市場流動性風險下或有償權之評價 - 政大學術集成

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(1)國立政治大學商學院金融學系碩士班碩士學位論文. 政治大. 立市場流動性風險下或有償權之評價. ‧ 國. 學. Contingent Claim Valuation in the Presence of Market Illiquidity. ‧. n. Ch. Yi-Chia e n g cHo hi. er. io. 何奕嘉. sit. y. Nat. al. i Un. v. 指導教授：江彌修博士 Advisor: Mi-Hsiu Chiang, Ph.D.. 中華民國 104 年 6 月 June, 2015.

(2) 誌謝牛頓亦是站在巨人的肩膀上看世界的，學術研究的產生其背後必定藏著許多的支持，不論是理論上、學習上亦或是情感上之支持，唯有聚精會神地投入其中方有機會完成曠世巨作。於金融所兩年之學習路程，隨著論文的付梓，即將劃上句點，這段時間以來的點點滴滴，充滿著無數的回憶與不捨，回憶之情將於我的懷中日漸晶瑩光耀，不捨之心將使我的人生成就勇氣。本論文之所以能順利完成，感謝江彌修教授悉心指導與教誨，對於研究的方向、觀念的啟迪、架構的匡正以及求學的態度逐一斧正與細細關懷，往往於我迷. 政治大度，日後此對於我想必是受益無窮的，於此獻上最真摯的敬意與謝意。此外，於立. 失學習方向之時給予我指引，引領我朝向成功邁進，無形中培養出優良之求學態. ‧ 國. 學. 論文口試期間，亦承蒙口試委員的費心審閱，並惠賜諸多建議，使得本論文能夠更臻完備，於此謹深致謝忱。. ‧. 於研究所修業期間，承蒙博班學長提供寶貴的意見及資料，從最初之論文公. sit. y. Nat. 式推倒至論文內容之寫作，給予我諸多諮詢指導，同時非常感謝金融所眾多伙伴. n. al. er. io. 兩年來的切磋討論與鼓勵，以及同袍戰友賴冠宇的互相激發學習，由於各位的陪. i Un. v. 伴，讓我於兩年的研究所生涯不至迷惘，而能以積極的態度面對，心中滿懷感謝之情。. Ch. engchi. 最後，特感謝父母於經濟上與精神上的支持，讓我能專注於課業研究之中，願以此與家人共享。如同西方諺語所言『參透為何，才能迎接任何』。爾後，將所學實際應用於日常生活中以及工作求職上，發揮所學。. 何奕嘉謹誌於政治大學大學金融研究所中華民國一百零四年六月 I.

(3) 中文摘要欲透過流動性調整模型來探討流動性風險對或有償權的影響，但本篇研究著重於選擇權的分析。根據 Feng (2014)，流動性折現因子由市場流動性與股價對市場流動性敏感度所構成，而且此流動性之動態過程具有均數復歸的特性。根據本篇研究結果，價內選擇權和價平選擇權的評價表現比傳統 Black-Scholes 好，如果進一步將流動性之跳躍性質引入模型，除了價內選擇權和價平選擇權之外，價外選擇權的評價表現亦呈現大幅度的改善。於探討模型評價表現優劣之餘，本篇文章欲更進一步探究市場不流動性對選擇權避險參數的影響。. 治政關鍵字：流動性折現因子、選擇權評價、選擇權避險參數、流動性選擇權、大立跳躍擴散 ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. II. i Un. v.

(4) Abstract This study uses a liquidity-adjusted pricing model to discuss the impact of the liquidity risk on Contingent Claim. However, we focus on the analysis of option. The liquidity discount factor consists of market liquidity and the sensitivity of stock prices to market illiquidity. The dynamic process of market liquidity possesses mean-reversion. Our empirical results show the liquidity model will improve pricing performance for ITM and ATM options. After incorporating diffusive jumps in liquidity, marked improvements in pricing performance for OTM options are observed. In addition, we. 政治大. discuss the impacts of liquidity risk on hedging parameters.. 立. ‧. ‧ 國. io. sit. y. Nat. n. al. er. Jump diffusion. 學. Keywords：Liquidity discount factor, Option pricing, Greeks, Liquidity options,. Ch. engchi. III. i Un. v.

