Short Maturity Forward Start Asian Options in Local Volatility Models
SShort Maturity Forward Start Asian Options in LocalVolatility Models
Dan Pirjol ∗ , Jing Wang † , Lingjiong Zhu ‡ Abstract
We study the short maturity asymptotics for prices of forward start Asian optionsunder the assumption that the underlying asset follows a local volatility model. Weobtain asymptotics for the cases of out-of-the-money, in-the-money, and at-the-money,considering both fixed strike and floating Asian options. The exponential decay of theprice of an out-of-the-money forward start Asian option is handled using large devi-ations theory, and is controlled by a rate function which is given by a double-layeroptimization problem. In the Black-Scholes model, the calculation of the rate functionis simplified further to the solution of a non-linear equation. We obtain closed form forthe rate function, as well as its asymptotic behaviors when the strike is extremely large,small, or close to the initial price of the underlying asset.Keywords: Forward start Asian option, short maturity asymptotics, local volatilitymodel, large deviation, variational problems.2010 Mathematics Subject Classification Numbers: 91G20, 91G80, 60F10.
Contents ∗ Email: [email protected] † Department of Mathematics, University of Illinois Urbana–Champaign. Email: [email protected] ‡ Department of Mathematics, Florida State University. Email: [email protected] a r X i v : . [ q -f i n . P R ] O c t Notations 39
Asian options are among the most popular traded instruments in the equity and commoditymarkets. A great variety of numerical and exact methods have been proposed for theirpricing [12,16,18,20,21,25,33,38,45], see Boyle and Potapchik [8] for a survey. Most of thesemethods are numerically and computationally intensive. Monte Carlo methods require along time, and the calculation of the Greeks is delicate. Methods based on inverting Laplacetransforms by numerical integration [21, 25] require special attention in the small maturityand/or small volatility region.Recently the pricing of Asian options with continuous time averaging has been studied inthe short maturity asymptotic regime [1, 35, 36] in the local volatility model. This approachuses large deviations theory, and relates the short maturity asymptotics to a rate functionfor the time average of the asset price. Explicit results for the rate function can be obtainedfor the Black-Scholes model and the CEV model. This approach avoids the numerical issuesin the short maturity and/or volatility region noted for the methods mentioned above. Arelated approach considers the asymptotics of the Asian options with discrete time averagingin the limit of a very large number of averaging dates has been proposed in [37] under theBlack-Scholes model.In this paper, we are interested in the forward start Asian options in the short maturityasymptotic regime, assuming that the asset price follows the local volatility model. Aforward start option becomes active at a specified date in the future; however, the premiumis paid in advance, and the time to maturity and the underlying asset are established atthe time the forward start option is purchased, see e.g. [34]. Forward starting Europeanoptions have been studied in various works in the literature. Such options have the payoff( S T − kS T ) + where t < T < T with t the pricing date, and k is the strike. More exoticderivatives such as cliquets depend also on the joint distribution of the asset price at futuretimes, see [22] for a survey.The forward start options depend on the future level of volatility, and several approacheshave been proposed to describe this quantity and model its dynamics. Stochastic volatilitymodels are a popular approach. For the purpose of pricing volatility derivatives a convenientapproach is the the variance curve model [9]. Dynamical models for implied volatility[40] have been also proposed, although an arbitrage-free dynamical specification leads tocomplicated consistency conditions [39]. The paper [26] introduced different notions offorward volatilities for forecasting purposes. Empirical studies of the forward smile havebeen carried out in a series of papers by Bergomi [5]. The paper [19] empirically studiesthe forward smile in Sato models and runs comparisons with a suite of models includingHeston and local volatility models for forward smile sensitive products such as cliquets.More closely related to the approach followed here, the asymptotics of the forward startEuropean options has been studied in various settings of small- and large-maturity in [29,30]under the assumption that the asset price follows an exponential L´evy and Heston model,Asian options are defined with an averaging period [ T , T ] with T > T . The totalaveraging period is T − T and the option pays at time T . For example, a forward start(fixed strike) Asian call option with strike K pays (cid:16) T − T (cid:82) T T S t dt − K (cid:17) + at time T , hence2he price of this option is given by an expectation under risk-neutral measure C ( T , T ) = e − rT E (cid:34)(cid:18) T − T (cid:90) T T S t dt − K (cid:19) + (cid:35) . (1.1)Similarly, the price of the forward start put fixed strike Asian option is given by P ( T , T ) = e − rT E (cid:34)(cid:18) K − T − T (cid:90) T T S t dt (cid:19) + (cid:35) . (1.2)Another popular instrument are floating strike Asian options. The forward start floatingstrike Asian call option has the price C f ( T , T ) = e − rT E (cid:34)(cid:18) κS T − T − T (cid:90) T T S t dt (cid:19) + (cid:35) , (1.3)and the price of the forward start put floating strike Asian option is given by P f ( T , T ) = e − rT E (cid:34)(cid:18) T − T (cid:90) T T S t dt − κS T (cid:19) + (cid:35) . (1.4)The papers [1, 35, 36] assume that the averaging period starts at the valuation time( T = 0). However, in practice the averaging period of the Asian options may start also atsome time T > T − T and theoption pays at time T . Forward start Asian options have been considered in the literature,and analytical approximations have been proposed in Bouaziz et al. [7] and Tsao et al. [42],also including quanto effects, see Chang et al. [10]. Vanmaele et al. [43] considered forwardstarting Asian options with discrete time averaging under the Black-Scholes model andderived upper bounds on the prices of these instruments from comonotonicity.In this paper, we are interested in the limiting behavior of forward start Asian optionswhen T and T approach to 0 with a constant ratio. Let T = τ T and T = T for somefixed ratio τ ∈ (0 ,
1) and maturity
T >
0. Clearly when the call option is out-of-the-money( S < K ) or at-the-money ( S = K ), we have C ( T ) := C ( T , T ) → T →
0. Similarly,when the put option is out-of-the-money ( S > K ) or at-the-money ( S = K ), we have P ( T ) := P ( T , T ) → T →
0. However, the limiting behaviors are quite different. Infact the out-of-the-money case is governed by rare events which shall be captured by largedeviation techniques, whereas the at-the-money case is governed by the fluctuations aboutthe typical events.When τ = 0, this falls into the case of a standard Asian option, whose short maturityasymptotics are studied in [35]. In this paper, we consider the strict forward case, namelywhen τ >
0. This situation requires special consideration.We assume that the underlying asset follows a local volatility model. From (1.1) one canrealize that the short maturity pricing problem is equivalent to estimating the probabilityof the average asset price exceeding the strike price, i.e. P (cid:16) − τ (cid:82) τ S T t dt > K (cid:17) , which is arare event when
K > S and T →
0. A natural approach is to use large deviation theoryand the contraction principle [13]. For instance, we can obtain that the price of an forward3tart Asian call ( S , K , τ , T ) satisfies, when K > S ,lim T → T log C ( T ) = − inf − τ (cid:82) τ e g ( t ) dt = Kg (0)=log S ,g ∈AC [0 , (cid:90) (cid:18) g (cid:48) ( t ) σ ( e g ( t ) ) (cid:19) dt =: −I fwd ( S , K, τ ) . This suggests that the major contribution to the probability P (cid:16) − τ (cid:82) τ S T t dt > K (cid:17) is closelyrelated to the minimum energy of absolutely continuous paths started from S whose arith-metic average during ( τ,
1) is K . This minimum energy is known as the rate function of alarge deviation problem. Although it is rather complicated to look for a closed form for therate function, we are able to reduce it to a double-layer variational problem I fwd ( S , K, τ ) = inf c ∈ R (cid:26) c τ + 11 − τ I (cid:16) S e F − ( cτ ) , K (cid:17)(cid:27) , (1.5)where F ( · ) is given by F ( · ) = (cid:82) · dzσ ( S e z ) and I ( x, K ) is given by the variational problem I ( x, K ) = inf ϕ ∈AC [0 , ,ϕ (0)=0 (cid:82) e ϕ ( u ) du = K/x (cid:40) (cid:90) (cid:18) ϕ (cid:48) ( u ) σ ( xe ϕ ( u ) ) (cid:19) du (cid:41) . We find out that the optimal path corresponding to the minimal energy is indeed a patch ofthe optimal path of a European option from time 0 to τ and the optimal path of an Asianoption from time τ to 1. It also coincides with the intuition that a forward start Asianoption is an interpolation of a European option and a standard Asian option.Furthermore, under the assumption of the Black-Scholes model (where volatility of theunderlying asset is a constant σ ), we are able to compute the rate function explicitly andobtain for K > S ,lim T → T log C ( T ) = − σ (1 − τ ) (cid:18) τ β tanh β − τ ) β − − τ ) β tanh β (cid:19) , and for K < S ,lim T → T log P ( T ) = − σ (1 − τ ) (cid:0) τ ξ tan ξ − (1 − τ ) ξ + (1 − τ ) ξ tan ξ (cid:1) , where β ∈ (0 , ∞ ) and ξ ∈ (0 , π/
2) are the unique solutions ofsinh ββ = KS e − τ − τ β tanh β , sin(2 ξ )2 ξ = KS e τ − τ ξ tan ξ . We also obtain the optimal path explicitly by gluing the optimal path of an Europeanoption ( S , S e cστ ) over the period [0 , τ ) with the optimal path of a standard Asian option( S e cστ , K ) over the averaging period [ τ, f ( t ) = (cid:40) ct ≤ t ≤ τcτ + ϕ (cid:16) t − τ − τ (cid:17) τ < t ≤ , ϕ ( · ) is the optimal path of a standard Asian option under Black-Scholes model withmaturity 1, and c is uniquely determined by the C smoothness of f . This also holds forgeneral local volatility models, namely the unique arg min c of (1.5) is such that the optimalpath has continuous first derivative, which is guaranteed by (2.7).The paper is organized as follows. In Section 2, we study the asymptotics of forwardstart Asian options in the out-of-the-money, at-the-money and in-the-money regimes, underthe local volatility model. In Section 3, we focus on the special case of the Black-Scholesmodel, for which explicit results can be obtained. We also study the asymptotic expansionsof the rate functions in the cases of deep out-of-the-money and around at-the-money, andintroduce the notions of τ -AATM and τ -DOTM that are more suitable for forward startingAsian options. At last in Section 4 we discuss the asymptotics for short maturity forwardstart Asian options with floating strike, considering both cases of newly issued and seasonedfloating strike Asian option. These cases correspond to the valuation period being prior orduring the averaging period, respectively. We also present numerical tests for the asymp-totic formulas, by comparing with the analytical approximation of [42] and Monte Carlosimulation. An Appendix summarizes the notations used for the various rate functionsgiving the short maturity asymptotics. In this section we study the short maturity asymptotics for the price of a forward startAsian option, under the assumption that the underlying asset follows the local volatilitymodel, see e.g. [22] for an overview. We assume the underlying asset price is given by thesolution of the following stochastic differential equation dS t = ( r − q ) S t dt + σ ( S t ) S t dW t , S > , (2.6)where W t is a standard Brownian motion, r ≥ q ≥ σ ( · ) is the local volatility. We impose the following assumptions on thelocal volatility σ ( · ): there exist 0 < σ < σ < ∞ and M, α > < σ ≤ σ ( x ) ≤ σ < ∞ , ∀ x ∈ [0 , ∞ ) , (2.7) | σ ( e x ) − σ ( e y ) | ≤ M | x − y | α , ∀ x, y ∈ R . Under these assumptions it was shown in the paper of Varadhan [44] that the log asset pricepaths, and hence the asset price paths satisfy large deviations principles. This property canbe shown to hold in a wider class of local volatility models including the CEV model [4, 15],which do not satisfy the conditions (2.7). For simplicity we restrict ourselves here to theclass of functions σ ( · ) with property (2.7), with the understanding that the results can begeneralized appropriately.We are interested in the asymptotic behavior of the forward start Asian call option withparameters ( S , K, T, τ ) C ( T ) = e − rT E (cid:34)(cid:18) − τ ) T (cid:90) TτT S t dt − K (cid:19) + (cid:35) , S , K, T, τ ) P ( T ) = e − rT E (cid:34)(cid:18) K − − τ ) T (cid:90) TτT S t dt (cid:19) + (cid:35) , as T →
0. Obviously C ( T ) and P ( T ) converge to 0 in the out-of-the-money and at-the-money cases. However, the causes are very different, which then lead to very differentvanishing speed. When it is out-of-the-money, the rapid decrease to 0 of the option pricescomes from the extremely small probability of positive payoff when maturity T →
0. Thedecrease is exponentially fast. When it is at-the-money, the option prices drop to 0 comesfrom the drop of Gaussian fluctuation as T →
0. The speed is at the scale of √ T . First we study the asymptotic for out-of-the-money case. Recall the definitions for a forwardstart Asian option. Let A ( T, τ ) be the forward averaged asset price under the risk-neutralmeasure A ( τ T, τ ) := 1(1 − τ ) T (cid:90) TτT E [ S t ] dt, (2.8)then A ( τ T, τ ) = S ( e ( r − q ) T − e ( r − q ) τT ) (1 − τ )( r − q ) T when r − q (cid:54) = 0 ,S when r − q = 0 . A forward start Asian call option is said to be out-of-the-money if
K > A ( τ T, τ ) and in-the-money if K < A ( τ T, τ ). However, note that A ( τ T, τ ) = S + O ( T ) when T →
0. Thisimplies that in the small maturity regime,
K > A ( τ T, τ ) is equivalent to K > S . Hencethroughout the rest of this paper, we say a forward start Asian call option is out-of-the-money if K > S and in-the-money if K < S . Correspondingly we call a forward startAsian put option out-of-the-money if K < S and in-the-money if K > S .First we prove the following lemma that allows us to transfer the price estimate problemto probabilistic estimates of rare events. Lemma 2.1.
