Finance PRICING EXOTIC OPTION UNDER STOCHASTIC VOLATILITY MODEL Pengshi Li
Introduction Scholes model, the assumption of constant Exotic options are called “customer tailored volatility in the Black-Scholes model contradicts options” or “special purpose option” because to the existence of the non-fl at implied volatility they are fl exible to be tailored to the specifi c surface observed in empirical studies. Bernales needs of investors. Strategies based on and Guidolin (2015) prove the existence of exotic options are often employed to hedge predictability patterns in the movement of the specifi c risk exposures from the fi nancial implied volatility surface. Various methods markets. Because exotic options are more are hence proposed to remedy the unrealistic effi cient and less expensive than their standard constant volatility assumption, one of them is counterparts, they are playing a signifi cant the stochastic volatility (SV) model which is hedging role in cost effective ways. Unlike suitable for the medium and long term maturity the vanilla call and put options, exotic options options. are either variation on the payoff patterns of A typical stochastic volatility model plain vanilla options or they are totally different assumes the volatility is driven by a mean- kinds of derivatives with other features. While reverting diffusion process. The motivation of simple vanilla call and put options are traded this assumption is that empirical researches in the exchanges, most of the exotic options are fi nd evidences of mean-reverting future, which traded in the over-the-counter markets. was not taken into account in the traditional Supershare option and chooser option option pricing model, in many fi nancial assets are two typical kinds of exotic options which such as stock price; see Alizadeh et al. (2002). suggest a broad range of usage and application Another motivation of the stochastic volatility in different fi nancial market conditions. assumption is that by considering an instant Supershare option is one type of binary correlation between the standard Brownian option. Unlike the binary option which has only motions in the process driving the underlying one boundary, a supershare option has an asset price and the process driving upper and lower boundary. If the underlying the volatility, the stochastic volatility model can asset price is between these boundaries, explain the leverage effect observed in fi nancial the payoff of the supershare option is the ratio markets. Smith (2015) fi nds the leverage of the underlying asset price and the lower effects in the U.S. stock markets. The leverage boundary, otherwise the supershare option is effect suggests volatility shocks are negatively worthless. Chooser option gives the holder correlated with underlying asset price shock, the right to choose at some future date before it also accounts for a skewed distribution for maturity whether the option will fi nally be a put the asset price observed by Carmichael and or a call with a common exercise price. Chooser Coen (2013). options are suitable for investors who are Studies on option pricing under stochastic certain about a strong volatility of the underlying volatility model had been carried out for years. asset price but uncertain about the direction of Heston (1993) obtained a closed-form formula the change. for options on bonds and currency in terms of Exact analytical formulae for supershare characteristic functions; Fouque et al. (2003) option and chooser option can be derived performed singular perturbation analysis on through the Black-Scholes world. Despite the fast-mean reverting stochastic volatility the popularity and longevity of the Black- model. And the accuracy of the corresponding
