the optimal function g ( x ) = ρx √ β , see (135). For ρ = 0 the solution to the variational problem (50) simplifies and is given as follows. K/S r r=0.1r=0 Figure 1.
Plot of the rate function J ( K/S , ρ ) vs K/S for two values ofthe ρ parameter ρ = 0 , .
1. This is related to the rate function I ( x ) for thelarge deviations of the average of the geometric Brownian motion A n as in(51). Corollary 12.
For the special case ρ = 0 , (59) J ( x ) = (cid:40) δ − δ tanh (cid:0) δ (cid:1) xS ≥ , ξ (tan ξ − ξ ) 0 < xS ≤ . and J ( x ) = ∞ otherwise, where δ is the unique solution of the equation (60) 1 δ sinh δ = xS , and ξ is the unique solution in (0 , π ) of the equation ξ sin(2 ξ ) = xS . (61)It can be shown that this is identical to the rate function for the short maturity asymptoticsof Asian options with continuous time averaging in the Black-Scholes model [40]. The ratefunction J ( K/S , ρ ) can be evaluated numerically using the result of Proposition 9. The plotof J ( x/S , ρ ) is shown in Figure 1 for ρ = 0 , . P ( A n ∈ · ), we can obtain the asymptotics of theout-of-the-money Asian options prices. This is given by the following result. Proposition 13.
When
K < S ρ ( e ρ − , (62) P ( n ) = e − n I ( K )+ o ( n ) , as n → ∞ , and when K > S ρ ( e ρ − , (63) C ( n ) = e − n I ( K )+ o ( n ) , as n → ∞ , where I ( · ) was defined in (50) . SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 13
Proof.
For any 0 < (cid:15) < K , P ( n ) ≥ e − rr − q ρ E [( K − A n )1 A n ≤ K − (cid:15) ] ≥ e − rr − q ρ (cid:15) P ( A n ≤ K − (cid:15) ) . (64)Therefore, lim inf n →∞ n log P ( n ) ≥ −I ( K − (cid:15) ). Since it holds for any (cid:15) ∈ (0 , K ), we concludethat(65) lim inf n →∞ n log P ( n ) ≥ −I ( K ) . On the other hand,(66) P ( n ) = e − rr − q ρ E [( K − A n )1 A n ≤ K ] ≤ e − rr − q ρ K P ( A n ≤ K ) , which implies that lim sup n →∞ n log P ( n ) ≤ −I ( K ). Hence, we proved the (62).For any (cid:15) > C ( n ) ≥ e − rr − q ρ E [( A n − K )1 A n ≥ K + (cid:15) ] ≥ e − rr − q ρ (cid:15) P ( A n ≥ K + (cid:15) ) . (67)Therefore, lim inf n →∞ n log C ( n ) ≥ −I ( K + (cid:15) ). Since it holds for any (cid:15) >
0, we have(68) lim inf n →∞ n log C ( n ) ≥ −I ( K ) . For any p + q = 1, p, q >
1, by H¨older’s inequality, C ( n ) = e − rr − q ρ E (cid:2) ( A n − K ) + A n ≥ K (cid:3) (69) ≤ e − rr − q ρ (cid:0) E [[( A n − K ) + ] p ] (cid:1) p ( E [(1 A n ≥ K ) q ]) q ≤ e − rr − q ρ ( E [( A n + K ) p ]) p P ( A n ≥ K ) q . By Jensen’s inequality, for any x, y >
0, it is clear that for any p ≥
2, ( x + y ) p ≤ x p + y p .Therefore, for any p ≥ E [( A n + K ) p ] ≤ p − ( E [ A pn ] + K p ) . We can compute that E [ A pn ] = n − p E (cid:34)(cid:32) n (cid:88) i =1 S e σZ ti +( r − q − σ ) t i (cid:33) p (cid:35) (71) = n − p E (cid:34)(cid:32) n (cid:88) i =1 S e σ √ τZ i +( r − q − σ ) τi (cid:33) p (cid:35) ≤ n − p E (cid:34)(cid:32) n (cid:88) i =1 S e σ √ τ max ≤ i ≤ n Z i + | ρ | (cid:33) p (cid:35) ≤ S p e | ρ | p E (cid:104) e √ βn p max ≤ i ≤ n Z i (cid:105) = S p e | ρ | p E (cid:104) e √ βn p | Z n | (cid:105) = S p e | ρ | p E (cid:20) e √ β √ n p | Z | (cid:21) , where we used the reflection principle for the Brownian motion and the Brownian scalingproperty. Note that E [ e θ | Z | ] is finite for any θ >
0. Hence, from (69), (70), (71), we concludethat for any 1 < q < p >
2, where p + q = 1),(72) lim sup n →∞ n log C ( n ) ≤ − q lim n →∞ n log P ( A n ≥ K ) = − q I ( K ) . Since it holds for any 1 < q <
2, by letting q ↓
1, we proved (63). (cid:3)
In-the-Money Case.
We consider the case of in-the-money Asian options, that is
K > S ρ ( e ρ −
1) for the put option (and
K < S ρ ( e ρ −
1) for the call option). Since A n → A ∞ a.s.we get from the bounded convergence theorem and put-call parity, that P ( n ) → K − S ρ ( e ρ − C ( n ) → S ρ ( e ρ − − K . The next results concern the speed of the convergence. Proposition 14.
When
K < S ρ ( e ρ − and ρ (cid:54) = 0 , (73) C ( n ) = e − rr − q ρ (cid:18) S ρ ( e ρ − − K (cid:19) + e − rρr − q S ( e ρ − n + O ( n − ) . and when K > S ρ ( e ρ − and ρ (cid:54) = 0 , (74) P ( n ) = e − rr − q ρ (cid:18) K − S ρ ( e ρ − (cid:19) − e − rρr − q S ( e ρ − n + O ( n − ) . The case ρ = 0 is similar. When K < S , (75) C ( n ) = ( S − K ) + e − n I ( K )+ o ( n ) , and when K > S , (76) P ( n ) = ( K − S ) + e − n I ( K )+ o ( n ) . Proof.
