Option pricing and implied volatilities in a 2-hypergeometric stochastic volatility model

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1 Opion pricing and implied volailiies in a 2-hypergeomeric sochasic volailiy model Nicolas Privaul Qihao She Division of Mahemaical Sciences School of Physical and Mahemaical Sciences Nanyang Technological Universiy 21 Nanyang Link Singapore March 16, 216 Absrac We derive closed-form analyical approximaions in erms of series expansions for opion prices and implied volailiies in a 2-hypergeomeric sochasic volailiy model wih correlaed Brownian moions. As in [4], hese expansions allow us o recover he well-known skew and smile phenomena on implied volailiy surfaces, depending on he values of he correlaion parameer. Key words: Sochasic volailiy; 2-hypergeomeric model; implied volailiy; series expansions. Mahemaics Subjec Classificaion: 91G2; 91B7; 35A35. 1 Inroducion Sochasic volailiy models have been inroduced as realisic models for he moion of asse prices in financial markes. The mos well-known of such models is he Heson [7] model, which however has one major drawback as is sochasic volailiy may reach zero in finie ime unless one imposes he Feller condiion, and his poses poenial nprivaul@nu.edu.sg SHEQ2@e.nu.edu.sg 1

2 problems in model calibraion, cf. e.g of Janek e al. [8] and of Henry- Labordère [6]. In view of his, he α-hypergeomeric sochasic volailiy model has been inroduced by Da Fonseca and Marini [2] o ensure sric posiiviy of volailiy. In he α-hypergeomeric model he dynamics of he asse price S a ime and he volailiy V are governed by ds = S e V dw 1, dv = a c 2 eαv d + ηdw 2, 1 c >, η >, a R, α >, and W 1 and W 2 are correlaed Brownian moions saisfying W 1, W 2 = ρ. In his model he risk free rae r is aken o be equal o and he value of c can be used o se he price of volailiy risk. Sochasic volailiy models generally do no admi explici soluions, and his has moivaed he developmen of approximae expansions. In Fouque e al. [3] a mehod o obain series expansions for European opion prices has been proposed in he Heson model. The firs and second order erms in his expansion do no depend on he value of sochasic volailiy which is a key quaniy in he Heson model, and as a consequence i canno be used o reproduce he smile effec in model calibraion. A more accurae approximaion has been proposed in Han e al. [4] for European opion prices in he Heson model via a series expansion ha involves he underlying sochasic volailiy, allowing he auhors o recover he smile effec and o avoid he secular effec and erminal layer problems posed by he hird erm in he expansion of [3], see also Kim [9] under sochasic ineres raes. In his paper we exend he mehod of [4], see also [1], in order o derive series expansions based on approximaions of he 2-hypergeomeric model of [2]. In paricular, our analyical approximae soluion depends on he underlying sochasic volailiy. We check ha our approximae soluions agree wih Mone Carlo simulaions, including in he case of firs order approximaions. We also derive implied volailiy esimaes which display he well known phenomena of skew and smile. 2

3 2 Sochasic volailiy We sar wih a general class of sochasic volailiy models in which he dynamics of he asse price and volailiy processes are given by ds ε = S ε p, V ε dw 1, dv ε = u, V ε d + εh, V ε dw 2, where ε >. Noe ha by Brownian rescaling, small volailiy coefficiens can be used o derive small ime asympoics, cf. e.g. Secion of [2]. Recall ha under absence of arbirage, he vanilla opion price of an opion wih payoff g ST ε akes he form f, S ε, V ε := E [g S ε T F ] where F [,T ] is he filraion generaed by W 1, W 2 [,T ], and he funcion f, x, v solves he PDE f + u, v f v + x2 2 p2, v f x + ερxp, vh, v 2 f 2 x v + ε2 2 h2, v 2 f =, 2 v2 cf. e.g in [3], wih he erminal condiion ft, x, v = gx. We sar by expanding f, x, v as f, x, v = f, x, v + εf 1, x, v + oε. 3 By plugging in he expansion 3 ino he pricing PDE 2 we ge he sysem of equaions f n + L f n + L 1 f n 1 + L 2 f n 2 =, n N, wih f n =, n 1, f T, x, v = gx and f n T, x, v =, n 1. In paricular he operaors L, L 1 and L 2 are given by L = u, v v + x2 2 p2, v 2 x 2, L 1 = ρxp, vh, v 2 x v, L 2 = 1 2 h2, v 2 v Deerminisic volailiy When n = we have f + L f =, S [,T ] and V ds = S p, V dw 1, dv = u, V 3 [,T ] are given by d

