Asymptotic Smiles for an Affine Jump-Diffusion Model
aa r X i v : . [ q -f i n . M F ] M a y Asymptotic Smiles for an Affine Jump-Diffusion Model
Nian Yao , Zhiqiu Li , Zhichao Ling , and Junfeng Lin College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060,Guangdong, China Department of Mathematics, Florida State University, Tallahassee, FL 32304,USA * Corresponding author [email protected] [email protected], [email protected], [email protected] May 11, 2020
Abstract
In this paper, we study the asymptotic behaviors of implied volatility of an affinejump-diffusion model. Let log stock price under risk-neutral measure follow an affinejump-diffusion model, we show that an explicit form of moment generating functionfor log stock price can be obtained by solving a set of ordinary differential equa-tions. A large-time large deviation principle for log stock price is derived by ap-plying the G¨artner-Ellis theorem. We characterize the asymptotic behaviors of theimplied volatility in the large-maturity and large-strike regime using rate function inthe large deviation principle. The asymptotics of the Black-Scholes implied volatilityfor fixed-maturity, large-strike and fixed-maturity, small-strike regimes are also studied.Numerical results are provided to validate the theoretical work.
Keywords : Stochastic processes; affine jump-diffusion model; large deviation prin-ciple; asymptotic implied volatility smiles; Introduction
Point process models the arrival times of events in many applications. Affine point pro-cess (or affine jump-diffusion model, or affine point process driven by a jump-diffusion) is apoint process whose event arrival intensity is driven by an affine jump-diffusion (Duffie et al.(2000)). An affine point process can be further characterized as self-exciting or mutual-exciting. A self-exciting process means a jump increases the probabilities of occurrence offuture jumps in the same component; while a mutual-exciting process increases the jumpintensity in other components as well.Because the affine point process has computational tractability, there have been manyapplications in finance and economics, such as Errais et al. (2010); Zhang et al. (2015);A¨ıt-Sahalia et al. (2015); Zhang and Glynn (2018); Gao and Zhu (2019). Errais et al. (2010)used affine point processes to model the cumulative losses due to corporate defaults in aportfolio. They assumed jump occurrence times are default times; while the jump sizes arethe portfolio losses at defaults. They used index and tranche swap rates before and afterLehman Brothers’ bankruptcy to conduct a market calibration study. Their results indi-cated the empirical importance of self-exciting property of a loss process. Meanwhile, theyshowed a simple affine point process is able to capture the implied default correlations dur-ing the month when Lehman defaulted. A¨ıt-Sahalia et al. (2015) observed jumps in stockmarkets extend over hours or days and across multiple markets. They concluded that aself-exciting (in time) and mutual-exciting (in space) process is capable of capturing suchclustering patterns. Zhang et al. (2015) established a central limit theorem and a large de-viation principle for affine point processes. By using these limits, they derived closed-formapproximations to the distribution of an affine point process. The large deviation princi-ple helped to construct an importance sampling scheme for estimating tail probabilities.Zhang and Glynn (2018) developed stochastic stability conditions for affine jump-diffusionprocesses. By imposing a strong mean-reversion condition and a mild condition on the jumpdistribution, they established ergodicity for affine jump-diffusion processes. They proved2trong laws of large numbers and functional central limit theorems for additive functionalsfor this class of models. When a closed-form solution of the characteristic function of anaffine jump-diffusion process is not available, pricing evaluation requires a numerical solu-tion of a set of ODEs within a numerical inversion routine. However, it is computationallyexpensive as the numerical transform inversion evokes thousands of calculations and eachcalculation requires the solution of a system of ODEs. Later Gao and Zhu (2019) extendedthe large-time limit theorems in Zhang et al. (2015). They derived large-time asymptoticexpansions in large deviations and refined central limit theorem for affine point processes.They proposed a new approach based on the mod- φ convergence theory and they obtainedthe precise large deviations and refined central limits for an affine point process simultane-ously. By truncating the asymptotic expansions, they obtained an explicit approximationfor large deviation probabilities and tail expectations; meanwhile, such explicit approxi-mation can be used as importance sampling in Monte Carlo simulations.Affine point process includes the linear Markovian Hawkes process as a special caseHawkes (1971b,a). Hawkes process has wide range of applications in various domainssuch as seismology Ogata (1988), genome analysis Reynaud-Bouret et al. (2010), socialnetwork Crane and Sornette (2008), modeling of crimes Mohler et al. (2011) and financeBacry et al. (2015) (Bacry et al. (2015) provided