When
K < S ρ ( e ρ − P ( n ) = e − n I ( k )+ o ( n ) . From put-call parity,(77) C ( n ) − P ( n ) = e − rt n E [ A n − K ] = e − rr − q ρ (cid:34) S e ρ − n (1 − e − ρn ) − K (cid:35) . Therefore, C ( n ) − P ( n ) − e − rr − q ρ (cid:18) S ρ ( e ρ − − K (cid:19) (78) = e − rρr − q S ( e ρ − (cid:34) n (1 − e − ρn ) − ρ (cid:35) = e − rρr − q S ( e ρ − (cid:34) ρ − ρ n + O ( n − ) − ρ (cid:35) = e − rρr − q S ( e ρ − n + O ( n − ) . Since P ( n ) = e − n I ( k )+ o ( n ) , we proved (73). Similarly, we have (74). (cid:3) SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 15
At-the-Money Case.
Consider next the case of at-the-money Asian options, that is K = S ρ ( e ρ −
1) = A ∞ . Since A n → A ∞ a.s., using the bounded convergence theorem, wehave P ( n ) → n → ∞ . Put-call parity implies that C ( n ) → n → ∞ as well. Notethat in the case of out-of-the-money, we have already seen that both P ( n ) and C ( n ) decay tozero exponentially fast in n , where the exponent is given by I ( K ). The next result is aboutthe speed that P ( n ) and C ( n ) decay to zero as n → ∞ for at-the-money Asian options. Wewill see that, unlike the out-of-the-money Asian options, whose asymptotics are governedby the large deviations results, the asymptotics for at-the-money case are governed by thenormal fluctuations from the central limit theorem and non-uniform Berry-Esseen bound. Proposition 15.
When the Asian option is at-the-money, that is, K = S ρ ( e ρ −
1) = A ∞ , P ( n ) = e − rρr − q S (cid:114) βv ( ρ ) π √ n (1 + o (1)) , (79) C ( n ) = e − rρr − q S (cid:114) βv ( ρ ) π √ n (1 + o (1)) , (80) as n → ∞ .Proof. C ( n ) = e − rt n E [( K − A n ) + ] = e − rρr − q E [( A n − A ∞ )1 A n ≥ A ∞ ](81) = e − rρr − q S √ n E (cid:20) √ n ( A n − A ∞ ) S √ n ( An − A ∞ ) S ≥ (cid:21) . We have proved in Proposition 2 that √ n ( A n − A ∞ ) S → N (0 , βv ( ρ )) as n → ∞ . Intuitively,it is clear that E (cid:20) √ n ( A n − A ∞ ) S √ n ( An − A ∞ ) S ≥ (cid:21) → E [ Z Z ≥ ] where Z ∼ N (0 , βv ( ρ )). Butin order to prove this, the central limit theorem is not sufficient. We need a non-uniformBerry-Esseen bound [37, 5], which we recall next. See e.g. Pinelis [38] for a survey on thissubject. Theorem 16 (Non-uniform Berry-Esseen bound) . For any independent and not neces-sarily identically distributed random variables X , X , . . . , X n with zero means and finitevariances and Var ( W n ) = 1 , where W n = (cid:80) ni =1 X i , let F n be the cumulative distribu-tion function of W n and Φ the standard normal cumulative distribution function, that is Φ( x ) := √ π (cid:82) x −∞ e − y / dy .The difference between the two distributions is bounded as [37, 5](82) | F n ( x ) − Φ( x ) | ≤ C (cid:80) ni =1 E | X i | | x | , for any −∞ < x < ∞ , where C is a constant. The best known bound on this constant in thegeneral (non-identical X i ) case is C < .
935 [38] . We have proved that(83) √ n ( A n − A ∞ ) S = n (cid:88) i =1 X i + ξ n + ε n , where(84) X i := √ βn / V i e ρ ( n +1) n − e ρin e ρn − , ≤ i ≤ n, where V i are i.i.d. N (0 ,
1) random variables and(85) ε n := 1 √ n n (cid:88) i =1 e ρ in − √ n e ρ − ρ . The plan of the proof will be to show that the contributions from the second and third termsin (83) are negligible, and to apply the non-uniform Berry-Esseen bound to the first term in(83).From (83), we have(86) E (cid:20) √ n ( A n − A ∞ ) S √ n ( An − A ∞ ) S ≥ (cid:21) = E (cid:34)(cid:32) n (cid:88) i =1 X i + ξ n + ε n (cid:33) (cid:80) ni =1 X i + ξ n + ε n ≥ (cid:35) , which implies that(87) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:20) √ n ( A n − A ∞ ) S √ n ( An − A ∞ ) S ≥ (cid:21) − E (cid:34)(cid:32) n (cid:88) i =1 X i (cid:33) (cid:80) ni =1 X i + ξ n + ε n ≥ (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E | ξ n | + | ε n | . We have proved already that E | ξ n | and | ε n | → n → ∞ . Next, notice that E (cid:34)(cid:32) n (cid:88) i =1 X i (cid:33) (cid:80) ni =1 X i + ξ n + ε n ≥ (cid:35) (88) = (cid:118)(cid:117)(cid:117)(cid:116) n (cid:88) i =1 Var( X i ) E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ (cid:35) , where Y i = ( (cid:80) ni =1 Var( X i )) − / X i , ¯ ξ n = ( (cid:80) ni =1 Var( X i )) − / ξ n and ¯ ε n = ( (cid:80) ni =1 Var( X i )) − / ε n ,so that Var( (cid:80) ni =1 Y i ) = 1. Recall that we already proved that(89) lim n →∞ n (cid:88) i =1 Var( X i ) = 2 βv ( ρ ) . The expectation on the right-hand side of (88) can be written as E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ (cid:35) (90)= E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ , | ¯ ξ n +¯ ε n |≤ δ (cid:35) + E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ , | ¯ ξ n +¯ ε n | >δ (cid:35) , SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 17 for any δ >
0. The second term is bounded from above by the Cauchy-Schwarz inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ , | ¯ ξ n +¯ ε n | >δ (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (91) ≤ E (cid:32)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ (cid:33) / E [(1 | ¯ ξ n +¯ ε n | >δ ) ] / ≤ E (cid:32) n (cid:88) i =1 Y i (cid:33) / P ( | ¯ ξ n + ¯ ε n | > δ ) / = P ( | ¯ ξ n + ¯ ε n | > δ ) / → , as n → ∞ . The first term in (90) can be written furthermore as E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ , | ¯ ξ n +¯ ε n |≤ δ (cid:35) (92) = E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ , | ¯ ξ n +¯ ε n |≤ δ, (cid:80) ni =1 Y i ≥ (cid:35) + E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ , | ¯ ξ n +¯ ε n |≤ δ, (cid:80) ni =1 Y i ≤ (cid:35) . The second term in (92) is negative and is bounded in absolute value as0 < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ , | ¯ ξ n +¯ ε n |≤ δ, (cid:80) ni =1 Y i ≤ (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (93) ≤ E (cid:32) n (cid:88) i =1 Y i (cid:33) / P (cid:32) n (cid:88) i =1 Y i + ¯ ξ n + ¯ ε n ≥ , | ¯ ξ n + ¯ ε n | ≤ δ, n (cid:88) i =1 Y i ≤ (cid:33) / ≤ P (cid:32) − δ ≤ n (cid:88) i =1 Y i ≤ (cid:33) / → [Φ(0) − Φ( − δ )] / , as n → ∞ by the central limit theorem.Next, we need to estimate the first term in (92). We first give an upper bound,(94) E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ , | ¯ ξ n +¯ ε n |≤ δ, (cid:80) ni =1 Y i ≥ (cid:35) ≤ E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i ≥ (cid:35) . Next, we give a lower bound, E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ , | ¯ ξ n +¯ ε n |≤ δ, (cid:80) ni =1 Y i ≥ (cid:35) (95) ≥ E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i ≥ δ, | ¯ ξ n +¯ ε n |≤ δ (cid:35) . This can be written further as E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i ≥ δ, | ¯ ξ n +¯ ε n |≤ δ (cid:35) (96) = E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i ≥ δ (cid:35) − E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i ≥ δ, | ¯ ξ n +¯ ε n | >δ (cid:35) . By following the same argument as in (91), we have(97) lim n →∞ E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i ≥ δ, | ¯ ξ n +¯ ε n | >δ (cid:35) = 0 . The bounds (94) and (95) can be combined with the bounds (93) to obtain simpler boundson the expectation in (92) in the n → ∞ limit. By (90)-(97), these bounds translate intocorresponding bounds for the expectation (93). We get for any δ ≥ n →∞ E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ (cid:35) (98) ≥ lim inf n →∞ E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i ≥ δ (cid:35) − [Φ(0) − Φ( − δ )] / , and lim sup n →∞ E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ (cid:35) ≤ lim sup n →∞ E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i ≥ (cid:35) . (99)Finally, take the δ → n →∞ E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i +¯ ξ n +¯ ε n ≥ (cid:35) = lim n →∞ E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i ≥ (cid:35) . (100)The non-uniform Berry-Esseen bound can be applied to compute the expectation on theright-hand side.The sums of third moments appearing in the non-uniform Berry-Esseen bound are esti-mated as follows. Recalling that Y i = ( (cid:80) ni =1 Var( X i )) − / X i where X i are defined in termsof N (0 ,
1) i.i.d. random variables V i as given in (84), we find n (cid:88) i =1 E | Y i | = 1( (cid:80) ni =1 Var( X i )) / n (cid:88) i =1 E | X i | (101) = 1( (cid:80) ni =1 Var( X i )) / (2 β ) / n / E | V | n (cid:88) i =1 (cid:32) e ρ ( n +1) n − e ρin n ( e ρn − (cid:33) n ≤ C ( ρ ) n / , where C ( ρ ) > ρ . Therefore, by the non-uniform Berry-Esseen bound, wehave(102) | F n ( x ) − Φ( x ) | ≤ C ( ρ ) n /
11 + | x | , SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 19 for any −∞ < x < ∞ , where F n is the cumulative distribution function of (cid:80) ni =1 Y n , and C ( ρ ) > Z ∼ N (0 , βv ( ρ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E (cid:34)(cid:32) n (cid:88) i =1 Y i (cid:33) (cid:80) ni =1 Y i ≥ (cid:35) − E [ Z Z ≥ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ xdF n ( x ) − (cid:90) ∞ xd Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (103) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ F n ( x ) dx − (cid:90) ∞ Φ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) ∞ C ( ρ ) n /
11 + | x | dx. which goes to zero as n → ∞ . We conclude that we have(104) C ( n ) = e − rρr − q S E [ Z Z ≥ ] 1 √ n (1 + o (1)) , as n → ∞ , where Z ∼ N (0 , βv ( ρ )). The expectation is given explicitly by(105) E [ Z Z ≥ ] = (cid:112) βv ( ρ ) 1 √ π (cid:90) ∞ xe − x dx = (cid:114) βv ( ρ ) π . This completes the proof of the asymptotics for the at-the-money call option C ( n ). Theasymptotics for the price of the at-the-money put option P ( n ) can be obtained by usingput-call parity. The proof is complete. (cid:3) Asymptotics for Floating Strike Asian Options
We consider in this section the floating strike Asian options, which are a variation of thestandard Asian option. The floating strike Asian call option with strike K and weight κ haspayoff ( κS T − A T ) + at maturity T and the floating strike put option has payoff ( A T − κS T ) + at maturity T .The floating-strike Asian option is more difficult to price than the fixed-strike case becausethe joint law of S T and A T is needed. Also, the one-dimensional PDE that the floating-strikeAsian price satisfies after a change of num´eraire is difficult to solve numerically as the Diracdelta function appears as a coefficient, see e.g. [28], [42], [2]. See [41, 8, 27] for alternativemethods which have been proposed to deal with this problem.It has been shown by Henderson and Wojakowski [27] that the floating-strike Asian optionswith continuous time averaging can be related to fixed strike ones. These equivalence relationshave been extended to discrete time averaging Asian options in [47]. According to theserelations we have e − rt n E (cid:2) ( κS t n − A n ) + (cid:3) = e − qt n E ∗ (cid:2) ( κS − A n ) + (cid:3) , (106) e − rt n E (cid:2) ( A n − κS t n ) + (cid:3) = e − qt n E ∗ (cid:2) ( A n − κS ) + (cid:3) , (107)The expectations on the right-hand side are taken with respect to a different measure Q ∗ ,where the asset price S t follows the process dS t = ( q − r ) S t dt + σS t dW ∗ t , (108)with W ∗ t a standard Brownian motion in the Q ∗ measure. We are interested in the asymptotics of the price of the Asian call/put options with payoffs( κS t n − A n ) + and ( A n − κS t n ) + , C ( n ) := e − rt n E (cid:2) ( κS t n − A n ) + (cid:3) , (109) P ( n ) := e − rt n E (cid:2) ( A n − κS t n ) + (cid:3) . (110)As n → ∞ , κS t n − A n → κS e ρ − S e ρ − ρ a.s. When κ < ρ (1 − e − ρ ) the call option isout-of-the-money and the put option is in-the-money. When κ > ρ (1 − e − ρ ), the call optionis in-the-money and the put option is out-of-the-money. When κ = ρ (1 − e − ρ ), the call andput options are at-the-money.For the expectations on the right-hand side of the equivalence relations (106), (107) wehave that as n → ∞ , κS − A n → κS − S e − ρ − − ρ a.s. We conclude that for κ < ρ (1 − e − ρ )these equivalence