4 and he vanilla opion price f, S, V := E [g S T F ] can be compued by he Black-Scholes formula as [ f, S S, V = E T K ] + F = E [ S exp Zγ, V 1 ] + 2 γ2, V K F, where Z N, 1 is independen of F and γ T 2, V := p 2 u, Vu du, [, T ]. We noe ha in he α-hypergeomeric model 1 wih η = he inegral T e αv u du can be compued in closed form as T e αv 2 u du = αc log 1 + αc 2 eαv T e αas ds = 2αc log 1 + αc 2 eαv cf of [2], and his yields he following proposiion. e αat 1 αa Proposiion 1. In he 2-hypergeomeric model 1 wih η = he European call price [ f, S S, V = E T K ] + F under he erminal condiion f T, x, v = x K + is given by f, x, v = xφ d +, x, v KΦ d, x, v, where Φ is he sandard Gaussian cumulaive disribuion funcion, d ±, x, v = 1 x log ± γ2, v, and γ 2, v = 1 γ, v K 2 c log 1 + ce 2v e2at 1. 2a 5 In he case of a pu opion he funcion f, x, v can be obained as f, x, v = xφ d +, x, v + KΦ d, x, v, [, T ], by a sandard call-pu pariy argumen. In he remainder of his paper we work in he 2-hypergeomeric model wih α = 2., 4

5 4 Firs order expansion In his secion we consider small values of he volailiy of volailiy by replacing η in 1 wih εηe V ε, ε >, i.e. we have ds ε = S ε e V ε dw 1, dv ε = a c 2 e2v ε and from 4 he operaors L, L 1 and L 2 are given by d + εηe V ε dw 2, 6 L = a c 2 e2v v + x2 2 e2v 2 x, L 2 1 = ηρxe 2v 2 x v, L 2 = η2 2 e2v 2 v. 2 In paricular when n = 1 we ge f 1 + L f 1 + L 1 f =, wih f 1 T, x, v =. Noe ha our approximaion S ε, V ε [,T ] does no lie wihin he class of 2-hypergeomeric models. Proposiion 2. The soluion of f 1 + L f 1 + L 1 f = under he erminal condiion f 1 T, x, v = is given by f 1, x, v = ρk η,v c d, x, vφ d, x, v e cγ2 + cγ 2, v 1, [, T ], cγ 2, v where φx is he sandard Gaussian probabiliy densiy funcion and γ, v is defined in 5. Proof. From he relaion φ d +, x, v = 1 exp 12 d +, x, v 2 2π = K x φ d, x, v and using he Feynman-Kac formula wih locally Lipschiz coefficiens as in e.g. Theorem 1 of Heah and Schweizer [5], we have T f 1, S, V = T = η ρke 2V r γ r, Vr = ηρk d, S, V γ 2, V E [ ] L 1 f r, S r, Vr F dr γ [ r, V v r E d r, S r, Vr ] φ d r, S r, V F dr φ T d, S, V e 2V r γ r, V γ r r, V v r dr, r 5

6 by a sandard compuaion based on he Gaussian disribuion d r, S r, Vr 1 S N log γ2, V, γ2, V γ r, Vr K 2 γ 2 r, Vr 1, r [, T ], given F. Finally we noe ha from 5 we have T e 2V r γ r, V γ r r, V 1 v r dr = c 5 Second order expansion T e 2V r 1 e cγ2 r,v r dr = 1 e cγ2,v c 2 + cγ 2, V 1. 7 The compuaion of a second order correcion erm f 2, x, v requires us o replace η in 1 wih εηe V γ 4, V in order o involve only even powers of γr, v when exending he compuaion of 7 above. In his case, L 1 and L 2 are replaced by he operaors L 1 = ηρxe 2v γ 4, v 2 x v, and we look for an expansion of he form L2 = η2 2 e2v γ 8, v 2 v 2, f, x, v = f, x, v + ε f 1, x, v + ε 2 f2, x, v + oε 2, 8 where f T, x, v = x K +, f 1 T, x, v =, f 2 T, x, v =, and f + L f =, f 1 + L f 1 + L 1 f =, f 2 + L f 2 + L 1 f1 + L 2 f =, Proposiion 3. The firs and second order coefficiens appearing in he expansion 8 are given by f 1, x, v = ηρk d, x, v c 3 γ 2, v φ d, x, v e cγ2,v c 2 γ 4, v + c 2 γ 2, v c3 3 γ6, v, f 2, x, v = η2 c Kφ d A 3, v, x, v γ, v + d, x, vb 3, v + d, x, v 2 B 3, v γ, v + η 2 ρ 2 Kφ d, x, v C 3, v 2D, v d, x, v 4 + d 3c 7 γ 4, x, v + d, x, v 3, v 3γ, v 3 where he funcions A i, B i, C i, D, E i are given below for i = 1, 2, d, x, v 2 E γ 5 3, v, v, [, T ],

