Black-Scholes in a CEV random environment
aa r X i v : . [ q -f i n . P R ] N ov BLACK-SCHOLES IN A CEV RANDOM ENVIRONMENT
ANTOINE JACQUIER AND PATRICK ROOME
Abstract.
Classical (Itˆo diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential L´evy and affine models, which exhibitsmall-maturity exploding smiles, have historically been proposed to remedy this (see [65] for an overview), andmore recently rough volatility models [2, 33]. We suggest here a different route, randomising the Black-Scholesvariance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through theCEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar toexponential L´evy models and fractional stochastic volatility models. Introduction
We propose a simple model with continuous paths for stock prices that allows for small-maturity explosionof the implied volatility smile. It is indeed a well-documented fact on Equity markets (see for instance [34,Chapter 5]) that standard (Itˆo) stochastic models with continuous paths are not able to capture the observedsteepness of the left wing of the smile when the maturity becomes small. To remedy this, several authors havesuggested the addition of jumps, either in the form of an independent L´evy process or within the more generalframework of affine diffusions. Jumps (in the stock price dynamics) imply an explosive behaviour for the small-maturity smile and are better able to capture the observed steepness of the small-maturity implied volatilitysmile. In particular, Tankov [65] showed that, for exponential L´evy models with L´evy measure supported onthe whole real line, the squared implied volatility smile explodes as σ τ ( k ) ∼ − k / (2 τ log τ ), as the maturity τ tends to zero, where k represents the log-moneyness. Such a small-maturity behaviour of the smile is not onlycaptured by jump-based models, but rough volatility (non-Markovian) models, where the stochastic volatilitycomponent is driven by a fractional Brownian motion, are in fact also able to reflect this property of the data.In a series of papers several authors [2, 7, 31, 33, 37, 39, 47] have indeed proved that, when the Hurst index ofthe fractional Brownian motion lies within (0 , / τ H − / asthe maturity τ tends to zero.In this paper we propose an alternative framework: we suppose that the stock price follows a standard Black-Scholes model; however the instantaneous variance, instead of being constant, is sampled from a continuousdistribution. We first derive some general properties, interesting from a financial modelling point of view, anddevote a particular attention to a particular case of it, where the variance is generated from independent CEVdynamics. Assume that interest rates and dividends are null, and let S denote the stock price process starting at S = 1, the solution to the stochastic differential equation d S τ = S τ √V d W τ , for τ ≥
0, where W is a standardBrownian motion. Here, V is a random variable, which we assume to be distributed as V ∼ Y t , for some t > Date : November 11, 2018.2010
Mathematics Subject Classification.
Key words and phrases. volatility asymptotics, random environment, forward smile, large deviations.AJ acknowledges financial support from the EPSRC First Grant EP/M008436/1. where Y is the unique strong solution of the CEV dynamics d Y u = ξY pu d B u , Y > p ∈ R , ξ > B is an independent Brownian motion (see Section 2.1 for precise statements). The main result of this paper(Theorem 2.3) is that the implied volatility generated from this model exhibits the following behaviour as thematurity τ tends to zero: σ τ ( k ) ∼ − p )3 − p (cid:18) k ξ (1 − p ) t τ (cid:19) / (3 − p ) , if p < ,k ξ tτ (log τ ) , if p = 1 ,k p − τ | log τ | , if p > , (1.1)for all k = 0. Sampling the initial variance from the CEV process at time t induces different term structuresfor small-maturity spot smiles, thereby providing flexibility to match steep small-maturity smiles. For p > p ≤ / p therefore allows the user to modulate theshort-maturity steepness of the smile.We are not claiming here that this model should come as a replacement of fractional stochastic volatilitymodels or exponential L´evy models, notably because its dynamic structure looks too simple at first sight.However, we believe it can act as an efficient building block for more involved models, in particular for stochasticvolatility models with initial random distribution for the instantaneous variance. While we leave these extensionsfor future research, we shall highlight how our model comes naturally into play when pricing forward-startoptions in stochastic volatility models. In [50] the authors proved that the small-maturity forward impliedvolatility smile explodes in the Heston model when the remaining maturity (after the forward-start date)becomes small. This explosion rate corresponds precisely to the case p = 1 / t ).The paper is structured as follows: in Sections 2.1 and 2.2 we introduce our model and relate it to otherexisting approaches. In Section 2.3 we use the moment generating function to derive extreme strike asymptotics(for some special cases) and show why this approach is not readily applicable for small and large-maturityasymptotics. Sections 2.4 and 2.5 detail the main results, namely the small and large-maturity asymptotics ofoption prices and the corresponding implied volatility. Section 2.6 provides numerical examples, and Section 2.7describes the relationship between our model and the pricing of forward-start options in stochastic volatilitymodels. Finally, the proofs of the main results are gathered in Section 3. Notations : Throughout the paper, the ∼ symbol means asymptotic equivalence, namely, the ratio of theleft-hand side to the right-hand side tends to one.2. Model and main results
Model description.
We consider a filtered probability space (Ω , F , ( F s ) s ≥ , P ) supporting a standardBrownian motion, and let ( Z s ) s ≥ denote the solution to the following stochastic differential equation:(2.1) d Z s = − V d s + √V d W s , Z = 0 , where V is some random variable, independent of the Brownian motion W , and in particular of the Brownianfiltration at time zero (see [51, Remarks 2.2 and 2.3] for details about this). The process ( Z s ) s ≥ , in finance, LACK-SCHOLES IN A CEV RANDOM ENVIRONMENT 3 corresponds to the logarithm of the underlying stock price, and the coefficient − / V ) that (e Z s ) s ≥ is a true ( F s ) s ≥ -martingale. In the case where V is a discrete random variable,this model reduces to the mixture of distributions, analysed, in the Gaussian case by Brigo and Mercurio [14, 15].In a stochastic volatility model where the instantaneous variance process ( V t ) t ≥ is uncorrelated with the assetprice process, the mixing result by Romano and Touzi [63] implies that the price of a European option withmaturity τ is the same as the one evaluated from the SDE (2.1) with V = τ − R τ V s d s . As τ tends to zero,the distribution of V approaches a Dirac Delta centred at the initial variance V . Asymptotics of the impliedvolatility are well known and weaknesses of classical stochastic volatility models are well documented [34].Although such models fit into the framework of (2.1), we will not consider them further in this paper. Definepathwise the process M by M s := − s + W s and let ( T s ) s ≥ be given by T s := s V . Then T is an independentincreasing time-change process and Z = M T in distribution. In this way our model can be thought of as arandom time change. Let now N be a L´evy process such that (e N s ) s ≥ is a ( F s ) s ≥ -adapted martingale; define V := τ − R τ V s d s where V is a positive and independent process, then (e N T s ) s ≥ is a classical time-changedexponential L´evy process, and pricing vanilla options is standard [20, Section 15.5]. However, as the maturity τ tends to zero, V converges in distribution to a Dirac Delta, in which case asymptotics are well known [65].The model (2.1) is also related to the Uncertain Volatility Model of Avellaneda and Par´as [3] (see also [23, 45,56]), in which the Black-Scholes volatility is allowed to evolve randomly within two bounds. In this framework,sub-and super-hedging strategies (corresponding to best and worst case scenarios) are usually derived via theBlack-Scholes-Barenblatt equation, and Fouque and Ren [32] recently provided approximation results when thetwo bounds become close to each other. One can also, at least formally, look at (2.1) from the perspective offractional stochastic volatility models, first proposed by Comte et al. in [18], and later developed and revivedin [6, 19, 8, 5, 9, 27, 41, 33, 35, 37, 42, 46, 58]. In these models, standard stochastic volatility models aregeneralised by replacing the Brownian motion driving the instantaneous volatility by a fractional Brownianmotion. This preserves the martingale property of the stock price process, and allows, in the case of shortmemory (Hurst parameter H between 0 and 1 /
2) for short-maturity steep skew of the implied volatility smile.However, the Mandelbrot-van Ness representation [57] of the fractional Brownian motion reads W Ht := Z t d W s ( t − s ) γ + Z −∞ (cid:18) t − s ) γ − − s ) γ (cid:19) d W s , for all t ≥
0, where γ := 1 / − H . This representation in particular indicates that, at time zero, the instantaneousvariance, being driven by a fractional Brownian motion, incorporates some randomness (through the secondintegral). Finally, we agree that, at first sight, randomising the variance may sound unconventional. Asmentioned in the introduction, we see this model as a building block for more involved models, in particularstochastic volatility with random initial variance, the full study of which is the purpose of ongoing research.After all, market data only provides us with an initial value of the stock price, and the initial level of thevariance is unknown, usually left as a parameter to calibrate. In this sense, it becomes fairly natural to leavethe latter random.2.1.1. Moment generating function.