Let C ( T ) , P ( T ) , T > , τ ∈ (0 , be as given above. Then we have lim T → T log C ( T ) = lim T → T log P (cid:18) − τ ) T (cid:90) TτT S t dt ≥ K (cid:19) , (2.9) and lim T → T log P ( T ) = lim T → T log P (cid:18) − τ ) T (cid:90) TτT S t dt ≤ K (cid:19) . (2.10) Proof.
The proof of (2.10) is similar to (2.9). We only prove (2.9) here. The idea followssimilarly as in [35]. First by H¨older’s inequality, for any p, p (cid:48) > p + p (cid:48) = 1 we have C ( T ) ≤ e − rT E (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) − τ ) T (cid:90) TτT S t dt − K (cid:12)(cid:12)(cid:12)(cid:12) − τ ) T (cid:82) TτT S t dt ≥ K (cid:21) ≤ e − rT (cid:18) E (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) − τ ) T (cid:90) TτT S t dt − K (cid:12)(cid:12)(cid:12)(cid:12) p (cid:21)(cid:19) p P (cid:18) − τ ) T (cid:90) TτT S t dt ≥ K (cid:19) p (cid:48) .
6e claim that when T →
0, there exists a constant
C > E (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) − τ ) T (cid:90) TτT S t dt − K (cid:12)(cid:12)(cid:12)(cid:12) p (cid:21) ≤ C. (2.11)Assume the above estimate, we can easily observe for any 1 < p (cid:48) < T → T log C ( T ) ≤ p (cid:48) log P (cid:18) − τ ) T (cid:90) TτT S t dt ≥ K (cid:19) . By letting p (cid:48) → (cid:15) > C ( T ) ≥ e − rT E (cid:20)(cid:18) − τ ) T (cid:90) TτT S t dt − K (cid:19) − τ ) T (cid:82) TτT S t dt ≥ K + (cid:15) (cid:21) ≥ e − rT (cid:15) P (cid:18) − τ ) T (cid:90) TτT S t dt ≥ K + (cid:15) (cid:19) . By letting (cid:15) → T → T log C ( T ) ≥ log P (cid:16) − τ ) T (cid:82) TτT S t dt ≥ K (cid:17) ,which implies (2.9). Now we are let to prove (2.11). When p >
2, by convexity of x → x p on (0 , ∞ ) we have that E (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) − τ ) T (cid:90) TτT S t dt − K (cid:12)(cid:12)(cid:12)(cid:12) p (cid:21) ≤ p − (cid:18) E (cid:20)(cid:18) − τ ) T (cid:90) TτT S t dt (cid:19) p (cid:21) + K p (cid:19) . Moreover, E (cid:20)(cid:18) − τ ) T (cid:90) TτT S t dt (cid:19) p (cid:21) ≤ − τ ) T (cid:90) TτT E ( S pt ) dt, and E ( S pt ) solves the differential equation d E ( S pt ) = (cid:18) p ( r − q ) E ( S pt ) + 12 p ( p − E [ S pt σ ( S t ) ] (cid:19) dt. Using (2.7) we obtain that E ( S pt ) ≤ S p e ( p ( r − q )+ p ( p − σ ) t , hence1(1 − τ ) T (cid:90) TτT E ( S pt ) dt ≤ S p e ( p ( r − q )+ p ( p − σ ) T . This implies (2.11). We then complete the proof.From Lemma 2.1 we can reduce the logarithmic estimates for prices of out-of-the-moneyforward start Asian options to the probability estimate of the rare event that the underlyingasset price goes up from S to K within time T →
0. A classic tool is the large deviationtheory. For the definition and basic properties of the large deviation theory, we refer to [13].
Theorem 2.2.
Assume the asset price S t follows the local volatility model as in (2.6) and (2.7) . Then the price of an out-of-the-money forward start Asian call option ( K > S )satisfies lim T → T log C ( T ) = −I fwd ( S , K, τ ); (2.12)7 nd the price of the corresponding out-of-the-money forward start Asian put option ( K
From Lemma 2.1 we just need to have a large deviation estimate for − τ ) T (cid:82) TτT S t dt .The idea is to use the contraction principle from large deviations theory (see e.g. [13]). Let X t = log S t . First note1(1 − τ ) T (cid:90) TτT S t dt = T (1 − τ ) T (cid:90) τ S tT dt = 11 − τ (cid:90) τ e X tT dt. On the other hand we know that P ( X tT ∈ · , t ∈ [0 , L ∞ ([0 , , R ) with the rate function (see e.g. [44]) I ( g ) = (cid:82) (cid:16) g (cid:48) ( t ) σ ( e g ( t ) ) (cid:17) dt, for g (0) = log S , g ∈ AC [0 , ∞ otherwise , where AC [0 ,
1] is the space of absolutely continuous functions.Since the map g (cid:55)→ − τ (cid:82) τ e g ( x ) dx from L ∞ ([0 , , R ) to R + is a continuous map, bycontraction principle (see e.g. [13]), we havelim T → T log C ( T ) = lim T → T log P (cid:18) − τ ) T (cid:90) TτT S t dt ≥ K (cid:19) (2.14)= − inf − τ (cid:82) τ e g ( t ) dt = Kg (0)=log S ,g ∈AC [0 , (cid:90) (cid:18) g (cid:48) ( t ) σ ( e g ( t ) ) (cid:19) dt. Hence we obtain (2.12). Following the same argument we can easily obtain (2.13).As a by-product, we can also obtain the estimate for an in-the-money forward startAsian option, by using put-call parity. We present it in the following corollary.
Corollary 2.3.
Assume the asset price S t satisfies (2.6) and (2.7) . Then the price of anin-the-money forward start Asian call option ( K < S ) satisfies C ( T ) = S − K + (cid:18)
12 ( r − q )(1 + τ ) − ( S − K ) r (cid:19) T + O ( T ) , as T → , (2.15) and the price of the corresponding in-the-money forward start Asian put option ( K > S )satisfies P ( T ) = K − S − (cid:18)
12 ( r − q )(1 + τ ) + ( S − K ) r (cid:19) T + O ( T ) , as T → . (2.16)8 roof. From the put-call parity we have C ( T ) − P ( T ) = e − rT E (cid:20) − τ ) T (cid:90) TτT S t dt − K (cid:21) = e − rT (cid:18) S ( e ( r − q ) T − e ( r − q ) τT ) (1 − τ )( r − q ) T − K (cid:19) if r − q (cid:54) = 0 e − rT ( S − K ) if r − q = 0= (cid:40) S − K + (cid:0) ( r − q )(1 + τ ) − ( S − K ) r (cid:1) T + O ( T ) if r − q (cid:54) = 0( S − K )(1 − rT ) + O ( T ) if r − q = 0 , as T →
0. On the other hand, when
K < S , from (2.13) we know that P ( T ) (cid:28) O ( T ) when T →
0, hence C ( T ) = S − K + (cid:18)
12 ( r − q )(1 + τ ) − ( S − K ) r (cid:19) T + O ( T ) . We can obtain (2.16) using the same argument.
In this section we consider at-the-money case, namely when S = K . First we notice thatit is much more likely for the asset price to hit K = S after an extremely short timecomparing to out-of-the-money case. Though the probability of this event is still small, itis at the scale of Gaussian fluctuation, namely √ T , which is significantly larger than e −I /T in the out-of-the-money case. In the following theorem, we look for the exact asymptoticof an at-the-money forward start Asian option when T → Theorem 2.4.
Consider an at-the-money forward start Asian option ( S = K, T, τ ) underthe local volatility model (2.6) and (2.7) . If in addition we assume the volatility satisfiesLipschitz continuity conditions: there exist α, β > such that for all x, y ≥ , | σ ( x ) x − σ ( y ) y | ≤ α | x − y | , | σ ( x ) − σ ( y ) | ≤ β | x − y | . Then the prices of the forward start Asian call and put options satisfy lim T → √ T C ( T ) = lim T → √ T P ( T ) = σ ( S ) S (cid:114) τ π . Proof.
