REFINED WING ASYMPTOTICS FOR THE MERTON AND KOU JUMP DIFFUSION MODELS

Transcription

1 ADVANCES IN MATHEMATICS OF FINANCE BANACH CENTER PUBLICATIONS, VOLUME 04 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 05 REFINED WING ASYMPTOTICS FOR THE MERTON AND KOU JUMP DIFFUSION MODELS STEFAN GERHOLD Vienna University of Technology Wiedner Hauptstraße 8 0/05-, A-040 Vienna, Austria sgerhold@fam.tuwien.ac.at JOHANNES F. MORGENBESSER University of Vienna Oskar-Morgenstern-Platz, A-090 Vienna, Austria johannes.morgenbesser@univie.ac.at AXEL ZRUNEK Vienna University of Technology Wiedner Hauptstraße 8 0/05-, A-040 Vienna, Austria axel.zrunek@aon.at Abstract. Refining previously known estimates, we give large-strike asymptotics for the implied volatility of Merton s and Kou s jump diffusion models. They are deduced from call price approximations by transfer results of Gao and Lee. For the Merton model, we also analyse the density of the underlying and show that it features an interesting almost power law tail.. Introduction. In recent years, the literature on asymptotic approximations of option prices and implied volatilities has grown at a fast pace. Important papers on large-strike asymptotics include [, 3,, 3]. Of particular relevance to the present note is the approach of Gao and Lee [8], who translate call price asymptotics to implied volatility asymptotics, robustly with respect to choice of model and asymptotic regime. So far, relatively few models have been analyzed in sufficient detail to use the full power of their transfer results. We extend concrete applicability of some results of [8] by presenting 00 Mathematics Subject Classification: Primary 9G0; Secondary 4A60. Key words and phrases: Implied volatility, jump diffusion, Kou model, Merton model, saddle point method. The paper is in final form and no version of it will be published elsewhere. DOI: /bc [85] c Instytut Matematyczny PAN, 05

2 86 S. GERHOLD, J. F. MORGENBESSER AND A. ZRUNEK refined strike asymptotics for the well-known jump diffusion models of Merton [5] and Kou []. Potential practical consequences of our expansions concern fast calibration and implied volatility parametrization resp. extrapolation. As we are in a fixed-maturity regime, we can set the interest rate to zero throughout. Also, initial spot is normalized to S 0 =. Log-returns are modeled by a Lévy jump diffusion N t X t = bt + σw t + with drift b R and diffusion volatility σ > 0. The process W is a standard Brownian motion, N is a Poisson process with intensity λ > 0, and the jumps Y j are i.i.d. real random variables. As for the law of the Y j, we focus on the double exponential Kou and Gaussian Merton cases. The dimensionless implied volatility V k is the solution of j= Y j c BS k, V k = E[S T + ], where c BS k, v = Φd Φd is the Black Scholes call price, with d, = k/v ± v/ and Φ the standard Gaussian cdf. Our interest is in the growth order of V k as k. While first order asymptotics are known for both models we treat, they suffer from limited practical applicability. Higher order terms typically increase accuracy significantly, even for moderate values of the log-strik. We observe that the large-strike behavior of the Kou model is of the same shape in terms of the asymptotic elements involved as for the Heston model [7]. Not obvious from the respective model dynamics, this fact is an immediate consequence of the local behavior of the moment generating function mgf at the critical moments. This behavior was analyzed in [7] from affine principles, whereas the present analysis for the Kou model profits from its very simple explicit mgf. For the Merton model, we include an approximation of the density Theorem 3.3. The reason is that it implies an interesting almost power law tail for the marginals of the underlying, of order k log log k. Throughout the paper, F k Gk means that the functions F and G satisfy F k = OGk for k.. Kou jump diffusion model. In the Kou model [], the Y j are double exponentially distributed, and thus have the common density fy = pλ + e λ+y [0, y + pλ e λ y,0 y with parameters λ + >, λ > 0 and p 0,. One of the advantages of this model is the memoryless property of the double exponential distribution, which leads to analytical formulas for several types of options. The moment generating function of the log-price X T