relations map an out-of-money floating strike call (put) Asian option ontoan out-of-money fixed strike put (call) Asian option. For κ > ρ (1 − e − ρ ) a similar relationholds between the respective in-the-money Asian options.Let us derive the asymptotics of the price of the floating strike Asian options. This couldbe expressed in terms of the asymptotics of the fixed strike Asian options obtained in theprevious sections, with the help of the equivalence relations. An alternative way is to derivedirectly the large deviation result for the floating strike Asian options. Then we will relatethe rate function to that for the fixed strike Asian options, and show that this is consistentwith the equivalence relations.We have the following result for the asymptotics of floating strike Asian options. Proposition 17. (i) When κ < ρ (1 − e − ρ ) , the call option is out-of-the-money, (111) C ( n ) = e − n H (0)+ o ( n ) , as n → ∞ , and the put option is in-the-money, (112) P ( n ) = − κS e − rr − q ρ + S e − rr − q ρ e ρ − ρ + e − rρr − q S ( e ρ − n + O ( n − ) ρ (cid:54) = 0 , (1 − κ ) S + e − n H (0)+ o ( n ) ρ = 0 . (ii) When κ > ρ (1 − e − ρ ) , the put option is out-of-the-money (113) P ( n ) = e − n H (0)+ o ( n ) , as n → ∞ , and the call option is in-the-money, (114) C ( n ) = κS e − rr − q ρ − S e − rr − q ρ e ρ − ρ − e − rρr − q S ( e ρ − n + O ( n − ) ρ (cid:54) = 0 ,S ( κ −
1) + e − n H (0)+ o ( n ) ρ = 0 . The rate function in (i) and (ii) is given by (115) H (0) := inf g ∈AC [0 , ,κe √ βg (1) − (cid:82) e √ βg ( y ) dy =0 (cid:90) (cid:18) g (cid:48) ( x ) − ρ √ β (cid:19) dx . (iii) When κ = ρ (1 − e − ρ ) , the call and put options are in-the-money, (116) lim n →∞ √ nC ( n ) = lim n →∞ √ nP ( n ) = S e − rρr − q E [ Z Z ≥ ] , SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 21 where Z = N (0 , s ) is a normal random variable with mean and variance (117) s = 2 βρ (cid:20) − ρ ( e ρ −
1) + e ρ − ρ (cid:21) . Proof.
The proof is similar to the fixed-strike case. The sketch of the proof will be given inthe Appendix. (cid:3)
We show next that the rate function H (0) can be simply related to I ( x ) defined in (50).Recall that we showed explicitly the dependence of ρ of the respective rate functions H ( · ) and I ( · ). Abusing the notations a bit to emphasize the dependence on ρ , let H ( · ; ρ ) := H ( · ) and I ( · ; ρ ) := I ( · ). We have the following result, which is clearly consistent with the equivalencerelations (106), (107). Proposition 18.
The rate functions for the fixed strike and floating strike Asian options arerelated as (118) H (0; ρ ) = I ( κS ; − ρ ) . Proof.
The functionals in the variational problems for H (0) and I ( x ) are identical, and theonly difference is in the constraints on g ( x ). The constraints can be related as follows.Let us express g ( x ) in the variational problem for H (0) in terms of a new function h ( x )defined as g ( x ) = g (1) + h (1 − x ). This function satisfies the constraint h (0) = 0. The ratefunction is now given by(119) H (0) := inf h ∈AC [0 , ,κ − (cid:82) e √ βh ( y ) dy =0 (cid:90) (cid:18) h (cid:48) ( x ) + ρ √ β (cid:19) dx, It is easy to see that this variational problem is identical to that for the rate function I ( x ),identifying K/S = κ and ρ → − ρ . This concludes the proof of the relation (118). (cid:3) Implied volatility and numerical tests
It has become accepted market practice to quote European option prices in terms of theirimplied volatility. This is defined as that value of the log-normal volatility which, uponsubstitution into the Black-Scholes formula, reproduces the market option prices. A similarnormal implied volatility can be defined in terms of the Bachelier formula.Although Asian options are quoted in practice by price, and not by implied volatility, itis convenient to define an equivalent implied volatility also for these options. We will definethe equivalent log-normal implied volatility of an Asian option with strike K and maturity T as that value of the volatility Σ LN ( K, T ) which reproduces the Asian option price whensubstituted into the Black-Scholes formula for an European option with the same parameters(
K, T ) C ( K, S , T ) = e − rT ( A ∞ Φ( d ) − K Φ( d )) , (120) P ( K, S , T ) = e − rT ( K Φ( − d ) − A ∞ Φ( − d )) , where(121) A ∞ = S ρ ( e ρ −
1) = 1( r − q ) T ( e ( r − q ) T − , and(122) d , = 1Σ LN ( K, T ) √ T (cid:18) log A ∞ K ±
12 Σ ( K, T ) T (cid:19) . The equivalent log-normal volatility Σ LN defined in this way exists for any Asian option callprice C ( K, S , T ) satisfying the Merton bounds ( A ∞ − K ) + ≤ e rT C ( K, S , T ) ≤ A ∞ [43]. Forfinite n the price of the Asian option is bounded as ( E [ A n ] − K ) + ≤ e rT C ( K, S , T ) ≤ E [ A n ]with E [ A n ] = n e ρ − − e − ρ/n , so the required bounds are satisfied for n → ∞ .One can define also a normal equivalent volatility Σ N ( K, T ) of an Asian option, as thatvolatility which reproduces the Asian option price when substituted into the Bachelier optionpricing formula.We would like to study the implications of the asymptotic results for Asian option pricesderived in Section 4 for the equivalent log-normal volatility Σ LN , and for the equivalentnormal volatility Σ N . This is given by the following result. Proposition 19. i) The asymptotic normal and log-normal equivalent implied volatilities ofan OTM Asian option in the n → ∞ limit at constant β = σ t n n are given by lim n →∞ Σ ( K, n ) σ = 12 log ( K/A ∞ ) J ( K/S , ρ ) , (123) lim n →∞ Σ N ( K, n ) σ = 12 ( K − A ∞ ) J ( K/S , ρ ) , (124) where J ( K/S , ρ ) is related to the rate function I ( x ) as in (51), and is given by Proposition 9.ii) The equivalent log-normal implied volatility for n → ∞ of an at-the-money Asian optionis (125) lim n →∞ Σ LN ( A ∞ , n ) σ = S A ∞ (cid:112) v ( ρ ) , and the corresponding result for the equivalent normal implied volatility is (126) lim n →∞ Σ N ( A ∞ , n ) σ = S (cid:112) v ( ρ ) . Proof.