(5) Table of Contents 誌謝 ................................................................................................................................... I 中文摘要 .......................................................................................................................... II Abstract ........................................................................................................................... III Table of Contents ............................................................................................................ IV List of Figures ................................................................................................................. VI List of Tables ................................................................................................................VIII Chapter 1. Introduction .............................................................................................. 1. Chapter 2. Methodology............................................................................................. 5 2.1. Parameter Estimation ............................................................................ 6. 2.3. Merton Jump-Diffusion Model ............................................................. 6. 2.4. Heston Stochastic Volatility Model ...................................................... 8. 2.5. Liquidity Model .................................................................................... 9. 2.6. Liquidity-Jump Diffusion Model........................................................ 17. ‧. ‧ 國. 學. 2.2. n. al. er. io. sit. y. Nat. Chapter 3. Chapter 4. 政治大 Measuring Market Illiquidity ............................................................... 5 立. i Un. v. Theory and Empirical Analysis .............................................................. 20. Ch. engchi. 3.1. Characteristic of Market Liquidity ..................................................... 20. 3.2. Comparison of Pricing Performance Across Models ......................... 21. 3.3. Hedging Parameters ............................................................................ 22 3.3.1. Greeks of Comparison Between Different Pricing Models ........ 22. 3.3.2. Effect of Liquidity and Liquidity-Jump Parameters on Greeks . 25. Conclusions ............................................................................................ 28. References ...................................................................................................................... 30 Appendix ........................................................................................................................ 32 Appendix I .............................................................................................. 32 IV.

(6) Appendix II ............................................................................................. 33 Appendix III ........................................................................................... 34. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. V. i Un. v.

(7) List of Figures Figure 1 The Dynamics of the Market Liquidity Measure ............................................. 20 Figure 2 Effects of Liquidity Parameter on Option Price (1) ......................................... 36 Figure 3 Effects of Liquidity Parameter on Option Price (2) ......................................... 36 Figure 4 Effects of Liquidity Jump Parameter on Option Price ..................................... 37 Figure 5 Delta in Different Models ................................................................................ 37 Figure 6 Gamma in Different Model .............................................................................. 38 Figure 7 Rho in Different Model .................................................................................... 38. 政治大 Figure 9 Vega in Different Model ................................................................................... 39 立 Figure 8 Theta in Different Model.................................................................................. 39. ‧ 國. 學. Figure 10 Delta in Different Model ( 2 Dimension ) ...................................................... 40 Figure 11 Gamma in Different Model ( 2 Dimension ) .................................................. 40. ‧. Figure 12 Rho in Different Model ( 2 Dimension ) ........................................................ 41. Nat. sit. y. Figure 13 Theta in Different Model ( 2 Dimension ) ..................................................... 41. n. al. er. io. Figure 14 Vega in Different Model ( 2 Dimension ) ...................................................... 42. i Un. v. Figure 15 Effect of Sensitivity to Liquidity on Delta ..................................................... 42. Ch. engchi. Figure 16 Effect of Sensitivity to Liquidity on Gamma ................................................. 43 Figure 17 Effect of Sensitivity to Liquidity on Rho ....................................................... 43 Figure 18 Effect of Sensitivity to Liquidity on Theta..................................................... 44 Figure 19 Effect of Sensitivity to Liquidity on Vega ...................................................... 44 Figure 20 Effect of Correlation Between Return and Market Illiquidity on Gamma ..... 45 Figure 21 Effect of Correlation Between Return and Market Illiquidity on Theta ........ 45 Figure 22 Effect of Correlation Between Return and Market Illiquidity on Vega ......... 46 Figure 23 Effect of Liquidity Jump Frequency on Theta ............................................... 46 Figure 24 Effect of Correlation Between Liquidity Jump and Stock Jump on Theta .... 47 VI.

(8) Figure 25 Effect of Liquidity Jump Frequency on Delta................................................ 51 Figure 26 Effect of Liquidity Jump Frequency on Gamma............................................ 52 Figure 27 Effect of Liquidity Jump Frequency on Rho.................................................. 52 Figure 28 Effect of Liquidity Jump Frequency on Vega ................................................ 53 Figure 29 Effect of Correlation Between Liquidity Jump and Stock Jump on Delta ..... 53 Figure 30 Effect of Correlation Between Liquidity and Stock’s Jump on Gamma ........ 54 Figure 31 Effect of Correlation Between Liquidity Jump and Stock Jump on Rho ....... 54 Figure 32 Effect of Correlation Between Liquidity Jump and Stock Jump on Vega ..... 55. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. VII. i Un. v.

(9) List of Tables Table 1 Model’s Parameter Estimation ........................................................................... 34 Table 2 Estimation Error In the Sample ......................................................................... 34 Table 3 Pricing Error Out of the Sample ........................................................................ 34 Table 4 Pricing Error Out of Sample With Different Situation ...................................... 35 Table 5 Improvement Rate of Pricing............................................................................. 35 Table 6 Monte Carlo Simulation .................................................................................... 47 Table 7 Effect of Market Liquidity on Delta .................................................................. 49. 政治大 Table 9 Effect of Market Liquidity on Rho .................................................................... 49 立 Table 8 Effect of Market Liquidity on Gamma .............................................................. 49. ‧ 國. 學. Table 10 Effect of Market Liquidity on Theta ................................................................ 50 Table 11 Effect of Market Liquidity on Vega ................................................................. 50. ‧. Table 12 Effect of Correlation Between Return and Market Illiquidity on Delta .......... 50. Nat. n. al. er. io. sit. y. Table 13 Correlation of Stock Return and Market Illiquidity Effect on Rho ................. 51. Ch. engchi. VIII. i Un. v.