In [28, 29, 44], the authors used the theory of large deviations, and inparticular the G¨artner-Ellis theorem, to prove small-and large-maturity behaviours of the implied volatility inthe Heston model and more generally (in [44]) for affine stochastic volatility models. This approach relies solelyon the knowledge of the cumulant generating function of the underlying stock price, and its rescaled limiting
ANTOINE JACQUIER AND PATRICK ROOME behaviour. For any τ ≥
0, let Λ Z ( u, τ ) := log E (e uZ τ ) denote the cumulant generating function of Z τ , definedon the effective domain D Zτ := { u ∈ R : | Λ Z ( u, τ ) | < ∞} ; similarly denote Λ V ( u ) ≡ log E (e u V ), whenever it iswell defined. A direct application of the tower property for expectations yields(2.2) Λ Z ( u, τ ) = Λ V (cid:18) u ( u − τ (cid:19) , for all u ∈ D Zτ . Unfortunately, the cumulant generating function of V is not available in closed-form in general. In Section 2.3below, we shall see some examples where such a closed-form solution is available, and where direct computationsare therefore possible. We note in passing that this simple representation allows, at least in principle, forstraightforward (numerical) computations of the slopes of the wings of the implied volatility using Roger Lee’sMoment Formula [53] (see also Section 2.3.2). The latter are indeed given directly by the boundaries (in R ) ofthe effective domain of Λ V . Note further that the model (2.1) could be seen as a time-changed Brownian motion(with drift); the representation (2.2) clearly rules out the case where Z is a simple exponential L´evy process (inwhich case Λ Z ( u, τ ) would be linear in τ ). In view of Roger Lee’s formula, this also implies that, contrary tothe L´evy case, the slopes of the implied volatility wings are not constant over time in our model.2.2. CEV randomisation.
As mentioned above, this paper is a first step towards the introduction of ‘randomenvironment’ into the realm of option pricing, and we believe that, seeing it ‘at work’ through a specific, yetnon-trivial, example, will speak for its potential prowess. We assume from now on that V corresponds to thedistribution of the random variable generated, at some time t , by the solution to the CEV stochastic differentialequation d Y u = ξY pu d B u , Y = y > p ∈ R , ξ > B is a standard Brownian motion, independentof W . The CEV process [13, 52] is the unique strong solution to this stochastic differential equation, up tothe stopping time τ Y := inf u> { Y u = 0 } . The behaviour of the process after τ Y depends on the value of p ,and shall be discussed below. We let Γ( n ; x ) := Γ( n ) − R x t n − e − t d t denote the normalised lower incompleteGamma function, and m t := P ( Y t = 0) = P ( V = 0) represent the mass at the origin. Define the constants(2.3) η := 12( p − , µ := log( y ) − ξ t . Straightforward computations show that, whenever the origin is an absorbing boundary, the density ζ p ( y ) ≡ P ( Y t ∈ d y ) / d y is norm decreasing and(2.4) m t = 1 − Γ − η ; y − p )0 ξ (1 − p ) t ! > t = 0 and the density ζ p is norm preserving. When p ∈ [1 / , p ≥
1, the process Y never hits zero P -almost surely.Finally, when p < /
2, the origin is an attainable boundary, and can be chosen to be either absorbing orreflecting. Absorption is compulsory if Y is required to be a martingale [43, Chapter III, Lemma 3.6]. Here itis only used as a building block for the instantaneous variance, and such a requirement is therefore not needed,so that both cases (absorption and reflection) will be treated. Introduce the function ϕ η : (0 , ∞ ) → (0 , ∞ ) by ϕ η ( y ) := y / y / − p | − p | ξ t exp − y − p ) + y − p )0 ξ t (1 − p ) ! I η (cid:18) ( y y ) − p (1 − p ) ξ t (cid:19) , LACK-SCHOLES IN A CEV RANDOM ENVIRONMENT 5 where I η is the modified Bessel function of the first kind of order η [1, Section 9.6]. The CEV density, ζ p ( y ) := P ( Y t ∈ d y ) / d y , then reads ζ p ( y ) = ϕ − η ( y ) , if p ∈ [1 / ,
1) or p < with absorption ,ϕ η ( y ) , if p > p < with reflection , yξ √ πt exp (cid:18) − (log( y ) − µ ) ξ t (cid:19) , if p = 1 , (2.5)valid for y ∈ (0 , ∞ ). When p ≥
1, the density ζ p converges to zero around the origin, implying that paths arebeing pushed away from the origin. On the other hand ζ p diverges to infinity at the origin when p < /
2, sothat the paths have a propensity towards the vicinity of the origin.It is clear from all the quantities above that the precise horizon t itself is not fundamental, as it only appearswith the multiplicative constant factor ξ . By scaling of the Brownian motion, t can be taken equal to unity,and is therefore rather irrelevant here; we shall keep it explicit in the notations, however, since it will turn outuseful when applying this framework to forward-start derivatives in Section 2.7.2.3. The moment generating function approach.
In the literature on implied volatility asymptotics, themoment generating function of the stock price has proved to be an extremely useful tool to obtain sharpestimates. This is obviously the case for the wings of the smile (small and large strikes) via Roger Lee’s formula,mentioned in Section 2.1.1, but also to describe short-and large-maturity asymptotics, as developed for instancein [44] or [48], via the use of (a refined version of) the G¨artner-Ellis theorem. In [51], the authors used thisproperty to study a generalised version of the Heston model, where the starting value of the instantaneousvolatility is randomised according to some distribution. It it closed to the present model, yet does not supersedeit, and makes full use of the knowledge of the moment generating function of the Heston model. As shown inSection 2.1.1, the moment generating function of a stock price satisfying (2.1) is fully determined by that of therandom variable V . However, even though the density of the latter is known in closed form (Equation (2.5)),the moment generating function is not so for general values of p . In the cases p = 0 (with either reflecting orabsorbing boundary) and p = 1 /
2, a closed-form expression is available and direct computations are possible.2.3.1.
Computation of the moment generating function.
Denote by Λ V ,r , Λ V ,a and Λ V / the moment generatingfunction of the random variable V when p = 0 (the subscript ‘r’ / ‘a’ denotes the reflecting / absorbing behaviourat the origin) and p = 1 /
2. The following quantities can be computed directly from [52, Part I, Section 6.4]:(2.6) Λ V ,a ( u ) = log (cid:20) m t + 12 exp (cid:18) ( uξ t − y ) u (cid:19) (cid:26) e uy E (cid:18) uξ t + y ξ √ t (cid:19) + e uy − − E (cid:18) uξ t − y ξ √ t (cid:19)(cid:27)(cid:21) , Λ V ,r ( u ) = log (cid:20)
12 exp (cid:18) ( uξ t − y ) u (cid:19) (cid:26) e uy E (cid:18) uξ t + y ξ √ t (cid:19) + e uy + 1 + E (cid:18) uξ t − y ξ √ t (cid:19)(cid:27)(cid:21) , Λ V / ( u ) = 2 y u − uξ t , where E ( z ) ≡ √ π R z exp( − x )d x is the error function. Note that when p = 1 / p = 0 in the absorptioncase, one needs to take into account the mass at zero in (2.4) when computing these expectations.2.3.2. Roger Lee’s wing formula.
In [53], Roger Lee provided a precise link between the slope of the total impliedvariance in the wings and the boundaries of the domain of the moment generating function of the stock price.More precisely, for any τ ≥
0, let u + ( τ ) and u − ( τ ) be defined as u + ( τ ) := sup { u ≥ | Λ Z ( u, τ ) | < ∞} and u − ( τ ) := sup { u ≥ | Λ Z ( − u, τ ) | < ∞} . ANTOINE JACQUIER AND PATRICK ROOME
The implied volatility σ τ ( k ) then satisfieslim sup k ↑∞ σ τ ( k ) τk = ψ ( u + ( τ ) −
1) =: β + ( τ ) and lim sup k ↓−∞ σ τ ( k ) τ | k | = ψ ( u − ( τ ))) =: β − ( τ ) , where the function ψ is defined by ψ ( u ) = 2 − (cid:16)p u ( u + 1) − u (cid:17) . Combining (2.6) and (2.2) yields a closed-form expression for the moment generating function of the stock price when p ∈ { , / } . It is clear that, when p = 0, u ± ( τ ) = ±∞ for any τ ≥
0, and hence the slopes of the left and right wings are equal to zero (thetotal variance flattens for small and large strikes). In the case where p = 1 /
2, explosion will occur as soon as (cid:0) u ( u − τ ξ t − (cid:1) = 0, so that u ± ( τ ) = 12 ± r ξ tτ , and β − ( τ ) = β + ( τ ) = 2 ξ √ tτ (cid:16)p ξ tτ + 16 − (cid:17) , for all τ > . The left and right slopes are the same, but the product ξ t can be directly calibrated on the observed wings.Note that the map τ β ± ( τ ) is concave and increasing from 0 to 2. In [24, 25], the authors highlightedsome symmetry properties between the small-time behaviour of the smile and its tail asymptotics. We obtainhere some interesting asymmetry, in the sense that one can observe the same type of rate of explosion (powerbehaviour, given by (1.1) in the case p < τ tends toinfinity, β ± ( τ ) converges to 2, so that the implied volatility smile does not ‘flatten out’, as is usually the casefor Itˆo diffusions or affine stochastic volatility models (see for instance [44]). In Section 2.5 below, we makethis more precise by investigating the large-time behaviour of the implied volatility using the density of theCEV-distributed variance.2.3.3. Small-time asymptotics.