3 ASYMPTOTICS FOR JUMP DIFFUSION MODELS 87 is given by Ms, T = E[expsX T ] σ s = exp T λ+ p + bs + λ λ + s + λ p. λ + s From Benaim and Friz refinement of Lee s moment formula Example 5.3 in [], it is known that V has the first order asymptotics where Ψx is defined by V k lim k k = / Ψ/ λ +, Ψx = 4 x + x x. 3 To formulate our refined expansion for the Kou call price, define α = λ +, α / = λλ + pt /, α 0 = T σ λ + bλ + λλ p λλ + pt /4 + λ log λ + λ + πλ + λ +. For better readability, the notation here α i, and β i below is as in Corollary 7. of [8]; coefficient indices mimic the asymptotic terms they belong to. Theorem.. The price of a call option in the Kou model satisfies as k. Ck, T = exp k α / k / α 0 k 3/4 + Ok /4 4 The call price expansion 4 is amenable to the general transfer results of Gao and Lee [8]. Indeed, their Corollary 7. immediately implies the following implied volatility expansion. Corollary.. The implied volatility of the Kou model satisfies, as k, V k = β / k / log k + β 0 + β l / k + β / / k + / Ok 3/4, 5 where and β / = γ α + α = Ψ / λ +, β 0 = γα /, β l / = γ 4, β / = α 0 + log + /α / γ + 4πα γ = α +. α α 3/ α + 3/ α /, The gain in numerical precision over the first order approximation depends on the model parameters. See Figures and for examples. It is an interesting fact that the expansion 5 has the same shape as the implied vol expansion of the Heston model see Theorem.3 in [7]. While this is not obvious from the model specifications, it is clear from an asymptotic analysis of the respective mgfs: Call price and density asymptotics are

4 88 S. GERHOLD, J. F. MORGENBESSER AND A. ZRUNEK governed by the type of singularity found at the critical moment, and this is exponential of a first order pole in both cases. For other papers analyzing functions with such behavior, see, e.g., [6, 9]. Fig.. Implied volatility of the Kou model solid compared with the fourth-order expansion 5 dotted and Lee s formula dashed. The parameters are T = 6, σ = 0.4, λ =, p = 0., λ =, and λ + = 3. Fig.. Implied volatility of the Kou model solid compared with the fourth-order expansion 5 dotted and Lee s formula dashed. The parameters are T =, σ = 0., λ = 5, λ + = 5, λ = 5, p = 0.5. We briefly comment on the qualitative implications of Corollary.. The dominating term depends only on λ +, i.e., the parameter that governs the propensity to jump in the in-the-money direction. If λ + increases, upward jumps become smaller, and the call clearly becomes cheaper, thus lowering implied volatility. Note that the function Ψ is decreasing. Second order asymptotics of implied vol, i.e., β 0 in 5, depend additionally on λ, p, and T. It is remarkable that the smile wings are very insensitive to the diffusion volatility σ and the downwards-jump parameter λ, which appear only in th / -term in 5.