(10) Chapter 1. Introduction. In the real financial market, it is impossible that investors understand clearly all financial derivatives. Because the investors are unfamiliar with financial derivatives, the financial commodities’ volume of trade is not prosperous in the market. Not only that, the unique characteristic of financial derivatives or the entire investment environment of financial market probably causes the market illiquidity. In a real stock market, as such, investors trade with liquidity risk. Many prior literatures provide evidence that investors ask for illiquidity premium due to the liquidity risk and stocks with imperfect liquidity. 政治大 the liquidity risk of the underlying asset directly affects option prices. 立. are priced at a liquidity discount compared to otherwise identical liquid stock. Clearly,. ‧ 國. 學. Market declines causing underlying asset illiquidity gradually receive much attention. Market illiquidity plays an important role on underlying asset as well as. ‧. options. Prior literatures investigate the issues concerning liquidity. For example,. Nat. sit. y. market microstructure literature indicates that liquidity factors are important. n. al. i Un. or turnover ratio are indicators of measuring liquidity risk.. Ch. engchi. er. io. determinants of stock and bond returns. Bid-ask spread、price impact of trades、volume. v. Traditional Black-Scholes assumes that market of underlying asset is frictionless as well as competitive and thus investors can trade quickly any amount at the market price with no additional cost. However, once assumptions are relaxed, the Black-Scholes pricing theory is not readily applicable. In addition, literatures concerning market microstructure literature show that liquidity factors are important determinants of stock returns. Consequently, this article will construct a model that incorporates the liquidity discount factor which is comprised of market liquidity and the sensitivity of stock prices to market illiquidity into the dynamic process of underlying asset based on stochastic market liquidity to discuss the impact of imperfect liquidity on option prices and the 1.

(11) Greeks. It is impossible that liquidity model with both diffusive stochastic liquidity and jump in return captures fully the feature of underlying asset. Intuitively, what is it that jumps in liquidity provide that jumps in return? A jump in returns has no impact on the future distribution of returns. We guess that jumps in liquidity fill the gap between jumps in returns and diffusive liquidity by providing a rapidly moving but persistent factor that drives the conditional liquidity of returns. Hence, this article finally focus on the role of jump in liquidity and returns in Dow-Jones index. In so doing, we expect pricing performance for out-of the-money or longer term options due to more. 治政 reasonable setting which allows flexibility on skewness大 and kurtosis. 立 According to Celso Brunetti and Alessio Caldarera, the definition of liquidity is the ‧ 國. 學. ability to trade quickly any amount at the market price with no additional cost. In this. ‧. article, we adopt this definition to quantify market liquidity risk. As a result of prior. sit. y. Nat. literatures such Garleanu et al. (2009) indicate that liquidity of the market is positively. io. er. related to demand pressure, this article will incorporate the liquidity risk into the option. al. pricing model from demand aspect (e.g. Celso Brunetti). Moreover, Robin K. Chou et al.. n. iv n C (2011) find that a reduction (increase) spot (option) U liquidity, there is a corresponding h ein n gchi increase in the level of the implied volatility curve. Arbitrage pricing theory asserts that the liquidity of the underlying asset is also of relevance to the pricing of options.. In addition, Cetin et al. (2006), who note that the Black-Scholes hedging strategy results in a positive liquidity cost find options become more expensive when the spot asset is less liquid. Illiquidity of underlying asset directly affecting the option prices, hedging strategies also affect the liquidity of option market. Frey (1998) also studies how a large agent, whose trades result in price moves, can replicate the payoff of a derivative security, thereby deriving a non-linear partial differential equation for the 2.

(12) hedging strategy. Engle (1999) thinks if market makers in the derivative markets can hedge their positions using the underlying asset, then the liquidity and spread within the derivative markets will be determined by the liquidity in the spot market, rather than by the activities of the derivative market itself. Cho and Engle (1999) even demonstrate that the hedging activities of option market makers through the underlying asset market leads to a positive correlation of the spread between the two markets. In this article, we will adopt the model referred by Shih-Ping Feng et al. (2013) to adjust option pricing model that demonstrates the impact of the liquidity risk on stock prices using a liquidity discount factor. Furthermore, we value the option prices under. European call option on an asset with stochastic volatility.. 學. ‧ 國. 治政 framework of Steven L. Heston (1993) using a new technique 大 to solve the disadvantage 立 that their models do not have closed form solutions. The solution is for the price of a ‧. In addition, this model allows arbitrary correlation between liquidity and spot asset. sit. y. Nat. returns. Precious literature finds liquidity cost is a significant component of the option. io. er. price and that the impact of illiquidity is dependent upon the moneyness of the option.. al. That is, the impact is more (less) significant for out-of-the-money (in-the-money). n. iv n C options, underlying stocks with a higher quoted half-spread ultimately lead to a h e average ngchi U higher level of implied volatility (a higher option price).. Robert C. Merton (1976) thinks the underlying stock returns are generated by a mixture of both continuous and jump processes. Hence, this article finally incorporates the jump diffusion into the original liquidity model in order to discuss the characteristic of jump diffusion. By so doing, we expect to improve the option pricing performance out-of-sample. The rest of this study is organized as follows. Section II presents the literature reviews which introduce the economy along with the role of liquidity and what impact 3.