In order to study the small-maturity behaviour of the implied volatility, onecould, whenever the moment generating function of the stock price is available in closed form (e.g. in the case p ∈ { , / } ), apply the methodology developed in [28]. The latter is based on the G¨artner-Ellis theorem, which,essentially, consists of finding a smooth convex pointwise limit (as τ tends to zero) of some rescaled version ofthe cumulant generating function. In the case where p = 1 /
2, it is easy to show that(2.7) Λ Z ( u ) := lim τ ↓ τ / Λ Z (cid:18) u √ τ , τ (cid:19) = , if u ∈ (cid:18) − ξ √ t , ξ √ t (cid:19) , + ∞ , otherwise . The nature of this limiting behaviour falls outside the scope of the G¨artner-Ellis theorem, which requires | Λ Z ( u ) | to diverge to infinity as u approaches the boundaries ± / ( ξ √ t ). It is easy to see that any other rescaling wouldyield even more degenerate behaviour. One could adapt the proof of the G¨artner-Ellis theorem, as was donein [50] for the small-maturity behaviour of the forward implied volatility smile in the Heston model (see also [21]and references therein for more examples of this kind). In the case (2.7), we are exactly as in the frameworkof [50], in which the small-maturity smile (squared) indeed explodes as τ − / , precisely the same explosion asthe one in (1.1). Unfortunately, as we mentioned above, the moment generating function of the stock price isnot available in general, and this approach is hence not amenable here.2.3.4. Large-time asymptotics.
The analysis above, based on the moment generating function of the stock price,can be carried over to study the large-time behaviour of the latter. In the case p = 1 /
2, computations are fully
LACK-SCHOLES IN A CEV RANDOM ENVIRONMENT 7 explicit, and the following pointwise limit follows from simple straightforward manipulations:lim τ ↑∞ τ − Λ Z ( u, τ ) = ( , if u ∈ [0 , , + ∞ , otherwise . The nature of this asymptotic behaviour, again, falls outside the scope of standard large deviations analysis,and tedious work, in the spirit of [10, 50], would be needed to pursue this route.2.4.
Small-time behaviour of option prices and implied volatility.
In the Black-Scholes model d S t = √ wS t d W t starting at S = 1, a European call option with strike e k and maturity T > k, w, T ) = N − k √ wT + √ wT ! − e k N − k √ wT − √ wT ! . For any k ∈ R \ { } , T >
0, and p >
1, the quantityJ p ( k ) := Z ∞ BS (cid:16) k, yT , T (cid:17) y − p d y, if k > , Z ∞ (cid:16) e k − (cid:16) k, yT , T (cid:17) (cid:17) y − p d y, if k < , (2.9)is well defined and independent of T . Indeed, since the stock price is a martingale starting at one, Call optionsare always bounded above by one, and hence, for k >
0, J p ( k ) ≤ R BS( k, y/T, T ) y − p d y + R ∞ y − p d y . Thesecond integral is finite since p >
1. When k >
0, the asymptotic behaviourBS (cid:16) k, yT , T (cid:17) ∼ exp (cid:18) − k y + k (cid:19) y / k √ π holds as y tends to zero, so that lim y ↓ BS( k, y/T, T ) y − p = 0, and hence the integral is finite. A similar analysisholds when k < β p := 13 − p , y p := (cid:18) k ξ t (1 − p )2 (cid:19) β p , y ∗ := k ξ t , the first two only when p <
1, and note that β p ∈ (0 , , ∞ ) to R :(2.11) f ( y ) := k y + y − p ) ξ t (1 − p ) , f ( y ) := ( yy ) (1 − p ) ξ t (1 − p ) ,g ( y ) := k y + log( y ) ξ t , g ( y ) := log( y ) ξ t , ANTOINE JACQUIER AND PATRICK ROOME as well as the following ones, parameterised by p : p < p = 1 p > c ( t, p ) f ( y p ) 1 / (2 ξ t ) 0 c ( t, p ) f ( y p ) 1 / (2 ξ t ) 0 c ( t, p ) 6 − p − p g ( y ∗ ) − µξ t p − c ( t, p ) 0 g ( y ∗ ) − µξ t − c ( t, p ) y p y (1 − p ) p exp (cid:18) k − y − p )0 ξ t (1 − p ) + f ′ ( y p ) f ′′ ( y p ) (cid:19) k ξ q πf ′′ ( y p ) t exp (cid:16) k − µ ξ t + µ log( y ∗ ) ξ t (cid:17) √ π | k | − ξ − t − / p − − y p − ξ t (1 − p )2 J p ( k )(2(1 − p ) ξ t ) η +1 Γ( η + 1) h ( τ, p ) τ p − / (3 − p ) (cid:0) log( τ ) + log log (cid:0) τ − (cid:1)(cid:1) h ( τ, p ) τ ( p − / (3 − p ) (log | log( τ ) | ) | log( τ ) | R ( τ, p ) O (cid:16) τ (1 − p ) / (3 − p ) (cid:17) O (cid:18) | log( τ ) | (cid:19) O ( τ p − ) Table 1.
List of constants and functionsThe following theorem (proved in Section 3.1) is the central result of this paper (although its equivalentbelow, in terms of implied volatility, is more informative for practical purposes):
Theorem 2.1.
The following expansion holds for all k ∈ R \ { } as τ tends to zero: E (cid:0) e Z τ − e k (cid:1) + = (1 − e k ) + + exp (cid:16) − c ( t, p ) h ( τ, p ) + c ( t, p ) h ( τ, p ) (cid:17) τ c ( t,p ) | log( τ ) | c ( t,p ) c ( t, p ) [1 + R ( τ, p )] . Remark 2.2. (i) Whenever p ≤ c and c are strictly positive; the function c is always strictly positive; when p < c is strictly positive; when p = 1, the functions c and c can take positive and negative values;(ii) Whenever p ≤ h ( τ, p ) ≤ h ( τ, p ) for τ small enough, so that the leading order is provided by h ;(iii) In the lognormal case p = 1, h ( τ, ∼ (log τ ) as τ tends to zero, so that the exponential decay of optionprices is governed at leading order by exp( − c ( t, τ ) ).Using Theorem 2.1 and small-maturity asymptotics for the Black-Scholes model (see [30, Corollary 3.5]or [36]), it is straightforward to translate option price asymptotics into asymptotics of the implied volatility: Theorem 2.3.
For any k ∈ R \ { } , the small-maturity implied volatility smile behaves as follows: σ τ ( k ) ∼ (1 − β p ) (cid:18) k ξ t (1 − p )2 τ (cid:19) β p , if p < ,k ξ tτ log( τ ) , if p = 1 ,k p − τ | log( τ ) | , if p > . This theorem only presents the leading-order asymptotic behaviour of the implied volatility as the maturitybecomes small. One could in principle (following [17] or [36, 38, 60]) derive higher-order terms, but these
LACK-SCHOLES IN A CEV RANDOM ENVIRONMENT 9 additional computations would impact the clarity of this singular behaviour. In the at-the-money k = 0 case,the implied volatility converges to a constant: Lemma 2.4.
The at-the-money implied volatility σ τ (0) converges to E ( √V ) as τ tends to zero. The proof of the lemma follows steps analogous to [50, Lemma 4.3], and we omit the details here. It in factdoes not depend on any particular form of distribution of √V , as long as the expectation exists. Note that,from Theorem 2.3, as p approaches 1 from below, the rate of explosion approaches τ − . When p tends to 1from above, the explosion rate is 1 / ( τ | log τ | ) instead. So there is a ”discontinuity” at p = 1 and the actual rateof explosion is less than both these limits. As an immediate consequence of Theorem 2.1 we have the followingcorollary. Define the following functions: h ∗ ( τ, p ) := τ − β p , if p < , (cid:12)(cid:12) log( τ ) − (cid:12)(cid:12) , if p > , log( τ ) − , if p = 1 , and Λ ∗ p ( k ) := ( c ( t, p ) , if p ≤ , p − , if p > , where c ( t, p ) is defined in Table 1, and depends on k (through y p ). Corollary 2.5.
For any p ∈ R , the sequence ( Z τ ) τ ≥ satisfies a large deviations principle with speed h ∗ ( τ, p ) and rate function Λ ∗ p as τ tends to zero. Furthermore, the rate function is good only when p < . Recall that a real-valued sequence ( Z n ) n ≥ satisfies a large deviations principle (see [22] for a precise intro-duction to the topic) with speed n and rate function Λ ∗ if, for any Borel subset B ⊂ R , the inequalities − inf z ∈ B o Λ ∗ ( z ) ≤ lim inf n ↑∞ n − log P ( Z n ∈ B ) ≤ lim sup n ↑∞ n − log P ( Z n ∈ B ) ≤ − inf z ∈ B Λ ∗ ( z )hold, where B and B o denote the closure and interior of B in R . The rate function Λ ∗ : R → R ∪ { + ∞} , bydefinition, is a lower semi-continuous, non-negative and not identically infinite, function such that the level sets { x ∈ R : Λ ∗ ( x ) ≤ α } are closed for all α ≥
0. It is said to be a good rate function when these level sets arecompact (in R ). Proof.
The proof of Theorem 2.1 holds with only minor modifications for digital options, which are equivalentto probabilities of the form P ( Z τ ≤ k ) or P ( Z τ ≥ k ). For p ∈ ( −∞ , τ ↓ h ∗ ( τ, p ) log P ( Z τ ≤ k ) = − inf (cid:8) Λ ∗ p ( x ) : x ≤ k (cid:9) . The infimum is null whenever k > p <
1, and Λ ∗ ( x ) ≡ / (2 ξ t ) is constant. Consider now an openinterval ( a, b ) ⊂ R . Since ( a, b ) = ( −∞ , b ) \ ( −∞ , a ], then by continuity and convexity of Λ ∗ p , we obtainlim τ ↓ h ∗ ( τ, p ) log P ( Z τ ∈ ( a, b )) = − inf x ∈ ( a,b ) Λ ∗ p ( x ) . The corollary then follows from the definition of the large deviations principle [22, Section 1.2]. When p ∈ (1 , ∞ ),the only non-trivial choice of speed is | (log τ ) − | , in which case lim τ ↓ | (log τ ) − | log P ( Z τ ≤ k ) = − (2 p − R ), and the corollary follows. (cid:3) Remark 2.6.
