In this paper, we prove an approximate formula expressed in terms of elementary functions for the implied volatility in the Heston model. The formula consists of the constant and first-order terms in the large maturity expansion of the implied volatility function. The proof is based on saddlepoint methods and classical properties of holomorphic functions.
In financial markets stochastic models are widely used by traders and risk managers to price and hedge financial products. The models are chosen on economic grounds to reflect the observed characteristics of market data, such as leptokurtic returns of asset prices and random instantaneous volatility of these returns. Pricing and hedging in realistic models are numerically intensive procedures that need to be performed very quickly because future investment decisions depend on the outcomes of these computations. Therefore, market practitioners pay particular attention to tractability when deciding which model to use. Among the plethora of possible choices, stochastic volatility models are extremely popular particularly in equity, foreign exchange and interest rate markets since (i) they feature most of the characteristics of these markets, and (ii) they are numerically tractable.
In practice, stochastic volatility models are first calibrated on market data, then used for pricing. Pricing financial products is mathematically tantamount either to solving a partial differential equation problem with boundary conditions (the final payoff of the product) or to calculating the expectation of this payoff using probabilistic tools such as Monte Carlo simulation or stochastic integration. For most models closed-form formulae are scarcely available, and accurate algorithms have been extended and used such as finite-differences (Kluge 2002), alternative direction implicit schemes (Hout & Foulon 2010) or quadrature (Albrecher et al. 2007) methods. The calibration step involves a proper selection of the data to be fitted by a model. A common practice is to calibrate the so-called implied volatility rather than option prices directly. The implied volatility is a standardized measure of option prices which makes them comparable even though the underlying assets are not the same. Since this calibration step is based on optimization algorithms, the lack of a closed-form formula for the implied volatility makes it very time consuming. For instance, the SABR stochastic volatility model has become very popular because a closed-form approximation formula for the implied volatility was derived in Hagan et al. (2002) and hence made the model easily tractable. Likewise, perturbation methods as developed in Fouque et al. (2000) have proved to be very useful for obtaining a closed-form approximation formula of option prices. Although these methods only hold under some constraints on the parameters, they provide useful initial reference points for calibration.
The Heston model (Heston 1993) introduced in 1993 has become one of the most widely used stochastic volatility models in the derivatives market (see Lewis 2000; Gatheral 2006; Albrecher et al. 2007; Andersen & Piterbarg 2007; Lord & Kahl in press). In this paper, we provide a closed-form approximation for the implied volatility in this model. The idea behind this result is the following: suppose one wants to calibrate the Heston model on market data. This can be performed in two different ways: (i) one uses a global optimization algorithm, which involves computing the implied volatility at each observed point until the algorithm converges; (ii) one specifies an initial set of parameters for the model and runs a local optimization algorithm such as the least-squares method. The latter solution is the most widely used in practice since it is less computer-intensive. However, its robustness heavily relies on the initial set of parameters to be specified. Simple closed-form approximations for the Heston model make this choice robust and accurate. One first calibrates the approximation on market data—which is straightforward since this is a closed-form—then one uses this calibrated set of parameters as a starting point in the whole calibration process.
Let us consider an European option with maturity t and maturity-dependent strike , then our main result is the following asymptotic closed-form formula for the implied volatility :1.1as the maturity t tends to infinity, where is defined in equation (3.1), A in equation (2.10) and ABS in equation (2.15). For a constant strike , we obtain the following formula:1.2as the maturity t tends to infinity, where V* is given by equation (2.6) and p* by equation (2.7).
It is a well-known fact that for a fixed strike, the implied volatility flattens as the maturity increases (Rogers & Tehranchi 2010); this is confirmed by formula (1.2) above, the zeroth-order term of which was already known (see Lewis 2000; Forde & Jacquier in press). However, the maturity-dependent strike formulation in formula (1.1) above reveals that the implied volatility smile does not flatten but rather spreads out in a very specific way as the maturity increases.
In the fixed-strike case, Lewis (2000) pioneered the research on large-time asymptotics of implied volatility in stochastic volatility models by studying the first eigenvalue and eigenfunction of the generator of the underlying stochastic process.
Recently, Tehranchi (2009) studied the large-time behaviour of the implied volatility when the stock price is a non-negative local martingale and obtained an analogue of formula (1.2) in that setting. Comparatively, there has been a profusion of work on small-time asymptotics, based on differential geometry techniques (Henry-Labordère 2005), partial differential equation methods (Berestycki et al. 2004) or large deviations techniques (Feng et al. 2009; Forde & Jacquier 2009). Likewise, many papers have studied the behaviour of the implied volatility smile in the wings (Lee 2004a; Benaim & Friz 2008, 2009; Gulisashvili 2009; Gulisashvili & Stein 2010).
