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PRICING EXOTIC OPTION UNDER STOCHASTIC VOLATILITY MODEL Pengshi Li

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PRICING EXOTIC OPTION UNDER STOCHASTIC VOLATILITY MODEL Pengshi Li 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 134 2019, XXII, 4 DOI: 10.15240/tul/001/2019-4-009 EEM_4_2019.inddM_4_2019.indd 113434 225.11.20195.11.2019 111:02:331:02:33 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. 4, XXII, 2019 135 EEM_4_2019.inddM_4_2019.indd 113535 225.11.20195.11.2019 111:02:331:02:33 Finance 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.
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