5 ASYMPTOTICS FOR JUMP DIFFUSION MODELS 89 As the transfer from call price to implied vol asymptotics is handled by [8] in a mechanical way, the rest of this section is devoted to the proof of Theorem.. The mgf is analytic in the strip Rs λ, λ +. By the exponential decay of Ms, T for Is, the Fourier representation Ck, T = ek πi η+i η i ks Ms, T e ds 6 ss of the call price is valid; see Lee [4]. The real part of the integration contour satisfies < η < λ +. We prove Theorem. by a saddle point analysis of 6. The series representation of the call price from Kou s paper [] seems less amenable to asymptotic analysis. The integrand blows up as s λ +. Identifying the dominating term /λ + s in, we define the approximate saddle point ŝ = ŝk as the solution of which is given by where ξ = λλ + pt. s e ks exp T λ λ +p λ + s ŝ = λ + ξ / k /, = 0, Lemma.3. The cumulant generating function ms, T = log Ms, T of X T satisfies mŝ, T = T σ λ + m ŝ, T = k + O, m ŝ, T = ξ / k 3/ + O, m ŝ + it, T = Ok for t < k, α > 0, where all derivatives are with respect to s. + bλ + T + ξ / k / + T λλ p λ + λ + λt + Ok /, Proof. The estimates follow by straightforward computations from. We now move the integration contour so that it passes through the saddle point ŝ. A small part of the contour, within distance Ok of the saddle point, captures the asymptotics of the full integral. Any exponent α 3, 3 4 is suitable. The integration variable thus becomes s = ŝ + it, t < k. From the estimates for m and m in Lemma.3, we have the local expansion Mŝ + it, T = exp mŝ, T + itk m ŝ, T t + Ot + t 3 k. The rational function /ss is locally constant, to first order: ŝ + itŝ + it = λ + λ + + Ok /. The integral close to the saddle point thus becomes πi ŝ+ik ŝ ik e ks Ms, T ss ds = ek ŝ Mŝ, T πλ + λ + k exp m ŝ, T t + Ok 3α+ dt. 7 k

6 90 S. GERHOLD, J. F. MORGENBESSER AND A. ZRUNEK Setting u := m ŝ, T /, we get from Lemma.3 k 3/4 u = + Ok 3/ and ξ /4 u = ξ/4 + k 3/4 Ok 3/. By substituting ω = ut and using the fact that Gaussian integrals have exponentially decaying tails, we find k exp k m ŝ, T t dt = u uk exp ω uk dω = πk 3/4 ξ /4 + Ok 3/. 8 Use this in 7, and replace Mŝ, T by the estimates in Lemma.3, to get the right hand side of 4. Note that to obtain a relative error k /4, and not just k /4+ε, it suffices to take one further term in the local expansion, keeping in mind the well-known fact that the saddle point method usually yields a full asymptotic expansion. See [7] for a detailed discussion of this issue in a similar analysis. The same remark applies in case of the Merton model below. To prove Theorem., it thus remains to show that the integral over t > k can be dropped, so that 7 asymptotically equals 6. This tail estimate is done in the following lemma. By symmetry, it suffices to treat t > k. Lemma.4. Let 3 < α < 3 4. Then we have πi ŝ+i e ŝ+ik α ks Ms, T ss ds exp k λ + + ξ / k / ξ / k 3/. Proof. Let s = ŝ + it = λ + ξ / k / + it, where t k. Then ξ Ms, T exp R ξ / k / it ξ / k / ξ / k / = exp + t ξ exp k + k ξ. Using the fact / + x x/ for x and that k is smaller than for sufficiently larg, we get Ms, T exp ξ / k / ξ / k 3/. 9 From ss + t and 9, we obtain Ms, T e ks π ss exp k λ + + ξ / k / ξ / k 3/ + t, and thus πi ŝ+i ŝ+ik e ks Ms, T ts ds exp k λ + + ξ / k / ξ / k 3/ k dt + t exp k λ + + ξ / k / ξ / k 3/. The proof of Theorem. is complete. Finally, we mention that tail asymptotics for the distribution of X T for σ = 0 can be found in [], Example 7.5. They can also be deduced from earlier work of Embrechts et al. [5].