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EEM_4_2019.inddM_4_2019.indd 134134 225.11.20195.11.2019 11:02:3311:02:33 Finance approximation in the presence of the 1. Research Methodology nonsmoothness of payoff functions (such as Denoting St as the price of the underlying call option) is also studied; Their asymptotic asset; the driving volatility Yt follows an O-U option pricing method has been applied to process under real world probability measure the research of turbo warrants pricing by P. The stochastic volatility model considered Wong et al. (2008); Park and Kim (2013) in this section is described by fi ve parameters investigate a semi-analytic pricing method which are: m the long run mean level of for lookback options in a general stochastic the driving volatility Yt ; α the mean-reverting rate volatility framework; Yang et al. (2014) derive of Yt ; ν the standard deviation of the invariant an analytic approximation formula for the price distribution of Yt ; ρ the instant correlation of the vulnerable options when the underlying captures the skewness, and γ the market price asset follows a stochastic volatility; Zhu and of the volatility risk. Assumes that the risk-free Lian (2015) work out analytical closed-form interest rate r is constant, the market selects formulae for the price of forward-start variance a constant market price of volatility risk and swap based on the Heston stochastic volatility price derivatives under a risk-neutral world model; Cao et al. (2016) investigate the pricing probability measure P*, then the stochastic problem of variance swaps under stochastic volatility model under such probability measure volatility and stochastic interest rate; Lee et al. can be written as: (2016) obtain a closed-form analytic formula of the price of the European vulnerable option dS rS dt f() Y S dW * (1) assuming that the underlying asset price follows tt ttt the Heston dynamics. Although many studies are devoted to option pricing under stochastic dY()2() m Y v Y dt tt t volatility model in recent years, to the best of (2) our knowledge, research on exotic option such *2* vdWdZ2(tt 1 ) as supershare and chooser option pricing have not been carried out in the stochastic volatility case. () r 2 ()y 1 (3) By extending the constant volatility in Black- fy() Scholes model, this paper studies the pricing of the supershare option and chooser option in a fast mean-reverting stochastic volatility where f(y) is assumed to be a positive, bounded * * scenario. This work may contribute to the exotic and suffi ciently regular function. Wt and Zt are option pricing literature in several ways. two independent standard Brownian motions Supershare and chooser options are both under risk-neutral world probability measure important fi nancial instruments, research on P*. Y has a invariant normal distribution with these two exotic options in stochastic volatility mean m and variance ν2. Λ(y) is defi ned as model may give more insights on the pricing of the combined market price of risk. supershare and chooser options. Further, this Consider a European exotic option with an paper presents the semi-analytical formulae expiry date T and payoff g(s) at expiry date. of the supershare and chooser option. These Since under the risk-neutral world probability * formulae are critical to the determination of measure P , the processes (St,Yt) is a Markov option delta and hedge ratio of the replicate process, the price of this European exotic option portfolio which is important to investors who at time t < T is a function of the present value use these exotic options to hedge their risk of the underlying asset price and the present exposure. value of the process driving the volatility. Then The rest of this paper is organized as the price of the European exotic option, which follows. Section 1 formulate the model setting denote as V(t,s,y), at time t is given by: of the underlying stochastic volatility model; Section 2 derives the zero-order and fi rst-order Vtsy(, ,) E*()egSrT t () | T approximation of the supershare option and (4) chooser option; A numerical result of these two |,SsYytt exotic options are carried out in Section 3; And the fi nal section concludes this paper.
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It can also be obtained as the solution of volatility economy can be approximated by the following partial differential equation by the combination of the zero-order approximation Feyman-Kac formula: and the fi rst-order approximation:
VV1 2 V 1 rs()() V f22 ys VV V O() (10) 2 01 ts2 s 1 VV2 [(my ) v2 ] (5) yy2 2. Analysis 1 2VV 2.1 European Supershare Option [2vfysv () 2()]0 y The payoff of a supershare option at expiry date sy y is defi ned as:
VTsy(,,) gs () (6) S SS T 1 (11) TKSKK {}12T where ε is a small parameter and we defi ne 1 α = 1 / ε. When the mean-reverting rate α is large, for positive constants K1 < K2. In the fast i.e. 0 < ε ≤ 1. The price of the European exotic mean-reverting stochastic volatility economy, option satisfying a problem of and can be the approximate price of the supershare option solved in asymptotic expansions which was is . The fi rst-order s suggested by Fouque et al. (2003). The main approximation Vts0 (, ) is simply the standard results of their research are presented as follow. Black-Scholes option price with a constant Result 1 The zero-order approximation volatility Ϭ. Lemma 1 The price of the supershare V0(t,s) for the price of the European option in the fast-mean reverting stochastic volatility option in the Black-Scholes model with constant economy is independent of the present volatility volatility is given by: and it is the standard Black-Scholes price at a constant volatility Ϭ defi ned as below: St SSt Nd()12 Nd () 2 K1 VVV1 2 2 000 srsrV 2 2 0 tss2 (7) lnsK / ( r )( T t ) 1 2 VTs0 ( , ) gs ( ) d ; 1 Tt ()ym 2 2 1 2 22 2v (8) fy() e dy lnsK /2 ( r )( T t ) 2 d 2 2 Tt Result 2 The fi rst-order approximation
Vts1(, ) for the price of the European option where K1, K2, r, Ϭ are constants. St is in the fast-mean reverting stochastic volatility the underlying asset price at time t, T is economy is independent of the present volatility the expiry date. N(x) denotes the cumulative and it is explicitly given as below: distribution function of a standard normal probability density function. 2V 2 0 Proof of Lemma1: Vts11(, ) ( T t )( Cs 2 s The payoff of a supershare option is defi ned as: 3 (9) 3 V0 Cs2 3 ) S s SS T 1 TKSKK {}12T where C1 and C2 are two parameters which 1 can be calibrated from the observed implied
volatility surface. for positive constants K1 < K2 . According to Result 3 The price of the European the risk-neutral formula the price of a supershare option in the fast-mean reverting stochastic option at time t is a function f (t,s) such that:
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fs(,t SSS ) E*()eSrT t | the property of the Brownian motion, we may t Tt write the formula above as: 2 * SSexp ZTtr ( )( Tt ) where Wt is a standard Brownian motion Tt under the risk-neutral measure. Because of 2
where Z is a standard normal random variable: WW** Z Tt Tt Therefore:
s 2 fts(, )E*()erT t exp Z T t ( r )( T t ) 1 {}KS12T K K1 2 2 z2 s 1 etrTtedzrT()t exp zT ( )( ) 1 2 {K12ST K } K1 2 2 2 Since , then the integrand KS121T K Ksexp zTtr ( )( Tt ) K 2 2 is nonzero if and only if:
22 lnsK /21 ( r )( T t ) ln sK / ( r )( T t ) dzd22 KK2 Tt1 Tt
Therefore:
22 dK sz fts( , )1 exp( zTt ( Ttdz )) d K 2 K1 2 22
dK s 1 =1 exp( (zTtdzTt )2 ) ( ) d K 2 K1 2 2 t2 ssdTtK 1 =1 edt2 Nd ( ) Nd ( ) dTt 12 K 2 KK112 2 lnsK /1 ( r )( T t ) dd Tt 2 1 K1 Tt 2 lnsK /2 ( r )( T t ) dd Tt 2 2 K2 Tt
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thus the price of a supershare option at time t 2 is equal to: lnsK / ( r )( T t ) 1 2 d12 ; St Tt SStt f(, t S ) N ( d12 ) N ( d ) K1 where N(x) denotes the standard normal cumulative distribution function. Denotes n(x) Hence according to Result 1 in Chapter 1 as the standard normal probability density and Lemma1, V s(t,s) is given by: 0 function, we can easily obtain that N'(x) = n(x) and N'(x) = –xn(x). From the defi nition of d s s 1 Vts(, ) Nd ( ) Nd ( ) dd 1 012(12) 12 K1 and d2, we have sssTt . Hence, the second order and third order derivative of 2 the value of the zero-order approximation with lnsK /1 ( r )( T t ) respect to the underlying asset price are given d 2 ; by: 1 Tt
2 s V0 11 22nd()12 nd () ( d )()()() nd 122 d nd ssKTtsK1 T t 1 ()
3V s 1 0 nd()( d22 ( T t )1)()( nd d 22 ( T t )1) 323 3/211 2 2 ssKTt1 ()
s According to Result 2 in section 1, the solution of the fi rst-order approximation Vts1 (, ) is explicitly given by the following:
23VVss Vtss (, ) ( TtCs )(2300 Cs ) 112ss23 Cs Cs 11nd() nd () ( d )()()() nd d nd 122 122 (13) K1 KTt1 Cs 2 nd()( d22 ( T t )1)()( nd d 22 ( T t )1) KTt 3 () 11 2 2 1
Combining the result of (12) and (13) we obtain the approximate price for the price of the supershare option under fast mean-reverting stochastic model.