The proof of our main result, theorem 2.2, is based on two methods: first, we use saddlepoint approximation methods to study the behaviour of the call price function as an inverse Fourier transform. This idea has already been applied by several authors, including Carr & Madan (2009), Glasserman & Kim (2009), Ait-Sahalia & Yu (2006) and Rogers & Zane (1999) in order to speed up the computation of option pricing algorithms based on inverse Fourier transforms. We are also able to obtain the saddlepoint in closed form, thus avoiding any numerical approximations in determining it. The second step in our proof relies on Cauchy’s integral theorem and contour integration for holomorphic functions in order to obtain precise estimates of call option prices in the large maturity limit.
The paper is organized as follows. Section 2 contains the large-time asymptotic formula for call options under the Heston and the Black–Scholes models, both in the maturity-dependent and in the fixed-strike case. The proof of the main theorem, theorem 2.2, is given in §5. In §3, we translate these results into implied volatility asymptotics and prove formulae (1.1) and (1.2) above. In §4, we calibrate the Heston model and provide numerical examples based on formulae (1.1) and (1.2).
2. Large-time behaviour of call options
Throughout this article, we work on a model with a filtration supporting two Brownian motions, and satisfying the usual conditions. Let (St)t≥0 denote a stock price process and we let . Interest rates and dividends are considered null. We assume the following Heston dynamics for the log-stock price:2.1with κ,θ,σ,y0>0,|ρ|<1.
The Feller condition 2κθ>σ2 ensures that 0 is an unattainable boundary for the process Y . If this condition is violated zero is an attainable, regular and reflecting boundary (see Karlin & Taylor 1981, ch. 15 for the classification of boundary points of one-dimensional diffusions). Since the analysis in this paper relies solely on the study of the behaviour of the Laplace transform of the process X, which remains well defined even if the Feller condition is not satisfied, we do not assume that the inequality 2κθ>σ2 holds.
Let us now define , , and . Throughout the whole paper, we will assume . This assumption ensures (see theorem 2.1 in Forde & Jacquier in press) that moments of S greater than 1 exist for all times t. This condition is fundamental for the analysis in the paper and is usually assumed in the literature (Andersen & Piterbarg 2007; Keller-Ressel in press). When this condition is violated the limiting logarithmic Laplace transform V defined in equation (2.2) of the process S does not have the same properties, and further research is needed to understand how the implied volatility behaves in this case. We know from Andersen & Piterbarg (2007) and Forde & Jacquier (in press) that is the mean-reversion level of the process Y under the so-called Share measure that is equivalent to the original probability measure with the Radon–Nikodym derivative given by the share price itself. If , the process Y will be neither mean-reverting nor ergodic under the Share measure. In the equities market this does not constitute a problem since the calibrated correlation ρ is always negative. However, this assumption may be restrictive in markets such as foreign exchange, and further research is required to relax it. Let V be the limiting logarithmic moment generating function of X defined as2.2for all p such that the limit exists and is finite. It follows from Andersen & Piterbarg (2007) that V is a well-defined and strictly convex function on (p−,p+) and is infinite outside, where2.3with p−<0 and p+>1. Furthermore, the function V takes the following form:2.4where2.5and we take the principal branch for the complex square root function in equation (2.5).
Let us now define the Fenchel–Legendre transform of V, which was computed in Forde & Jacquier (in press) and is given by the formula2.6where the function is defined by2.7Tedious but straightforward calculations using the explicit formulae above yield the following proposition.
The function , where p− and p+ are defined in equation (2.3), is strictly increasing, infinitely differentiable and satisfies the following properties:as well as the equation2.8
Since the image of p* is (p−,p+), the function V* is well defined on . The following properties of V* are easy to prove and will be used throughout the paper:
(a) V*′(x)=p*(x) for all ;
(b) V*′′(x)>0 for all ;
(c) x↦V*(x) is non-negative, has a unique minimum at −θ/2 and V*(−θ/2)=0;
(d) x↦V*(x)−x is non-negative, has a unique minimum at and .