(13) liquidity possesses. Section III constructs suitable model to value the liquidity-adjusted option prices as well as introduces the estimation procedure in our empirical analysis. Section IV summarizes the empirical results. Section V presents our conclusions of the studies.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 4. i Un. v.

(14) Chapter 2. Methodology. In this chapter, we will define the market liquidity and descript separately the Merton jump diffusion model as well as Heston stochastic volatility model. Then, we construct the liquidity model under the frame of Heston. Finally, we incorporate the jump diffusion into the original liquidity model.. 2.1 Measuring Market Illiquidity While the liquidity is an elusive concept, there is, nonetheless, consensus on the definition that liquidity is the ability to trade quickly any amount at the market price. 政治大. with no additional cost. The illiquid market is characterized by the possibility of a. 立. sudden rise or drop of underlying asset’s price with modest trading volume. In order to. ‧ 國. 學. capture the phenomenon of market illiquidity, we define the indicator at which measure the market liquidity.. ‧. ri ,t. (2.1). sit. Nat. Vi ,t. y. at . io. n. al. er. where ri ,t and Vi ,t denote the return and trading volume for underlying assets. i Un. v. respectively. This measure not only inherits the sign of the return, but also normalizes. Ch. engchi. the return by the trading activity. While the value of at is zero, it indicates a situation in which the market is fairly liquid and there is not any unbalances between demand and supply. However, absolute value of at which is not equal to zero indicates the market illiquidity occurs; at  0 when the market is in short supply, at < 0 when the market is in surplus.. 5.

(15) 2.2 Parameter Estimation In this article, we adopt the nonlinear optimization method for estimation of parameter. In other words, it minimizes the sum of the absolute value of price differences between the theoretical option prices and market prices which are available in the real financial market. Nt. SSE  t   . i,n i ,n Cmarket  t , n,t , K n,t   Cmodel  t , n , t , K n , t . (2.2). Nt. n 1. Ideally, all of the parameters can be simultaneously estimated. Therefore, we use the optimization procedure in formula (2.2) to estimate the parameters. Furthermore, we. 政治大. use the parameters estimated from the market’s data1 on the previous day as inputs for. 立. our option pricing formula to compute the option prices for the current day. Then, we. ‧ 國. 學. compare these computed option prices with the actual market prices to evaluate the performance of our option pricing model.. ‧. sit. y. Nat. 2.3 Merton Jump-Diffusion Model. io. er. The two basic components of Merton jump-diffusion model are the Brownian. al. motion and the Poisson process. The Brownian motion is similar to the Black-Scholes. n. iv n C model. The definition of the Poissonhprocess is given inUthe following. engchi The Poisson process The gaps  i  i  1. between successive jumps are independent. identically distributed random variables with the exponential distribution with parameter λ, that is, with cumulative distribution function P  i  x   e   x and let n. Tn   i . The process i 1. 1. We use the DJIA index and option prices on the DJIA index to estimate parameters. 6.

(16) Nt   It Tn  n 1. is called the Poisson process with parameter λ. Compound Poisson process2 The compound Poisson process is a generalization where the waiting times between jumps are exponential but the jump sizes can have an arbitrary distribution. More precisely, let N be a Poisson process with parameter λ and. Yi . i  1 be a sequence of independent random variables. The process Nt. X T   Yi i 1. 政治大. is called compound Poisson process.. 立. 學. ‧ 國. The underlying stock price, S t , follows the dynamic process under the risk-measure Q is represented as (2.3).. ‧. dSt     k  dt  σdWt P ,Q  d N  t  St. (2.3). sit. y. Nat. where dWt P,Q and dN  t  are independent. In addition,  is mean number of arrivals. n. al. er. io. per unit time, N  t  is Poisson process with an intensity of  , Y  1 is random. Ch. i Un. v. variable percentage change in the stock price if Poisson event occurs. If Y has a. engchi. log-normal distribution, then the price can be written as . F  S ,   . e       . n 0. n!. n. f n  S , . (2.4). where     1  k  . f n  S ,  is the value of the option, conditional on knowing that exactly n Poisson jumps will occur during the life of the option. The actual value of the option is just the weighted sum of each of these prices where each weight equals the probability that a Poisson random variable. 2. This definition refer Peter Tankov and Ekaterina Voltchkova article which is “Jump-diffusion. models: a practitioner’s guide” 7.