The idea is to look for a linear estimate of the underlying asset S t , which is enoughto capture the small maturity asymptotic of C ( T ) and P ( T ) up to order √ T . The proof issimilar as in [35] (see Theorem 6). Here we only sketch the major steps. Step 1 : We first consider an approximate process X t := e − ( r − q ) t S t , which is a martingaleand satisfies the SDE dX t = σ ( X t e ( r − q ) t ) X t dW t , X = S . For any underlying process x t ∈ C ([0 , , R + ), we denote C ( T, x t ) := e − rT E (cid:34)(cid:18) − τ ) T (cid:90) TτT x t dt − S (cid:19) + (cid:35) . For a, b >
0, we say a (cid:29) b if ab → + ∞ , and a (cid:28) b if ab → e rT | C ( T, S t ) − C ( T, X t ) | ≤ E (cid:20) − τ ) T (cid:90) TτT (cid:12)(cid:12)(cid:12) e ( r − q ) t − (cid:12)(cid:12)(cid:12) X t dt (cid:21) = S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ( r − q ) T − e ( r − q ) τT ( r − q )(1 − τ ) T − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( T ) . Hence we have | C ( T, S t ) − C ( T, X t ) | = O ( T ) . Step 2 : Next we consider a further approximation of X t by a Gaussian process ˆ X t = S + σ ( S ) S W t , where W t is a standard Brownian motion. Note e rT (cid:12)(cid:12)(cid:12) C ( T, X t ) − C (cid:16) T, ˆ X t (cid:17)(cid:12)(cid:12)(cid:12) ≤ E (cid:20) max τT ≤ t ≤ T | X t − ˆ X t | (cid:21) . We claim that E (cid:104) max τT ≤ t ≤ T | X t − ˆ X t | (cid:105) = O ( T ). This is due to Doob’s martingale inequal-ity and the fact that E (cid:20) max τT ≤ t ≤ T | X t − ˆ X t | (cid:21) ≤ (cid:16) E [( X T − ˆ X T ) ] (cid:17) ≤ √ M T (2.17)for some constant
M >
0. The detailed proof of the last inequality in (2.17) can be foundin [35] (see page 21-22). At the end, combining Step 1 and Step 2, we obtain that C ( T ) = C (cid:16) T, ˆ X t (cid:17) + O ( T ) . At the end we just need to compute C (cid:16) T, ˆ X t (cid:17) for the Gaussian process ˆ X t . Note1(1 − τ ) T (cid:90) TτT ˆ X t dt − S = σ ( S ) S − τ ) T (cid:90) TτT W t dt, which is a Gaussian random variable with mean 0 and variance σ ( S ) S − τ ) T ) E (cid:34)(cid:18)(cid:90) TτT W t dt (cid:19) (cid:35) = 13 σ ( S ) S (1 + 2 τ ) T. Hence we have E (cid:34)(cid:18) − τ ) T (cid:90) TτT ˆ X t dt − S (cid:19) + (cid:35) = 1 √ σ ( S ) S √ τ √ T E [ Z Z> ]= σ ( S ) S (cid:114) (1 + 2 τ ) T π , where Z is a standard Gaussian random variable with mean zero and variance one. Theresult for the put option can be proved similarly.10 .3 Discussions on variational problem In this section we further discuss the variational problem for out-of-the-money case givenin Theorem 2.2. We want to analyze the rate function I fwd ( S , K, τ ) = inf f ∈A ( K/S ,τ ) (cid:90) (cid:18) f (cid:48) ( t ) σ ( S e f ( t ) ) (cid:19) dt, (2.18)where A ( K/S , τ ) = (cid:26) f ∈ AC [0 , (cid:12)(cid:12)(cid:12)(cid:12) f (0) = 0 , − τ (cid:90) τ e f ( t ) dt = KS (cid:27) . (2.19)We have the following result. Proposition 2.5.
The solution of the variational problem (2.18) is given by the extremumproblem I fwd ( S , K, τ ) = inf c ∈ R (cid:26) c τ + 11 − τ I (cid:16) S e F − ( cτ ) , K (cid:17)(cid:27) , (2.20) where I ( x, K ) is given by the variational problem I ( x, K ) = inf ϕ ∈A ( K/x, (cid:40) (cid:90) (cid:18) ϕ (cid:48) ( u ) σ ( S e ϕ ( u ) ) (cid:19) du (cid:41) , (2.21) and F ( · ) is defined by F ( · ) = (cid:90) · dzσ ( S e z ) . (2.22) Proof.
The variational problem (2.18) with the constraint (2.19) can be transformed into anunconstrained variational problem by introducing a Lagrange multiplier λ for the functionalΛ[ f ] := 12 (cid:90) (cid:18) f (cid:48) ( t ) σ ( S e f ( t ) ) (cid:19) dt − λ (cid:18)(cid:90) τ e f ( t ) dt − KS (1 − τ ) (cid:19) . (2.23)We split the variational problem (2.18) with the constraint (2.19) into two parts and analyzethem separately. Part 1:
When 0 ≤ t ≤ τ , the Euler-Lagrange equation reads ddt (cid:18) f (cid:48) ( t ) σ ( S e f ( t ) ) (cid:19) = 0 , ≤ t ≤ τ . This implies that f (cid:48) ( t ) σ ( S e f ( t ) ) = c , from which we obtain that f is monotone. By integrationwe have that (cid:90) f ( t )0 dzσ ( S e z ) = F ( f ( t )) = ct. Clearly F is a strictly increasing function. By inverting F we then obtain the argmin of therate function I fwd ( S , K, τ ) in the period [0 , τ ]: f ( t ) = F − ( ct ) , ≤ t ≤ τ. (2.24)11he corresponding energy is 12 (cid:90) τ (cid:18) f (cid:48) ( t ) σ ( S e f ( t ) ) (cid:19) dt = 12 c τ. Part 2:
Now consider the region τ ≤ t ≤
1. We want to find f ( t ) ∈ AC [ τ, f ( τ ) = F − ( cτ ), − τ ) (cid:82) τ S e f ( t ) dt = K , and has the minimum energy (cid:82) τ (cid:16) f (cid:48) ( t ) σ ( S e f ( t ) ) (cid:17) dt .Let us reparametrize f ( t ), t ∈ [ τ,
1] as follows. Consider the variable u ∈ [0 ,
1] such that forall t ∈ [ τ,
1] we have t = τ + u (1 − τ ) . We also center the function f by letting ϕ : [0 , → R be such that F − ( cτ ) + ϕ ( u ) = f ( t ) = f ( τ + u (1 − τ )) . (2.25)Then clearly we have dt = (1 − τ ) du , f (cid:48) ( t ) = 11 − τ ϕ (cid:48) ( u ) . The function ϕ ( u ) satisfies the boundary condition ϕ (0) = 0 and (2.19) can be written as (cid:26) ϕ ∈ AC [0 , (cid:12)(cid:12)(cid:12)(cid:12) ϕ (0) = 0 , (cid:90) e ϕ ( u ) du = KS e − F − ( cτ ) (cid:27) . (2.26)From (2.18) we know for any c ∈ R , I fwd ( S , K, τ ) ≤ c τ + 1(1 − τ ) inf ϕ ∈A ( K/S ,τ ) (cid:26) (cid:90) (cid:18) ϕ (cid:48) ( u ) σ ( S e F − ( cτ ) e ϕ ( u ) ) (cid:19) du (cid:27) . Let the contribution from F − ( cτ ) be absorbed into a redefinition of S , then the righthand side of the above inequality is indeed12 c τ + 1(1 − τ ) I ( S e F − ( cτ ) , K ) := G c Hence we obtain I fwd ( S , K, τ ) ≤ inf c ∈ R G c . On the other hand, let G = inf c ∈ R G c . Then for any (cid:15) >
0, there exists a c (cid:48) ∈ R such that G > G c (cid:48) − (cid:15) ≥ I fwd ( S , K, τ ) − (cid:15). Let (cid:15) → G ≥ I fwd ( S , K, τ ). This yields (2.20) and concludes the statement. Remark 2.6.
We know that when S = K (at-the-money), the option price is of scale √ T as T →
0, hence the rate function I fwd ( S , K, τ ) vanishes for K = S . This can be easilychecked by noting that c = 0 and ϕ = 0 solves the minimization problem.12e call a path f ( t ) ∈ AC [0 ,
1] the optimal path of I fwd ( S , K, τ ) if it is the minimizerof the variational problem (2.18) with the constraint (2.19).Next, we study the continuity of the derivative of the optimal path f ( t ). From the aboveproposition we know that f ( t ) is a C function on both (0 , τ ) and ( τ, τ is the so-called corner point. In fact we know that f (cid:48) ( t ) is continuousat t = τ as well, hence f ∈ C ([0 , , R ). This is guaranteed by Erdmann-Weierstrasscondition (see for example § Theorem 2.7 (Erdmann-Weierstrass corner conditions) . If a function y ( t ) is an extremumof the variational problem δδy ( t ) (cid:90) ba L ( y, y (cid:48) ) dt = 0 , (2.27) then ∂ y (cid:48) L and y (cid:48) ∂ y (cid:48) L − L must be continuous at each corner point τ of y ( t ) , that is pointswhere y (cid:48) ( t ) may have different values on each side of τ . In our case the function L ( y, y (cid:48) ) of the variational problem (2.23) is, up to an unessentialconstant, L ( f, f (cid:48) ) = 12 (cid:18) f (cid:48) ( t ) σ ( S e f ( t ) ) (cid:19) − λ { τ ≤ t ≤ } e f ( t ) , (2.28)such that ∂ f (cid:48) L ( f, f (cid:48) ) = f (cid:48) ( t ) σ ( S e f ( t ) ) . (2.29)Since σ ( x ) is continuous everywhere, it follows by the Erdmann-Weierstrass condition for ∂ f (cid:48) L ( f, f (cid:48) ) that f (cid:48) ( t ) is continuous at the corner point τ . This proves that the optimalsolution f ( t ) of this variational problem is C on (0 , In this section we consider forward start Asian options under Black-Scholes model, which isa special case of the local volatility model with σ ( · ) in (2.6) being a constant function. Thevariational problem significantly simplifies in this case and we are able to obtain a closedform for the rate function I fwd ( S , K, τ ). Let σ ( · ) = σ >
0. The result of Proposition 2.5 simplifies as follows.
Proposition 3.1.
Assume the underlying asset price S t follows Black-Scholes model. Thenthe price of an out-of-the-money forward start Asian call option ( K > S ) satisfies lim T → T log C ( T ) = − σ J (BS)fwd ( K/S , τ ) , (3.30) and the price of an out-of-the-money forward start Asian put option ( K < S ) satisfies lim T → T log P ( T ) = − σ J (BS)fwd ( K/S , τ ) , (3.31)13 here J (BS)fwd ( K/S , τ ) = inf c ∈ R (cid:26) c τ + 11 − τ J BS (cid:18) KS e − cτ (cid:19)(cid:27) . (3.32) Here J BS ( · ) is the rate function for an Asian option with averaging starting at time zero,which is given by the variational problem J BS ( x ) = inf (cid:82) e ϕ ( u ) du = xϕ ∈AC [0 , ,ϕ (0)=0 (cid:26) (cid:90) ( ϕ (cid:48) ( u )) du (cid:27) . (3.33) Proof.
This easily follows from Proposition 2.5. We just need to show when σ ( · ) = σ , wehave I fwd ( S , K, τ ) = 1 σ J BSfwd ( K/S , τ ) . This is straightforward by noting I BS ( · , K ) = σ J BS ( K/ · ) and F ( x ) = x/σ . Remark 3.2.
An alternative way to look at the forward start Asian call option withparameters ( S , K, T, τ ) under the Black-Scholes model is as follows. The underlying assetprice is given by S t = S e σW t +( r − q − σ ) t . Then the averaging over [ τ T, T ] is given by S τT A [ τT,T ] , where A [ τT,T ] = 1(1 − τ ) T (cid:90) (1 − τ ) T e σW t +( r − q − σ ) t dt. The underlying of the forward start Asian is therefore a product of two uncorrelated randomvariables: a log-normally distributed variable S τT := X and A [ τT,T ] . The call option pricecan be written as C ( T ) = e − rT E [( XA [ τT,T ] − K ) + ] = e − rT E (cid:34) X E (cid:34)(cid:18) A [ τT,T ] − KX (cid:19) + (cid:12)(cid:12)(cid:12)(cid:12) X (cid:35)(cid:35) (3.34)= e − rT (cid:90) ∞−∞ e − x τT e σx +( r − q − σ ) τT C A ( K/S τT , (1 − τ ) T ) dx √ πτ T = e − rT +( r − q ) τT √ πτ T (cid:90) ∞−∞ e − τT ( x − στT ) C A ( K/S τT , (1 − τ ) T ) dx , where C A ( K/S τT , (1 − τ ) T ) is the undiscounted Asian call option price, with averagingstarting at time zero, underlying starting price at 1, and strike price at K/S τT . In con-clusion, the price of a forward start Asian option is a Gaussian-weighted average over theprices of Asian options starting at time zero.It is already known that (for instance see [35]) when T →
0, we have C A ( K/S τT , (1 − τ ) T ) = exp (cid:18) − T (1 − τ ) J BS ( K/S τT ) + o (1 /T ) (cid:19) . We plug this asymptotic result back into (3.34). Using the Laplace method we know that C ( T ) is dominated by the exponential termexp (cid:18) − x τ T − T (1 − τ ) J BS (cid:16) Ke − σx − ( r − q − σ ) τT /S (cid:17)(cid:19) , J BS ( · ) is the rate function for the Asian option in the Black-Scholes model. Hencewe have lim T → T log C ( T ) = − inf x ∈ R (cid:26) x τ + 11 − τ J BS (cid:0) Ke − σx /S (cid:1) (cid:27) . This coincides with (3.32) by letting x = cτ /σ . Corollary 3.3.