In the case p = 1 /
2, as discussed in Section 2.3.4, the moment generating function of Z is availablein closed form. However, the large deviations principle does not follow from the G¨artner-Ellis theorem, sincethe pointwise rescaled limit of the mgf is degenerate (in the sense of (2.7)). Small-maturity at-the-money skew and convexity.
The goal of this section is to compute asymptoticsfor the at-the-money skew and convexity of the smile as the maturity becomes small. These quantities areuseful to traders who actually observe them (or approximations thereof) on real data. We define the leftand right derivatives by ∂ − k σ τ (0) := lim k ↑ ∂ k σ τ ( k ) | k =0 and ∂ + k σ τ (0) := lim k ↓ ∂ k σ τ ( k ) | k =0 , and similarly ∂ − kk σ τ (0) := lim k ↑ ∂ kk σ τ ( k ) | k =0 and ∂ + kk σ τ (0) := lim k ↓ ∂ kk σ τ ( k ) | k =0 . The following lemma describes thisshort-maturity behaviour in the general case where V is any random variable supported on [0 , ∞ ). Lemma 2.7.
Consider (2.1) and assume that E ( V n/ ) < ∞ for n = − , , , and m t := P ( V = 0) < . As τ tends to zero, ∂ ± k σ τ (0) ∼ ± m t E ( √V ) √ π √ τ ,∂ − kk σ τ (0) ∼ ∂ + kk σ τ (0) ∼ E ( √V ) τ (cid:18) E (cid:16) V − / (cid:17) − E ( √V ) − (cid:18) − m t √ π (cid:19)(cid:19) . When m t >
0, the at-the-money left skew explodes to −∞ and the at-the-money right skew explodes to + ∞ .Furthermore, the small-maturity at-the-money convexity tends to infinity. Proof.
We first focus on the at-the-money skew. By definition the Call option price with log-moneyness k andmaturity τ reads C ( k, τ ) = BS( k, σ τ ( k ) , τ ), and therefore ∂ k C ( k, τ ) = ∂ k BS( k, σ τ ( k ) , τ ) + ∂ k σ τ ( k ) ∂ w BS( k, σ τ ( k ) , τ ) . Also by (3.1), an immediate application of Fubini yields(2.12) ∂ k C ( k, τ ) = Z ∞ ∂ k BS( k, y, τ ) P ( V ∈ d y ) + m t ∂ k (cid:0) − e k (cid:1) + , We first assume that m t = 0. The at-the-money skew is then given by(2.13) ∂ k σ τ ( k ) | k =0 = (cid:0) ∂ w BS( k, σ τ ( k ) , τ ) | k =0 (cid:1) − (cid:18)Z ∞ ∂ k BS( k, y, τ ) | k =0 P ( V ∈ d y ) − ∂ k BS( k, σ τ ( k ) , τ ) | k =0 (cid:19) . Note now that, for any fixed y > N ,BS(0 , y, τ ) = 1 − N (cid:18) √ yτ (cid:19) = 12 + N X n =0 α n y n +1 / τ n +1 / + O (cid:16) τ N +3 / (cid:17) , for some explicit sequence ( α n ) n ≥ , as τ tends to zero, and therefore, C (0 , τ ) = Z ∞ BS(0 , y, τ ) P ( V ∈ d y ) = 12 + N X n =0 α n τ n +1 / E (cid:16) V n +1 / (cid:17) + O (cid:16) τ N +3 / (cid:17) . This allows us to refine Lemma 2.4, so that, for any N , there exists a sequence ( σ n ) n =0 ,...,N such that σ τ (0) = N X n =0 σ n τ n + O (cid:0) τ N +1 (cid:1) . We are now interested in the at-the-money skew. Since ∂ k BS( k, σ τ ( k ) , τ ) | k =0 = −N (cid:18) − σ τ (0) √ τ (cid:19) , clearly admits the same (modulo signs) expansion as the Call price, it follows from (2.13) that the difference onthe right will always be zero, and the lemma follows. When m t >
0, we need to take left and right derivatives
LACK-SCHOLES IN A CEV RANDOM ENVIRONMENT 11 to account for the atomic term. Since ∂ − k (cid:0) − e k (cid:1) + | k =0 = ∂ − kk (cid:0) − e k (cid:1) + | k =0 = − ∂ + k (cid:0) − e k (cid:1) + | k =0 = ∂ + kk (cid:0) − e k (cid:1) + | k =0 = 0, the asymptotic skew stated in the lemma follows immediately.The small-maturity convexity follows similar arguments, which we only outline. Since ∂ kk C ( k, τ ) = ∂ kk BS( k, σ τ ( k ) , τ ) + 2 ∂ k σ τ ( k ) ∂ wk BS( k, σ τ ( k ) , τ ) + (cid:0) ∂ k σ τ ( k ) (cid:1) ∂ ww BS( k, σ τ ( k ) , τ )+ ∂ kk σ τ ( k ) ∂ w BS( k, σ τ ( k ) , τ ) ,∂ kk C ( k, τ ) = Z ∞ ∂ kk BS( k, y, τ ) P ( V ∈ d y ) + m t ∂ kk (cid:0) − e k (cid:1) + , then ∂ kk σ τ (0) = (cid:0) ∂ w BS(0 , σ τ (0) , τ ) (cid:1) − (cid:8) ∂ kk C (0 , τ ) − ∂ kk BS(0 , σ τ (0) , τ ) (cid:9) = (cid:0) ∂ w BS(0 , σ τ (0) , τ ) (cid:1) − (cid:26)Z ∞ ∂ kk BS( k, y, τ ) P ( V ∈ d y ) − ∂ kk BS(0 , σ τ (0) , τ ) + m t ∂ kk (cid:0) − e k (cid:1) + (cid:27) , and the lemma follows by straightforward computations similar to the skew case. (cid:3) Large-time behaviour of option prices and implied volatility.
In this section we compute the large-time behaviour of option prices and implied volatility. The proofs are given in Section 3.2. It turns out thatasymptotics are degenerate in the sense that option prices decay algebraically to their intrinsic values. Thestructure of the asymptotic depends on the CEV parameter p and whether the origin is reflecting or absorbing: Theorem 2.8.
Define the following quantity: M ( η ) := 2 − p − η Γ (cid:0) − p (cid:1) √ π Γ(1 + η ) | − p | η +1 ( ξ t ) η +1 exp − y − p )0 ξ t (1 − p ) ! , with η given in (2.3) . The following expansions hold for all k ∈ R as τ tends to infinity:(i) if p < / and the origin is absorbing then E (cid:0) e Z τ − e k (cid:1) + = 1 − m t + m t (1 − e k ) + − k/ y (cid:18) − p (cid:19) M ( − η ) 1 + O (cid:0) τ − (cid:1) τ − p ; (ii) if p < / and the origin is reflecting then E (cid:0) e Z τ − e k (cid:1) + = 1 − e k/ M ( η ) 1 + O (cid:0) τ − (cid:1) τ − p . For other values of p , asymptotics are more difficult to derive and we leave this for future research. Theasymptotic behaviour of option prices is fundamentally different to Black-Scholes asymptotics (Lemma A.2)and it is not clear that one can deduce asymptotics for the implied volatility. For example, the intrinsic valuesdo not necessarily match as τ tends to infinity because of the mass at the origin. The one exception is whenthe origin is reflecting, in which case the implied volatility tends to zero. This result is a direct translation ofTheorem 2.8 into implied volatility asymptotics: Theorem 2.9. If p < / and the origin is reflecting, the following pointwise limit holds for all k ∈ R : lim τ ↑∞ τ log τ σ τ ( k ) = 8(1 − p ) . Proof.
One could prove the statement directly by computing the asymptotic behaviour of the Black-ScholesCall price BS( k, y, τ ) as the maturity τ tends to infinity (pointwise, for any k > y ∈ R ), see Lemma A.2,and comparing it to the Call price expansion in Theorem 2.8(ii). Instead, we choose to apply Tehranchi’s result [66], which is the first fully model-independent study of the large-maturity behaviour of the impliedvolatility. Assuming (a) that the underlying stock price exp( Z ) is a non-negative local martingale under P , and(b) that exp( Z t ) converges almost surely to zero as time tends to infinity, Tehranchi [66, Theorem 3.1] provedthat the expansion(2.14) σ τ ( k ) = − E (cid:0) e Z τ ∧ e k (cid:1) − (cid:8) − log E (cid:0) e Z τ ∧ e k (cid:1)(cid:9) + 4 k − π + ε ( k, τ )then holds uniformly on compact subsets of the real line as τ tends to infinity, where the function ε ( · ) accountsfor higher-order error terms. It is clear here that the two assumptions above are satisfied in our model. Notethat (b) is equivalent to Call prices converging to one as the maturity tends to infinity (see [62, Lemma 3.3] forinstance). Using the almost sure equality (e Z τ − e k ) + = e Z τ − e Z τ ∧ e k , and the fact that (e Z τ ) τ ≥ is a truemartingale, it is then straightforward to show that Theorem 2.9 follows from Theorem 2.8(ii) and Tehranchi’sexpansion (2.14). (cid:3) Remark 2.10.