The proof is given in the Appendix. (cid:3)
We note that in (123) σ depends implicitly on n as the limit is taken at fixed β . Inparticular, in the fixed maturity regime τ n = T fixed, we have σ ∼ n − / such that both σ and Σ LN ( K ) approach 0 as n → ∞ , in such a way that their ratio approaches a finite non-zerovalue. We will use this relation for finite n to approximate the equivalent log-normal impliedvolatility Σ LN ( K ) as(127) Σ ( K, n ) = σ
12 log ( K/A ∞ ) J ( K/S , ρ ) . and analogously for Σ N ( K ). These volatilities can be used together with (120) to obtainapproximations for Asian option prices.We show in Table 2 numerical results for the asymptotic approximation for the Asianoptions obtained from (120), for a few scenarios proposed in [20]. They are compared againsta few alternative methods considered in the literature: the method of Linetsky [33], PDEmethods [19, 50], inversion of Laplace transform [11, 44], and the log-normal approximation[32] corresponding to continuous-time averaging.The numerical agreement of the asymptotic result with the precise results of the spectralexpansion [33] is very good, and the difference is always below 0 .
5% in relative value. A more The lower bound follows from the convexity of the payoff ( x − K ) + and the upper bound follows from( x − K ) + ≤ x . SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 23
Table 1.
The 7 benchmark scenarios considered for pricing Asian optionsin [20, 33], etc. Here, q = 0.Scenario r T S K σ V : the approximation error of theasymptotic result is always below 0 . V (compared with the log-normal approximation whichhas an error as large as 1 . V (for scenario 7)). This is smaller than the typical precision on σ around the ATM point, and compares well with typical bid-ask spreads for Asian optionswhich can be ∼ V for maturities up to 1-2Y. Remark 20.
We comment on the relation of the asymptotic implied volatility (123) to thelog-normal approximation [32] . The log-normal approximation [32] corresponds to a flat equiv-alent log-normal volatility Σ ( Levy )LN ( T ) . In contrast, the asymptotic equivalent log-normal im-plied volatility Σ LN ( K ) given by (123) has a non-trivial dependence on strike. It can be eas-ily shown that the log-normal implied volatility reproduces the asymptotic equivalent impliedvolatility at the ATM point in the limit lim σ T → ,rT = ρ Σ ( Levy )LN ( T ) = Σ LN ( K = A ∞ ) . The results of Table 2 show that the asymptotic result is an improvement over the log-normal approximation.
Remark 21.
The results of [20, 33] are obtained using continuous-time averaging, whileour result (123) was derived for discrete time Asian options. However, we note that theresult (123) does not depend on the size of the time step τ , so it should hold for arbitrarilysmall time step. It is shown elsewhere [40] that a result similar to (123) holds for the smallmaturity limit of continuous time Asian options at fixed σ, r, q , with the substitution ρ = 0 .The limiting procedure adopted in this paper, of taking n → ∞ at fixed β, ρ , allows one totake into account the dependence on r, q in the short maturity expansion. In order to address the performance of the asymptotic results in the small volatility andmaturity regime we compare our results against those in Table 4 of [19]. As pointed outin [44, 20], some of the methods proposed in the literature have numerical issues in theseregimes of the model parameters. The scenarios considered for this test correspond to σ =0 . , S = 100 , r = 0 . , q = 0, and three choices of maturity and strike as shown in Table 3.For reasons of space economy, we present only a subset of the test results in Table 4 of [19],which show the best agreement with a Monte Carlo calculation. The asymptotic results arein very good agreement with the alternative methods shown. We note that the computingtime required by the asymptotic method is very good, as it requires only the solution of asimple non-linear algebraic equation, and the evaluation of a function.We present in Table 4 a comparison with the test results for discretely sampled Asianoptions corresponding to the scenarios considered in Table B of [50]. These scenarios haveparameters r = 0 . , q = 0 , σ = 0 . , T = 1 , K = 100. The results are compared against Table 2.
Numerical results for Asian call options under the 7 scenariosconsidered in [20, 33], etc. FPP3: the 3rd order approximations in Foschi etal. [19], Vecer: the PDE method from [50], MAE3: the matched asymptoticexpansions from Dewynne and Shaw [11], Mellin500: the Mellin transformbased method in Shaw [44]. The last column shows the results from thespectral expansion in [33], and the LN column shows the result of the log-normal approximation [32]. The column PZ gives the results of the asymptoticresult of this paper using (120).Scenario FPP3 MAE3 Mellin500 Vecer PZ LN Linetsky1 0.055986 0.055986 0.056036 0.055986 0.055998 0.056054 0.0559862 0.218387 0.218369 0.218360 0.218388 0.218480 0.219829 0.2183873 0.172267 0.172263 0.172369 0.172269 0.172460 0.173490 0.1722694 0.193164 0.193188 0.192972 0.193174 0.193692 0.195379 0.1931745 0.246406 0.246382 0.246519 0.246416 0.246944 0.249791 0.2464166 0.306210 0.306139 0.306497 0.306220 0.306744 0.310646 0.3062207 0.350040 0.349909 0.348926 0.350095 0.351517 0.359204 0.350095
Table 3.