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2.2 European Chooser Option Where K, r, ϭ, T0 < T are constants. St is A chooser option is a contract in which the holder the underlying asset price at time t, T is of this contract has the right to choose at some the expiry date. future date whether the option is to be a put T0 Proof of Lemma2: or a call option with the same strike price K, and The payoff at T0 of a chooser option is: the expiration time T > T0. The payoff at T0 of a chooser option is: CHmax c ( T T , S , K ), p ( T T , S , K ) TTT00000 CHmax c ( T T , S , K ), TT00 0 (14) According to the call-put parity, a put option at time T can be written as: pT(,,) T S K 0 0 T0 pT(,,)(,,) T S K cT T S K 00TT00 c(τ,s,K) and p(τ,s,K) are the vanilla call option SKerT() T0 and put option respectively, these functions are T0 given by the following formula:
thus the payoff of a chooser option at time T0 r can be rewritten as: c(,, sK ) sNd (12 ) Ke Nd ( ) r psKKeNd(,, ) (21 ) sNd ( ) CHmax c ( T T , S , K ), c ( T T , S , K ) TTT000 00 1 2 ln(sK / ) ( r ) SKerT() T0 T0 d 2 1,2 The payoff above is equivalent to: where r, ϭ, K are constants, τ = T – t is time to CH c(,,)( T T S K KerT() T0 S ) expiry. TT000 T 0 In the fast mean-reverting stochastic volatility economy, the approximate price of which implies the standard chooser option is the chooser option is: equivalent to a portfolio composed of a long call option with strike K, expiration time T and a long put with strike KerT() T0 , expiration time T . So the arbitrage price of a standard chooser c 0 The fi rst-order approximation is V0 (t,s) option at time tT0, 0 is: simply the standard Black-Scholes option price
with a constant volatility . chtS(,tt ) CH cT ( tS , t , K ) Lemma 2 The price of the chooser option pT(,, tS KerT() T0 ) in the Black-Scholes model with constant 0 t volatility is given by: Using the vanilla call and put option formula, c * we have: Vts011(, ) sNd ( ) N ( d ) rT() t rT() t * ch(, t s ) sN ( d12 ) Ke N ( d ) Ke N() d22 N () d KerT()() T00 e rT t N() d** sN () d 1 21 ln(sK / ) ( r 2 )( T t ) *()*rT t sNd()()11 N d Ke N () d 22 Nd () d 2 ; 1,2 Tt where 1 2 ln(sK / ) ( r )( T0 t ) 1 d * 2 ln(sK / ) ( r 2 )( T t ) 1,2 2 Tt0 d1,2 Tt and
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Combining the result of (15) and (16) we 1 2 ln(sK / ) ( r )( T0 t ) obtain the approximate price * 2 d1,2 for the price of the chooser option under fast Tt 0 mean-reverting stochastic model.