In the limiting case of very small averaging period, τ → , (3.32) reducesto the rate function of an European option lim τ → J ( BS )fwd ( K/S , τ ) = 12 log (cid:18) KS (cid:19) . (3.35) When τ → , (3.32) reduces to the rate function of a standard Asian option with averagingstarting from time , namely lim τ → J ( BS )fwd ( K/S , τ ) = J BS ( K/S ) . Proof.
The second conclusion is obvious. We just show (3.35). When τ →
1, from (3.32)we have that lim τ → J ( BS )fwd ( K/S , τ ) = inf c ∈ R (cid:26) c + 11 − τ J BS (cid:18) KS e c (cid:19)(cid:27) . Since − τ → + ∞ as τ →
1, the optimal c ∈ R is the smallest real number for which J BS (cid:16) KS e c (cid:17) vanishes. Moreover, from (3.33) we know that J BS (cid:16) KS e c (cid:17) = 0 as soon as S e c = K . Hence we obtain the optimal c = log (cid:16) KS (cid:17) , which gives (3.35). This agrees withthe intuition that when τ is close to 1, almost no averaging takes place. It therefore fallsback to a standard European option. In this section we look for a closed form expression for the rate function σ J (BS)fwd ( K/S , τ )of an out-of-the-money forward start Asian option under the Black-Scholes model. From(3.32) this amounts to solving a double-layer variational problem. We know that when τ ∈ (0 , τ = 0 and τ = 1. It can also be observed by looking at its optimal path. Recall that a path f ( t ) ∈ AC [0 ,
1] is the optimal path of J ( BS )fwd ( S , K, τ ) if it satisfies f (0) = 0 , − τ (cid:90) τ e f ( t ) dt = KS , J ( BS )fwd ( K/S , τ ) = 12 (cid:90) [ f (cid:48) ( t )] dt. From the proof of Proposition 2.5 we know that f ( t ) is a combination of two continuouspaths, namely the optimal paths for the European option ( S , S e cτ , τ ) and the optimalpath for the Asian option ( S e cτ , K , 1 − τ ). Explicitly we have f ( t ) = (cid:40) ct ≤ t ≤ τcτ + ϕ c (cid:16) t − τ − τ (cid:17) τ < t ≤ , (3.36)15here ϕ c ( · ) ∈ AC [0 ,
1] is the argmin of J BS (cid:16) KS e c (cid:17) . Here c is uniquely determined by theminimizer of (3.32), which shall be discussed in detail later. The explicit forms of J BS ( x )and its argmin ϕ ( · , x ) were computed in [35] (see Proposition 12). We recall these resultsbelow. Lemma 3.4.
Let J BS ( x ) be given as in (3.33) . Then we have J BS ( x ) = (cid:40) β − β tanh( β/ x ≥ ξ (tan ξ − ξ ) x < , and for u ∈ [0 , ϕ ( u, x ) = βu − (cid:16) e βu + e β e β (cid:17) x ≥ (cid:16) cos ξ cos ( ξ ( u − (cid:17) x < , where β ∈ [0 , ∞ ) and ξ ∈ [0 , π/ are the unique solutions of the following equations β sinh β = x, ξ sin(2 ξ ) = x. t = 0 K/S = 1.5 t t = 0 K/S = 0.60K/S = 0.80 t Figure 1: Optimal paths ϕ ( · , x ) for x = K/S > K/S < , τ = 0. The valuesof K/S are as shown.The left plot in Figure 1 shows the optimal path ϕ ( u, x ) for two strikes with x = K/S > ϕ (0 , x ) = 0 and the transversality condition ϕ (cid:48) (1 , x ) = 0. Recall that ϕ ( u, x ) satisfies also (cid:82) e ϕ ( u,x ) du = K/S = x .By Erdmann-Weierstrass corner conditions we know that f is in C ([0 , , R ), which infact, uniquely determines the constant c . In the proposition below, we given a simple anddirect verification of f ∈ C ([0 , , R ) and obtain an explicit expression for c . For x = 1, we define β = 0 and ξ = 0. roposition 3.5. Let J ( BS )fwd ( K/S , τ ) be given as in (3.32) and f ∈ AC [0 , be the optimalpath as described in (3.36) . Then f ∈ C ([0 , , R ) . Moreover, we have c = (cid:40) − τ β c tanh (cid:16) β c (cid:17) K ≥ S e cτ − − τ ξ c tan ξ c K ≤ S e cτ , where β c ∈ [0 , ∞ ) and ξ c ∈ [0 , π/ are the unique solutions of β c sinh β c = KS e cτ , ξ c sin(2 ξ c ) = KS e cτ . (3.37) Proof.
To show f ∈ C ([0 , , R ), we just need to prove that f (cid:48) ( τ − ) = f (cid:48) ( τ +). By (3.36) itis equivalent to prove that c = 11 − τ ϕ (cid:48) c (0) . (3.38)From Lemma 3.4 and the definition of ϕ c we know that ϕ (cid:48) c (0) = (cid:40) β c tanh (cid:16) β c (cid:17) K ≥ S e cτ , − ξ c tan( ξ c ) K < S e cτ . (3.39)One the other hand, since c is the argmin of J fwd ( S , K, τ ) in (3.32), it is a critical point.Hence c = − τ (1 − τ ) ∂∂c J (cid:18) KS e cτ (cid:19) . (3.40)By Lemma 3.4 we can easily obtain that ∂∂c J BS (cid:18) KS e cτ (cid:19) = (cid:40) − τ β c tanh (cid:16) β c (cid:17) K ≥ S e cτ , τ ξ c tan ξ c K ≤ S e cτ . Combine the above two equations we have c = (cid:40) − τ β c tanh (cid:16) β c (cid:17) K ≥ S e cτ , − − τ ξ c tan ξ c K ≤ S e cτ . Comparing with (3.39) we immediately obtain (3.38). This completes the proof.From Proposition 3.5 we can easily obtain the following fact.
Proposition 3.6.
For a forward start Asian option with parameters ( K, S , τ ) and τ ∈ (0 , , let c = S e f ( τ ) be as described above. Then(1) If K = S , we have c = 0 ;(2) If K > S , then c > and S < S e cτ < K ;(3) If K < S , then c < and S > S e cτ > K . For K = S e cτ , we define β c = ξ c = 0. c Figure 2: Graphical representation of the equation (3.40) for c . This equation can berepresented as the intersection of the curve τ ∂∂c J (cid:16) KS e cτ (cid:17) and c (1 − τ ) plotted vs c . Thetwo cases shown correspond to K > S (red) and K < S (blue). The values of c in the twocases are positive and negative, respectively. Proof.
First note that the optimal path (3.36) is piecewisely monotone. This is becausethe optimal paths for a regular European option and a standard Asian option are bothmonotone (see [35], page 25). Moreover from (3.38) we know that c has the same sign as ϕ (cid:48) (0). This implies that the optimal path f is monotone in the whole interval [0 , c > K > S , c < K < S and c = 0 if K = S . Note S e f ( τ ) = S e cτ . By monotonicity of S e f ( t ) , t ∈ [0 ,
1] we can easily conclude that S e cτ always falls between S and K .Now we are able to derive the closed form of the rate function of an out-of-the-moneyforward start Asian option under Black-Scholes model. Theorem 3.7.
Let C ( T ) (resp. P ( T ) ) be the price of an out-of-the-money forward startAsian call (resp. put) option ( S , K, T, τ ) under Black-Scholes model with volatility σ . Thenwe have the short maturity asymptotics of the prices as follows. For K > S , lim T → T log C ( T ) = − σ (1 − τ ) (cid:18) τ β tanh β − τ ) β − − τ ) β tanh β (cid:19) , (3.41) and for K < S , lim T → T log P ( T ) = − σ (1 − τ ) (cid:0) τ ξ tan ξ − (1 − τ ) ξ + (1 − τ ) ξ tan ξ (cid:1) , (3.42) where β ∈ (0 , ∞ ) and ξ ∈ (0 , π ) are the unique solutions of sinh ββ = KS e − τ − τ β tanh β , K > S , (3.43)sin(2 ξ )2 ξ = KS e τ − τ ξ tan ξ , K < S . (3.44)18 roof. By combining Proposition 3.1 and Proposition 3.5 we immediately obtain the con-clusion.We show in Table 1 numerical results for c as given by the solution of the equation (3.40)for a forward start Asian option with several values of τ and different strikes K > S . Thesolution for β c of the equation (3.37) is also shown. Substituting these numerical valuesinto the expressions for f ( t ) shown above, gives the optimal paths shown in the left plot ofFigure 3.These optimal paths have two pieces: a linear piece (dashed blue) corresponding to thetime interval before the averaging period, and the concave piece (solid black) correspondingto the averaging period.Table 1: Numerical solution of the equation (3.40) for c , for the forward start Asian optionwith τ values shown and K > S . For each c we compute β c given by the solution of β c sinh β c = KS e − cτ . The last column shows the rate function of the forward start Asianoption in the Black-Scholes model. K/S τ c β c J ( BS )fwd ( K/S , τ )1.0 0.0 - 0.0 0.01.1 0.0 - 0.76340 0.013371.2 0.0 - 1.06487 0.048131.3 0.0 - 1.2873 0.098181.4 0.0 - 1.46814 0.159321.5 0.0 - 1.62213 0.228541.0 0.25 0 0 0.01.1 0.25 0.189261 0.5392 0.009041.2 0.25 0.35968 0.751447 0.032941.3 0.25 0.514475 0.907739 0.067941.4 0.25 0.65612 1.03462 0.111321.5 0.25 0.786554 1.14257 0.161091.0 0.5 0 0 0.01.1 0.5 0.142709 0.38003 0.006811.2 0.5 0.272543 0.52806 0.024871.3 0.5 0.391598 0.636173 0.051461.4 0.5 0.501497 0.723307 0.084551.5 0.5 0.603527 0.796955 0.122671.0 0.75 0 0 0.01.1 0.75 0.114339 0.239673 0.095311.2 0.75 0.218666 0.332169 0.182321.3 0.75 0.314588 0.399221 0.262361.4 0.75 0.403356 0.452895 0.336471.5 0.75 0.485962 0.497977 0.40546We also show numerical values for c and β c for a few values of K/S ≤ f ( t ) for forward start Asian options with19 t = 0.5 K/S = 1.50K/S = 1.20 t t = 0.5 K/S = 0.60K/S = 0.80 t Figure 3: Optimal path f ( t ) given by (3.36) for K/S = 1 . , . K/S = 0 . , . τ = 0 .
5. The two pieces ofthe path with different analytical form are shown as the dashed blue and solid black curves,respectively.averaging over [ τ,
1] with τ = 0 . K/S = 0 . , τ ] region, and a convex function piece in the [ τ,
1] region.