a comprehensive survey of applicationsof Hawkes process in finance).Option pricing problems have been well studied when the underlying follows a jump-diffusion process. Back to the 1970s, Merton (1976) proposed a jump-diffusion process andassumes the jump size follows a log-normal distribution. They showed a European optioncan be written as a weighted sum of Black-Scholes European option prices. Later Kou(2002) assumed the jump size follows a double exponential distribution and a closed-formsolution was provided.As to the underlying follows an affine jump-diffusion point process or has Hawkes jumps,option pricing problems are much less studied. This is because of the closed-form solution3f option pricing is no longer available. For instance, Ma et al. (2017) studied a vulnerableEuropean option pricing problem assuming underlying asset and option writer’s asset valueboth following the Hawkes processes. However, as the analytic solutions are unavailable,they implemented the thinning algorithm to compare the proposed model performanceversus other models.There have been studies on option pricing problems at asymptotic regimes. Forde and Jacquier(2011) studied the large-time asymptotic behaviors of European call and put options underthe Heston stochastic volatility model. They derived the large-time large deviation prin-ciple for the log return of underlying over time-to-maturity by applying the G¨artner-Ellistheorem. At the same time, they derived the asymptotic Black-Scholes implied volatility atlarge-time. Later Jacquier and Roome (2016) characterizes the forward implied volatilitysmiles for the same model. Similar work has been extended to other stochastic volatil-ity models, such as the SABR and CEV-Heston models (Forde and Pogudin (2013)), aclass of affine stochastic volatility models (Jacquier et al. (2013)) and multivariate Wishartstochastic volatility models (Alfonsi et al. (2019)).Lee (2004) studied the asymptotics of the Black-Scholes implied volatility in the regimewhere maturity T is fixed and strike is large or small. They showed the large-strike tail ofthe implied volatility skew is bounded by O ( | x | / ), where x is log-moneyness. They provedthe explicit moment formula that determines the smallest coefficient in that bound for agiven T . In addition, they pointed out the linkage between finite moments and tail slopesis model-independent. Some applications of moment formula such as skew extrapolationand model calibration were discussed.In this paper, we study the asymptotic behaviors of the implied volatility of an affinejump-diffusion model. This article is organized as follows: In Section 2.1, we express themoment generating function of the affine jump-diffusion model as solutions of a set ofordinary differential equations by using the Feynman-Kac formula. In Section 2.2, we ob-tain the large-time large deviation principle of the log return of the stock price under the4isk-neutral measure by using G¨artner-Ellis theorem. In Section 2.3, we characterize theasymptotic behaviors of the implied volatility in the large-maturity and large-strike regimeusing rate function in the large deviation principle. In Section 2.4, we study the asymp-totic of the implied volatility for fixed-maturity, large-strike and fixed-maturity small-strikeregimes. In Section 3, we conduct numerical studies to validate the theoretical work. Lastly,conclusion remarks are in Section 4. We assume the underlying stock S t under the risk-neutral measure Q follows an affinejump-diffusion model: dS t S t − = σdW Q t + ( dJ t − λ Nt µ Y dt ) , (2.1)where J t = N t X i =1 ( e Y i − , (2.2)where Y i are i.i.d. random jump sizes independent of N t and W Q t and µ Y = E [ e Y ] − Y i follows a probability distribution Q ( da ). We assume that N t is an affine point processwhich has intensity λ Nt = α + βλ t at t > λ t satisfies the dynamics: dλ t = b ( c − λ t ) dt + σ p λ t dB t + adN t . (2.3)We make following basic assumptions that are required for modelling an affine jump-diffusion model (Zhu (2014)): Assumption 1. a, b, c, α, β, σ > .2. b > aβ . This condition indicates that there exists a unique stationary process λ ∞ which satisfies the dynamics (2.3).3. bc ≥ σ . This condition implies that λ t ≥ with probability 1. B t is independent of W Q t . One should notice that, the point process N t reduces to a linear Hawkes process with an exponential decay kernel when the Brownianmotion term B t = 0. If adN t = 0, then the process λ t reduces to a CoxIngersollRoss process.The log stock price under the risk-neutral measure via S t = S e X t is X t = − σ t + σW Q t − µ Y Z t λ Ns ds + N t X i =1 Y i . (2.4)We can write N t = P i =1 { T i ≤ t } and L t = P i ≥ Y i { T i ≤ t } where T n is the n-th jump timeof N t . The two-dimensional process ( λ, L ) is Markovian on D = R + × R with an infinitegenerator given by L f ( λ, L ) = b ( c − λ ) ∂f∂λ + 12 σ λ ∂ f∂λ + ( α + βλ ) Z R ( f ( λ + a, L + y ) − f ( λ, L )) Q ( dy ) (2.5)for a given function f : R + × R → R with twice continuously differentiable and for all λ ∈ R + , | R R f ( L + y, λ + a ) Q ( dy ) | < ∞ . X t In this section, we compute the moment generating function for X t . The result issummarized in following Lemma 2. Lemma 2.