In fact, the proof of Theorem 2.9 actually provides higher-order terms, but we omit them herefor brevity. The case of Theorem 2.9(i) shows that the Call option price converges to1 − m t + m t (1 − e k ) + = ( − m t , if k ≥ , − m t e k , if k < , as the maturity tends to infinity. This clearly is never equal to zero since m t ∈ (0 , Numerics.
We provide here two types of numerical examples. In Section 2.6.1, we show how randomisingthe Black-Scholes model according to (2.1) distorts the standard flat Black-Scholes implied volatility surface,and generates a realistic-looking one. In Section 2.6.2, we compare numerically the asymptotic results for theimplied volatility smile to the true smile generated from(2.1).2.6.1.
Black-Scholes-CEV surface.
We consider here the following values:(2.15) t = 1 , ξ = 20% , p = 0 . , y = 10% , S = 1 . In Figure 1, we plot the implied volatility surface generated by (2.1) according to the values given in (2.15).First, note that, contrary to the standard Black-Scholes model, the surface is not flat. Second, and moreimportantly, the smile becomes steeper and more pronounced as the maturity becomes small. This is a widelyrecognised fact on Equity markets, and seems to validate the approach followed in this paper. Note that,
LACK-SCHOLES IN A CEV RANDOM ENVIRONMENT 13 following Section 2.3.2, one can taylor the parameters of the CEV component in order to match any desired(arbitrage-free) slope for the wings of the smile.
Figure 1.
BS-CEV implied volatility surface, with parameters being given in (2.15).2.6.2.
Asymptotics.
We calculate option prices using the representation (3.1) and a global adaptive Gauss-Kronrod quadrature scheme. We then compute the smile with a simple root-finding algorithm. In Figure 2(a),(b) and (c) we plot the smile for different maturities and values for the CEV power p . The model parametersare y = 0 . ξ = 0 . y / − p and t = 1 /
2. Note here that we set ξ to be a different value for each p . This isdone so that the models are comparable: ξ is then given in the same units and the quadratic variation of theCEV variance dynamics are approximately matched for different values of p . The graphs highlight the steepnessof the smiles as the maturity gets smaller and the role of p in the shape of the small-maturity smile. Note (asmentioned in previous sections) that the random variance acts as a shock to the small-maturity volatility surfaceand then flattens out. The shape of the shock depends on the CEV power, p . Out-of-the money volatilities(for K / ∈ [0 . , . p increases (this can be seen from Theorem 2.3). The volatilityfor strikes close to the money K ∈ [0 . , .
1] appears to be less explosive as one increases p , which might beexplained from the strike dependence of the coefficients of the asymptotic in Theorem 2.3. In order to compareour asymptotic to the true smile we use Theorem 2.1 to extend Theorem 2.1 to higher order. For the case p < , k = 0 we find that σ τ ( k ) ∼ a ( k ) τ − β p + a ( k ) τ − pβ p as τ tends to zero with a ( k ) := (1 − β p ) (cid:18) k ξ t (1 − p )2 (cid:19) β p , a ( k ) = 2 f ( y p ) a ( k ) k , and β p , y p and f defined in (2.10)-(2.11). At first order we see a close match with the true smile in Figure 2(d).2.7. Application to forward smile asymptotics.
We now show how our model (2.1) and the asymptoticsderived above for the implied volatility can be directly translated into asymptotics of the forward impliedvolatility in stochastic volatility models. For a given martingale process e X , a forward-start option with resetdate t , maturity t + τ and strike e k is worth, at inception, E (exp( X t + τ − X t ) − e k ) + . In the Black-Scholesmodel, the stationarity property of the increments imply that this option is simply equal to a standard Calloption on e X (started at X = 0) with strike e k and maturity τ ; therefore, one can define the forward impliedvolatility σ t,τ ( k ), similarly to the standard implied volatility (see [48] for more details). Suppose now that the (a) p=0.2. (b) p=1(c) p=1.3 (d) Actual vs asymptotic Figure 2.
In (a), (b), (c) we plot K σ τ (log K ) for maturities of 1/12 (circles),1/2 (squares),1(diamonds),2 (triangles) and 5 (backwards triangles) for increasing values of the CEV power p .In (d) we plot the actual small maturity smile for p = 0 . τ = 1 /
100 (circles) and thezeroth (squares) and first order (diamonds) smile using Theorem 2.3. Parameters of the modelare given in the text.log stock price process X satisfies the following SDE:(2.16) d X s = − Y s d s + p Y s d W s , X = 0 , d Y s = ξ s Y ps d B s , Y = y > , d h W, B i s = ρ d s, with p ∈ R , | ρ | < W, B are two standard Brownian motions. Fix the forward-start date t > ξ u := ( ξ, if 0 ≤ u ≤ t, ¯ ξ, if u > t, where ξ > ξ ≥
0. This includes the Heston ( p = 1 /
2) and 3/2 ( p = 3 /
2) models with zero mean reversionas well as the SABR model ( p = 1). Let X ( t ) τ := X t + τ − X t denote the forward price process and let CEV( t, ξ, p )be the distribution such that Law( Y t ) = Law( V ) = CEV( t, ξ, p ). Then the following lemma holds: Lemma 2.11.
In the model (2.16) the forward price process X ( t ) · solves the following system of SDEs: (2.17) d X ( t ) τ = − Y ( t ) τ d τ + q Y ( t ) τ d W τ , X ( t )0 = 0 , d Y ( t ) τ = ¯ ξ (cid:16) Y ( t ) τ (cid:17) p d B τ , Y ( t )0 ∼ CEV( t, ξ, p ) , LACK-SCHOLES IN A CEV RANDOM ENVIRONMENT 15 where Y ( t )0 is independent to the Brownian motions ( W τ ) τ ≥ and ( B τ ) τ ≥ . This lemma makes it clear that forward-start options in stochastic volatility models are European options ona stock price with similar dynamics to (2.16), but with initial variance sampled from the variance distributionat the forward-start date. When ¯ ξ = 0, then X ( t ) · = Z and forward smile asymptotics follow immediately: Corollary 2.12. If ¯ ξ = 0 , Theorem 2.1, Theorem 2.3 and Lemma 2.4 hold with Z = X ( t ) · and σ τ = σ t,τ . Remark 2.13. (i) Corollary 2.12 explicitly links the shape and fatness of the right tail of the variance distribution at theforward-start date and the asymptotic form and explosion rate of the small-maturity forward smile. Takefor example p >
1: the density of the variance in the right wing is dominated by the polynomial y − p and the exponential dependence on y is irrelevant. So the smaller p in this case, the fatter the right tailand hence the larger the coefficient of the expansion. This also explains the algebraic (not exponential)dependence for forward-start option prices.(ii) The asymptotics in the p > τ is similar to small-maturity exponential L´evy models. This extreme nature is related to the fatness of the right tail of thevariance distribution: for example, the 3 / p = 3 /
2) allows for the occurrence of extreme pathswith periods of very high instantaneous volatility (see [26, Figure 3 ]).(iii) The asymptotics in Theorems 2.1 and 2.3 remain the same (at this order) regardless of whether the varianceprocess is absorbing or reflecting at zero when p ∈ ( −∞ , / p = 1 / p = 1 /
2) produces small-maturity forward smilesthat are too convex and ”U-shaped” and inconsistent with observations, but that SABR-or lognormal-based models ( p = 1) produce less convex or ”U-shaped” small-maturity forward smiles. Our resultsprovide theoretical insight into this effect. We observed in Section 2.6 and Figure 2 that the explosioneffect was more stable for strikes close to the money as one increased p . The strike dependence of theasymptotic implied volatility in Theorem 2.3 is given by K p | log K | for p = 1 / K
7→ | log K | for p = 1. It is clear from the figures that the forward implied volatility is more U-shaped for p ≥ Proofs
Proof of Theorem 2.1.