7 Proof. The expression of f1 is obained by he same argumen as in he proof of Proposiion 2. For f 2 we have f 2 + L f 2 + L 1 f1 + L 2 f = wih f 2 T, x, v =, hence f 2 can be compued by similar argumens from he Feynman-Kac formula and he expeced value f 2, S, V = T [ E L1 f1 r, S r, V r + L ] 2 f r, S r, V r F dr. For simpliciy of exposiion we skip he corresponding compuaions, which are significanly longer han in he proof of Proposiion 2. We have A 1, v = γ8, v 2c A 2, v = γ8, v c A 3, v = γ8, v 1c + 5γ6, v + 2γ4, v + 9γ2, v + 3 4c 2 c 3 4c 4 2c + 3c 6 5 8γ 2, v, 5γ6, v c 2 16γ4, v c 3 24 c 4 γ2, v 48 c 5 24c 6 γ 2, v, + e 2cγ2,v A 1, v + A 2, ve cγ2,v c c 6 8γ 2, v, B 1, v = γ6, v γ4, v 3γ2, v 3 4c 2 2c 3 4c 4 4c 3c 6 5 8γ 2, v, B 2, v = γ6, v c 2 + 4γ4, v c γ2, v c c c 6 γ 2, v, B 3, v = γ8, v 1c + e 2cγ2,v B 1, v + B 2, ve cγ2,v C 1, v = γ5, v c 2 + γ3, v 2c 3 + 3γ2, v c c 6 γ 3, v + 9c 7 4γ 4, v + 15c 7 4γ 5, v, C 2, v = 3γ5, v c 2 9γ3, v c 3 6γ2, v c c 6 γ 3, v 36c 7 γ 4, v + 24c 7 γ 5, v, C 3, v = 7γ7, v 3c 2γ, v c 4 4γ, v c 4 + 9c c 6 γ 2, v, + 21c 5 2γ, v + 9c 6 2γ 2, v 64γ, v c 4 36 c 5 12c 5 γ, v 36c 6 γ 2, v + C 1, ve 2cγ2,v + C 2, ve cγ2,v + 189c 6 2γ 3, v + 135c 7 4γ 4, v 111c 7 4γ 5, v, D, v = e 2cγ2,v e cγ2,v c 3 γ 6, v 3 c + 3cγ 2, v 2 γ2, v ,

8 E 1, v = γ1, v c 2 + γ8, v c 3 15γ6, v c 4 27γ4, v c 5 51γ2, v 2c 6 12 c 7, E 2, v = 2γ1, v c 2 + 2γ8, v c γ6, v c γ4, v c γ2, v c c 7, E 3, v = γ12, v 15c + 4γ6, v c 4 189γ2, v 2c 6 + E 1, ve 2cγ2,v + e cγ2,v E 2, v 492 c 7. Noe ha in he case of pu opions, only he funcion f, x, v is modified by he sandard call-pu pariy argumen, while higher order erms such as f 1, x, v, f 1, x, v and f 2, x, v remain unchanged. In Figure 1 we plo he opion price agains he sochasic variance v wih correlaion ρ =.5 and parameers x = K, T =.1, =, a = c/2 = 1, η = 2 and ε =.1. The Mone Carlo curve required 3, samples based on 3, ime seps. Opion Prices Mone Carlo f f + ε f ~ 1 f + ε f ~ 1 + ε 2 f ~ Figure 1: Opion price f ploed agains v wih ρ =.5. v Figure 1 shows ha our asympoic soluions are in agreemen wih he Mone Carlo soluion, even if we only use f of our analyical approximae soluions when v is in he inerval [, 4], while he Mone Carlo esimae hovers around he analyical approximae soluions for larger values of v. For larger v, furher increasing he numbers of simulaions and ime seps would yield a smooher Mone Carlo graph. In he nex Table 1 we presen he approximaed values obained from f, f + ɛ f 1, f + ɛ f 1 + ɛ 2 f2, wih he parameers S = K, T =.1, =, a = c/2 = 1, η = 2, ɛ =.1 and ρ =.5. The corresponding Mone Carlo esimaes required 1,, samples based on 1, ime seps, while he evaluaion of he approximaions is insananeous. The large number of ime seps is due o insabiliies in he soluion of sochasic differenial equaions SDEs wih non-lipschiz here exponenial 8