The moment generating function for X t is E [ e θX t ] = e ( − θσ + θ σ − θµ Y α ) t + D ( t ;Θ) λ + θ L + F ( t ;Θ) (2.6) where θ ∈ R , Θ = ( θ , θ , θ ) ∈ R and D ( t ; Θ) , F ( t ; Θ) satisfy the following ordinarydifferential equations D ′ ( t ; Θ) + bD ( t ; Θ) − σ D ( t ; Θ) − β R R ( e D ( t ;Θ) a + θ y − Q ( dy ) − θ = 0 ,F ′ ( t ; Θ) − bcD ( t ; Θ) − α R R ( e D ( t ;Θ) a + θ y − Q ( dy ) = 0 ,D (0; Θ) = θ , F (0; Θ) = 0 . (2.7)6 roof. Given any θ in R , the moment generating function for X t is E [ e θX t ] = E (cid:20) e θ (cid:16) − σ t + σW Q t − µ Y R t λ Ns ds + P Nti =1 Y i (cid:17) (cid:21) = e ( − θσ + θ σ − θµ Y α ) t E [ e − θµ Y β R t λ s ds + θL t ] . (2.8)For any Θ = ( θ , θ , θ ) ∈ R , we assume E [ e θ R Tt λ s ds + θ λ T + θ L T | λ t = λ, L t = L ] = u ( t, λ, L ) := u ( t, λ, L, Θ) . (2.9)By applying Feynman-Kac formula, we have ∂u∂t + b ( c − λ ) ∂u∂λ + σ λ ∂ u∂λ + ( α + βλ ) R R ( u ( t, λ + a, L + y ) − u ( t, λ, L )) Q ( dy ) + θ λu = 0 ,u ( T, λ, L,
Θ) = e θ λ + θ L . (2.10)Let us try a solution in the form of u ( t, λ, L ) = e A ( t ;Θ) λ + B ( t ;Θ) L + C ( t ;Θ) , then A ( t ; Θ) , B ( t ; Θ) , C ( t ; Θ)satisfy the following ordinary differential equations A ′ ( t ; Θ) − bA ( t ; Θ) + σ A ( t ; Θ) + β R R ( e A ( t ;Θ) a + B ( t ;Θ) y − Q ( dy ) + θ = 0 ,B ′ ( t ; Θ) = 0 ,C ′ + bcA ( t ; Θ) + α R R ( e A ( t ;Θ) a + B ( t ;Θ) y − Q ( dy ) = 0 ,A ( T ; Θ) = θ , B ( T ; Θ) = θ , C ( T ; Θ) = 0 . (2.11)Then we have u ( s, λ, L ) = e A ( s ;Θ) λ + θ L + C ( s ;Θ) and A ( s ; Θ) , C ( s ; Θ) satisfy the followingordinary differential equations A ′ ( t ; Θ) − bA ( t ; Θ) + σ A ( t ; Θ) + β R R ( e A ( t ;Θ) a + θ y − Q ( dy ) + θ = 0 ,C ′ + bcA ( t ; Θ) + α R R ( e A ( t ;Θ) a + θ y − Q ( dy ) = 0 ,A ( T ; Θ) = θ , C ( T ; Θ) = 0 . (2.12)Let f ( t, λ, L ) := f ( t, λ, L, Θ) := E [ e θ R t λ s ds + θ λ t + θ L t | λ = λ, L = L ]. Let u ( t, λ, L ) = f ( T − t, λ, L ) and make the time change t T − t to change the backward equation to the7orward equation, we have − ∂f∂s + b ( c − λ ) ∂f∂λ + σ λ ∂ f∂λ + ( α + βλ ) R R ( f ( s, λ + a, L + y ) − f ( s, λ, L )) Q ( dy ) + θ λf = 0 ,f (0 , λ, L, Θ) = e θ λ + θ L . (2.13)We try f ( s, λ, L ) = e D ( s ;Θ) λ + E ( s ;Θ) L + F ( s ;Θ) , then we have D ( s ; Θ) , E ( s ; Θ) , F ( s ; Θ) satisfythe following ordinary differential equations D ′ ( t ; Θ) + bD ( t ; Θ) − σ D ( t ; Θ) − β R R ( e D ( t ;Θ) a + E ( t ;Θ) y − Q ( dy ) − θ = 0 ,E ′ ( t ; Θ) = 0 ,F ′ − bcD ( t ; Θ) − α R R ( e D ( t ;Θ) a + E ( t ;Θ) y − Q ( dy ) = 0 ,D (0; Θ) = θ , E (0; Θ) = θ , F (0; Θ) = 0 . (2.14)Finally we have f ( s, λ, L ) = e D ( s ;Θ) λ + θ L + F ( s ;Θ) and D ( s ; Θ), F ( s ; Θ) satisfy the followingordinary differential equations D ′ ( s ; Θ) + bD ( s ; Θ) − σ D ( s ; Θ) − β R R ( e D ( s ;Θ) a + θ y − Q ( dy ) − θ = 0 ,F ′ ( s ; Θ) − bcD ( s ; Θ) − α R R ( e D ( s ;Θ) a + θ y − Q ( dy ) = 0 ,D (0; Θ) = θ , F (0; Θ) = 0 . (2.15) X t In this section, we derive the following theorem which describes the large-time largedeviation asymptotic behaviors of the log stock price. This result will be used later toderive the asymptotics for option pricing and implied volatility smiles in the regime wherethe maturity is large and the log-moneyness is of the same order as the maturity. We referreaders to Dembo and Zeitouni (1998) for formal definition of large deviation principle andthe applications. 