Let C ( k, τ ) := E (e Z τ − e k ) + . This function clearly depends on the parameter t ,but we omit this dependence in the notations. The tower property implies(3.1) C ( k, τ ) = Z ∞ BS( k, y, τ ) ζ p ( y )d y + m t (cid:0) − e k (cid:1) + , where BS is defined in (2.8), ζ p is density of V given in (2.5) and m t is the mass at the origin (2.4). Our goalis to understand the asymptotics of this integral as τ tends to zero. We break the proof of Theorem 2.1 intothree parts: in Section 3.1.1 we prove the case p >
1, in Section 3.1.2 we prove the case p ∈ ( −∞ ,
1) and inSection 3.1.3 we prove the case p = 1. We only prove the result for k >
0, the arguments being completely analogous when k <
0. The key insight is that one has to re-scale the variance in terms of the maturity τ before asymptotics can be computed. The nature of the re-scaling depends critically on the CEV power p andfundamentally different asymptotics result in each case. Note that for k > (cid:0) − e k (cid:1) + = 0, so that the atomicterm in (3.1) is irrelevant for the analysis. When k < Case: p > . In Lemma 3.1 we prove a bound on the CEV density. This is sufficient to allow us to proveasymptotics for option prices in Lemma 3.2 after rescaling the variance by τ . This rescaling is critical becauseit is the only one making BS( k, y/τ, τ ) independent of τ . Let χ ( τ, p ) := τ p | − p | ξ t Γ(1 + | η | ) (cid:16) − p ) ξ t (cid:17) | η | exp − y − p )0 ξ t (1 − p ) ! , and we have the following lemma: Lemma 3.1.
The following bounds hold for the CEV density for all y, τ > when p > : ζ p (cid:16) yτ (cid:17) ≥ χ ( τ, p ) y p ( − ξ t (1 − p ) (cid:18) τy (cid:19) p − ) ,ζ p (cid:16) yτ (cid:17) ≤ χ ( τ, p ) y p ( y − p p − tξ ! " ξ t (1 − p ) (cid:18) τy (cid:19) p − + 1 ξ t (1 − p ) (cid:18) τyy (cid:19) p − . Proof.
From [55, Equation (6.25)] we know that for x > ν > − /
2, the modified Bessel function satisfies(3.2) 1Γ( ν + 1) (cid:16) x (cid:17) ν ≤ I ν ( x ) ≤ e x Γ( ν + 1) (cid:16) x (cid:17) ν , so that the expression for the CEV density in (2.5) implies that for p > χ ( τ, p ) y p exp − ξ t (1 − p ) (cid:18) τy (cid:19) p − ! ≤ ζ p (cid:16) yτ (cid:17) ≤ χ ( τ, p ) y p e m ( y,τ ) , where m ( y, τ ) := − ξ t (1 − p ) (cid:18) τy (cid:19) p − + 1 ξ t (1 − p ) (cid:18) τyy (cid:19) p − . For fixed τ >
0, note that m ( · , τ ) : R + R + takes a maximum positive value at y = y τ with m ( y τ, τ ) = y − p / (2( p − tξ ). When m ( · ) > m ( y,τ ) = 1 + e γ m ( y, τ ) forsome γ ∈ (0 , m ( y, τ )), and hence e m ( y,τ ) ≤ m ( y τ,τ ) m ( y, τ ). If m ( · ) < m ( y,τ ) ≤ | m ( y, τ ) | ≤ m ( y τ,τ ) | m ( y, τ ) | . The result for the upper bound then follows by the triangle inequality for | m ( y, τ ) | . Thelower bound simply follows from the inequality 1 − x ≤ e − x , valid for x >
0, and1 − ξ t (1 − p ) (cid:18) τy (cid:19) p − ≤ exp − ξ t (1 − p ) (cid:18) τy (cid:19) p − ! . (cid:3) Lemma 3.2.
When p > , Theorem 2.1 holds. LACK-SCHOLES IN A CEV RANDOM ENVIRONMENT 17
Proof.
The substitution y → y/τ and (3.1) imply that the option price reads C ( k, τ ) = R ∞ BS( k, y, τ ) ζ p ( y )d y = τ − R ∞ BS( k, y/τ, τ ) ζ p ( y/τ )d y . Using Lemma 3.1 and Definition (2.9), we obtain the following bounds: χ ( τ, p ) τ (cid:20) J p ( k ) − τ p − ξ t (1 − p ) J p − ( k ) (cid:21) ≤ C ( k, τ ) ,χ ( τ, p ) τ " J p ( k ) + exp y − p p − tξ ! τ p − ξ t (1 − p ) J p − ( k ) + τ p − ξ t (1 − p ) y p − J p − ( k ) ! ≥ C ( k, τ ) . Hence for τ < (cid:12)(cid:12)(cid:12)(cid:12) C ( k, τ ) τχ ( τ, p )J p ( k ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ exp y − p p − tξ ! J p − ( k )2 ξ t (1 − p ) J p ( k ) + J p − ( k ) ξ t (1 − p ) y p − J p ( k ) ! τ p − , which proves the lemma since J p ( k ) is strictly positive, finite and independent of τ . (cid:3) Case: p < . We use the representation in (3.1) and break the domain of the integral up into a compactpart and an infinite (tail) one. We prove in Lemma 3.4 that the tail integral is exponentially sub-dominant(compared to the compact part) and derive asymptotics for the integral in Lemma 3.5. This allows us to applythe Laplace method to the integral. We start with the following bound for the modified Bessel function of thefirst kind and then prove a tail estimate in Lemma 3.4.
Lemma 3.3.
The following bound holds for all x > and ν > − / : I ν ( x ) < ν + 2Γ( ν + 2) (cid:16) x (cid:17) ν e x . Proof.
Let x >
0. From [64, Theorem 7, page 522], the inequality I ν ( x ) < I ν +1 ( x ) / I ν +2 ( x ) holds whenever ν ≥ −
2, and hence combining it with (3.2) (valid only for ν > − / ν ( x ) < Γ( ν + 3)Γ( ν + 2) (cid:16) x (cid:17) ν e x , when ν > − /
2. The lemma then follows from the trivial identity Γ( ν + 3) = ( ν + 2)Γ( ν + 2). (cid:3) Lemma 3.4.
Let
L > and p < . Then the following tail estimate holds as τ tends to zero: Z ∞ L BS (cid:16) k, yτ β p , τ (cid:17) ζ p (cid:16) yτ β p (cid:17) d y = O exp − ξ t (1 − p ) (cid:20) L − p τ (1 − β p ) / − y − p (cid:21) !! . Proof.
Lemma 3.3 and the density in (2.5) imply ζ p (cid:16) yτ β p (cid:17) ≤ b τ − pβ p y − p exp − ξ t (1 − p ) (cid:26) y − p τ β p (1 − p ) − y − p (cid:27) + ( yy ) − p τ β p (1 − p ) ξ t (1 − p ) ! , where the constant b is given by( η + 2) | − p | ξ t Γ( η + 2) (cid:16) − p ) ξ t (cid:17) η , resp. ( | η | + 2) | − p | ξ t Γ( | η | + 2) (cid:16) − p ) ξ t (cid:17) | η | , if the origin is reflecting (resp. absorbing) when p < /
2; the exact value of b is however irrelevant for theanalysis. Set now L >
1. Using this upper bound and the no-arbitrage inequality BS( · ) ≤
1, we find Z ∞ L BS (cid:16) k, yτ β p , τ (cid:17) ζ p (cid:16) yτ β p (cid:17) d y ≤ Z ∞ L ζ p (cid:16) yτ β p (cid:17) d y ≤ b τ − pβ p Z ∞ L y − p exp − ξ t (1 − p ) (cid:26) y − p τ β p (1 − p ) − y − p (cid:27) + ( yy ) − p τ β p (1 − p ) ξ t (1 − p ) ! d y ≤ b τ − pβ p Z ∞ L y − p exp − ξ t (1 − p ) (cid:26) y − p τ β p (1 − p ) − y − p (cid:27) + ( yy ) − p τ β p (1 − p ) ξ t (1 − p ) ! d y, where the last line follows since y − p > y − p . Setting q = (cid:16) y − p /τ β p (1 − p ) − y − p (cid:17) / ( ξ √ t (1 − p )) yields Z ∞ L y − p exp − (cid:16) y − p τ βp (1 − p ) − y − p (cid:17) ξ t (1 − p ) + ( yy ) − p τ β p (1 − p ) ξ t (1 − p ) d y = ξ √ t (1 − p ) τ β p ( p − " ξ √ t (1 − p ) Z ∞ L τ q exp " − q y − p qξ √ t (1 − p ) d q + y − p Z ∞ L τ exp " − q y − p qξ √ t (1 − p ) d q , (3.3)with L τ := (cid:16) L − p /τ β p (1 − p ) − y − p (cid:17) / ( ξ √ t (1 − p )) > τ since L > p ∈ ( −∞ , τ ∗ := L − p y − p ! ( β p (1 − p )) − , so that, for τ < τ ∗ we have L τ > y − p / ( ξ √ t (1 − p )) and hence for q > L τ : y − p qξ √ t (1 − p ≤ q . In particular, for the integrals in (3.3) we have the following bounds for τ < τ ∗ : Z ∞ L τ q exp − q y − p qξ √ t (1 − p ) ! d q ≤ Z ∞ L τ q exp (cid:18) − q (cid:19) d q = 2 exp (cid:18) − L τ (cid:19) , Z ∞ L τ exp − q y − p qξ √ t (1 − p ) ! d q ≤ Z ∞ L τ exp (cid:18) − q (cid:19) d q ≤ L τ exp (cid:18) − L τ (cid:19) , where the last inequality follows from the upper bound for the complementary normal distribution functionin [67, Section 14.8]. The lemma then follows from noting that 1 − β p = 2 β p (1 − p ). (cid:3) Lemma 3.5.