Under Black-Scholes model, we have a closed form for the logarithmic estimate for the priceof an out-of-the-money forward start Asian option I fwd ( S , K, τ ). It has different behaviorwhen the option strike takes extreme values. In this section, we take a further step andestimate these asymptotic behaviors. For convenience we use log-strike x = log( K/S ) inthe rest of this section.Recall for a standard Asian option, we call it around ATM (AATM) if | x | (cid:28) | x | (cid:29)
1. For a forward start Asian option, this is the case only if τ ∼ O (1). To include also the limit values of τ we define the following regions in ( τ, x )plane: • τ -almost-ATM ( τ − AATM) region. This is the region (1 − τ ) | x | (cid:28)
1. It includes theAATM region. • τ -deep-OTM ( τ − DOTM) region. This is the region (1 − τ ) | x | (cid:29)
1, and includes theregion of very large x → ∞ and very small − x → ∞ strikes. The deep-OTM regionwith x < − τ ) x and the form of theexpansion depends on the relative size of this parameter to 1. In order to see this we recallthe relation (3.43) determining β for K > S (1 − τ ) log (cid:18) sinh ββ (cid:19) + τ β tanh (cid:18) β (cid:19) = (1 − τ ) x . The relative size of the product (1 − τ ) x on the right hand side and 1 decides if the expansionof the left hand side is done around β = 0 or β → ∞ . We recall that a similar result holds20able 2: Numerical solution of the equation (3.40) for c , for the forward start Asian optionwith τ values shown and K < S . For each c we compute ξ c given by the solution of ξ c sin(2 ξ c ) = KS e − cτ . The last column shows the rate function of the forward start Asianoption in the Black-Scholes model. K/S τ c ξ c J ( BS )fwd ( K/S , τ )1.0 0.0 - 0.0 00.9 0.0 - 0.39334 0.017010.8 0.0 - 0.56555 0.078230.7 0.0 - 0.70509 0.205800.6 0.0 - 0.83002 0.437350.5 0.0 - 0.94775 0.841611.0 0.25 0 0 00.9 0.25 -0.21239 0.278525 0.011160.8 0.25 -0.453789 0.401176 0.050350.7 0.25 -0.732556 0.5013 0.129500.6 0.25 -1.06111 0.591885 0.267650.5 0.25 -1.45904 0.678576 0.497241.0 0.5 0 0 00.9 0.5 -0.158352 0.197664 0.008340.8 0.5 -0.336107 0.285876 0.037450.7 0.5 -0.538556 0.358898 0.095840.6 0.5 -0.773475 0.426057 0.196940.5 0.5 -1.05297 0.491616 0.363421.0 0.75 0 0 0.00.9 0.75 -0.126473 0.125404 0.006670.8 0.75 -0.267951 0.181998 0.029890.7 0.75 -0.428465 0.229381 0.076390.6 0.75 -0.613921 0.273526 0.156720.5 0.75 -0.833483 0.317279 0.28867for the equation (3.44) determining ξ for K < S (1 − τ ) log (cid:18) sin 2 ξ ξ (cid:19) + 2 τ ξ tan ( ξ ) = (1 − τ ) x . We consider the asymptotic expansion of the rate function for forward starting Asian op-tions in the Black-Scholes model in each of these regions. We first consider the asymptoticsof the rate function in the τ − AATM region (1 − τ ) | x | (cid:28) Proposition 3.8.
In the τ − AATM region (1 − τ ) | x | (cid:28) we have the following asymptotic xpansion in powers of − ττ x in the Black-Scholes model. I ( BS )fwd ( x, τ, σ ) = 1(1 − τ ) σ (cid:26) τ τ ) (cid:18) x (1 − τ ) τ (cid:19) − τ (1 − τ )10(1 + 2 τ ) (cid:18) x (1 − τ ) τ (cid:19) + τ (109 − τ )1400(1 + 2 τ ) (cid:18) x (1 − τ ) τ (cid:19) + O (cid:16) ( x (1 − τ )) (cid:17) (cid:27) . Proof. (1) Consider first the case
K > S . We have(1 − τ ) log (cid:18) sinh ββ (cid:19) + τ β tanh (cid:18) β (cid:19) (cid:28) . This implies that β (cid:28) − ττ x = β tanh β − ττ log (cid:18) sinh ββ (cid:19) = β − β
24 + O ( β ) + 1 − ττ (cid:18) β − β
180 + O ( β ) (cid:19) Hence 12 β = 3 τ τ (cid:18) − ττ x (cid:19) + 3 τ (2 + 13 τ )10(1 + 2 τ ) (cid:18) − ττ x (cid:19) + O (cid:32)(cid:18) − ττ x (cid:19) (cid:33) and I ( BS )fwd ( x, τ, σ ) = 12 σ (1 − τ ) (cid:18) (1 + 2 τ ) β − τ β + 17(1 + 6 τ )20160 β + O ( β ) (cid:19) = 3 τ σ (1 − τ ) (1 + 2 τ ) (cid:18) − ττ x (cid:19) − τ σ (1 − τ )(1 + 2 τ ) (cid:18) − ττ x (cid:19) + τ (109 − τ )1400 σ (1 − τ ) (1 + 2 τ ) (cid:18) − ττ x (cid:19) + O (cid:32)(cid:18) − ττ x (cid:19) (cid:33) . (2) Next consider K < S . When − (1 − τ ) x (cid:28)
1, from (3.44) we know that ξ (cid:28)
1, andby expanding it we obtain − x (1 − τ ) τ = 2(1 + 2 τ )3 τ ξ + 2(2 + 13 τ )45 τ ξ + O ( ξ ) , hence we have ξ = 3 τ τ ) (cid:18) − − ττ x (cid:19) − τ (2 + 13 τ )20(1 + 2 τ ) (cid:18) − − ττ x (cid:19) + O (( − (1 − τ ) x ) )Therefore we obtain the asymptotic expansion of the rate function I ( BS )fwd ( x, τ, σ ) = 2 σ (1 − τ ) (cid:18) τ ξ + 2(4 τ + 1)15 ξ + 17(1 + 6 τ )315 ξ + O ( ξ ) (cid:19) = 1 σ (1 − τ ) (cid:26) τ τ ) (cid:18) − x (1 − τ ) τ (cid:19) + 3 τ (1 − τ )10(1 + 2 τ ) (cid:18) − x (1 − τ ) τ (cid:19) + τ (109 − τ )1400(1 + 2 τ ) (cid:18) − x (1 − τ ) τ (cid:19) + O (cid:16) ( − x (1 − τ )) (cid:17) (cid:27) . Hence the conclusion follows. 22 emark 3.9.
Recall that when log(
K/S ) = o (1), the rate function of a standard Asianoption with averaging starting time zero J BS ( K/S ) is approximately log (cid:16) KS (cid:17) (see [35],page 7). Plug it into (3.32) and find the minimum of the quadratic equation of c τ c + 32(1 − τ ) (cid:18) log (cid:18) KS (cid:19) − cτ (cid:19) . We can easily compute the minimizer c ∗ = τ log (cid:16) KS (cid:17) and the minimum τ ) log (cid:16) KS (cid:17) .This provides us a simple way to estimate the rate function of an Asian forward option,which indeed coincides with the first term of (3.45). However, this approach is not accurateenough to provide further terms in the asymptotic expansion.Since AATM region is included in the τ –AATM region, it follows immediately fromProposition 3.8 that we have: Corollary 3.10.
In the AATM region x → we have the following expansion for the ratefunction of a forward start Asian option under the Black-Scholes model with parameters ( x = log( K/S ) , τ, σ ) I ( BS )fwd ( x, τ, σ ) = 32 σ (cid:26) x τ − (1 − τ ) τ ) x (3.45)+ (1 − τ ) (109 − τ )2100(1 + 2 τ ) x + O ( x ) (cid:27) . Remark 3.11.
In the limit τ = 0 this reduces to I ( BS )fwd ( x, , σ ) = 1 σ (cid:26) x − x + 1091400 x + O ( x ) (cid:27) , (3.46)which reproduces Eq.(35) in [35]. In the opposite limit τ → x ∼ O (1) we get fromproposition 3.8 the simple limit I ( BS )fwd ( x, , σ ) = 12 σ x , (3.47)which is the rate function for the large deviations of a sum of iid Gaussian random variables,which gives also the rate function for an out-of-the-money European option in the Black-Scholes model.Next let us consider the asymptotic behaviors of the rate functions in the τ − deep out-of-the-money regime. Proposition 3.12.
Consider an out-of-the-money forward start Asian call option underthe Black-Scholes model with parameter ( x, τ, σ ) . When the option is in the deep out-of-the-money region, namely (1 − τ ) x (cid:29) , its rate function has the following asymptoticexpansion I ( BS )fwd ( x, τ, σ ) = 12 σ (cid:26) x + 2 x log(2(1 − τ ) x ) − x + (log(2(1 − τ ) x )) (3.48)+ O (cid:18) log (2(1 − τ ) x ) x (cid:19)(cid:27) . roof. Clearly when x = log( K/S ) (cid:29)
1, from (3.43) we havesinh ββ = e x − τ − τ β tanh β . We have x = log( K/S ) (cid:29) − τ ) x (cid:29)
1. From the above equation we know that(1 − τ ) log (cid:18) sinh ββ (cid:19) + τ β tanh (cid:18) β (cid:19) (cid:29) . This implies that β (cid:29)
1. Now from (3.43) we obtain x = β − log(2 β ) + τ − τ β + O (cid:18) e − β + τ − τ e − β (cid:19) . By inverting the series we have β = (1 − τ ) x + (1 − τ ) log(2(1 − τ ) x ) + (1 − τ ) log(2(1 − τ ) x ) x + O (cid:18) (1 − τ ) log (2(1 − τ ) x ) x (cid:19) . By plugging in (3.41) we have I ( BS )fwd ( x, τ, σ )= 12 σ (1 − τ ) (cid:16) β − − τ ) β + O ( e − β ) (cid:17) = 12 σ (cid:26) x + 2 x log(2(1 − τ ) x ) − x + log (2(1 − τ ) x ) + O (cid:18) log (2(1 − τ ) x ) x (cid:19)(cid:27) . Remark 3.13.
Consider the fixed x and τ → τ → I ( BS )fwd ( x, τ, σ ) = 1 σ (cid:26) x + x log(2 x ) − x + 12 log (2 x ) + O (cid:18) log (2 x ) x (cid:19)(cid:27) . This can be compared with the large strike asymptotic result for the rate function of astandard Asian option with averaging starting from time 0 in Eq. (36) of [35] .Next we consider the case of a deep out-of-the-money put Asian option, which has smallstrike, namely x = log( K/S ) (cid:28) −
1. This case is more complex, and we distinguish tworegions in ( τ, x ) with distinct asymptotics.
Proposition 3.14.
Let I ( BS )fwd ( x, τ, σ ) be the rate function of an out-of-the-money forwardstart put Asian option under the Black-Scholes model. Assume the put option is in the deepout-of-the-money region, namely (1 − τ ) x (cid:28) − .(1) If − (1 − τ ) x (cid:29) and τ − τ (cid:28) e x ( − x ) , namely e x ( − x ) τ → + ∞ , we have τ → and I ( BS )fwd ( x, τ, σ ) = 2 σ (1 − τ ) (cid:26) e − x τ + (1 − τ ) e − x − π − o ( e − x τ + e − x ) (cid:27) . (3.49) Note that in Eq. (36) of [35] the coefficient of log (2 x ) should be instead of , and there is no log(2 x )term to the order considered. With these corrections, the τ = 0 result is reproduced indeed.
2) If − (1 − τ ) x (cid:29) and τ − τ (cid:29) e x ( − x ) , namely e x ( − x ) τ → , we have I ( BS )fwd ( x, τ, σ ) = 2 σ (cid:26) x τ − − x τ log (cid:18) − x (1 − τ )2 τ (cid:19) + − x τ + 14 τ log (cid:18) − x (1 − τ )2 τ (cid:19) − π
12 3 − τ (1 − τ ) + o (1) (cid:27) . Remark 3.15.