8 heorem 3. (Large Deviation Principle for X t ). Under Assumption 1, Q ( t X t ∈ · ) satis-fies a large deviation principle on R with the rate function: I ( x ) = sup θ ∈ R { θx − Λ( θ ) } , (2.16) where Λ( θ ) = (cid:18) σ θ − (cid:18) σ + µ Y α (cid:19) θ + bcy ( θ ) + α (cid:16) e ay ( θ ) E [ e θY ] − (cid:17)(cid:19) and y ( θ ) is the smaller solution of the equation − by + 12 σ y + β ( E [ e ay + θY ] − − θµ Y β = 0 , (2.17) if solution exists. Otherwise y ( θ ) = + ∞ .Proof. From (2.8) and (2.15) we know ( θ , θ , θ ) = ( − θµ Y β, , θ ) and, for any θ ∈ R , wehave: E [ e θX t ] = e ( − θσ + θ σ − θµ Y α ) t E [ e − θµ Y β R t λ s ds + θL t ]= e ( − θσ + θ σ − θµ Y α ) t + ¯ D ( t,θ ) λ + θL + ¯ F ( t,θ ) . (2.18)where ¯ D ( t ; θ ) and ¯ F ( t ; θ ) satisfy the following ordinary differential equations ¯ D ′ ( t ; θ ) + b ¯ D ( t ; θ ) − σ ¯ D ( t ; θ ) − β R R ( e ¯ D ( t ; θ ) a + θy − Q ( dy ) + θµ Y β = 0 , ¯ F ′ ( t ; θ ) − bc ¯ D ( t ; θ ) − α R R ( e ¯ D ( t ; θ ) a + θy − Q ( dy ) = 0 , ¯ D (0; θ ) = 0 , ¯ F (0; θ ) = 0 . (2.19)Thus, from (2.18) we haveΛ( θ ) : = lim t →∞ t log E [ e θX t ]= 12 σ θ − (cid:18) σ + µ Y α (cid:19) θ + λ lim t →∞ ¯ D ( t ; θ ) t + lim t →∞ ¯ F ( t ; θ ) t , From (2.19), one can see thatΓ(
D, θ ) := − bD + σ D + β R R ( e aD + θy − Q ( dy ) − θµ Y β = − bD + σ D + β ( E [ e aD + θY ] − − θµ Y β. θ such thatΓ( y, θ ) = − by + 12 σ y + β ( E [ e ay + θY ] − − θµ Y β = 0 (2.20)has a solution of y ( θ ). We know thatΓ ′ y ( y, θ ) = − b + σ y + aβe ay E [ e θY ] , Γ ′′ y ( y, θ ) = σ + a βe ay E [ e θY ]and we find that Γ ′′ y ( y, θ ) >
0, so Γ( y, θ ) is convex and Γ ′ y ( y, θ ) is increasing in y . Clearlywe have lim y →−∞ Γ ′ y ( y, θ ) = −∞ and lim y → + ∞ Γ ′ y ( y, θ ) = + ∞ , so there exists a unique y c ( θ )which satisfies the following equation, − b + σ y c + aβe ay c E [ e θY ] = 0 . (2.21)We take the derivative of y c ( θ ) on θ , y ′ c ( θ ) = − aβe ay c ( θ ) E [ Y e θY ] σ + a βe ay c ( θ ) E [ e θY ] (2.22)And we can rewrite Γ( y c ( θ ) , θ )Γ( y c ( θ ) , θ ) = G ( θ ) := − by c ( θ ) + σ y c ( θ ) + βe ay c ( θ ) E [ e θY ] − β ( θµ Y + 1) (2.23)Now we arrive at find the scope of θ such that G ( θ ) ≤
0. Take the derivative of G ( θ ) on θ , G ′ ( θ ) = β (cid:16) e ay c ( θ ) E [ Y e θY ] − µ Y (cid:17) (2.24) G ′′ ( θ ) = σ βe ay c ( θ ) E [ Y e θY ] + a β e ay c ( θ ) ( E [ Y e θY ] E [ e θY ] − E [ Y e θY ] ) σ + a βe ay c ( θ ) E [ e θY ] (2.25)By Cauchy-Schwarz inequality we can get G ′′ ( θ ) >
0, so G ( θ ) is convex, and G ′ ( θ ) isincreasing. Further, with the fact that lim θ →−∞ y c ( θ ) = bσ from (2.21), we can easily seethat lim θ →−∞ G ′ ( θ ) <
0, so we just need to judge whether θ c exist such that G ′ ( θ c ) = 0. Wediscusses in two cases. 10 ase one : lim θ → + ∞ G ′ ( θ ) ≤