Hence according to Result 1 in section 2 c 3. Results and Discussion and lemma 2, V (t,s) is given by: 0 From the result of Section 2, an approximation for the price of supershare option and chooser Vtsc (, ) sNd ( ) N ( d* ) 011 option under fast mean-reverting stochastic (15) volatility economy has been formally obtained, KerT() t N() d * N () d 22 and the parameters in the stochastic volatility model are reduced to C1 and C2. These two parameters can be calibrated from implied 1 2 ln(sK / ) ( r )( T t ) volatility surface. In order to illustrate with 2 a numerical example, we consider a supershare d1,2 ; Tt option with the strike prices of K = 400 and 1 K2 = 500; we also consider a chooser option 1 2 with strike price K = 460 and T = 0.5. ln(sK / ) ( r )( T0 t ) 0 d * 2 Following parameters are chosen throughout 1,2 Tt the numerical computation: the (constant) 0 effective volatility Ϭ = 0.1; the time to maturity ; the risk-free interest rate * T – t = 1 r = 2%; From the defi nition of d1,2 and d 1,2, we * the underlying asset price at time 0 is chosen as d 1 d1,2 1 1,2 s [400, 500]; two calibrated parameters are s sTt s sTt 0 have and . Hence, C1 = –0.0044 and C2 = 0.000154 respectively. the second order and third order derivative of The numerical result of the zero-order the value of the zero-orde r approximation with approximation (B-S price) and approximation respect to the underlying asset price are given by: price of the supershare option price at time 0 is shown in Fig. 1; Fig. 2 shows the ratio of the fi rst- 2 c * order approximation and the approximation V0 nd()11 nd ( ) 2 price of the supershare option. s sTtsTt 0 From the numerical results of the supershare 3V c nd() d option, one can observe that ratio of the fi rst- 0 111 order approximation and the approximation 3 2 s sTtTt price is always negative, hence the supershare option under the fast mean-reverting stochastic nd()** d 111 volatility is underpriced compared to its B-S 2 price which can be observed in Fig. 1; Fig. 2 sTtTt00 shows the ratio of the fi rst-order approximation and the approximation price of the supershare According to the Results 2 in Section 1, option. the solution of the fi rst-order approximation The numerical result of the zero-order c is explicitly given by the following: Vts1 (, ) approximation (B-S price) and approximation price of the chooser option price at time 0 is 23cc c 23VV00 shown in Fig. 3; Fig. 4 shows the ratio of the fi rst- Vts112(, ) ( TtCs )(23 Cs ) ss order approximation and the approximation * price of the chooser option. Cs nd() nd ( ) 11 1 From the numerical results of the chooser Tt T t 0 (16) option, one can observe that ratio of the fi rst- order approximation and the approximation nd()11 d ()Tt 1 price is always positive, hence the price of CsTt Tt 2 the chooser option under the fast mean- ** nd() d 111 reverting stochastic volatility is higher than its Tt Tt B-S price which can be observed in Fig. 3. 00
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The zero-order approximation price (solid) and the stochastic volatility Fig. 1: approximation price (dashed) for supershare options
Source: own
The ratio of the fi rst-order approximation and the approximation price Fig. 2: for supershare options
Source: own
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The zero-order approximation price (solid) and the stochastic volatility Fig. 3: approximation price (dashed) for chooser options
Source: own
The ratio of the fi rst-order approximation and the approximation price Fig. 4: for chooser options
Source: own
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EEM_4_2019.inddM_4_2019.indd 142142 225.11.20195.11.2019 11:02:3511:02:35 Finance Conclusions Carmichael, B., & Coen, A. (2013). Asset In this paper, we derive the zero-order pricing with skewed-normal return. Finance approximation term and the fi rst-order Research Letters, 10(2), 50-57. http://dx.doi. approximation term for the suppershare org/10.1016/j.frl.2013.01.001. option and chooser option under a fast mean- Fouque, J. P., Papanicolaou, G., Sircar, reverting stochastic volatility scenario. By R., & Solna, K. (2003). Singular perturbations deriving the approximation term, we obtained in option pricing. SIAM Journal on Applied the analytic approximation formulae for these Mathematics, 63(5), 1648-1665. http://dx.doi. two European exotic option under a stochastic org/10.1137/S0036139902401550. volatility model. The numerical analysis in Heston, S. L. (1993). A closed-form our model shows that stochastic volatility of solution for options with stochastic volatility underlying asset underprices the supershare with applications to bond and currency options. options, while in the case of the chooser