Case (1) corresponds to the left wing asymptotics ( − x ) → ∞ in a regionof small τ , below the curves shown in Fig. 4. Case (2) corresponds to τ ∼ O (1) and largelog-strike − x (cid:29) -4 -2 0 2 40.20.40.60.81 AATM DOTMDOTM t- t-t- x t Figure 4: Regions in the ( x, τ ) plane with distinct asymptotic expansion. The interior of thecentral funnel-shaped region corresponds to the τ -AATM region (represented as (1 − τ ) | x | ≤ . τ − DOTM corresponds to (1 − τ ) | x | (cid:29) − τ ) | x | ≥ x < κ defined as τ − τ = κe x ( − x ): κ = 1(blue), 0.5 (black), 0.1 (red). Region (1) of Proposition 3.14 corresponds to values of ( x, τ )below these curves which are also in the τ − DOTM region, and region (2) corresponds tovalues of ( x, τ ) above these curves which are also in the τ − DOTM region.
Proof.
Clearly when x <
0, from (3.44) we have ξ > − x = 2 τ − τ ξ tan ξ + log (cid:18) ξ sin 2 ξ (cid:19) . (1) Since − (1 − τ ) x (cid:29) ξ → π . Let ζ = π − ξ where ζ →
0. Expand theabove equation at ζ = 0 we obtain − x = 2 τ − τ π ζ + log (cid:18) π ζ (cid:19) − τ − τ + O ( ζ ) . (3.50)Now since τ − τ (cid:28) e x ( − x ), we claim that2 τ − τ π ζ (cid:28) log (cid:18) π ζ (cid:19) . (3.51)25therwise if τ − τ π ζ (cid:38) log (cid:16) π ζ (cid:17) , (3.50) implies that − x (cid:16) τ − τ π ζ . Plug this back into theprevious inequality we obtain − x (cid:38) log (cid:0) − τ τ ( − x ) (cid:1) , which conflicts with the assumption τ − τ (cid:28) e x ( − x ). Hence claim (3.51) holds. Plug into (3.50), we have − x = log (cid:18) π ζ (cid:19) + o (log(1 /ζ )) . Hence the asymptotic expansion of the rate function is I ( BS )fwd ( x, τ, σ ) = 2 σ (1 − τ ) (cid:32) τ (cid:18) π ζ (cid:19) + (1 − τ ) π ζ − π − O ( τ + ζ ) (cid:33) = 2 σ (1 − τ ) (cid:26) e − x τ + (1 − τ ) e − x − π − o ( e − x τ + e − x ) (cid:27) . (2) Again we have ξ → π and ζ = π − ξ , ζ →
0. When τ − τ (cid:29) e x ( − x ), (3.50) implies that2 τ − τ π ζ (cid:29) log (cid:18) π ζ (cid:19) . (3.52)The proof of the above claim is similar as in (1) by reversing the signs. Hence − x = 2 τ − τ π ζ + log (cid:18) π ζ (cid:19) + O ( τ + ζ ) . By inverting the series we have π ζ = − x (1 − τ )2 τ − − τ τ log (cid:18) − x (1 − τ )2 τ (cid:19) +1+ 1 − τ τ · log (cid:16) − x (1 − τ )2 τ (cid:17) − x + o log (cid:16) − x (1 − τ )2 τ (cid:17) − x . Hence the asymptotic expansion of the rate function is I ( BS )fwd ( x, τ, σ )= 2 σ (1 − τ ) (cid:32) τ (cid:18) π ζ (cid:19) + (1 − τ ) π ζ + (cid:18)
112 (24 + π ) τ − π − (cid:19) + O ( ζ ) (cid:33) = 2 σ (cid:26) x τ − − x τ log (cid:18) − x (1 − τ )2 τ (cid:19) + − x τ + 14 τ log (cid:18) − x (1 − τ )2 τ (cid:19) − π
12 3 − τ (1 − τ ) + o (1) (cid:27) . Remark 3.16.
For a fixed x if τ →
0, it falls into the first case that − (1 − τ ) x (cid:29) τ − τ (cid:28) e x ( − x ). By taking (3.49) to the limit τ → I BS ( x, τ, σ ) = 2 σ (cid:18) e − x − π − o (1) (cid:19) , which agrees with the asymptotic result for a standard Asian option with averaging startingfrom time 0 (see [35], page 8). For a, b >
0, we say a (cid:38) b if there exists a constant C > a ≥ Cb . We denote by a (cid:16) b if there exists constant C , C > C a ≤ b ≤ C a . t = 0.5t = 0t = 0.25t = 0.75 x Figure 5: The equivalent log-normal volatility Σ LN ( x, τ ) /σ of the forward start Asian optionin the Black-Scholes model, vs x = log( K/S ), for τ = 0 , . , . , .
75 (from bottomto top). The dots show numerical values obtained from Eq. (3.54) and the solid curvesshow the numerical approximation obtained keeping three terms in the series expansionapproximation (3.57).
It is convenient to introduce the so-called equivalent log-normal (Black–Scholes) volatilityΣ LN of a forward start Asian option. This is defined as the constant volatility for which theBlack–Scholes price of an European (vanilla) option with maturity T and underlying value A ( τ T, T ) = 1(1 − τ ) T (cid:90) TτT E ( S t ) dt = S ( r − q )(1 − τ ) T ( e ( r − q ) T − e ( r − q ) τT ) (3.53)reproduces the price of the forward start Asian option with parameters ( S , K, T, τ ), whichwe denote by C ( T ) ( P ( T ) resp.). Analogously we define the normal (Bachelier) equivalentvolatility as that volatility Σ N for which the Bachelier price of an European option withmaturity T and strike K reproduces the Asian option price.Take an Asian call option as example. We have C BS ( S , K, Σ LN , T ) = C ( T ) , where C BS ( S , K, Σ LN , T ) = e − rT ( A ( T )Φ( d ) − K Φ( d )) , with d , = LN √ T (cid:0) log( A ( τ T, T ) /K ) ± Σ T (cid:1) , where A ( τ T, T ) is given in (3.53) and C ( T ) = e − rT E (cid:34)(cid:18) − τ ) T (cid:90) TτT S t dt − K (cid:19) + (cid:35) , with S t following the Black–Scholes model with volatility σ .We have the analog of Proposition 18 in [35].27 roposition 3.17. Assume r = q = 0 and the stated assumptions on the local volatilityfunction.(1) The short-maturity T → limit of the log-normal equivalent volatility of a forwardstart out-of-the-money Asian option is lim T → Σ ( K, S , T ) = log ( K/S )2 I fwd ( K, S , τ ) . (3.54) The corresponding result for the normal (Bachelier) equivalent volatility is lim T → Σ ( K, S , T ) = ( K − S ) I fwd ( K, S , τ ) . (3.55) (2) The short-maturity T → limit of the log-normal equivalent volatility of a forwardstart ATM Asian option is lim T → Σ LN ( S , S , T ) = σ ( S ) (cid:114) τ . (3.56) Remark 3.18.
Substituting here the expansion of the forward start rate function in theBlack-Scholes model and expanding in powers of x = log( K/S ) we getlim T → Σ LN ( K, S , T ) = σ (cid:114) τ (cid:18) − τ ) τ ) x − (1 − τ ) (23 − τ )2100(1 + 2 τ ) x + O ( x ) (cid:19) . (3.57)In the limit τ → O ( x ) term gives the ATM skew, and the O ( x ) term gives the ATM curvature of the Asian implied volatility smile. Proof of Proposition 3.17.
The proof is analogous to the proof of Proposition 18 in [35] andis omitted here.
Remark 3.19.
The result in (3.56) corresponds to the log-normal implied volatility of anat-the-money forward start Asian option paying ( A [ T ,T ] − K ) + at time T . From this resultwe observe that the price of this option is the same as that of an European option withvariance σ ( T + ( T − T )). The variance can be also written as σ (cid:18) τ T + 13 (1 − τ ) T (cid:19) = 13 (1 + 2 τ ) σ T = Σ ( S , S ) T , (3.58)we get that this is also equal to the price of an European option with volatility Σ LN ( S , S ) :=lim T → Σ LN ( S , S , T ) and maturity T . We correct here a typo in Eq. (52) of [35], which should have ( K − S ) in the numerator. This result is informally known to quants as a general rule of thumb for pricing forward start Asianoptions [47]. Floating strike Asian options
An important class of Asian options that are encountered in practice are the floating Asianoptions. The strike of these options is defined with respect to the average of the asset priceover a forward period [ T , T ] with T < T .Such an option pays at time T the payoff of the formCall : (cid:18) κS T − T − T (cid:90) T T S t dt (cid:19) + (4.59)Put : (cid:18) T − T (cid:90) T T S t dt − κS T (cid:19) + . (4.60)Here κ > t the valuation time, we distinguish two cases for t :(i) t < T . The entire averaging period is in the future. The option is forward start andthis case is studied in Sec. 4.1.(ii) T < t < T . Namely the valuation time t enters into the averaging period, and theintegral over the asset price includes a deterministic contribution and a stochastic contri-bution (cid:90) T T S u du = (cid:90) tT S u du + (cid:90) T t S u du . (4.61)Part of the averaging period is deterministic, and can be absorbed into a constant strikein the payoff. This corresponds to so-called “seasoned Asian option”, or generalized Asianoptions. This case will be treated below in Sec. 4.2. We consider in this section the forward start floating strike Asian options with t < T < T ,and take t = 0 for simplicity. Let T = T > T = τ T with τ ∈ (0 , C f ( κ, T, τ ) = e − rT E (cid:20) (cid:18) κS T − − τ ) T (cid:90) TτT S t dt (cid:19) + (cid:21) ,P f ( κ, T, τ ) = e − rT E (cid:20) (cid:18) − τ ) T (cid:90) TτT S t dt − κS T (cid:19) + (cid:21) . We will assume that the underlying asset price S t follows the local volatility model (2.6)and (2.7).The floating strike call Asian option is in-the-money (floating strike put Asian optionis out-of-the-money) when κS e ( r − q ) T > A ( τ T, T ) with A ( τ T, T ) = r − q )(1 − τ ) T ( e ( r − q ) T − e ( r − q ) τT ), and they are both at-the-money when κS e ( r − q ) T = A ( τ T, T ). In the T → T → A ( τ T, T ) = S and we get that for κ <
1, the call option is out-of-the-money and the put option is in-the-money; when κ = 1, both the call and put optionsare at-the-money; when κ >
1, the call option is in-the-money and the put option is out-of-the-money.Similar to the case of the fixed strike Asian options, as the maturity T →
0, the pricesof out-of-the-money floating-strike forward start Asian options decrease to 0 exponentially29ast, and the rate can be captured using large deviation theory. The prices of at-the-moneyoptions, on the other hand, decrease at the speed of √ T , and the rates can be obtainedby linear approximation with a Gaussian random variable. Also as before, the estimateson the prices of in-the-money floating-strike forward start Asian options follow easily fromput-call parity. In the following theorem, we study these asymptotic estimates in detail. Theorem 4.1.