Test results for Asian call options under small volatility σ =0 . , S = 100 , r = 0 . , q = 0. The column FPP3 shows the 3rd orderapproximations in Foschi et al. [19]. The column MAE3 shows the resultsusing the matched asymptotic expansions from Dewynne and Shaw [11]. Thecolumn Mellin500 shows the results of the Mellin transform based method inShaw [44]. The column PZ shows the asymptotic results of this paper. T K
PZ FPP3 MAE3 Mellin5000.25 99 1.60739 1 . × . × . × . × − . × − . × − . × − . × − . × − . × . × . × . × . × . × . × − . × − . × − . × . × . × . × . × . × . · − . × − . × − . × − those obtained in [50, 45, 9]. The asymptotic results agree with the alternative methods upto about 1%-1.5% in relative error.Finally, in order to test the asymptotic relation (123) for the equivalent log-normal impliedvolatility we show in Figure 2 the equivalent log-normal implied volatility of several Asianoptions obtained by numerical simulation (black dots). These results are obtained by MonteCarlo pricing of Asian options with parameters σ = 0 . , r = q = 0 , τ = 0 . , (128)and n = 50 , ,
200 averaging dates. The Monte Carlo calculation used N MC = 10 samples.The strikes considered cover a region around the ATM point K = S ; the numerical precision SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 25
Table 4.
Asymptotic results for discretely sampled Asian call options underthe scenarios considered in Table B of [50], comparing with the results of[50, 45, 9]. S = 95 S = 100 S = 105Vecer n = 250 8.4001 11.1600 14.3073 n = 500 8.3826 11.1416 14.2881 n = 1000 8.3741 11.1322 14.2786 ∞ n = 250 8.3972 11.1573 14.3054 n = 500 8.3804 11.1392 14.2866 n = 1000 8.3719 11.1300 14.2771 ∞ n = 250 8.3972 11.1572 14.3048 n = 500 8.3801 11.1388 14.2857 n = 1000 8.3715 11.1296 14.2762 ∞ − − − PZ 8.3789 11.1362 14.2818of the simulation decreases rapidly outside of this region. We note very good agreement withthe asymptotic result of Proposition 19, even for n as low as 50. Acknowledgements
We are grateful to an anonymous referee and the editor for their helpful comments andsuggestions. D. P. would like to thank Dyutiman Das and Roussen Roussev for useful dis-cussions about Asian options in financial practice. L. Z. is partially supported by NSF GrantDMS-1613164. 7.
Appendix
Proof of Proposition 4.
The variational problem appearing in equation (39) can be writtenequivalently by introducing the function f ( x ) = bg ( x ) as(129) λ ( a, b ; ρ ) = 1 b sup f ∈AC [0 , (cid:26) − ab (cid:90) e f ( x ) dx − (cid:90) (cid:0) f (cid:48) ( x ) − ρ (cid:1) dx (cid:27) . The functional Λ[ f ] appearing in this variational problem can be rewritten asΛ[ f ] = − ab (cid:90) dxe f ( x ) − (cid:90) (cid:0) f (cid:48) ( x ) − ρ (cid:1) dx (130) = − ab (cid:90) e f ( x ) dx − (cid:90) [ f (cid:48) ( x )] dx + f (1) ρ − ρ . In the second line we integrated by parts and wrote (cid:82) f (cid:48) ( x ) dx = f (1) where we took intoaccount the constraint f (0) = 0. Although in Proposition 4 we have a >
0, the variational problems in Section 4 require also the case of negative a . For this reason we will treat hereboth cases of positive and negative a .The optimal function f ( x ) satisfies the Euler-Lagrange equation(131) f (cid:48)(cid:48) ( x ) = ab e f ( x ) , with the boundary conditions(132) f (0) = 0 , f (cid:48) (1) = ρ . The second boundary condition (at x = 1) is a transversality condition.We observe that the quantity(133) E = − ab e f ( x ) + 12 [ f (cid:48) ( x )] = − ab e f (1) + 12 ρ is a constant of motion of the differential equation (131). Its value was expressed in termsof f (1) by taking x = 1 and using the boundary condition (132). Taking the integral of thisrelation over x : (0 ,
1) can be used to eliminate the integral of [ f (cid:48) ( x )] in the functional Λ[ f ].This can be put into the equivalent formΛ[ f ] = − ab (cid:90) e f ( x ) dx + ab e f (1) + f (1) ρ − ρ . (134)The Euler-Lagrange equation (131) can be solved exactly. Two independent solutions ofthis equation are f ( x ) = δx − (cid:18) e δx + γ γ (cid:19) , (135) f ( x ) = − | cos( ξx + η ) | + 2 log | cos η | . (136)The first solution was given in [26] where a related differential equation appears in the contextof optimal sampling for Monte Carlo pricing of Asian options. It is easy to see by direct sub-stitution into (131) that these functions satisfy this equation, with the appropriate boundarycondition at x = 0. Requiring that the coefficient in this equation and the boundary condition f (cid:48) (1) = ρ are satisfied gives two conditions.For f ( x ) we have the conditions2 γδ = − ab (1 + γ ) , δ γ − e δ γ + e δ = ρ . (137)Eliminating γ between these two equations as γ = δ + ρδ − ρ e δ gives an equation for δ :(138) δ − ρ = − ab (cid:18) cosh (cid:18) δ (cid:19) + ρδ sinh (cid:18) δ (cid:19)(cid:19) . For f ( x ) we obtain the conditions2 ξ = ab cos η , ξ tan( ξ + η ) = ρ . (139)The second relation allows one to eliminate η as(140) tan η = ρ − ξ tan ξξ + ρ tan ξ . We obtain the equation for ξ (141) 2 ξ (4 ξ + ρ ) = ab (2 ξ cos ξ + ρ sin ξ ) . SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 27
K/S S T=0.5(K,T)
K/S S T=1(K,T)
K/S S T=2(K,T)
Figure 2.