From definition (2.6) of V* and relation (2.8), the equality in (a) follows. The inequality in (b) is a consequence of (a) and proposition 2.1. Now, (a), (b) and proposition 2.1 imply that −θ/2 is the only local minimum of the function V* and is therefore a global minimum. The definition of V* given in equation (2.6) implies V*(−θ/2)=−V(0)=0. Since the stock price S is a true martingale (Andersen & Piterbarg 2007), we have V(1)=0 and proposition 2.1 implies that . This proves (c). From (a) and proposition 2.1, we know that the function x↦V*(x)−x has a unique minimum attained at and . Therefore (b) implies (d).
(a) Large-time behaviour of call options under the Heston model
In this section, we derive the asymptotic behaviour of call option prices under the Heston dynamics (2.1) as the maturity t tends to infinity, both for maturity-dependent and for fixed strikes. Before diving into the core of this paper, let us introduce the function by2.9which will feature in the main formulae of theorem 2.2 and proposition 2.7. The next theorem is the main result of the paper and its proof is given in §5.
For the Heston model and the assumptions above, we have the following asymptotic behaviour for the price of a call option with strike for all ,as , where2.10where2.11V is defined in equation (2.4), p* in equation (2.7), V* in equation (2.6) and d in equation (2.5) and sgn(x) equals 1 if x is positive and −1 otherwise.
Property (b) on page 5 implies that the square root is a strictly positive real number. Note also that the function A defined in equation (2.10) is not continuous at the points −θ/2 and by proposition 2.1 (see also figure 1).
It is proved in proposition 2.1 that p*(−θ/2)=0 and . Note further that U′(0)=(θ−y0)/(2κ) and .
Theorem 2.2 is similar in spirit to the saddlepoint approximation for a density of random variable X given in Butler (2007)where , and p*(x) is the unique solution to K′(p*(x))=x. Here X:=Xt−x0, K(p)=tV (p)+O(1), and we substitute x to xt so that
In order to precisely compare our result to the existing literature, we prove the following lemma, which gives the asymptotic behaviour of vanilla call options when the strike K is fixed, independent of the maturity. The following lemma was derived in Lewis (2000, ch. 6); a rigorous proof is detailed here in appendix A.
Under the same assumptions as theorem 2.2, for any , we have the following behaviour for a call option with fixed strike ,
(b) Large-time behaviour of the Black–Scholes call option formula
By a similar analysis, we can deduce the large-time asymptotic call price for the Black–Scholes model. This result is of fundamental importance for us since it will allow us to compute the implied volatility by comparing the Black–Scholes and the Heston call option prices. Throughout the rest of the paper, we let CBS(S0,K,t,Σ) denote the Black–Scholes price of a European call option written on a reference stock price S, with strike K>0, initial stock price S0>0, time-to-maturity t≥0, and volatility Σ>0 (with zero interest rate and zero dividend). Similar to §2, let us define the function as in equation (2.2), where . In the Black–Scholes case, it reads2.12Similarly to equation (2.6), we can define the functions and , by2.13and2.14The following proposition, proved in appendix B, gives the behaviour of the Black–Scholes price as the maturity tends to infinity.
Let σ>0 and let the real number a1 satisfy a1>−σ2t for large times t. Then for all , we have the following asymptotic behaviour for the Black–Scholes call option formula in the large-strike, large-time casewhere2.15and where the function is defined in equation (2.9).
If we set a1=0 in proposition 2.7, we obtain the large-time expansion for a call option under the standard Black–Scholes model with volatility σ and log-moneyness equal to xt.
As in the Heston model above, we derive here the equivalent of proposition 2.7 when the strike does not depend on the maturity anymore.
With the assumptions above, we have the following behaviour for the Black–Scholes call option formula in the fixed-strike, large-time case
This lemma is immediate from the Black–Scholes formula given in appendix B and the approximation (B2) for the Gaussian cumulative distribution function.
3. Large-time behaviour of implied volatility
The previous section dealt with large-time asymptotics for call option prices. In this section, we translate these results into asymptotics for the implied volatility. Recall that Forde & Jacquier (in press) and Lewis (2000) have already derived the leading-order term for the implied volatility in the large-time, fixed-strike case. Our goal here is to obtain the leading order and the correction term in the large-time, large-strike case. Theorem 3.1 provides the main result, i.e. the large-time behaviour of the implied volatility in the large strike case. In the following, will denote the implied volatility corresponding to a vanilla call option with maturity t and (maturity-dependent) strike in the Heston model (2.1). We now define the functions and by3.1and3.2where A is defined in equation (2.10), ABS in equation (2.15), U in equation (2.11), V in equation (2.4), p* in equation (2.7) and V* in equation (2.6). They are all completely explicit, so that the functions and are also explicit. From the properties of V* proved earlier V*(x) and V*(x)−x are non-negative, so that is a well-defined real number for all . Then the following theorem holds.