Assume the asset price S t follows the local volatility model as in (2.6) and (2.7) . Then we have the following short maturity asymptotic estimates for the prices offloating-strike forward start Asian options as T → .(1) When κ < , we have lim T → T log C f ( κ, T, τ ) = −I f ( S , κ, τ ) , (4.62) and P f ( κ, T, τ ) = S (1 − κ ) − S T (cid:18)
12 ( r + q ) − κq −
12 ( r − q ) τ (cid:19) + O ( T ) , (4.63) where I f ( S , κ, τ ) = inf − τ (cid:82) τ e g ( t ) dt = κe g (1) g (0)=log S ,g ∈AC [0 , (cid:90) (cid:18) g (cid:48) ( t ) σ ( e g ( t ) ) (cid:19) dt. (2) When κ > , we have C f ( κ, T, τ ) = S ( κ −
1) + S T (cid:18)
12 ( r + q ) − κq −
12 ( r − q ) τ (cid:19) + O ( T ) , (4.64) and lim T → T log P f ( κ, T, τ ) = −I f ( S , κ, τ ) . (4.65) (3) When κ = 1 , we have lim T → √ T C f ( κ, T, τ ) = lim T → √ T P f ( κ, T, τ ) = σ ( S ) S √ π √ − τ . (4.66) Proof. (1) Following the same argument as in Lemma 2.1, we immediately obtain thatlim T → T log C f ( κ, T, τ ) = lim T → T log P (cid:18) κS T ≥ − τ (cid:90) τ S tT dt (cid:19) . Then by applying the sample path large deviation of P ( S t · ∈ · , t ∈ [0 , L ∞ [0 ,
1] and thecontraction principle, we obtain (4.62). To see (4.63), we just need to use put-call parity C f ( κ, T, τ ) − P f ( κ, T, τ ) = e − rT (cid:18) κ E ( S T ) − − τ ) T (cid:90) TτT E ( S t ) dt (cid:19) = e − rT S (cid:18) κe ( r − q ) T − e ( r − q ) τT (cid:18) r − q )(1 − τ ) T + O ( T ) (cid:19)(cid:19) = S ( κ −
1) + S T (cid:18)
12 ( r + q ) − κq −
12 ( r − q ) τ (cid:19) + O ( T ) . C f ( κ, T, τ ) = O ( e −I f /T ) = o ( T ) when κ <
1, then (4.65) followsimmediately.(2) We can easily obtain (4.64) and (4.65) using the same arguments as in (1).(3) Following similar arguments as in Theorem 2.4, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C f ( κ, T, τ ) − E (cid:34)(cid:18) κ ˆ S T − − τ (cid:90) τ ˆ S tT dt (cid:19) + (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O ( T ) , where ˆ S t is a Gaussian process given by ˆ S t = S + σ ( S ) S W t . When κ = 1, since ˆ S T − − τ (cid:82) τ ˆ S tT dt is a normal random variable with mean zero and variance σ ( S ) S E (cid:34)(cid:18) W T − − τ (cid:90) τ W tT dt (cid:19) (cid:35) = 13 (1 − τ ) σ ( S ) S T. Hence we have E (cid:34)(cid:18) ˆ S T − − τ (cid:90) τ ˆ S tT dt (cid:19) + (cid:35) = 1 √ √ − τ σ ( S ) S √ T E [ Z Z> ] , where Z is a standard normal random variable. We then obtain (4.66).In the rest of this section we further investigate the variational problems of the ratefunctions. Proposition 4.2.
Assume the asset price S t follows the local volatility model as in (2.6) and (2.7) . Consider an out-of-the-money floating-strike forward start Asian option and let I f ( S , κ, τ ) be the rate function as in Theorem 4.1. We have I f ( S , κ, τ ) = inf c ∈ R (cid:26) c τ + 11 − τ I (cid:16) S e F − ( cτ ) , κ (cid:17)(cid:27) , (4.67) where F ( · ) = (cid:82) · dzσ ( S e z ) and I ( x, κ ) = inf (cid:82) e ϕ ( u ) du = κe ϕ (1) ϕ (0)=0 ,ϕ ∈AC [0 , (cid:40) (cid:90) (cid:18) ϕ (cid:48) ( u ) σ ( xe ϕ ( u ) ) (cid:19) du (cid:41) . (4.68) Proof.
Recall I f ( S , κ, τ ) = inf − τ (cid:82) τ e g ( t ) dt = κe g (1) g (0)=log S ,g ∈AC [0 , (cid:90) (cid:18) g (cid:48) ( t ) σ ( e g ( t ) ) (cid:19) dt. If we consider function f ∈ AC [0 ,
1] such that f ( t ) = g ( t ) − log S . Then we can rewrite I f ( S , κ, τ ) as I f ( S , κ, τ ) = inf − τ (cid:82) τ e f ( t ) dt = κe f (1) f (0)=0 ,f ∈AC [0 , (cid:90) (cid:18) f (cid:48) ( t ) σ ( S e f ( t ) ) (cid:19) dt. f ( t ) comes from the family of absolutely continuous paths { f c ( t ) } c ∈ R given by f c ( t ) = (cid:40) F − ( ct ) 0 < t < τF − ( cτ ) + ϕ c (cid:16) t − τ − τ (cid:17) τ ≤ t < , where ϕ c ( u ), 0 < u < I (cid:16) S e F − ( cτ ) , κ (cid:17) . Here I ( x, κ ) is given as in (4.68)The energy associated to each f c ( t ) is given by12 (cid:90) (cid:18) f (cid:48) c ( t ) σ ( S e f c ( t ) ) (cid:19) dt = 12 c τ + 11 − τ I (cid:16) S e F − ( cτ ) , κ (cid:17) . Hence we obtain the rate function I f ( S , κ, τ ) = inf c ∈ R I c = inf c ∈ R (cid:26) c τ + 11 − τ I (cid:16) S e F − ( cτ ) , κ (cid:17) (cid:27) . and c is chosen to minimize I c .Under the Black-Scholes model the above result can be simplified significantly. Proposition 4.3.
Assume the asset price S t follows the Black-Scholes model, namely σ ( · ) in (2.6) in a constant σ > . Then the rate function of a out-of-the-money floating-strikeforward start Asian option is given by I ( BS ) f ( κ, τ, σ ) = J BS ( κ )(1 − τ ) σ , (4.69) where J BS ( · ) is given by (3.33) and is given explicitly in Lemma 3.4.Proof. From Proposition 4.2 we know that when σ ( · ) = σ , the rate function is independentof S and given by I f ( S , κ, τ ) = inf c ∈ R (cid:26) c τ + 1(1 − τ ) σ inf ϕ ∈D ( κ ) (cid:26) (cid:90) (cid:0) ϕ (cid:48) ( u ) (cid:1) du (cid:27)(cid:27) , where D ( κ ) = (cid:26) ϕ ∈ AC [0 , (cid:12)(cid:12)(cid:12)(cid:12) ϕ (0) = 0 , (cid:82) e ϕ ( u ) du = κe ϕ (1) (cid:27) . First we claim thatinf ϕ ∈D ( κ ) (cid:26) (cid:90) (cid:0) ϕ (cid:48) ( u ) (cid:1) du (cid:27) = J BS ( κ ) . (4.70)The proof can be found in [35]. For completeness we briefly sketch it here. The main ideais to use the method of the Lagrange multiplier. Consider the auxiliary variational problemΛ( ϕ ) := 12 σ (cid:90) (cid:0) ϕ (cid:48) ( t ) (cid:1) dt + λ (cid:18)(cid:90) e ϕ ( t ) dt − κe ϕ (1) (cid:19) , over all functions in AC [0 ,
1] satisfying ϕ (0) = 0. The problem has been reduced to solvingthe Euler-Lagrange equation ϕ (cid:48)(cid:48) ( t ) = λσ e ϕ ( t ) , ϕ (0) = 0 , ϕ (cid:48) (0) = 0 , ϕ (cid:48) (1) = λσ κe ϕ (1) . Let h ∈ AC [0 ,
1] be such that h ( t ) = ϕ (1 − t ) − ϕ (1) , hence ϕ ( t ) = h (1 − t ) + ϕ (1). The left hand side of (4.70) is then given byinf (cid:82) e h ( t ) dt = κh (0)=0 , h ∈AC [0 , (cid:26) (cid:90) (cid:0) h (cid:48) ( u ) (cid:1) du (cid:27) , which is exactly J BS ( κ ). Hence we have (4.70), and I ( BS ) f ( κ, τ, σ ) = inf c ∈ R (cid:26) c τ + 1(1 − τ ) σ J BS ( κ ) (cid:27) = 1(1 − τ ) σ J BS ( κ ) . Remark 4.4.
Notice that under Black-Scholes model, the optimization of the rate function I f ( S , κ, τ ) in (4.67) with respect to c is trivial. This comes from the fact that I ( x, κ ) in(4.68) is independent of x when σ ( · ) is a constant function. Hence the optimal path remainsflat when 0 < t < τ in order to achieve smallest energy.At last, it is worth investigating an equivalence of floating-strike and fixed-strike for-ward starting Asian options under the Black-Scholes model, which was first introduced byHenderson and Wojakowski in [27] for regular Asian options. In the proposition below, westudy the equivalence for forward start Asian options. Similar results for Asian optionswith discrete time averaging have been discussed in [43], see Theorem 9 in [43].Denote by C f ( S , κ, r, q, T , T ) the price of a floating-strike call option with floating-strike κ , forward starting time T , maturity T , and underlying asset dynamics S t = S e ( r − q − σ ) t + σW t , t ≥ , S > , (4.71)where W t is a standard Brownian motion. Denote by C x ( S , K, r, q, T , T ) the price of afixed-strike forward start call option with strike price K , forward starting time T , maturity T , and underlying asset following (4.71). Analogously denote the prices for put-options: P f ( S , κ, r, q, T , T ) and P x ( S , κ, r, q, T , T ). Proposition 4.5.
Assume the asset price S t follows the Black-Scholes model with constantvolatility σ . Then we have that C f ( S , κ, r, q, τ T, T ) = P x ( S , κS , q, r, , (1 − τ ) T ) , (4.72) and P f ( S , κ, r, q, τ T, T ) = C x ( S , κS , q, r, , (1 − τ ) T ) . (4.73) We also have C f ( S , κ, r, q, , (1 − τ ) T ) = P x ( S , κS , q, r, τ T, T ) , (4.74) and P f ( S , κ, r, q, , (1 − τ ) T ) = C x ( S , κS , q, r, τ T, T ) . (4.75)33 roof. We prove (4.72). Since C f ( S , κ, r, q, τ T, T ) = e − rT E (cid:34)(cid:18) κS T − − τ ) T (cid:90) TτT S t dt (cid:19) + (cid:35) = e − rT E (cid:34) S e ( r − q ) T − σ T + σW T (cid:18) κ − − τ ) T (cid:90) TτT S t S T dt (cid:19) + (cid:35) = e − qT S E (cid:34) e − σ T + σW T (cid:18) κ − − τ ) T (cid:90) TτT S t S T dt (cid:19) + (cid:35) . We change the probability measure such that dP ∗ dP = e − σ T + σW T . By Girsanov’s theorem we know that W ∗ t = W t − σt is a Brownian motion under theprobability measure P . Hence we have C f ( S , κ, r, q, τ T, T ) = e − qT S E ∗ (cid:34)(cid:18) κ − − τ ) T (cid:90) TτT S t S T dt (cid:19) + (cid:35) . (4.76)Moreover, note that S t S T = e ( r − q + σ )( t − T )+ σ ( W ∗ t − W ∗ T ) D = e ( r − q + σ )( t − T )+ σ ˆ W T − t , where ˆ W is a standard Brownian motion under P ∗ . By changing the integration variable t → T − t we obtain that C f ( S , κ, r, q, τ T, T ) = e − qT E ∗ (cid:34)(cid:32) κS − − τ ) T (cid:90) (1 − τ ) T S e ( q − r − σ ) t + σ ˆ W t dt (cid:33) + (cid:35) . (4.77)We then obtain (4.72) by realizing the right hand side of the above equation is exactly P x ( S , κS , q, r, , (1 − τ ) T ).Similarly, for a floating-strike forward start Asian put option we have P f ( S , κ, r, q, τ T, T ) = e − rT E (cid:34)(cid:18) − τ ) T (cid:90) TτT S t dt − κS T (cid:19) + (cid:35) = e − qT S E (cid:34) e − σ T + σW T (cid:18) − τ ) T (cid:90) TτT S t S T dt − κ (cid:19) + (cid:35) = e − qT E ∗ (cid:34)(cid:32) − τ ) T (cid:90) (1 − τ ) T S e ( q − r − σ ) t + σ ˆ W t dt − κS (cid:33) + (cid:35) . This is indeed (4.73). The remaining symmetry relations (4.74), (4.75) are proved in asimilar way. 34 .2 Generalized Asian options
We consider here a more general type of Asian options with payoffCall : ( κS T − A T + K ) + , (4.78)Put : ( A T − κS T − K ) + , (4.79)where the asset price average A T is defined with respect to the time period [0 , T ]. Althoughthese options are not forward start, they appear naturally when pricing so-called seasonedfloating-strike Asian options. These options are initially forward start, but their evaluationrequires the study of these payoffs when the valuation date enters the averaging period. Wewill call them generalized Asian options, following Linetsky [32].The prices of these options are given by expectations in the risk-neutral measure. Forexample C gen ( S , κ, K, T ) = e − rT E (cid:34)(cid:18) κS T − T (cid:90) T S t dt + K (cid:19) + (cid:35) . (4.80)The generalized Asian call option is in-the-money if κ > − K/S , (4.81)and at-the-money if κ = 1 − K/S and out-of-the-money otherwise.We study here the short maturity T → Theorem 4.6.