7 ASYMPTOTICS FOR JUMP DIFFUSION MODELS 9 3. Merton jump diffusion model. In one of his classical papers, Merton [5] proposed a Lévy jump diffusion with Gaussian jumps as a model for log-returns. If mean and variance of the jump size distribution are µ resp. δ, then the mgf is the entire function Ms, T = exp T σ s + bs + λe δ s /+µs. Benaim and Friz [3] gave first order logarithmic asymptotics for the call price, L := log Ck, T δ k log k, k, 0 and first order asymptotics for implied volatility: V k 3/4 δk/log k /4. Our refined results are best formulated using the implicitly defined saddle point ŝ = ŝk, satisfying m ŝ, T = k. As in the Kou model, we write m = log M for the cumulant generating function. Theorem 3.. For k, the call price in the Merton model satisfies Ck, T = δ ŝ Mŝ, T log k + O πm ŝ, T log k = δ exp k ŝ + T σ ŝ / + bŝ + λe δ log k πt σ + λδ ŝ + µδ ŝ + µ + δ e + O. δ ŝ /+µŝ log k 3 Fig. 3. The solid curve is the implied vol of the Merton model, with parameters T = 0., σ = 0.4, λ = 0., µ = 0.3, and δ = 0.4. The dashed line is the first order approximation, whereas the dotted line is our refined approximation 4. Corollary 3.. The implied volatility of the Merton model satisfies, for k, V k = G k, L 3 k log L + log 4 + O k / log k 5/4 4 π = 3/4 δklog k /4 + c klog k 3/4 k log log k + O, 5 log k 5/4

8 9 S. GERHOLD, J. F. MORGENBESSER AND A. ZRUNEK where L = log Ck, T is the absolute log of the call price, G k, u = u + k u, and c = 9/4 δ / µ + δ 3/4 δ 3/. Theorem 3.3. The density of the Merton log-return X T satisfies, for k, f XT k = e kŝ Mŝ, T πm ŝ, T + Ok / 6 = exp kŝ + T σ /ŝ + bŝ + λe δ /ŝ +µŝ πt σ + λδ ŝ + µ δ ŝ + µ + δ e + Ok / 7 δ ŝ /+µŝ = exp δ k log k + Ok. 8 Formula 3 is just with M replaced by its explicit form, and the same holds for 7 and 6. For numerical accuracy, it is preferable to use 4 and 7, and not the more explicit variants 5 and 8. We begin the proofs by deducing 5 and 8 from the semi-explicit formulas 4 and 7, using asymptotic approximations of ŝ. The saddle point satisfies log k ŝ = δ µ δ + O log log k log k. 9 This estimate follows from the saddle point equation m ŝ, T = k by a tedious, but straightforward application of the classical bootstrapping technique see, e.g., Chapter of [0]. From the saddle point equation, wnow that e δ ŝ /+µŝ = k/t σ ŝ b λδ ŝ + µ λδt k. 0 log k Using these properties in 7 yields 8. For the density of the underlying itself we thus obtain f ST k = f XT log k/k = k /δ log log k e Olog k. The marginal law of the underlying has almost a power law tail, due to the very slow increase of log log k, but it is still asymptotically lighter than that of any power law. The influence of the model parameters seems a bit surprising here: Neither the jump size nor the Poisson intensity appear in the main factor in, but only the jump size variance. To obtain the explicit refined volatility expansion 5, note that, by 3, 9, and 0, we can refine 0 to L = δ k log k µ + δ δ k log log k k + O. log k Since G k, u = ku / 8 k u 3/ + Ok 3 u 5/, using in 4 yields 5. We proceed to the proofs of the first equalities in Theorem 3. and Corollary 3.. The proof of Theorem 3.3 is very similar to that of Theorem 3., using the Fourier representation of the density, and is omitted see [6]. Alternatively, the density asymptotics could be deduced from its series representation, 4. in [4], by the Laplace method.