9 coefficiens such as 6. v f f + ɛ f 1 f + ɛ f 1 + ɛ 2 f2 Mone Carlo Table 1: Values of f, f + ɛ f 1, f + ɛ f 1 + ɛ 2 f2 compared o Mone Carlo esimaes. We check ha our approximae soluions using up o he second correcion erms gives values closer o he Mone Carlo esimaes. 6 Implied volailiy In his secion we provide an esimaion of he implied volailiy. σ imp which is deermined by he equaion f BS, x, T, K, σ imp = f, x, v, where f BS, x, T, K, σ imp is he classical Black-Scholes funcion, cf. e.g. Da Fonseca and Grasselli [1] in muli-facor models. Theorem 4. The implied volailiy σ imp admis he series expansion σ imp, x, v = σ, x, v + εσ 1, x, v + ε 2 σ 2, x, v + oε 2, where σ, x, v := γ, v/ T, and σ 2, x, v := σ 1, x, v := f 1, x, v K T φ d, x, v, f 2, x, v K T φ d, x, v d +, x, vd, x, v σ2 1, x, v 2σ, x, v. 9

10 Proof. The implied volailiy σ imp is deermined by equaing f BS, x, T, K, σ imp = f, x, v = f, x, v + ε f 1, x, v + ε 2 f2, x, v + oε 2, where f BS is he classical Black-Scholes funcion wih implied volailiy σ imp. Expressing he implied volailiy as a power series σ imp, x, v = σ, x, v + εσ 1, x, v + ε 2 σ 2, x, v + oε 2 in ε, we expand f BS, x, T, K, σ imp and using a Taylor expansion in erms of ε o obain f BS, x, T, K, σ imp = f BS, x, T, K, σ, x, v + εσ 1, x, v + ε 2 σ 2, x, v BS f, x, T, K, σ, x, v σ ε2 σ1, 2 x, v 2 BS f, x, T, K, σ σ 2, x, v + The firs hree erms of he implied volailiy expansion are obained by idenificaion of coefficiens in he above expressions. In Figure 2 we plo he esimaion of implied volailiy agains he raio moneyness K/x of he srike price o he asse price, wih he parameers T = 1, =, a = c/2 = v = 1, η = 3.5 and ε =.1. σ imp σ σ + εσ 1 σ + εσ 1 + ε 2 σ 2 σ imp σ σ + εσ 1 σ + εσ 1 + ε 2 σ k/s k/s a ρ =.8 b ρ = Figure 2: Implied volailiy σ imp ploed agains he moneyness K/x. Our implied volailiy esimae σ, x, v + εσ 1, x, v + ε 2 σ 2, x, v exhibis he wellknown skew and smile phenomena. In addiion hey show ha i can be necessary o 1

11 ake ino accoun he correcion erms f 1 and f 2 for improved calibraion. Mone Carlo esimaes of volailiy are no available due o he insabiliies observed in Figure 1 for he numerical soluion of SDEs such as 6. References [1] J. Da Fonseca and M. Grasselli. Riding on he smiles. Quan. Finance, 1111: , 211. [2] J. Da Fonseca and C. Marini. The α-hypergeomeric sochasic volailiy model. Preprin arxiv: , 214. [3] J.P. Fouque, G. Papanicolaou, K.R. Sircar, and K. Sølna. Muliscale Sochasic Volailiy for Equiy, Ineres Rae Derivaives, and Credi Derivaives. Cambridge Universiy Press, Cambridge, 211. [4] J. Han, M. Gao, Q. Zhang, and Y. Li. Opion prices under sochasic volailiy. Appl. Mah. Le., 261:1 4, 213. [5] D. Heah and M. Schweizer. Maringales versus PDEs in finance: an equivalence resul wih examples. J. Appl. Probab., 374: , 2. [6] P. Henry-Labordère. Analysis, Geomery, and Modeling in Finance. Chapman & Hall/CRC Financial Mahemaics Series. CRC Press, Boca Raon, FL, 29. [7] S.L. Heson. A closed-form soluion for opions wih sochasic volailiy wih applicaions o bond and currency opions. The Review of Financial Sudies, 62: , [8] A. Janek, T. Kluge, R. Weron, and U. Wysup. FX smile in he Heson model. SFB 649 Discussion Paper 47, 21. [9] Y.J. Kim. Opion pricing under sochasic ineres rae: An empirical invesigaion. Asia-Pacific Financial Markes, 9:23 44, 22. [1] Q. Zhang, J. Han, and M. Gao. Opion price wih sochasic volailiy for boh fas and slow mean-revering regimes. C. R. Mah. Acad. Sci. Paris, : ,