The equivalent log-normal volatility Σ LN ( K, S ) of Asian optionsin the Black-Scholes model given by (123) (black curve). The red line isat √ σ and corresponds to the ATM equivalent volatility. The dots showthe log-normal equivalent volatility obtained by Monte Carlo pricing of theAsian options with maturity T = 0 . , ,
2. The BS model parameters are r = q = 0 , σ = 0 .
2. The time step of the MC simulation is τ = 0 .
01 and thenumber of paths N MC = 1 m .Finally, the integral appearing in Λ[ f ] can be computed in closed form for each solution,and we have T ( δ, ρ ) = (cid:90) dxe f ( x ) = 1 δ sinh δ + 2 ρδ sinh (cid:18) δ (cid:19) , (142) T ( ξ, ρ ) = (cid:90) dxe f ( x ) = 12 ξ sin(2 ξ ) (cid:18) ρ ξ tan ξ (cid:19) . (143)Substituting these results into (134), we find the following results for the function λ ( a, b ; ρ ) λ ( a, b ; ρ )(144)= − aT ( δ ) + ae δ (cid:18) γe δ + γ (cid:19) + 1 b ρ (cid:18) δ − (cid:18) e δ + γ γ (cid:19)(cid:19) − ρ b = a (cid:26) δ (cid:18) − ρδ + ρ δ (cid:19) − − ρδ sinh δ (cid:27) + 2 b ρ log (cid:20) cosh δ ρδ sinh δ (cid:21) − ρ b ,λ ( a, b ; ρ )(145)= − aT ( ξ ) + a cos η cos ( ξ + η ) + 1 b ρ log cos η cos ( ξ + η ) − ρ b = a (cid:26) − sin ξ (cid:18) ρξ − ρ ξ (cid:19) + ρ − ξ sin(2 ξ ) (cid:27) + 2 ρb log (cid:20) cos ξ + ρ ξ sin ξ (cid:21) − ρ b , where δ and ξ are given by the solutions of the equations (138) and (141), respectively.For given ( a > , b, ρ ), only one of these two equations has a solution, which determinesthe optimal function f ( x ) uniquely, and the function λ ( a, b ; ρ ). This completes the proof ofProposition 4. (cid:3) Proof of Proposition 9.
The variational problem (50) can be written equivalently in terms of J ( x, ρ ) defined as in (51), by introducing the function f ( y ) = √ βg ( y ) as(146) J ( x, ρ ) = inf f ∈AC [0 , , (cid:82) e f ( y ) dy = xS (cid:90) ( f (cid:48) ( y ) − ρ ) dy . The integral constraint on f ( y ) is taken into account by introducing a Lagrange multiplier a and defining an auxiliary functionalΛ[ f ] = 12 (cid:90) dy (cid:0) f (cid:48) ( y ) − ρ (cid:1) + a (cid:18)(cid:90) dye f ( y ) − xS (cid:19) (147) = 12 (cid:90) dy [ f (cid:48) ( y )] + a (cid:90) dye f ( y ) − ρf (1) + 12 ρ − a xS . The solution of this variational problem f ( y ) satisfies the Euler-Lagrange equation(148) f (cid:48)(cid:48) ( y ) = ae f ( y ) , with boundary conditions (the condition at y = 1 is a transversality condition)(149) f (0) = 0 , f (cid:48) (1) = ρ . This differential equation and the associated boundary conditions are identical to the equationappearing in the proof of Proposition 4. As shown, this can be solved exactly, and thesolutions are given in (135), (136). The details of the proof will be slightly different, as inthe present case the coefficient a (the Lagrange multiplier) is not known, but is one of theunknowns of the variational problem. However, we will show that it can be determined usingthe integral constraint(150) (cid:90) e f ( y ) dy = KS . Before proceeding with the solution of the variational problem, we give a preliminary resultwhich expresses the rate function only in terms of a, f (1).
Lemma 22.
The rate function J ( x, ρ ) is given by J ( K/S , ρ ) = a (cid:18) KS − e f (1) (cid:19) − ρf (1) + ρ . (151) Proof.
The Euler-Lagrange equation (148) conserves the following quantity(152) E = 12 ( f (cid:48) ( y )) − ae f ( y ) , which gives(153) 12 [ f (cid:48) ( y )] − ae f ( y ) = 12 ρ − ae f (1) . Taking the integral of this relation over x : (0 , (cid:3) The only remaining part of the proof is determining a, f (1). This can be done from theconstraint (150). Substituting (135) into this constraint gives(154) (cid:90) dxe f ( x ) = 1 δ sinh δ + 2 ρδ sinh ( δ/
2) = KS . SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 29 which is an equation for δ . This equation has solutions only for K/S ≥ ρ/
2. Once δ, γ are known, the Lagrange multiplier a is determined using the relation (137). Substitutinginto (151) we find the rate function J ( K/S , ρ ) = 12 ( β − ρ ) (cid:18) − β/ β + ρ tanh( β/ (cid:19) (155) − ρ log (cid:20) cosh( β/
2) + ρβ sinh( β/ (cid:21) + ρ . A similar calculation using f ( x ) gives(156) (cid:90) dxe f ( x ) = 12 ξ sin(2 ξ ) (cid:18) ρ ξξ (cid:19) = KS . Both η and ξ + η must be in the ( − π/ , π/
2) range. The equation (156) has solutions onlyfor
K/S ≤ ρ/ ξ , the Lagrange multiplier a is found from (139). Substituting into(151) we find the rate function(157) J ( K/S , ρ ) = 2 (cid:18) ξ + 14 ρ (cid:19) (cid:26) tan ξξ + ρ tan ξ − (cid:27) − ρ log (cid:18) cos ξ + ρ ξ sin ξ (cid:19) + ρ . This completes the proof of Proposition 9. (cid:3)
Proof of Proposition 17.