0, in this case, only lim θ → + ∞ G ( θ ) < θ min satisfies θ = aβ + σ ) y c + ασ y c − aβ +2 baβµ Y , − b + σ y c + aβe ay c E [ e θY ] = 0 . (2.26) G () min lim ( + ) G( )<0lim ( + )
G( ) 0
Figure 1: Case one
Case two : lim θ → + ∞ G ′ ( θ ) >
0, in this case G ′ ( θ c ) = 0 has a unique solution θ c . And G ( θ c ) is the minimum of G ( θ ). We write θ min and θ max for the two solutions for equation G ( θ ) = − by c ( θ ) + σ y c ( θ ) + βe ay c ( θ ) E [ e θY ] − β ( θµ Y + 1) = 0 , − b + σ y c + αβe αy c E ( e θY ) = 0 . (2.27)11 G () cmin max G( c )>0G( c )=0G( c )<0 Figure 2: Case two1. If lim θ → + ∞ G ( θ ) <
0, then when θ ≥ θ min in (2.26), G ( θ ) ≤ θ → + ∞ G ′ ( θ ) >
0, then when θ ∈ [ θ min , θ max ], G ( θ ) ≤ θ ∈ [ θ min , θ max ] (in Case one , θ max −→ + ∞ ), we haveΛ( θ ) = lim t →∞ t log E [ e θX t ] = 12 σ θ − (cid:18) σ + µ Y α (cid:19) θ + bcy ( θ ) + α (cid:16) e ay ( θ ) E [ e θY ] − (cid:17) . When θ / ∈ [ θ min , θ max ], this limit is ∞ .We are to check two conditions for G¨artner-Ellis theorem. The first condition is essentialsmoothness. By differentiating the equation (2.20) with respect to θ , that is when θ → θ min(max) , then y → y c , and ∂y∂θ = β ( µ Y − e ay E [ Y e θY ]) − b + σ y + aβe ay E [ e θY ] → + ∞ . The second is 0 ∈ [ θ min , θ max ]. As [ θ min , θ max ] is the range of θ such that equation (2.20)has a solution of y ( θ ). When θ = 0, the equation becomesΓ( y,
0) = − by + 12 σ y + βe ay − β = 0 . (2.28)It is straightforward to see that y = 0 is the solution, therefore 0 ∈ [ θ min , θ max ].12pon applying G¨artner-Ellis theorem (refer to Dembo and Zeitouni (1998) for the defi-nition of essential smoothness and statement of G¨artner-Ellis theorem), Q ( t X t ∈ · ) satisfiesa large deviation principle with rate function I ( x ) = sup θ ∈ R (cid:26) θx − (cid:18) σ θ − (cid:18) σ + µ Y α (cid:19) θ + bcy ( θ ) + α (cid:16) e ay ( θ ) E [ e θY ] − (cid:17)(cid:19)(cid:27) . In this section, we use the rate function in the large deviation principle for X t tocharacterize the asymptotic behaviours of implied volatility in large-maturity and large-strike regime.Consider an European call option with maturity T and strike K is given as C ( K, T ) := D ( T ) E (cid:2) ( S T − K ) + (cid:3) , where S T is the underlying stock price at maturity T and D ( T ) is the discount factor. Oneshould notice the corresponding put option price P ( K, T ) can be found straightforwardlyusing call-put parity. C ( K, T ) indicates the dependence on the maturity T and strike K .Let F = E S T be the forward price of underlying stock. For a given F , the log moneyness k is related to strike by k := log( K/F ) , (2.29)so K ( k ) = F e k is the strike at log moneyness k . The Black-Scholes implied volatility withlog moneyness k and at maturity T is defined as σ BS ( k, T ) which uniquely solves C ( K ( k ) , T ) = C BS ( k, σ BS ( k, T )) , (2.30)where C BS ( k, σ ) = D ( T ) ( F Φ( d + ) − K ( k )Φ( d − )) and d ± = − kσ √ T ± σ √ T , σ BS ( k, T ) uniquely solves P ( K ( k ) , T ) = P BS ( k, σ BS ( k, T )) , (2.31)where P BS ( k, σ ) = D ( T )( K ( k )Φ( − d − ) − F Φ( − d + )) . Theorem 4.