The functions and are continuous on andFurthermore, the error term tends to zero as t goes to infinity uniformly in x on compact subsets of .
We first prove that the functions and are continuous. In fact, the continuity of the function follows from properties (c) and (d) on page 5. We have already observed that the function A defined in equation (2.10) is discontinuous at and −θ/2 (see remark 2.3). Elementary calculations show that in the neighbourhood of we have the following expansion (where ):This expansion, together with formula (2.15), implies that the function is also discontinuous at and −θ/2. Since the quotient is strictly positive on , the function is therefore continuous on this set. The expansion above can also be used to obtain the following expression:which implies the equality . A similar argument shows continuity of at −θ/2.
We now prove the formula in the theorem in the case (recall that ). Note that as defined in equation (3.1) satisfies the following quadratic equation:3.3where V* is given by equation (2.6) and V*BS by equation (2.13). The proof of the theorem is in two steps: first, we prove the uniform convergence (on small intervals) of the implied variance to , then we derive a similar result for the first-order correction term. As a first step, we have to prove that, for all δ>0, there exists t*(δ)>0 such that for all t>t*(δ) and all x in some small neighbourhood of we have . By theorem 2.2 and equation (3.3) we know that for all ϵ>0, there exists t*(ϵ) and an interval containing x′ such that for all x in this interval and all t>t*(ϵ) we have the lower bound3.4and the upper bound3.5Note thatfor all , since V*(x)−x>0 by property (d). For x fixed, the function Σ↦V*BS(x,Σ)−x defined on is continuous and strictly decreasing, where V*BS is given in equation (2.13). Thus, for any δ>0 such that for all x in the small neighbourhood of x′ defineand
Note that the limits are uniform in x on the chosen neighbourhood of x′. Combining equations (3.4), (3.5) and proposition 2.7, there exists t*(δ) such that for all t>t*(δ) and all x near x′ we haveandThus, by the monotonicity of the Black–Scholes call option formula as a function of the volatility, we have the following bounds for the implied volatility at maturity t:
In the second step of the proof we show that for all δ>0 there exist t*(δ) and a small neighbourhood of x′ such that, for all t>t*(δ) and all x in this neighbourhood, the following holds:3.6Note that the definition in equation (3.2) impliesTheorem 2.2 implies that for any ϵ>0 there exist a small interval containing x′ and t*(ϵ) such that for all x in this interval and all t>t*(ϵ) we have3.7andLet δ>0 and define . The reason for this definition lies in the following identitywhich holds by definition (2.15). By equation (3.7) there exist t*(δ) and a neighbourhood of x′ such that for all x in this neighbourhood and all t>t*(δ) we haveandBy strict monotonicity of the Black–Scholes price as a function of the volatility, we obtainfor all x in some interval containing x′ and all t>t*(δ). This proves equation (3.6). An analogous argument implies the inequality in equation (3.6) for since the functions A and are continuous on this set and by equation (3.3) the following identity holds:(the function is defined in equation (2.9)). In the cases a similar argument proves equation (3.6) at the point x′ but not necessarily on its neighbourhood.
All that is left to prove is that the inequality in equation (3.6) holds uniformly on compact subsets of the complement . To every point in a compact set we can associate a small interval that contains it such that inequality (3.6) holds for all x in that interval and all large times t. This defines a cover of the compact set. We can therefore find a finite collection of such intervals that also covers our compact set. It now follows that the inequality in equation (3.6) holds for any x in the compact set and all times t that are larger than the maximum of the finite number of t*(δ) that correspond to the intervals in the finite family that covers the original set. ■
(a) The large-time, fixed-strike case
This section is the translation of lemmas 2.6 and 2.9 in terms of implied volatility asymptotics, and improves the understanding of the behaviour of the Heston implied volatility in the long term. Let σt(x) denote the implied volatility corresponding to a vanilla call option with maturity t and fixed strike in the Heston model (2.1). Let us define the function by3.8where A is defined in equation (2.10), V* in equation (2.6) and p* in equation (2.7). Elementary calculations show that A(0)<0. From the properties of V*, a1(x) is then well defined as a real number for all .
With the assumptions above, we have the following behaviour for the implied volatility in the fixed-strike case:where V* is given by equation (2.6) and a1 by equation (3.8). The error term tends to zero as t goes to infinity uniformly on compact subsets of .