Assume that S t follows the local volatility model (2.6) and (2.7) . Then theshort maturity asymptotics T → for the prices of OTM generalized Asian options are asfollows.(1) When κ < − K/S , we have lim T → T log C gen ( S , κ, K, T ) = −I g ( S , κ, K ) , (4.82) where I g ( S , κ, K ) = inf (cid:82) e g ( t ) dt = κe g (1) + Kg (0)=log S ,g ∈AC [0 , (cid:90) (cid:18) g (cid:48) ( t ) σ ( e g ( t ) ) (cid:19) dt. (2) When κ > − K/S , we have lim T → T log P gen ( S , κ, K, T ) = −I g ( S , κ, K ) . (4.83) (3) When κ = 1 − K/S , we have lim T → √ T C gen ( κ, K, T ) = lim T → √ T P gen ( κ, K, T ) = σ ( S ) S √ π (cid:114) κ − κ + 13 . (4.84)In the Black-Scholes model we can give a more explicit result for the rate function ofthe generalized Asian options. 35 .2 0.4 0.6 0.8 1 1.2 1.40.20.40.60.811.21.4 AB AA AK/S k Figure 6: Regions in the (
K/S , κ ) plane for the rate function of the generalized Asianoption. The points on the blue line correspond to ATM generalized Asian options K/S + κ = 1. The rate function J g ( κ, K/S ) vanishes along this line. Proposition 4.7.
The rate function for the generalized Asian option in the Black-Scholesmodel has the following form.(1) In the region A of the ( K/S , κ ) plane (above the blue line K/S + κ = 1 in Fig. 6),we have for the rate function of an OTM call J g ( κ, K/S ) = σ I g ( κ, K/S ) = 12 β − β γγ + 1 e β − e β + γ , (4.85) where γ = e β κβ + (cid:112) κ β , (4.86) and β is the solution of the equation γ + 1 γ + e β e β − β = κe β (cid:18) γ + 1 γ + e β (cid:19) + KS . (4.87) (2) in the region B of the ( K/S , κ ) plane (below the blue line K/S + κ = 1 in Fig. 6),we have the rate function of an OTM put J g ( κ, K/S ) = σ I g ( κ, K/S ) = 2 ξ (tan( ξ + η ) − tan η − ξ ) , (4.88) where η is the solution of the equation ξ sin(2( ξ + η )) = κ . (4.89)36 his determines η up to a discrete ambiguity η = (cid:40) − ξ + arcsin(2 ξκ ) + nπ − ξ − arcsin(2 ξκ ) + ( n + ) π , n ∈ N . (4.90) Finally, ξ is given by the solution of the equation ξ cos η (tan( ξ + η ) − tan η ) = κ cos η cos ( ξ + η ) + KS . (4.91) Proof.
The rate function in the Black-Scholes model is given by the solution of the varia-tional problem I g ( κ, K ) = inf f σ (cid:90) [ f (cid:48) ( t )] dt , (4.92)where the infimum is taken over all functions f ( t ) satisfying f (0) = 0 and (cid:90) e f ( t ) dt = κe f (1) + KS . (4.93)The proof follows closely that of Prop. 23 in [35] and will be omitted.We note a few properties of the rate function for generalized Asian options. Proposition 4.8.
The rate function J g ( κ, K/S ) for the generalized Asian options in theBlack-Scholes model has the following properties.(i) The rate function vanishes along the ATM line κ + K/S = 1 .(ii) It is symmetric under the exchange of its arguments J g ( κ, K/S ) = J g ( K/S , κ ) . (4.94) Proof. (i) It is easy to see that f ( t ) = 0 satisfies the integral constraint (4.93) providedthat κ + KS = 1. Thus the optimal path along this line is f ( t ) = 0 and the rate functionvanishes. In fact it is easy to see that this holds also in the more general local volatilitymodel and is not specific to the Black-Scholes model.(ii) The integral constraint (4.93) for J g ( κ, K/S ) can be written equivalently in termsof the function h ( t ) defined by f ( t ) = h (1 − t ) + f (1) as e f (1) (cid:90) e h ( t ) dt = κe f (1) + KS . (4.95)Noting that h (0) = 0 , h (1) = − f (1), this can be expressed further as (cid:90) e h ( t ) dt = κ + KS e h (1) , (4.96)which is just the integral constraint for J g ( K/S , κ ). The rate function has the same formwhen expressed in terms of h ( t ) by noting that f (cid:48) ( t ) = − h (cid:48) (1 − t ). Thus the optimal pathfor J f ( κ, K/S ) is mapped by this change of variable into the optimal path for J g ( K/S , κ ),and the rate function takes the same value for both cases. This concludes the proof of theresult. 37 .3 Numerical tests for forward start floating strike Asian options We consider here the pricing of floating strike forward start Asian options. Followingthe same approach as in [36], we consider them as call/put options on the underlying B [ τT,T ] := κS T − A [ τT,T ] . For κ ≥ B [ τT,T ] is the entire real axis. Anormal approximation is thus more appropriate than a log-normal approximation for thedistribution of this random variable.The forward price of the B [ τT,T ] asset is F f ( τ T, T ) := E [ B [ τT,T ] ] = S (cid:32) κe ( r − q ) T − e ( r − q ) T − e ( r − q ) τT ( r − q )(1 − τ ) T (cid:33) . (4.97)We propose the price approximation expressed as the Bachelier formula C f ( κ, τ ) = e − rT (cid:18) F f ( τ T, T ) N ( d ) − √ π Σ N √ T e − d (cid:19) , (4.98)with d = F f ( τT,T )Σ N √ T . This corresponds to a call option on the underlying B [ τT,T ] with zerostrike, which is the payoff of the option considered ( B [ τT,T ] ) + .Requiring agreement with the short maturity option pricing asymptotics gives the fol-lowing short maturity asymptotics for the normal equivalent volatility Σ N ( K, T ). Proposition 4.9.
Assume r = q = 0 .(1) the short-maturity limit T → of the normal (Bachelier) equivalent volatility of anOTM forward start floating strike Asian option in the BS model is lim T → Σ ( κ, T ) = σ ( K − S ) J BS ( κ ) (1 − τ ) . (4.99) (2) the short-maturity limit T → of the normal (Bachelier) equivalent volatility of anATM κ = 1 forward start floating strike Asian option in the BS model is lim T → Σ N (1 , T ) = σS √ √ − τ . (4.100) Proof.
The proof is analogous to the proof of Prop. 17 in [36] and will be omitted.We consider three benchmark cases, corresponding to a forward start floating strike calloptions with κ = 1 with payoff (cid:0) S T − A [ τT,T ] (cid:1) + and maturity T = 1 year. The averagingperiod is as followsCase I: (1 − τ ) T = 60 days , τ = 1 − − τ ) T = 100 days , τ = 1 − − τ ) T = 182 days , τ = 1 − . (4.103)They correspond to the benchmark case considered in Tables II,III and IV of [42]. Theinitial asset price is S = 100, the interest rate is r = 10% and dividend rate q = 0.38he forward F f ( τ T, T ) for the three cases is F f ( τ T, T ) = 0 . , . , .
71, which ispositive. Thus the options are in-the-money. However in the short maturity limit T → κ = 1. For this reason we will use the ATM normalimplied volatility (4.100) in the numerical estimates below.In Table 3 we show results for the benchmark cases considered in Tables II,III,IV of [42].The results of the analytical approximation of [42], and the result of MC simulation, arecompared against the short maturity asymptotic result following from Eq. (4.98). Theasymptotic result is lower than the MC simulation result. The asymptotic result is sensitiveto the interest rate r through the forward rate F f ( τ T, T ). The deviation from the exactresult increases with r , as seen in the numerical tests in [35]. For the case considered herethe interest rate r = 10% is rather large, which explains the larger difference comparingwith the benchmark results. However we note qualitative agreement with the benchmarkscenarios, as the differences with the MC simulation is always below 10% in relative value.Table 3: Numerical tests for the forward start floating strike Asian options, comparing withbenchmark test results in [42]. C f (1 , τ ) gives the result of the short maturity asymptoticapproximation (4.98). TCL shows the analytical approximation of [42] and MC shows theresult of a Monte Carlo simulation. σ τ C f (1 , τ ) TCL MC0.2 305/365 2.13005 2.304166 2.2646160.3 305/365 2.96462 3.223310 3.1714570.4 305/365 3.80438 4.142445 4.0845900.5 305/365 4.64622 5.058229 4.9999820.6 305/365 5.48911 5.969642 5.9086690.2 265/365 2.92733 3.147266 3.1054130.3 265/365 3.99604 4.320233 4.2686040.4 265/365 5.07577 5.493439 5.4934390.5 265/365 6.15994 6.660697 6.6207610.6 265/365 7.24634 7.821191 7.7904360.2 183/365 4.33054 4.609475 4.5547390.3 183/365 5.74906 6.152161 6.0960580.4 183/365 7.19388 7.696422 7.6615320.5 183/365 8.64938 9.231254 9.2284350.6 183/365 10.1102 10.76042 10.798441 A Notations
We summarize here the notations for the rate functions used in the main text. • I ( S , K, τ ) is the rate function for an Asian option with averaging starting at timezero in the local volatility model. • I fwd ( S , K, τ ) is the rate function for a forward start Asian option in the local volatilitymodel. This depends on S , K in a non-trivial way.39 I ( BS )fwd ( K/S , σ, τ ) = σ J ( BS )fwd ( K/S , τ ) is the rate function for a forward start Asianoption in the Black-Scholes model. This depends only on the ratio K/S . • I BS ( K/S , σ ) = σ J BS ( K/S ) is the rate function for an Asian option in the Black-Scholes model with averaging starting at time 0. This corresponds to τ = 0 and is re-lated to the forward start rate functions defined above as I BS ( K/S , σ ) = I ( BS )fwd ( K/S , σ, J BS ( K/S ) = J ( BS )fwd ( K/S , • I f ( S , κ, τ ) is the rate function for a forward start floating-strike Asian option in thelocal volatility model. • I ( BS ) f ( κ, τ, σ ) is the rate function for a forward start floating-strike Asian option inthe Black-Scholes model. • I g ( S , κ, K ) is the rate function for the generalized Asian option. • J g ( κ, K/S ) is the rate function for the generalized Asian option in the Black-Scholesmodel. Acknowledgements
Lingjiong Zhu is partially supported by NSF Grant DMS-1613164.
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