In the joint regime of large-maturity, large-strike with k = log( K/S ) ( T →∞ , | k | → ∞ ) , the implied volatility σ BS ( k, T ) approaches the limit lim T →∞ σ BS ( xT, T ) = σ ∞ ( x ) , (2.32) where σ ∞ ( x ) = I ( x ) − x − p I ( x ) − xI ( x )) x ∈ ( −∞ , x L ) ∪ ( x R , ∞ )2(2 I ( x ) − x + 2 p I ( x ) − xI ( x )) x ∈ [ x L , x R ] (2.33) where I ( x ) is defined in (2.16) and x L = − (cid:18) σ + µ Y α (cid:19) + ( bc + aα ) β ( µ Y − E [ Y ]) aβ − b + α E [ Y ] , (2.34) and x R = (cid:18) σ − µ Y E [ e Y ] α (cid:19) + (cid:0) bc + a E [ e Y ] α (cid:1) E [ e Y ] β (cid:0) µ Y − E [ ¯ Y ] (cid:1) a E [ e Y ] β − b + E [ e Y ] α E [ ¯ Y ] , (2.35) where ¯ Y follows the probability distribution e Y E [ e Y ] d Q .Proof. First, let us give a more explicit expression for I ( x ) in (2.16). Note that I ( x ) = θ ∗ x − Λ( θ ∗ ) , Let ddθ I ( x ) = 0, where x = Λ ′ ( θ ∗ ) so that σ θ ∗ − (cid:18) σ + µ Y α (cid:19) + bcD ′ ( θ ∗ ) + αD ′ ( θ ∗ ) e aD E [ e θ ∗ Y ] + α E [ Y e aD + θ ∗ Y ] = x, D ′ ( θ ∗ ) = x + σ + µ Y α − θ ∗ σ − α E [ Y e aD + θ ∗ Y ] bc + αe aD E [ e θ ∗ Y ] . On the other hand, take the derivative of equation Γ( D ( θ ) , θ ) = 0 on θ , − bD ′ ( θ ) + σ D ( θ ) D ′ ( θ ) + β E h ( aD ′ ( θ ) + Y ) e aD ( θ )+ θY i − µ Y β = 0 , that is D ′ ( θ ) (cid:16) σ D ( θ ) − b + aβ E [ e aD ( θ )+ θY ] (cid:17) = µ Y β − β E [ Y e aD ( θ )+ θY ] . Therefore we can solve for θ ∗ and D ( θ ∗ ) from the following equations: x + σ + µ Y α − θ ∗ σ − α E [ Y e aD + θ ∗ Y ] bc + αe aD E [ e θ ∗ Y ] (cid:0) σ D ( θ ∗ ) − b + aβ E [ e aD ( θ ∗ )+ θ ∗ Y ] (cid:1) = β (cid:0) µ Y − E [ Y e aD ( θ ∗ )+ θ ∗ Y ] (cid:1) − bD ( θ ∗ ) + σ D ( θ ∗ ) + β (cid:0) E [ e aD ( θ ∗ )+ θ ∗ Y ] − (cid:1) − θ ∗ µ Y β = 0 . (2.36)Second, let us define the share measure ¯ Q as d ¯ Q d Q (cid:12)(cid:12)(cid:12)(cid:12) F t = S t S = e X t . (2.37)Note that S t S = e − σ t + σW Q t − µ Y R t λ Ns ds + P Nti =1 Y i = e − σ t + σW Q t · N t Y i =1 e Y i E [ e Y ] · e log E [ e Y ] N t − µ Y R t λ Ns ds . Thus, under the share measure ¯ Q ,¯ X t = 12 σ t + σW ¯ Q t − µ Y Z t ¯ λ ¯ Ns ds + ¯ N t X i =1 ¯ Y i , (2.38)where ¯ Y i are i.i.d. and according to ¯ Q so that it has the probability distribution e Y E [ e Y ] d Q N t is an affine point process with intensity¯ λ ¯ Nt = E [ e Y ] λ Nt . Thus, ¯ Q ( t ¯ X t ∈ · ) satisfies a large deviation principle with¯ I ( x ) := sup θ ∈ R { θx − ¯Λ( θ ) } , here¯Λ( θ ) := lim t →∞ t log E [ e θ ¯ X t ] = 12 σ θ + (cid:18) σ − µ Y E [ e Y ] α (cid:19) θ + bc ¯ D ( θ )+ E [ e Y ] α (cid:16) e a ¯ D ( θ ) E [ e θ ¯ Y ] − (cid:17) , where ¯ D ( θ ) is the smaller solution of the equation − b ¯ D ( θ ) + 12 σ ¯ D ( θ ) + E [ e Y ] β (cid:16) E [ e a ¯ D ( θ )+ θ ¯ Y ] − (cid:17) − θµ Y E [ e Y ] β = 0 . (2.39)As a corollary, ¯ Q ( − t ¯ X t ∈ · ) satisfies a large deviation principle with the rate function¯ I ( − x ). Moreover, for any x ∈ R and for any sufficiently small δ > Q (cid:18) x − δ < ¯ X t t < x + δ (cid:19) = E h e X t x − δ< Xtt 0) regimes.Define ˜ p := sup n p : E Q [ S pT ] < ∞ o , (2.46)and ˜ q := sup n q : E Q [ S − qT ] < ∞ o . (2.47)The following lemma gives an explicit formula relating the right-hand (or large- K orpositive- x ) tail slope and the left-hand (or small- K or negative- x ) tail slope to how manyfinite moments the underlying possesses. Lemma 5. (Lee (2004)) For k = log( K/S ) . Let β R := lim sup k → + ∞ σ BS ( k ) | k | /T and β L := lim sup k →−∞ σ BS ( k ) | k | /T .Then β R ∈ [0 , and β L ∈ [0 , and ˜ p = 12 β R + β R − , ˜ q = 12 β L + β L − , here := ∞ . Equivalently, β R = 2 − p ˜ p + ˜ p − ˜ p ) ,β L = 2 − p ˜ q + ˜ q − ˜ q ) , where the right-hand expression is to be read as zero, in the case ˜ p = ∞ or ˜ q = ∞ . Applying Lee’s moment formula, we obtain the following results for our model: Theorem 6. In the joint regime of