The proof of this theorem is similar to the proof of theorem 3.1. It is in fact simpler because we do not have to consider special cases for . We therefore only give an argument that implies the inequality3.9holds on some small neighbourhood of x′ for all large t. Lemma 2.6 and equation (3.8) imply that for all ϵ>0, there exists a t*(ϵ) and a neighbourhood of x′ such that for all x in this neighbourhood and all t>t*(ϵ) we haveFor any δ>0 a continuity argument implies the existence of ϵ(δ)>0, which tends to zero uniformly on a neighbourhood of x′, such thatTherefore, there exists t*(δ)>0 and an interval containing x′ such that for all x in this interval and all t>t*(δ) we haveThe monotonicity of the Black–Scholes formula as a function of the volatility implies equation (3.9). The proof can now be concluded in the same way as in theorem 3.1. ■
4. Numerical results
We present here some numerical evidence of the validity of the asymptotic formula for the implied volatility obtained in theorem 3.1. We calibrated the Heston model on the European vanilla options on the Eurostoxx 50 on 15 February 2006. The maturities range from one year to nine years, the strikes from 1460 up to 7300, and the initial spot S0 equals 3729.79. The calibration, performed using the Zeliade Quant Framework, by Zeliade Systems, on the whole implied volatility surface gives the following parameters: κ=1.7609, θ=0.0494, σ=0.4086, y0=0.0464 and ρ=−0.5195. We then use these calibrated parameters to generate an implied volatility smile for two different maturities, 5 and 9 years (note that for convenience the smiles were generated with interest rates and dividends equal to zero). The two plots below contain the generated implied volatility smile for the Heston model for the two maturities as well as the zeroth- and first-order approximations obtained in equation (3.1) and in theorem 3.1. The errors plotted are the differences between the generated Heston implied volatility and the implied volatility obtained by the two asymptotic approximations (figures 2 and 3).
5. Proof of theorem 2.2
The proof of the theorem is divided into a series of steps: we first write the Heston call price in terms of an inverse Fourier transform of the characteristic function of the stock price (5.1). Then we prove a large-time estimate for the characteristic function (lemma 5.1). The next step is to deform the contour of integration of the inverse Fourier transform through the saddlepoint of the integrand (equation (2.7) and proposition 2.1). Finally, studying the behaviour of the integral around this saddlepoint (proposition 5.8) and bounding the remaining terms (lemma 5.6) completes the proof. The special cases x=−θ/2 and in formula (2.10) are proved in §§5c and d.
(a) The Lee–Fourier inversion formula for call options
Using similar notation to Lee (2004b), setand define the characteristic function of Xt−x0 byFrom theorem 1 in Lord & Kahl (in press) and proposition 3.1 in Andersen & Piterbarg (2007), we know that ϕt(z) can be analytically extended for any such that −ℑ(z)∈At,X, and from our assumptions on the parameters in §2, for all t≥0. By theorem 5.1 in Lee (2004b), for any α∈(p−,p+), we have the following Fourier inversion formula for the price of a call option on St:where and is a contour such that γ+(u):=u − i(α − 1). The first four terms on the right-hand side are complex residues that arise when we cross the poles of (iz−z2)−1 at z=0 and z=i. We now set k=i−z, substitute x to xt, and use the fact that St is a true martingale for all t≥0 (Andersen & Piterbarg 2007, prop. 2.5). From now on, as k will always denote a complex number, we use the notation k=kr+iki for . Note thatthat kr↦ℜ(eikxte−ik(Xt−x0)/(ik−k2)) is an even function and kr↦ℑ(eikxte−ik(Xt−x0)/(ik−k2)) an odd function. Clearly the normalized call price is real, so if we take the real part of both sides and break up the integral, we obtain5.1for any R>0, where, for any , we define the contours5.2and5.3For ease of notation, we do not write explicitly the dependence of these contours on R. We will see later how to choose R. In the following lemma, we characterize the large-time asymptotic behaviour of the characteristic function ϕt.
For all such that −ki∈(p−,p+), we haveℜ(d(k))>0, and ϵ(k,t)=O(e−tℜ(d(k))), where U is defined in equation (2.11), p−, p+ in equation (2.3), d in equation (2.5) and V is the analytic continuation of formula (2.4).
If −ki is not in (p−,p+), this large-time behaviour of ϕt still holds, but ℜ(d(k)) might be null (for instance if kr=0) so that ϵ(k,t) does not tend to 0 as .