When p < , Theorem 2.1 holds.Proof. Let e τ := τ β p , with β p defined in (2.10). Applying the substitution y → y/ e τ to (3.1) yields C ( k, τ ) = Z ∞ BS( k, y, τ ) ζ p ( y )d y = 1 e τ Z ∞ BS (cid:16) k, y e τ , τ (cid:17) ζ p (cid:16) y e τ (cid:17) d y = 1 e τ Z L BS (cid:16) k, y e τ , τ (cid:17) ζ p (cid:16) y e τ (cid:17) d y + 1 e τ Z ∞ L BS (cid:16) k, y e τ , τ (cid:17) ζ p (cid:16) y e τ (cid:17) d y, LACK-SCHOLES IN A CEV RANDOM ENVIRONMENT 19 for some
L > τ tends to zero, we obtain ζ p (cid:16) y e τ (cid:17) = τ pβ p / y p/ e − y − p )02 ξ t (1 − p )2 ξy p/ √ πt exp (cid:18) − τ β p (1 − p ) y − p ) ξ t (1 − p ) + 1 τ β p (1 − p ) ( yy ) (1 − p ) ξ t (1 − p ) (cid:19) h O (cid:16) τ (1 − p ) β p (cid:17)i . Note that this expansion does not depend on the sign of η and so the same asymptotics hold regardless of whetherthe origin is reflecting or absorbing. In the Black-Scholes model, Call option prices satisfy (Lemma A.1):BS (cid:16) k, y e τ , τ (cid:17) = y / k √ π (cid:16) τ e τ (cid:17) / exp (cid:18) − k y e ττ + k (cid:19) (cid:16) O (cid:16) τ e τ (cid:17)(cid:17) , as τ tends to zero. Using the identity 1 − β p = 2 β p (1 − p ) we then compute1 τ β p Z L BS (cid:16) k, yτ β p , τ (cid:17) ζ p (cid:16) yτ β p (cid:17) d y = τ β p (4 − p ) / y p/ e − y − p )02 ξ t (1 − p )2 + k πk ξ √ t Z L y (1 − p ) e − f y ) τ − βp + f y ) τ (1 − βp ) / d y h O (cid:16) τ (1 − β p ) / (cid:17)i , where f , f are defined in (2.11). Solving the equation f ′ ( y ) = 0 gives y = y p with y p defined in (2.10) and wealways choose the positive root and set L > y p . Let I ( τ ) := R L y (1 − p ) exp (cid:16) − f ( y ) τ − βp + f ( y ) τ (1 − βp ) / (cid:17) d y . Then forsome ε > τ tends to zero, the asymptotic equivalences I ( τ ) ∼ exp − f ( y p ) τ − β p + f ( y p ) τ (1 − β p ) / + f ′ ( y p ) f ′′ ( y p ) ! y (1 − p ) p Z y p + εy p − ε exp − q f ′′ ( y p )( y − y p ) τ (1 − β p ) / − f ′ ( y p ) q f ′′ ( y p ) d y ∼ exp − f ( y p ) τ − β p + f ( y p ) τ (1 − β p ) / + f ′ ( y p ) f ′′ ( y p ) ! y (1 − p ) p Z ∞−∞ exp − q f ′′ ( y p )( y − y p ) τ (1 − β p ) / − f ′ ( y p ) q f ′′ ( y p ) d y = exp − f ( y p ) τ − β p + f ( y p ) τ (1 − β p ) / + f ′ ( y p ) f ′′ ( y p ) ! τ (1 − β p ) / y (1 − p ) p s πf ′′ ( y p ) . hold. It follows that as τ tends to zero:1 τ β p Z L BS (cid:16) k, yτ β p , τ (cid:17) ζ p (cid:18) yβ p (cid:19) d y = exp (cid:18) − c ( t, p ) τ − β p + c ( t, p ) τ (1 − β p ) / (cid:19) c ( t, p ) τ c ( t,p ) h O (cid:16) τ − βp (cid:17)i . From Lemma 3.4 we know that1 τ β p Z ∞ L BS (cid:16) k, yτ β p , τ (cid:17) ζ p ( y/β p )d y = O exp − ξ t (1 − p ) (cid:18) L − p τ (1 − β p ) / − y − p (cid:19) !! . Choosing
L > max (cid:16) , (cid:0) ξ t (1 − p ) f ( y p ) (cid:1) / (2 − p ) , y p (cid:17) makes this tail term exponentially subdominant to τ − β p R L BS( k, y/τ β p , τ ) ζ p ( y/β p )d y , which completes the proof of the lemma. (cid:3) Case: p = 1 . In the lognormal case p = 1, the random variable log( V ) is Gaussian with mean µ (definedin (2.3)) and variance ξ t . The proof is similar to Section 3.1.2, but we need to re-scale the variance by τ | log( τ ) | .We prove a tail estimate in Lemma 3.6 and derive asymptotics for option prices in Lemma 3.7. Lemma 3.6.
The following tail estimate holds for p = 1 and L > as τ tends to zero ( µ defined in (2.3) ): Z ∞ L BS (cid:18) k, yτ | log( τ ) | , τ (cid:19) ζ (cid:18) yτ | log( τ ) | (cid:19) d y = O exp ( − ξ t (cid:20) log (cid:18) Lτ | log( τ ) | (cid:19) − µ (cid:21) )! . Proof.
By no-arbitrage arguments, the Call price is always bounded above by one, so that Z ∞ L BS (cid:18) k, yτ | log( τ ) | , τ (cid:19) ζ (cid:18) yτ | log( τ ) | (cid:19) d y ≤ Z ∞ L ζ (cid:18) yτ | log( τ ) | (cid:19) d y. With the substitution q = ξ √ t [log( y/ ( τ | log( τ ) | )) − µ ], the lemma follows from the bound for the complementaryGaussian distribution function [67, Section 14.8]. (cid:3) Lemma 3.7.
Let p = 1 . The following expansion holds for option prices as τ tends to zero: C ( k, τ ) = c ( t,
1) exp (cid:16) − c ( t, h ( τ, p ) + c ( t, h ( τ, p ) (cid:17) τ c ( t, | log( τ ) | c ( t, (cid:18) O (cid:18) | log( τ ) | (cid:19)(cid:19) , with the functions c , c , ..., c , h and h given in Table 1.Proof. Let e τ := τ | log( τ ) | . With the substitution y → y/ e τ and using (3.1), the option price is given by C ( k, τ ) = Z ∞ BS( k, y, τ ) ζ ( y )d y = 1 e τ Z ∞ BS (cid:16) k, y e τ , τ (cid:17) ζ (cid:16) y e τ (cid:17) d y = 1 e τ (Z L BS (cid:16) k, y e τ , τ (cid:17) ζ (cid:16) y e τ (cid:17) d y + Z ∞ L BS (cid:16) k, y e τ , τ (cid:17) ζ (cid:16) y e τ (cid:17) d y ) =: C( k, τ ) + C ( k, τ ) , for some L >
0. Consider the first term. Using Lemma A.1 with e τ = τ | log( τ ) | , we have, as τ tends to zero,BS (cid:18) k, yτ | log( τ ) | , τ (cid:19) = exp (cid:18) − k | log( τ ) | y + k (cid:19) y / k | log( τ ) | / √ π (cid:20) O (cid:18) | log( τ ) | (cid:19)(cid:21) . ThereforeC( k, τ ) = e k/ (cid:16) O (cid:16) | log( τ ) | (cid:17)(cid:17) | log( τ ) | / ξk π √ t Z L exp − k | log( τ ) | y − (cid:16) log (cid:16) yτ | log( τ ) | (cid:17) − µ (cid:17) ξ t y / d y = exp (cid:18) k − (log( τ ) + log | log( τ ) | ) + µ ξ t − µ (log( τ ) + log | log( τ ) | ) ξ t (cid:19) I ( τ ) h O (cid:16) | log( τ ) | (cid:17)i ξk π √ t | log( τ ) | / , where I ( τ ) := R L g ( y ) exp ( − g ( y ) | log τ | + g ( y ) log | log( τ ) | ) d y with g and g defined in (2.11) and g ( y ) := √ y exp (cid:18) µ log( y ) ξ t (cid:19) . The dominant contribution from the integrand is the | log( τ ) | term; the minimum of g is attained at y ∗ givenin (2.10), and g ′′ ( y ∗ ) = 4 / ( ξ t k ) >
0. Set I ( τ ) := Z ∞−∞ exp − ( y − y ∗ ) q | log( τ ) | g ′′ ( y ∗ ) − g ′ ( y ∗ ) log | log( τ ) | p | log( τ ) | g ′′ ( y ∗ ) ! d y = s πg ′′ ( y ∗ ) | log( τ ) | . LACK-SCHOLES IN A CEV RANDOM ENVIRONMENT 21
Then for some ε > τ tends to zero, the asymptotic equivalences with L > y ∗ , I ( τ ) ∼ Z y ∗ + ǫy ∗ − ǫ g ( y ) exp n − g ( y ) | log( τ ) | + g ( y ) log | log( τ ) | o d y ∼ g ( y ∗ )e − g ( y ∗ ) | log( τ ) | + g ( y ∗ ) log | log( τ ) | Z y ∗ + ǫy ∗ − ǫ e − g ′′ ( y ∗ )( y − y ∗ ) | log( τ ) | + g ′ ( y ∗ )( y − y ∗ ) log | log( τ ) | d y ∼ g ( y ∗ ) exp (cid:18) − g ( y ∗ ) | log( τ ) | + g ( y ∗ ) log | log( τ ) | + ( g ′ ( y ∗ ) log | log( τ ) | ) g ′′ ( y ∗ ) | log( τ ) | (cid:19) I ( τ )= g ( y ∗ ) exp (cid:18) − g ( y ∗ ) | log( τ ) | + g ( y ∗ ) log | log( τ ) | + ( g ′ ( y ∗ ) log | log( τ ) | ) g ′′ ( y ∗ ) | log( τ ) | (cid:19) s πg ′′ ( y ∗ ) | log( τ ) | . hold. Therefore as τ tends to zero:C( k, τ ) = c ( t,
1) exp (cid:16) − c ( t, h ( τ,
1) + c ( t, h ( τ, (cid:17) τ c ( t, | log( τ ) | c ( t, (cid:20) O (cid:18) | log( τ ) | (cid:19)(cid:21) , with the functions c , c , ..., c , h and h given in Table 1. For ease of computation we note that c ( t,
1) = √ y ∗ exp (cid:16) k − µ ξ t + µ log( y ∗ ) ξ t (cid:17) k ξ √ πt p g ′′ ( y ∗ ) = | k | ξ t / exp (cid:16) k − µ ξ t + µ log( y ∗ ) ξ t (cid:17) √ π . Now by Lemma 3.6, C ( k, τ ) = 1 τ | log( τ ) | Z ∞ L BS (cid:18) k, yτ | log( τ ) | , τ (cid:19) ζ (cid:18) yτ | log( τ ) | (cid:19) d y = 1 τ | log( τ ) | O exp ( − ξ t (cid:20) log (cid:18) Lτ | log( τ ) | (cid:19) − µ (cid:21) )! . Since for some
B > − ξ t (cid:20) log (cid:18) Lτ | log( τ ) | (cid:19) − µ (cid:21) ! ≤ B ( τ | log( τ ) | ) ξ t (log( L ) − µ ) exp (cid:18) − ξ t h ( τ, (cid:19) , choosing L such that log( L ) > µ yields O exp ( − ξ t (cid:20) log (cid:18) Lτ | log( τ ) | (cid:19) − µ (cid:21) )! = O (cid:18) exp (cid:18) − ξ t h ( τ, (cid:19)(cid:19) . Hence C ( k, τ ) is then exponentially subdominant to the compact part sinceexp (cid:16) c ( t, h ( τ, − c ( t, h ( τ, (cid:17) O exp ( − ξ t (cid:20) log (cid:18) Lτ | log( τ ) | (cid:19) − µ (cid:21) )! = O (cid:16) e − c ( t, h ( τ, (cid:17) , and the result follows. (cid:3) Proof of Theorem 2.8.