options The Review of Financial Studies, 6(2), 327-343. its price in stochastic volatility model is higher http://dx.doi.org/10.1093/rfs/6.2.327. than the price in the constant volatility model. Lee, M. K., Yang, S. J., & Kim, J. H. (2016). Further studies will be carried out on other type A closed form solution for vulnerable options of exotic options and calibrating the parameters with Heston’s stochastic volatility. Chaos, from the implied volatility surface will be our Solitons & Fractals, 86(1), 23-27. http://dx.doi. subsequent research. org/10.1016/j.chaos.2016.01.026. Park, S. H., & Kim, J. H. (2013). A semi- Guangdong Provincial Humanities analytic pricing formula for lookback options under and Social Science Platform-Pearl River a general stochastic volatility model. Statistics Delta Industrial Ecology Research Center & Probability Letters, 83(11), 2537-2543. http:// (2016WZJD005). dx.doi.org/10.1016/j.spl.2013.08.002. Guangdong Social Science Research Smith, G. P. (2015). New Evidence on Base – Quality and Brand Development and Sources of Leverage Effects in Individual Research Center of Dongguan University of Stocks. Financial Review, 50(3), 331-340. Technology (GB200101). http://dx.doi.org/10.1111/fi re.12069. "Supported by Research start-up funds of Wong, H. Y., & Chan, C. M. (2008). Turbo DGUT (GC300501-148)". warrants under stochastic volatility. Quantitative Finance, 8(7), 739-751. http://dx.doi. References org/10.1080/14697680701691469. Alizaden, S., Brandt, M., & Diebold, F. Yang, S. J., Lee, M. K., & Kim, J. H. (2014). (2002). Range-based estimation of stochastic Pricing vulnerable options under a stochastic volatility models. Journal of Finance, 57(3), volatility model. Applied Mathematics Letters, 34(1), 1047-1091. https://doi.org/10.1111/1540- 7-12. http://dx.doi.org/10.1016/j.aml.2014.03.007. 6261.00454. Zhu, S. P., & Lian, G. H. (2015). Bernales, A., & Guidolin, M. (2015). Pricing forward-start variance swaps with Learning to smile: Can rational learning stochastic volatility. Applied Mathematics and explain predictable dynamics in the implied Computation, 250(1), 920-933. http://dx.doi. volatility surface. Journal of Financial org/10.1016/j.amc.2014.10.050. Markets, 26(1), 1-37. https://doi.org/10.1016/j. fi nmar.2015.10.002. Pengshi Li, Ph.D. Cao, J. L., Lian, G. H., & Roslan, T. (2016). Dong Guan University of Technology Pricing variance swaps under stochastic School of Economics and Management volatility and stochastic interest rate. Applied Department of Accounting and Finance Mathematics and Computation, 277(20), 72-81. China http://dx.doi.org/10.1016/j.amc.2015.12.027. [email protected]
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Abstract
PRICING EXOTIC OPTION UNDER STOCHASTIC VOLATILITY MODEL Pengshi Li
This paper studies supershare and chooser options in a stochastic volatility economy. These two options are typical exotic options which suggest a broad range of usage and application in different fi nancial market conditions. Despite the popularity and longevity of the Black-Scholes model, the assumption of constant volatility in the Black-Scholes model contradicts to the existence of the non-fl at implied volatility surface observed in empirical studies. Although many studies are devoted to option pricing under stochastic volatility model in recent years, to the best of our knowledge, research on exotic option such as supershare and chooser option pricing have not been carried out in the stochastic volatility case. Supershare and chooser options are both important fi nancial instruments, research on these two exotic options in stochastic volatility model may give more insights on the pricing of supershare and chooser options. By extending the constant volatility in the Black-Scholes model, this paper studies the pricing problem of the supershare option and chooser options in a fast mean-reverting stochastic volatility scenario. Analytic approximation formulae for these two exotic options in fast mean-reverting stochastic volatility model are derived according to the method of asymptotic expansion which shows the approximation option price can be expressed as the combination of the zero-order and fi rst-order approximations. By incorporating the stochastic volatility effect, the numerical analysis in our model shows that stochastic volatility of underlying asset underprices the supershare options, while in the case of the chooser options its price in stochastic volatility model is higher than the price in the constant volatility model.
Keywords: Exotic options, supershare, chooser, stochastic volatility, mean-reverting.
JEL Classifi cation: C40.
DOI: 10.15240/tul/001/2019-4-009
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