fixed-maturity, large-strike (small-strike) with k =log( K/S ) ( | k | → ∞ ) , the implied volatility σ BS ( k, T ) approaches the limit lim sup k → + ∞ σ BS ( k, T ) | k | /T = 2 − p ˜ p + ˜ p − ˜ p ) , (large strike) , lim sup k →−∞ σ BS ( k, T ) | k | /T = 2 − p ˜ q + ˜ q − ˜ q ) , (small strike) , (2.48) where ˜ p and ˜ q are defined via Z ∞ d ¯ DH ( ¯ D ; ˜ p − 1) = T, Z ∞ d ¯ DH ( ¯ D ; − ˜ q ) = T, and H ( ¯ D ; p ) := − b ¯ D + 12 σ ¯ D + β Z R ( e ¯ Da + py − Q ( dy ) − pµ Y β. Proof. Let us determine the ˜ p and ˜ q in (2.46) and (2.47) for S T in (2.1). Recall that ˜ p + 1is the largest p such that E [ e pX T ] < ∞ . From (2.18), we know E [ e pX T ] = e ( − pσ + p σ − pµ Y α ) T + ¯ D ( T ; p ) λ + pL + ¯ F ( T ; p ) , where ¯ D ( T ; p ) and ¯ F ( T ; p ) solve a set of ODEs. According to the ODEs (2.19), we see¯ F ( T ; p ) is determined by ¯ D ( T ; p ), so E [ e pX T ] < ∞ ⇐⇒ ¯ D ( T ; p ) < ∞ and the critical ˜ p isthe value of p such that ¯ D ( T ; p ) = ∞ . Recall that ¯ D ( t ; p ) solves the ODE in (2.19) ¯ D ′ ( t ; p ) = − b ¯ D ( t ; p ) + σ ¯ D ( t ; p ) + β R R ( e ¯ D ( t ; p ) a + py − Q ( dy ) − pµ Y β := H ( ¯ D ; p ) , ¯ D (0; p ) = 0 . (2.49)19efine ¯ D ′ ( t ; p ) = H ( ¯ D ; p ), Z ¯ D ( T ; p )¯ D (0; p ) d ¯ DH ( ¯ D ; p ) = Z T dt = T. (2.50)Therefore the critical p = ˜ p − R ∞ d ¯ D/H ( ¯ D, p ) = T as ¯ D ( T ; p ) = ∞ . For a givenmaturity T , we can find a p which satisfies Z ∞ dx − bx + σ x + βe ax E [ e pY ] − β − pµ Y β = T. (2.51)Similarly, the critical ˜ q = − q satisfies R ∞ d ¯ D/H ( ¯ D, q ) = T . Remark 7. Numerical examples are provided in later sections to verify the existence of p and q values for different T ’s in (2.51). In this section, we provide some numerical study results. The strength of the self-exciting process is controlled by a in (2.3) and β in the intensity function λ Nt . Hence wevary a and β values to study how these two parameters affect the rate function and theasymptotic implied volatility. a is chosen to be 0 . , . β is chosen to be0 . , . 25 and 0 . 5. For all numerical studies, we define the jump size Y ∼ N (0 , σ ). Otherparameters are b = 1, c = 0 . α = 1, σ = 0 . δ = 0 . a values. One should notice as a increases,the growth rate of I ( x ) increases. This is expected as more rare events occur when a increases, so the rate function I ( x ) tends to be smaller. The right figure is the zoom-in ofthe left figure and it shows the minimums do not coincide. Rate function ¯ I ( x ) is shown inFigure 4 and it has similar behaviors as I ( x ) in Figure 3. Figure 5 shows the asymptotic ofimplied volatility in the large-maturity and large-strike regime for different a values. Theaffine point jump-diffusion model can capture the implied volatility smiles in this regime.Forde and Jacquier (2011) found similar implied volatility smiles for the Heston model20n the same regime. Consider the At-The-Money cases when x = 0, the ATM volatilityincreases as a increases. It is because as more rare events occur, the implied volatilityis higher. Besides, the growth rate of the implied volatility into In-The-Money/Out-The-Money increases as a increases.Numerical results for different β values are shown in Figures 6, 7 and 8. Because theparameter β controls the strength of the self-exciting process intensity, so varying β hassimilar effects as varying a . x -1 -0.5 0 0.5 1 I ( x ) I ( x ) = sup θ ∈ R { θ x − Λ ( θ ) } for di ff erent a ’s a=1a=0.5a=0.05 x -0.25 -0.2 -0.15 -0.1 -0.05 0 I ( x ) I ( x ) = sup θ ∈ R { θ x − Λ ( θ ) } for di ff erent a ’s a=1a=0.5a=0.05 Figure 3: Left: I ( x ) for a = 0 . , . I ( x ) = 0. x -1 -0.5 0 0.5 1 ¯ I ( x ) ¯ I ( x ) = I ( x ) − x for di ff erent a ’s a=1a=0.5a=0.05 x ¯ I ( x ) ¯ I ( x ) = I ( x ) − x for di ff erent a ’s a=1a=0.5a=0.05 Figure 4: Left: ¯ I ( x ) for a = 0 . , . I ( x ) = 0.21 -1 -0.5 0 0.5 1 σ ∞ ( x ) σ ∞ ( x ) for di ff erent a ’s a=1a=0.5a=0.05 Figure 5: σ ∞ ( x ) for a = 0 . , . x -1 -0.5 0 0.5 1 I ( x ) I ( x ) = sup θ ∈ R { θ x − Λ ( θ ) } for di ff erent β ’s β =0.5 β =0.25 β =0.1 x -0.25 -0.2 -0.15 -0.1 -0.05 0 I ( x ) I ( x ) = sup θ ∈ R { θ x − Λ ( θ ) } for di ff erent β ’s β =0.5 β =0.25 β =0.1 Figure 6: Left: I ( x ) for β = 0 . , . 