Lemma A.2 and (3.1) yield the following asymptotics as τ tends to infinity:(3.4) C ( k, τ ) = 1 − m t + m t (1 − e k ) + + τ − / e k/ L ( τ )(1 + O ( τ − )) , where L ( τ ) := R ∞ q ( z )e − τz d z , and we set q ( z ) ≡ − ζ p (8 z ) / √ πz. As z tends to zero recall the followingasymptotics for the modified Bessel function of the first kind of order η [1, Section 9.6.10]:I η ( z ) = 1Γ( η + 1) (cid:16) z (cid:17) η (cid:0) O (cid:0) z (cid:1)(cid:1) . Using this asymptotic and the definition of the density in (2.5) we obtain the following asymptotics for thedensity as y tends to zero when p < p < / ζ p ( y ) = y y − p | − p | ξ t Γ( | η | + 1) (2(1 − p ) ξ t ) | η | exp − y − p )0 ξ t (1 − p ) ! (cid:16) O (cid:16) y − p ) (cid:17)(cid:17) . Analogous arguments yield that when p < / y tends to zero,(3.6) ζ p ( y ) = y − p | − p | ξ t Γ( η + 1) (2(1 − p ) ξ t ) η exp − y − p )0 ξ t (1 − p ) ! (cid:16) O (cid:16) y − p ) (cid:17)(cid:17) . In order to apply Watson’s lemma [61, Part 2, Chapter 2] to L , it suffices to require that q ( z ) = O (e cz )for some c > z tends to infinity. This clearly holds here since lim z ↑∞ ζ p ( z ) = 0. We also require that q ( z ) = a z l (1 + O ( z n )) as z tends to zero for some l > − n >
0. When p ≥
1, it can be shown that ζ p isexponentially small, and a different method needs to be used. When p < l = 1 − p − and so we require p < . Analogously, when p < / l = − p − and we require p < . An application of Watson’s Lemma in conjunction with (3.4) yields Theorem 2.8. Appendix A. Black-Scholes asymptotics
This appendix gathers some useful expansions for the Black-Scholes Call price function BS defined in (2.8).
Lemma A.1.
Let k, y > and e τ : (0 , ∞ ) → (0 , ∞ ) be a continuous function such that lim τ ↓ τ e τ ( τ ) = 0 . Then BS (cid:18) k, y e τ ( τ ) , τ (cid:19) = y / k √ π (cid:18) τ e τ ( τ ) (cid:19) / exp (cid:18) − k y e τ ( τ ) τ + k (cid:19) (cid:26) O (cid:18) τ e τ ( τ ) (cid:19)(cid:27) , as τ tends to zero . Proof.
Let k, y > τ ∗ ( τ ) ≡ τ / e τ ( τ ). By assumption, τ ∗ ( τ ) tends to zero, and (2.8) impliesBS (cid:16) k, y e τ , τ (cid:17) = BS ( k, y, τ ∗ ( τ )) = N ( d ∗ + ( τ )) − e k N ( d ∗− ( τ )) , where we set d ∗± ( τ ) := − k/ ( p yτ ∗ ( τ )) ± p yτ ∗ ( τ ). Note that d ∗± tends to −∞ as τ tends to zero. Theasymptotic expansion 1 − N ( z ) = (2 π ) − / e − z / (cid:0) z − − z − + O ( z − ) (cid:1) , valid for large z ([1, page 932]), yieldsBS (cid:18) k, y e τ ( τ ) , τ (cid:19) = N (cid:0) d ∗ + ( τ ) (cid:1) − e k N (cid:0) d ∗− ( τ ) (cid:1) = 1 − N (cid:0) − d ∗ + ( τ ) (cid:1) − e k (1 − N (cid:0) − d ∗− ( τ ) (cid:1) )= 1 √ π exp (cid:18) − d ∗ + ( τ ) / (cid:19) (cid:26) d ∗− ( τ ) − d ∗ + ( τ ) + 1 d ∗ + ( τ ) − d ∗− ( τ ) + O (cid:18) d ∗ + ( τ ) (cid:19)(cid:27) , as τ tends to zero, where we used the identity d ∗− ( τ ) − k = d ∗ + ( τ ) . The lemma then follows from thefollowing expansions as τ tends to zero:exp (cid:18) − d ∗ + ( τ ) (cid:19) = exp (cid:18) − k yτ ∗ + k (cid:19) (1 + O ( τ ∗ ( τ ))) , d ∗− ( τ ) − d ∗ + ( τ ) + 1 d ∗ + ( τ ) − d ∗− ( τ ) = y / τ ∗ ( τ ) / k (1 + O ( τ ∗ ( τ ))) . (cid:3) Lemma A.2.
Let y > and k ∈ R . Then, as τ tends to infinity, BS( k, y, τ ) = 1 − √ πτ y exp (cid:18) − yτ k (cid:19) (cid:0) O ( τ − ) (cid:1) . LACK-SCHOLES IN A CEV RANDOM ENVIRONMENT 23
Proof.
Let y >
0. Then BS( k, y, τ ) = N (cid:0) d ∗ + ( τ ) (cid:1) − e k N (cid:0) d ∗− ( τ ) (cid:1) , where d ∗± ( τ ) := − k/ ( √ yτ ) ± √ yτ , Hence d ∗± tends to ±∞ as τ tends to infinity. Similarly to the proof of the previous lemma,BS( k, y, τ ) = N (cid:0) d ∗ + ( τ ) (cid:1) − e k (cid:0) − N (cid:0) − d ∗− ( τ ) (cid:1)(cid:1) = 1 − √ π exp (cid:18) − d ∗ + ( τ ) (cid:19) (cid:26) d ∗ + ( τ ) − d ∗− ( τ ) + 1 d ∗− ( τ ) − d ∗ + ( τ ) + O (cid:18) d ∗ + ( τ ) (cid:19)(cid:27) , as τ tends to infinity, where we used the identity d ∗− ( τ ) − k = d ∗ + ( τ ) . The lemma then follows from thefollowing expansions as τ tends to infinity:exp (cid:18) − d ∗ + ( τ ) (cid:19) = exp (cid:18) − yτ k (cid:19) (cid:0) O ( τ − ) (cid:1) , d ∗ + ( τ ) − d ∗− ( τ ) + 1 d ∗− ( τ ) − d ∗ + ( τ ) = 4 √ πτ y (cid:0) O ( τ − ) (cid:1) . (cid:3) References [1] M. Abramowitz and I. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. NewYork: Dover Publications, 1972.[2] E. Al`os, J. Le´on and J. Vives. On the short-time behavior of the implied volatility for jump-diffusion models with stochasticvolatility.
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