From Albrecher et al. (2007), we have, for all such that −ki∈(p−,p+),5.4where d is defined in equation (2.5), V is the analytic extension of formula (2.4) and the correct branch for the complex logarithm and the complex square root function is the principal branch (see also Albrecher et al. 2007; Lord & Kahl in press) and is defined by5.5For all such that −ki∈(p−,p+), we have ℜ(d(k))>0. LetThen we haveThen, as t tends to infinity we havefor some constant c. Set ϵ(k,t):=ϵ1(k,t)ϵ2(k,t)−1 and the lemma follows. ■
(b) The saddlepoint and its properties
We first recall the definition of a saddlepoint in the complex plane (Bleistein & Handelsman 1975):
Let be an analytic complex function on an open set . A point such that the complex derivative dF/dz vanishes is called a saddlepoint.
Note that the function defined in equation (2.2) can be analytically extended and we define for every the function by5.6Note that the exponent of the integrand in equation (5.1) has the form −Fx(k)t by lemma 5.1, therefore the saddlepoint properties of Fx given in the following elementary lemma are fundamental.
The saddlepoints of the complex function are given bywhere .
Since we are looking for the saddlepoint in , we can use the representation in equation (2.4) for the function V . Therefore, the equation Fx′(z)=0 is quadratic and hence has the two purely imaginary solutions since the expression (x2σ2+ 2xκθρσ+κ2θ2)=(xσ+κθρ)2+κ2θ2(1−ρ2) is strictly positive for any . It is also clear from the definition of p− and p+ given in equation (2.3) and the assumptions on the coefficients that , and therefore are saddlepoints of Fx. ■
The next task is to choose the saddlepoint of the function Fx in such a way that it converges to the saddlepoint of the function for all , where VBS is given by equation (2.12), in the Black–Scholes model as both the volatility of volatility and the correlation in model (2.1) tend to zero. It is easy to see that the saddlepoint of equals ip*BS(x) for any , were p*BS is given by equation (2.14). We can rewrite defined in lemma 5.3 as5.7where is defined on page 4. The first term converges to and the last one to ±x/θ as (ρ,σ) tends to 0. When both ρ and σ tend to 0, a Taylor expansion at first order of the third term gives5.8Take now the positive sign in equation (5.7), then the second and third terms cancel out in the limit because η converges to 2κ as (ρ,σ) tends to 0. In that case we have for all , where the Black–Scholes variance is equal to θ. If we were to take for the saddlepoint, in the limit we would not recover p*BS(x) by equation (5.8), since the function (ρ,σ)↦ρ/σ has no limit as the pair (ρ,σ) tends to 0. Therefore, we define the saddlepoint to be . Moreover, we observe the following equality:where p*(x) is defined in equation (2.7).
In the fixed strike case, i.e. when x=0, we obtainThe corresponding saddlepoint ip*(0) is the same as the one in Lewis (2000, ch. 6).
The following lemma is of fundamental importance and will be the key tool for proposition 5.8.
Let . Then, for any ki∈(p−,p+), the function kr↦ℜ(−ikx−V (−ik)) has a unique minimum at 0 and is strictly decreasing (resp. increasing) for (resp. .
Note that the statement in the lemma is equivalent to the map kr↦−ℜ (V (−i(kr+iki))) having a unique minimum at kr=0 for any ki∈(p−,p+) and being increasing (resp. decreasing) on the positive (resp. negative) halfline. Let ki∈(p−,p+), thenwhereFrom the identity and the fact that the principal value of the square-root is used, we get5.9is monotonically increasing in u, u2 and v2. First, note that , hence u is a parabola with a unique minimum at kr=0, so that, from equation (5.9), it suffices to prove the following claim:
For every ki∈(p−,p+), the function g:=u2+v2 has a unique (strictly positive) minimum attained at kr=0 and is strictly increasing (resp. decreasing) for kr>0 (resp. kr<0).
Let us write , for all , where . We have5.10The coefficient and the constant κ2 are strictly positive, so the claim follows if χ(ki)>0 for all ki∈(p−,p+), whereThe discriminant is , so that χ has no real root and is hence always strictly positive. This proves the claim and concludes the proof of the lemma. ■
The following two results complete the proof of theorem 2.2, by studying the behaviour of the two integrals in equation (5.1) as the time to maturity tends to infinity. The following lemma proves that the integral along γp*(x) is negligible and proposition 5.8 hereafter provides the asymptotic behaviour of the integral along the contour ζp*(x).