25 and 0 . 5; Right: Zoom-in of left figure near I ( x ) = 0.22 -1 -0.5 0 0.5 1 ¯ I ( x ) ¯ I ( x ) = I ( x ) − x for di ff erent β ’s β =0.5 β =0.25 β =0.1 x ¯ I ( x ) ¯ I ( x ) = I ( x ) − x for di ff erent β ’s β =0.5 β =0.25 β =0.1 Figure 7: Left: ¯ I ( x ) for β = 0 . , . 25 and 0 . 5; Right: Zoom-in of left figure near ¯ I ( x ) = 0. x -1 -0.5 0 0.5 1 σ ∞ ( x ) σ ∞ ( x ) for di ff erent β ’s β =0.5 β =0.25 β =0.1 Figure 8: σ ∞ ( x ) for β = 0 . , . 25 and 0 . a values; while rightfigure displays the ratio in the fixed-maturity and small-strike regime. The maturity T is23hosen within a reasonable range. In both figures, we observe that, for a given T , the ratioof implied volatility to log-moneyness increases as the self-exciting intensity parameter a increases. It is interesting to point out that, in these regimes, the ratio of Black-Scholesimplied volatility to log-moneyness decreases as maturity increases. This is practicallyobserved on an implied volatility surface. Results for various values of β ’s are provided inFigure 10. We obtain similar results because β controls the strength of the self-excitingprocess as well. T ( − p ˜ p + ˜ p − ˜ p ) / T limsup k → + ∞ σ BS ( k,T ) / | k | ( fi xed-T large-strike) for di ff erent a ’s a=1a=0.5a=0.05 T ( − p ˜ q + ˜ q − ˜ q ) / T limsup k → − ∞ σ BS ( k,T ) / | k | ( fi xed-T small-strike) for di ff erent a ’s a=1a=0.5a=0.05 Figure 9: Left: lim sup k → + ∞ σ ( k,T ) | k | (fixed-maturity large-strike) for a = 0 . , . k →−∞ σ ( k,T ) | k | (fixed-maturity small-strike) for a = 0 . , . ( − p ˜ p + ˜ p − ˜ p ) / T lim sup k → + ∞ σ BS ( k, T ) / | k | ( fi xed-T large-strike) for di ff erent β ’s β =0.5 β =0.25 β =0.1 T ( − p ˜ q + ˜ q − ˜ q ) / T lim sup k → − ∞ σ BS ( k, T ) / | k | ( fi xed-T small-strike) for di ff erent β ’s β =0.5 β =0.25 β =0.1 Figure 10: Left: lim sup k → + ∞ σ ( k,T ) | k | (fixed-maturity large-strike) for β = 0 . , . 25 and 0 . k →−∞ σ ( k,T ) | k | (fixed-maturity small-strike) for β = 0 . , . 25 and 0 . In this paper, we study the asymptotic behaviors of the implied volatility of an affinejump-diffusion model. Let X t = log( S t /S ) and S t follows an affine jump-diffusion modelunder risk-neutral measure. By applying the Feynman-Kac formula, we compute the mo-ment generating function for X t . An explicit form of the moment generating function canbe found by solving a set of ordinary differential equations. A large-maturity large devi-ation principle for X t is obtained by using the G¨artner-Ellis Theorem. We characterizethe asymptotic behaviors of implied volatility for X t in the joint regime of large-maturityand large-strike regime. We use Lee’s moment formula to derive the asymptotics for Black-Scholes implied volatility in the fixed-maturity, large-strike and fixed-maturity, small-strikeregimes. Numerical studies are provided to validate the theoretical work. We observe thevolatility smiles in the joint regime of large-maturity and large-strike. As the self-excitingintensity parameter ( a or β ) increases, which means more rare events tending to occur,the ATM volatility increases and volatility smile tends to be more convex. Ratios ofBlack-Scholes implied volatility to log-moneyness in fixed-maturity large, small-strike and25arge-strike regimes are shown. For a given maturity T , as the self-exciting parameter( a or β ) increases, the ratio of implied volatility to log-moneyness increases. 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