Start by noting that S t n = S e σZ tn +( r − q − σ ) t n can be writtenequivalently as S e √ βn (cid:80) nj =1 ( V j + ρ √ β ) − βn in distribution where V j are i.i.d. N (0 ,
1) randomvariables. Let us also recall that A n = n S (cid:80) n − k =0 e √ βn (cid:80) kj =1 ( V j + ρ √ β ) − βkn . The terms βn , βkn are uniformly bounded and negligible and if we let g ( x ) = n (cid:80) (cid:98) xn (cid:99) j =1 ( V j + ρ √ β ), then κe √ βn (cid:80) nj =1 ( V j + ρ √ β ) − n (cid:80) n − k =0 e √ βn (cid:80) kj =1 ( V j + ρ √ β ) = κe √ βg (1) − (cid:82) e √ βg ( x ) dx . The map g (cid:55)→ κe √ βg (1) − (cid:82) e √ βg ( x ) dx is continuous in the supremum norm and by contraction principle, P ( κS t n − A n ∈ · ) satisfies a large deviation principle with the rate function(158) H ( x ) = inf g ∈AC [0 , ,κe √ βg (1) − (cid:82) e √ βg ( y ) dy = xS (cid:90) (cid:18) g (cid:48) ( y ) − ρ √ β (cid:19) dy. As n → ∞ , κS t n − A n → κS e ρ − S e ρ − ρ a.s. When κ < ρ (1 − e − ρ ), the call option isout-of-the-money and(159) C ( n ) = e − n H (0)+ o ( n ) , as n → ∞ , and by put-call parity, when ρ (cid:54) = 0, C ( n ) − P ( n ) = e − rt n E [ κS t n − A n ](160) = κS e − rr − q ρ − e − rr − q ρ S e ρ − n (1 − e − ρn )= κS e − rr − q ρ − S e − rr − q ρ e ρ − ρ − e − rρr − q S ( e ρ − n + O ( n − ) . Therefore, as n → ∞ , the asymptotics for in-the-money put option is(161) P ( n ) = − κS e − rr − q ρ + S e − rr − q ρ e ρ − ρ + e − rρr − q S ( e ρ − n + O ( n − ) . When ρ = 0,(162) P ( n ) = (1 − κ ) S + e − n H (0)+ o ( n ) , as n → ∞ . When κ = ρ (1 − e − ρ ), i.e., at-the-money, the asymptotics for C ( n ) and P ( n ) are governedby the central limit theorem. √ nS ( κS t n − A n ) can be approximated by(163) κe ρ √ β √ n n − (cid:88) j =0 V j − √ βn / n − (cid:88) j =0 V j n (cid:88) i = j +1 e ρ in , with V j = N (0 ,
1) i.i.d. random variables. The variance of this expression converges to (cid:90) (cid:20) κe ρ (cid:112) β − √ βρ ( e ρ − e ρx ) (cid:21) dx = 2 βρ (cid:90) (1 − e ρx ) dx (164) = 2 βρ (cid:20) − ρ ( e ρ −
1) + e ρ − ρ (cid:21) . We can further use the nonuniform Berry-Esseen bound for the central limit theorem toobtain the following asymptotics,(165) lim n →∞ √ nC ( n ) = lim n →∞ √ nP ( n ) = S e − rρr − q E [ Z Z ≥ ] , where Z is a normal random variable with mean 0 and variance(166) 2 βρ (cid:20) − ρ ( e ρ −
1) + e ρ − ρ (cid:21) . When κ > ρ (1 − e − ρ ), the put option is out-of-the-money and(167) P ( n ) = e − n H (0)+ o ( n ) , as n → ∞ , and when ρ (cid:54) = 0, we have for in-the-money call option(168) C ( n ) = κS e − rr − q ρ − S e − rr − q ρ e ρ − ρ − e − rρr − q S ( e ρ − n + O ( n − ) , and when ρ = 0,(169) C ( n ) = S ( κ −
1) + e − n H (0)+ o ( n ) , as n → ∞ . (cid:3) Proof of Proposition 19. i) The price of an undiscounted European option in the Black-Scholes model depends only on σ T and K/F with F the forward asset price. In our casegiven by (120) we have F = A ∞ , and we denote this dependence as e − rT A ∞ ¯ C BS ( K/A ∞ , σ T ),with ¯ C BS ( k, v ) := Φ( √ v ( − log k + v )) − k Φ( √ v ( − log k − v )).By definition of the equivalent log-normal implied volatility we have C ( n ) = e − rT A ∞ ¯ C BS ( K/A ∞ , Σ T ) . (170)Consider an OTM Asian call option K > A ∞ . We have from Proposition 13lim n →∞ n log C ( n ) = − β J ( K/S , ρ ) . (171) SYMPTOTICS FOR THE AVERAGE OF GBM AND ASIAN OPTIONS 31
Also, we have lim T → (Σ T ) log (cid:0) A ∞ ¯ C BS ( K/A ∞ , Σ T ) (cid:1) = −
12 log ( K/A ∞ )(172)We get thus, setting T = t n ,lim n →∞ Σ ( K, n ) n τ = lim n →∞ Σ ( K, n ) nτ log[ A ∞ ¯ C BS ( K/A ∞ , Σ T )] n log C ( n )(173) = β log ( K/A ∞ ) J ( K/S , ρ ) . Recalling that β = σ n τ this is written equivalently as(174) lim n →∞ σ Σ ( K, n ) = 12 log ( K/A ∞ ) J ( K/S , ρ ) , which reproduces the result (123).ii) At-the-money Asian option. The Black-Scholes formula gives for this case¯ C BS (1 , Σ T ) = Φ (cid:18)
12 Σ LN √ T (cid:19) − Φ (cid:18) −
12 Σ LN √ T (cid:19) (175) = 1 √ π Σ LN √ T (cid:0) O (cid:0) Σ T (cid:1)(cid:1) . The large n asymptotics of the ATM Asian option given in Proposition 15 reads(176) C ( A ∞ , n ) = 1 √ π S e − rρr − q (cid:112) βv ( ρ ) 1 √ n . The two results are related as C ( A ∞ , n ) = e − rT A ∞ ¯ C BS (1 , Σ T ). Recalling that we have σ √ t n = √ n √ β we obtain the asymptotics of the equivalent implied volatility of the ATMAsian option(177) lim n →∞ Σ LN ( A ∞ , n ) σ = S A ∞ (cid:112) v ( ρ ) . This reproduces equation (125).The proof of (124) proceeds in a similar way, starting with the Bachelier formula for thecall option prices. (cid:3)
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