For any and any m>V*(x), there exists R(m)>0 such that for every with |kr|>R(m), we have5.11Therefore5.12where the contour γp*(x) is defined in equation (5.2).
(i) For every , we have V*(x)≥x by (d) hence m−x>0. Therefore, the modulus of the integral (5.12) tends to zero exponentially in time and in m.
We only need to prove equation (5.12). Recall from lemma 5.1, after some rearrangements, thatIt follows from equations (2.5), (2.4) and (5.5) thatandHence there exists a constant C>0, independent of k and t, such that the following inequality holds:Define . Then if , where the positive constant R is given in definition (5.2), the equality (5.12) follows. ■
For any R>0 and , we have as ,5.13where A is given in equation (2.10), V* in equation (2.6) and ζp*(x) in equation (5.3).
Let . Applying lemma 5.1 on the compact interval [−R,R], we havefor t large enough. By lemma 5.5, we know that kr↦−ℜ(i(kr+ip*(x))x+V (−i(kr+ip*(x)))) has a unique minimum at kr=0 and the value of the function at this minimum equals V*(x) by the definition of V*. The functions V and U are analytic along the contour of integration and thus, by Olver (1974, theorem 7.1, §7, ch. 4), we haveas t tends to infinity. The ϵ(k,t) term is a higher order term which we can ignore at the level we are interested in. ■
Lemma 5.6 and proposition 5.8 complete the proof of theorem 2.2 for the general case. Concerning the two special cases, we first introduce a new contour, the path of steepest descent, which represents the optimal (in a sense made precise below) path of integration. Note that, the general case can also be proved using this path, but lemma 5.5 simplifies the proof.
(c) Construction of the path of steepest descent
We first recall the definition of the path of steepest descent before computing it explicitly for the Heston model in the large time case.
Definition 5.9 (Stein & Sharkarchi 2003).
Let and be an analytic complex function. The steepest descent contour is a map such that
— ℜ(F) has a minimum at some point z0∈γ and ℜ(F′′(z0))>0 along γ.
— ℑ(F) is constant along γ.
These two conditions imply that F′(z0)=0.
The following lemma computes the path of steepest descent explicitly in the Heston case for the function Fx given in equation (5.6) passing through ip*(x).
The path of steepest descent γ in the Heston model is the map defined bywhere5.14β:=κθξ(2κρ−σ), and .
Note that ξ is strictly positive, so that the function ki is well defined.
By definition, the contour of steepest descent is such that the function ℑ(Fx°γ) remains constant. So we look for the map γ such that ℑ(Fx(γ(s)))=0, for all because Fx(γ(0)) is already real. Using the identity , for all , we find that the function Fx°γ is real along the contour γ:s↦s+iki(s). Note also that this contour is orthogonal to the imaginary axis at ki(0) (Stein & Sharkarchi 2003, exercise 2, ch. 8). ■
— The contour γ depends on x, but for clarity we do not write this dependence explicitly.
— The construction of γ is such that the saddlepoint defined in equation (2.7) satisfies ip*(x)=γ(0).
— We have ki(s)=ℑ(γ(s)) is an even function of s, i.e. γ is symmetric around the imaginary axis.
We now prove theorem 2.2 in the two special cases . In these cases, we need a result similar to proposition 5.8, as lemma 5.6 still holds, i.e. we need the asymptotic behaviour of the integral in equation (5.13) for the two special cases. The problem with these special cases is that (ik−k2)−1 in the integrand in equation (5.13) has a pole at the saddlepoint, so we need to deform the contour using Cauchy’s integral theorem and take the real part to remove the singularity, before we can use a saddlepoint expansion.
(d) Proof of the call price expansion for the special cases
We here prove theorem 2.2 in the case for which , , and for simplicity we also assume that κ<(σ−2ρ2σ)/(2ρ) (the other cases follows similarly). From equation (5.14), we see that in this case, γ lies below the horizontal contour such that γH(s):=s+i (in the other case, γ lies above γH). We want to construct a new contour leaving the pole outside. Let ϵ>0 and denote the clockwise oriented circular keyhole contour parameterized as γϵ(θ):=i+ϵ eiθ around the pole. To leave the pole outside the new contour of integration, we need to follow γ on , switch to the keyhole contour as soon as we touch it, follow it clockwise (above the pole), and get back to γ on . As γ is below γH, it intersects γϵ on its lower half, which can be analytically represented as such that . From equation (5.14), the two contours intersect at s*=±ϵ. Choose now 0<ϵ<δ<R (lemma 5.12 makes the choice of δ precise and ϵ must be such that 1+2ϵ<p+) and define the following contours (they are all considered anticlockwise, see figure 4)
— given by γδ,R(u)=u+iki(δ), with ki(δ) defined in equation (5.14);
— γϵ,δ is the restriction of γ to the union of the intervals ;
— is the portion of the circular keyhole contour γϵ which lies above γ, i.e. the upper half keyhole contour as well as the two sections of γϵ between γ and γH;
— are the two vertical strips joining ±R+i(1+2ϵ) to ±R−iki(δ).