(17) 2.4 Heston Stochastic Volatility Model The Heston model assumes that the underlying stock price, S t , follows a Black-Scholes type stochastic process, but with a stochastic variance vt that follows a Cox, Ingersoll, and Ross (1985) process. Hence, the Heston model is represented by the bivariate system of stochastic differential equations (SDEs) dSt  rt dt   t dWt S ,Q St. . (2.5). . d t     t dt    t dWt ,Q. (2.6). where dWt S ,Q dWt ,Q   dt . In addition, σ is level of volatility of the variance,  is. 政治大 mean reversion speed of the variance,  is long run level for the variance. 立 ‧. ‧ 國. can be written as. 學. Under the above assumption of dynamic process, we can know the pricing formula. Ct  St P1  Kert T t  P2. y. Nat.   1 1 ei ln K Pn  P  ln ST  ln K    Re  f n   2 0  i. n. Ch.   d . engchi U. When the characteristic functions. er. io. sit. with in-the-money probabilities P1 and P2. al. (2.7). v ni. n  1, 2. (2.8). f i  ; x, v  are known, each in-the-money. probability Pi can be recovered from the characteristic function via the inversion theorem3, as (2.8). In addition, the characteristic functions f n  ; x, v  is represented as follows:. f n    e AT v0  BT . n  1, 2. with. 3. Inversion theorems can be found in many textbooks, such as that by Stuart (2010). 8. (2.9).

(18) A T  . B T  . bn  d n   i  1  ednT   d nT  2  1  gne . (2.10).  1  g n ed T   b  d   i  T  2ln     2  n n  1  gn n.   . (2.11). where. gn . bn  d n   i bn  d n   i. d1 .   i  bn . 2.   2  2  i . d2 .   i  bn . 2.   2  2  i . 立. 治政 . b     , b  大 1. 2. ‧ 國. 學. 2.5 Liquidity Model. Modeling liquidity risk by employing the underlying asset’s price specification. ‧. process developed by Brunetti and Caldarera (2006). The liquidity discount factor  t. Nat. sit. y. is incorporated into the demand function of underlying asset to capture the impact of. n. al. er. io. liquidity on the prices. In this model, the stock possesses one and only price that makes. i Un. v. the demand of stock clear supply. Under the market-clearing condition, the demand for. Ch. engchi. stock depends on the following factors： 1.. Stock-specific information. 2.. Systematic factor. 3.. Level of aggregate liquidity. According to above assumptions, we assume that the demand function is a function of the stock price, the liquidity discount factor, and the stock-specific information process.  I  D  St ,  t , I t     t    t St  9. (2.12).

(19) where Ψ is a smooth, strictly increasing function, and v  0 is a constant.. y is constant. Therefore,. In addition, we also assume that the quantity supplied. the market clearing price is defined by the market-clearing condition  I Ψ t   t St.   y . (2.13). Through formula (2.13), we will show that stock price can be written as formula (2.14)： I t St   t Ψ 1  y . (2.14). 政治大 which captures the impact of liquidity on the stock price is a function of the level of 立. where S t is the clearing price and I t is defined above. The liquidity discount factor. ‧. ‧ 國. illiquidity.. 學. liquidity in the market as well as the sensitivity of the stock to the level of market. From this liquidity model, we know that when the market liquidity at and the. sit. y. Nat. sensitivity of stock prices to market illiquidity  are equal to zero, the liquidity. n. al. er. io. discount factor is equal to one. In this situation, the dynamic process of stock price is the same as dynamic process of Black-Scholes. St . Ch. engchi. i Un. v. at 1 &  t 1 I t I t BS  S    t St t  t Ψ 1  y  Ψ 1  y . (2.15). From formula (2.15),  t is the liquidity discount factor for stock and is defined by  t  e  X. t. (2.16). where dX t  at dt  at dWt  ,P. By Ito’s Lemma, the dynamic process of liquidity discount factor follows. 10. (2.17).

(20) d t. 1       at   2 at2  dt   at dWt  , P t  2 . (2.18). Meanwhile, the information process for stock follows. dI t   dt   dWt I , P It. (2.19). where dWt  , P dWt I , P  0 . In order to understand the properties of market liquidity, we adopt the market liquidity measure proposed by Amihud (2002), which is also adopted by Brunetti (2006),. 政治大 tends to fluctuate around the mean zero. Thus, we assume that market liquidity follows 立 to serve as a proxy for market liquidity. Accordingly, it supposes that market liquidity. ‧ 國. 學. a dynamic process of mean-reversion as dat     at  dt   dWt a , P. (2.20). ‧. where dWt  , P dWt a , P   . α is the mean reversion speed of the market liquidity, θ is. y. Nat. io. sit. the mean level of the market liquidity and ξ is the level of volatility in the market. n. al. er. liquidity. Specially, we also allow correlation between liquidity discount factor and market liquidity.. Ch. engchi. i Un. v. Under the market-clearing condition, the imperfect liquid stock price conforms to the following dynamic process:. dSt  1       at   2 at2  dt   dWt I , P   at dWt  , P St  2 . (2.21). 1 where        1 2 and κ  νη . (2.21) indicates that the stock price is 2. adjusted based on information relating to the stock price, the level of market liquidity, and the sensitivity of the stock to the level of market illiquidity. In order to facilitate analysis of the stock price, we decompose dWt  ,P into 11.