9 ASYMPTOTICS FOR JUMP DIFFUSION MODELS 93 Formula 4 in Corollary 3. is a special case of Corollary 7. in [8]; the error term follows from 0. Our Theorem 3. is then useful for approximating L in 4 numerically, or symbolically to obtain the explicit expansion 5; see above. Note that our refined call approximation yields the second order term and higher order terms as well in 5, which cannot be deduced from 0. It remains to prove. Again, we appeal to the representation 6, where η >, and shift the integration contour to the saddle point ŝ. For the central range, we let s = ŝ+it, where t < k with α 3,. Lemma 3.4. The cumulant generating function ms, T = log Ms, T satisfies for k, k log log k mŝ, T = δ + O, log k log k where t < k. m ŝ, T = k, m ŝ, T = δk log k + O, log k m ŝ + it, T = k log k + O, log k Proof. As for the Kou model, these expansions follow by a straightforward computation from the explicit mgf. For the second equation, note that we are using the exact saddle point, and not an approximation as we did in the Kou model. Since the saddle point tends to infinity, the rational function /ss locally tends to zero: ŝ + itŝ + it = δ + O. log k log k We have used 9 here. Using this and Lemma 3.4, we see that the integral over the central range has the asymptotics, after handling the Gaussian integral as in 8. To complete the proof, we need to provide a tail estimate, to the effect that the integral outside of s = ŝ + it with t < k is negligible. Again, it suffices to treat the upper portion, by symmetry. Lemma 3.5. Let 3 < α <. Then πi ŝ+i e ks Ms, T ŝ+ik ss ds ek ŝ Mŝ, T e δk / log k log k Proof. We first estimate Ms, T : σ Ms, T = exp R T s + bs + λ e δ s /+µs σ = exp T ŝ t + bŝ + λ cosδ tŝ + µte δ ŝ t /+µŝ σ exp T ŝ t + bŝ + λ e δ ŝ /+µŝ e δ k /. Using e δ k / = δ k / + Ok 4α.

10 94 S. GERHOLD, J. F. MORGENBESSER AND A. ZRUNEK and 0, we get Ms, T e T σ t / Mŝ, T e δk / log k. Since ss log k, this estimate implies πi ŝ+i e ks Ms, T ŝ+ik ss ds ek ŝ Mŝ, T e δk / log k log k ek ŝ Mŝ, T e δk / log k log k e σ t T/ dt k Acknowledgments. This note is based in part on the thesis [6], where some proofs are discussed in greater detail. S. Gerhold gratefully acknowledges financial support from the Austrian Science Fund FWF under grant P 4880-N5.. References [] J. M. P. Albin, M. Sundén, On the asymptotic behaviour of Lévy processes. I. Subexponential and exponential processes, Stochastic Process. Appl , [] S. Benaim, P. Friz, Smile asymptotics II: Models with known moment generating functions, J. Appl. Probab , 6 3. [3] S. Benaim, P. Friz, Regular variation and smile asymptotics, Math. Finance 9 009,. [4] R. Cont, P. Tankov, Financial Modelling with Jump Processes, Chapman Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 004. [5] P. Embrechts, J. L. Jensen, M. Maejima, J. L. Teugels, Approximations for compound Poisson and Pólya processes, Adv. in Appl. Probab , [6] P. Flajolet, S. Gerhold, B. Salvy, Lindelöf representations and non-holonomic sequences, Electron. J. Combin. 7 00, paper R3. [7] P. Friz, S. Gerhold, A. Gulisashvili, S. Sturm, On refined volatility smile expansion in the Heston model, Quant. Finance 0, [8] K. Gao, R. Lee, Asymptotics of implied volatility to arbitrary order, Finance Stoch. 8 04, [9] S. Gerhold, Counting finite languages by total word length, Integers 0, [0] R. L. Graham, M. Grötschel, L. Lovász eds., Handbook of Combinatorics, Vol.,, Elsevier, Amsterdam, 995. [] A. Gulisashvili, Asymptotic formulas with error estimates for call pricing functions and the implied volatility at extreme strikes, SIAM J. Financial Math. 00, [] S. Kou, A jump-diffusion model for option pricing, Management Sci , [3] R. W. Lee, The moment formula for implied volatility at extreme strikes, Math. Finance 4 004, [4] R. W. Lee, Option pricing by transform methods: extensions, unification, and error control, J. Comput. Finance 7 004, no. 3, [5] R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financ. Econ , [6] A. Zrunek, Volatility smile expansions in Lévy models, master s thesis, Vienna Univ. of Technology, December 03;