By Cauchy’s integral theorem, we now have5.15Recall that the curves ζ1+2ϵ and γ1+2ϵ are defined in equations (5.3) and (5.2), respectively, and rewrite equation (5.15) as5.16The integral on the left-hand side is equal to the normalized call price by theorem 5.1 in Lee (2004b), which is independent of ϵ (this holds because 1+2ϵ<p+). For k close to i, we haveso that5.17Lemma 5.12 gives the behaviour of the last integral on the right-hand side of equation (5.16) as ϵ tends to 0 for δ small enough.
The other integrals can be bounded as follows. By lemma 5.6, the integral along γ1+2ϵ is , for t>t*(m), R>R(m), as ϵ tends to 0.
The curves are both vertical strips of length δ and therefore their images are compact sets. Applying the tail estimate of lemma 5.6 along , we know that for any , there exist t(m) and R(m) such thatLemma 5.5 implies that the real function attains its global minimum at 0 for any fixed ki∈(p−,p+) and is strictly decreasing (resp. increasing) for kr<0 (resp. kr>0). It therefore follows that the function , where , attains its minimum value , where ki(δ) is defined in equation (5.14), at the points u=±δ. It can be checked directly that , g′(0)=0 and g′′(0)>0 and hence for every δ>0 there exists ϵ0>0 such that the following inequality holds:Therefore lemma 5.1 yields the following inequalityWe now prove the following lemma about the integral along γϵ,δ as ϵ tends to 0.
For δ>0 and sufficiently small we havewhere U is given by equation (2.11) and V by equation (2.4).
Recall that γ is the contour of steepest descent defined in lemma 5.10 and that the curve γϵ,δ is its restriction to the intervals . Note that is an even function and is an odd function. We therefore obtain5.18From equation (5.14), for s around 0, we have (iγ(s)−γ(s)2)−1=i/s−1+O(s), so ℜ((iγ(s)−γ(s)2)−1)=−1+O(s), i.e. taking the real part removes the singularity at k=i. Using lemma 5.1, we then havefor some m>0 large enough, where we define the function byThen, from equation (5.14), we have the following expansion:5.19We can, therefore, extend the function q to the map for some δ>0, where is an open disc of radius δ. Note that for we have ℜ(q(s))=−1−V ′′′(1)/(6V ′′(1))+U′(1)+O(s) and hence the function does not have a singularity at s=0.
Recall that if a function has a Laurent series expansionwith , then the function has an analytic continuation on the whole disc Bδ(0). It follows from equation (5.19) that there exists a holomorphic function such that Q(s)=ℜ(q(s)) for any s∈(−δ,δ). Thus by Olver (1974, theorem 7.1, ch. 4), we have ■
The authors would like to thank J. Appleby, J. Feng, J. P. Fouque, J. Gatheral, A. Gulisashvili, M. Keller-Ressel, A. Lewis, R. Lee and C. Martini for many useful discussions and the anonymous referees for useful comments.
Appendix A. Proof of lemma 2.6
The proof is analogous to the proof of theorem 2.2. The residue in theorem 2.2 is equal to 1 (arising from the term), and from equation (5.1), the integral part is equal, for R large enough, readsand the behaviour of these integrals follows exactly the lines of the proof of theorem 2.2.
Appendix B. Proof of proposition 2.7
Let us now consider a squared volatility of the form , then the Black–Scholes call option readsB1and let . Recall that (Olver 1974)B2
The case x>σ2/2. As , there exists t* such that for all t>t*, . From equation (A1), we have, using a Taylor expansion for z±,
The cases x<−σ2/2 and −σ2/2<x<σ2/2 follow likewise.
The case x=σ2/2. From equation (A1), we haveThe case x=−σ2/2 is analogous.
- Received November 27, 2009.
- Accepted April 29, 2010.
- © 2010 The Royal Society