(21) dWt a , P and dWt MKT , P . Hence, we can rewrite as follows. dWt  , P   dWt a , P  1   2 dWt MKT , P. (2.22). where dWt MKT , P is a Brownian motion and dWt a , P dWt MKT , P = 0 . Therefore, the stock price under measure P can be written as. dSt  1       at   2 at2  dt   dWt I , P   at 1   2 dWt MKT , P   at dWt a , P (2.23) St  2  where κ is stock volatility from idiosyncratic risk, β is sensitivity of the stock to. 政治大 is level of liquidity in the market,  立. level of market illiquidity, ρ is correlation between liquidity discount factor and market liquidity, at. t. ‧ 國. .   1     a 2. 2. 2 2 t. dWt I , P .  at 1   2.   1     a 2. 2. 2 2 t. dWt MKT , P. (2.24). ‧. dWt S , P . 學. To facilitate our analysis, we further derive the dWt S , P as. is liquidity discount factor.. Nat. sit. er. io.  2  1   2   2 at2 .. y. I ,P 2 MKT , P  xdWt S , P , so we will know x is equal to where we let  dWt   at 1   dWt. al. n. iv n C h price So, the dynamic process of stock i U 𝑃 is rewritten as e n gunder c hmeasure. (2.25). dSt  1       at   2 at2  dt   2  1   2   2 at2 dWt S , P   at dWt a , P St  2 . (2.25). where dWt S , P dWt a , P  0 . ω is the set of all possible equivalent martingale measures, and the equivalent martingale measure, Q   , is characterized by Girsanov density T. t ,1dWs dQ  T  et dP. S ,P. T. .  t ,1dWsa ,P  t. T. T. t. t. 1 2 1 t ,1ds  t2,2 ds 2 2. . . (2.26). By adopting Girsanov theorem to change P measure into Q measure, we will obtain formula (2.27) and formula (2.28) 12.

(22) dWt S ,Q  dWt S , P  t ,1dt. (2.27). dWt a ,Q  dWt a , P  t ,2 dt. (2.28). and. where dWt S ,Q and dWt a ,Q are Brownian motion under risk-neutral probability measure Q . To transform St / Bt into a martingale under the equivalent measure, both. 1 and 2 must satisfy the following formula (2.29): t ,1  2  1   2   2 at2  t ,2  at =    at   2 at2  rt 1 2. (2.29). 政治大. Eventually, the dynamic process of underlying asset price under Q measure is. 立. rewritten as. ‧ 國. (2.31). io. sit. y. dat     at  2  dt   dWt a ,Q. n. al. er. Nat. where dWt S ,Q dWt a ,Q  0 .. (2.30). ‧. and. 學. dSt  rt dt   2  1   2   2 at2 dWt S ,Q   at dWt a ,Q St. Ch. i Un. v. According to the idea proposed by Heston (1993), liquidity risk premium t ,2 is. engchi. proportional to market liquidity measure at and denoted as. t ,2 . wat. . (2.32). Through change of measure, market liquidity process can be rewritten as. . . dat     at dt   dWt a ,Q. where     w  and  .  .  w. 13. (2.33).

(23) Theorem 1.4 Under the equivalent martingale measure, Q , the theoretical price of the European call option with the risk-free interest rate at time t and the strike price for the option contract, K , is derived as: Ct  St P1  Ke  rt T t  P2. (2.34).   1 1 e i ln K  P1   Re  f1    d 2  0  i . and P2 .   1 1 ei ln K  Re  f 2   2  0  i.   d . . Proof. The European call option can be written as. . 政治  大. Ct  E Q e rt T t   ST  K  | Ft . 立. . .  e rt T t  E Q ST IST  K | Ft  e  rt T t  KE Q IST  K | Ft. . (2.35). ‧ 國. 學. For the first term we use the stock price as a numeraire to change the measure Q to Q1 . The Radon-Nikodym derivatives are denoted as. ‧ y. sit. io. n. al. i Un. Therefore, the European call option will be rewritten as. . Ch. e n g c h i. Ct  E Q e  rt T t   ST  K  | Ft. . . . v. .  e  rt T t  E Q ST IST  K | Ft  e  rt T t  KE Q IST  K | Ft. . . (2.36). er. Nat. ST T dQ1 St t rs ds ST  e dQ BT St Bt. .  St E Q1 IST  K | Ft  e  rt T t  KE Q IST  K | Ft. . . (2.37). Now let us consider the first characteristic function of X T  ln  ST  is defined as. f1    E Q1  ei X T  In addition, the X T is demoted by Ito’s Lemma as (2.39) 4. Shih-Ping Feng, Mao-Wei Hung, Yaw-Huei Wang. “Option pricing with stochastic liquidity risk:. Theory and evidence.” 14. (2.38).

(24) 1   ln ST  ln St    rt   2   2 at2   dt    2  1   2   2 at2 dWt S ,Q    at dWt a ,Q 2  . (2.39). Hence, the characteristic function f1   can be rewritten as f1    e. where. 3 .  2 1 i   at2 1 i   1 i  r  X t    2 1 i i   2 2 2. 1  . 1 at2 dt  2  at dt  3 aT2  E Q  e    .  1 2   , 1  i   1   2   2 1  i    2  2   . 2 . (2.40). 2 1  i  and 2.  1  i  . The detailed procedure of derivations see the Appendix I. 2  a dt   a dt  a   a Now we let F1 is denoted as E Q  e   with F1  a, T , T   e 2 t. 政治大 satisfies the following differential equation 立 1. 2. t. 2 3 T. 3 T. . . F1. . 2 1 F1 2    at     1a  2 a F  0 1 2 2 a a. The solution of (2.41) has the form. sit. io. with A1 (T, T)  2 3 , A2 (T,T)  0 and A3 (T,T)  0 .. n. al. Ch. y. 1 A1 t ,T at2  A2  t ,T at  A3  t ,T  2. (2.42). er. Nat. F1  , t , T   e. (2.41). ‧. ‧ 國. t. . 學. F1. . i Un. v. The system of ordinary differential equations is detailed as. engchi. A1   2 A12  2 A1  2 2  0 t. (2.43). A2     2 A1  A2   A1   2  0 t. (2.44). A3 1 2 2 1   A2   A2   2 A1  0 t 2 2. (2.45). and. Based on above equations, we can derive the solution as follows. A1  t , T  . sinh 1    2 cosh  1   1     1 2    cosh 1    2 sinh 1   15. (2.46).

(25) A2  t , T  .    sinh 1    2 cosh 1   1   2 3 1    1   23    cosh     sinh    1  cosh 1    2 sinh 1        1 2 1 1    2. (2.47) and 1 1 A3  t , T    ln  cosh  1    2 sinh  1     2 2 2 2 2 2  sinh  1    1   3        1  cosh      sinh    2 2 13 1 2 1  . . . .   2 3  3   cosh  1   1   2 3  1  cosh  1    2 sinh  1  . 1. where  1  2 21   2 ,  2 . 立. (2.48). 1 . 1 治and        . 政大   2  2. 3. 2. . 2. 3. 2. ‧ 國. (2.49). io. y. E Q  e . . . 1 at2 dt 2 at dt 3aT2.   . (2.50). er. 1  2 i at2 i i  r  X t    2 1i i   2 2 2. sit. Nat. f 2    e. . ‧. and. f 2    E Q  ei X T. 學. Similarly, the second characteristic function of X T  ln  ST  is defined as:.  i 2 i 1  2   where 1   i  1   2   2 i   2  and 3  .  , 2  2 2 2   . n. al. Ch. engchi. Now we let F2 is denoted as E Q  e. . i Un. . v. 1 at2dt 2 at dt 3aT2. .  with F a, T , T  e a   2 . 3 T. satisfies the following differential equation. F2 t. . F2 a.    at  . . . 2 1 F2 2   1a  2 a F  0 2 2 2 a. (2.51). The solution of (2.51) has the form. F2  , t , T   e. 1 B1 t ,T at2  B2  t ,T at  B3  t ,T  2. with B1 (T,T)  23 , B2 (T,T)  0 and B3 (T,T)  0 . 16. (2.52).

(26) Under the same framework, we also can obtain the following solution. B1  t , T  . B2  t , T  . sinh 1   2 cosh 1   1      1  2  cosh 1   2 sinh 1  . (2.53).    sinh 1   2 cosh 1   1  1  23  1   2 3     1  cosh 1   2 sinh 1    1  cosh 1    2 sinh 1   2. (2.54) and 1 1 B3  t , T    ln  cosh 1   2 sinh 1     2 2  sinh 1   2 212  32    1   2 3 2 1  cosh 1   2 sinh 1  . 政治大 2 3.  213. where 1  2 21   2 , 2 . 3.   cosh 1   1    cosh 1   2 sinh 1  . 1 and 3   2   2 2 . 2 1   2 3 . ‧. ‧ 國. 1. 學. 立       . (2.55). y. Nat. io. sit. By methods for the numerical integration of inverse Fourier transform in the option. n. al. er. pricing formula, we finally can obtain the European call option price as Theorem 1.. iv. Ch 2.6 Liquidity-Jump Diffusion U n Model engchi. Based on pricing model of liquidity proposed by Feng (2014), this article further incorporates the concept of jump diffusion into the model in order to describe perfectly market liquidity. Therefore, the dynamic process, under the Q measure, of stock price will be written as 2   S   S  2  1 2 e   2 2 d ln St   rt     at      1  dt   2  1   2   2 at2 dWt S ,Q   at dWt a ,Q  J S dNt 2   1   J a    . (2.56) where λ is Poisson processes with constant intensities, J S which is jump sizes in 17.

(27) . . 2 stock has a normal distribution, that is, J S ~ N  S , S . In addition, the dynamic. process, under the Q measure, of liquidity will be written as. . . dat     at dt   dWt a ,Q  J a dN t. (2.57). where J a which is jump sizes in liquidity has a normal distribution, that is,. J a ~ N   a , a2  . Furthermore, liquidity-jump model possesses correlated jump sizes. . . a 2 and J S | J a ~ N  S   J  ,  S .. By generalized Feynman–Kac theorem, it implies that f  ; x, a,  solves the. 政治大. following partial integro-differential equation (PIDE):. 立. ‧ 國 . (2.58). ‧. . 學. 2   S   S  2  f 1 2 2 f 2 f  e   f 1 2 2   rt     at   S   1    2   2 a 2  2  a  t  2 ax x  1   J a   x 2    f 1 2  2 f    at    S   f  , x  J S , a  J a   f  , x, a   d  J S , J a   0 a 2 a 2. n. al. option with maturity T and strike K  e. Ch. CT  k . denote the price of European call. er. io. Fourier Transform in section 2.6. First, we let. sit. y. Nat. In order to derive the European call option price, we adopt the method of Fast. k. engchi. . i Un. CT  k   e  rT  e x  e k  fT  x  dx. v. (2.59). k. where fT is the risk-neutral density of square-integrable because. CT  k . X T  log  ST  . The function CT is not. converges to S 0 for k    . Hence, we consider a. modified function cT  k   e k CT  k . (2.60). which is square-integrable for a suitable α  0 . The choice of α may depend on the model for ST . The Fourier transform of CT is defined by 18.

(28) e rT T  v    1 i . .  T  v    eivk  e x  ek  cT  k  dk =.  2    v 2  i  2  1 v. . (2.61). where T is the Fourier transform of fT . A sufficient condition for CT to be square-integrable is given by  T  0  being finite. This is equivalent to E  ST 1   . (2.62). Now, we get the desired option price in terms of  T using Fourier inversion CT  k  . e  k. . . e.   v  dv.  ivk. (2.63). 0. Finally, we can derive the option price as. 政e 治大 C k   e    d    k . 立. e rT f       1 i . ‧ 國. where. (2.64). 0.  2     2  i  2  1. 學.    .  i k. T. .. ‧. In addition to FFT, Appendix II also introduce another mothed to derive the. Nat. sit. y. European call option price. However, we will adopt Monte Carlo simulation to obtain. al. n. model in this article.. er. io. the European call option prices and hedging parameters under liquidity-jump diffusion. Ch. engchi. 19. i Un. v.

(29) Chapter 3. Theory and Empirical Analysis. In this chapter, we will show the market liquidity and compare the pricing performance with different pricing model. Then, we discuss the impact of liquidity on option prices and hedging parameters.. 3.1 Characteristic of Market Liquidity 0.6. 0.4. 政治大. 0.2. 立. 0. ‧ 國. sit. y. Nat. -0.6. ‧. -0.4. 學. -0.2. al. er. io. Figure 1 The Dynamics of the Market Liquidity Measure. v. n. This figure illustrates the dynamics of the market liquidity measured. Ch. i Un. by dividing the S&P 500 index return by the volume of the component. engchi. stocks in the S&P 500 index. The sample period runs from January 1 to December 31, 2011.. The pattern of the liquidity proxy compiled from real market data in Figure 1 indicates that there is a phenomenon of mean-reversion for market liquidity. It then tends to fluctuate around the mean level which is approximately equal to zero and the volatility of market liquidity becomes larger gradually at the end of 2011. Therefore, we adopt the concept of Vasicek to model dynamic process of market liquidity. In addition, Lubos Pastor & Robert F. Stambaugh (2003) consider that stocks whose returns are more exposed to marketwide liquidity fluctuations command higher expected returns. 20.

(30) So, the worse market liquidity is, the higher the option price is.. 3.2 Comparison of Pricing Performance Across Models The long term average level of liquidity is negative but approximately zero. There is a negative correlation between the discount of liquidity and market liquidity. It interprets that the higher the market liquidity, the larger the discount of liquidity. Under the stochastic volatility model, the long term average level of volatility is about 0.02 which interprets that the stock’s volatility is a little in the long term. In addition, comparing to stochastic volatility and Black-Scholes’ model, the stock’s volatility of. 政治大 diffusion. This means the real market exists a phenomenon of jump diffusion. Thus, we 立 jump diffusion model is lower. However, it is pretty high that the frequency of jump. ‧ 國. 學. discuss the jump diffusion is from stock or liquidity by adding the jump diffusion term into the liquidity model. From Table 1, we know the jump diffusion mainly comes from. ‧. liquidity instead of stock.. sit. y. Nat. Regardless of stochastic volatility, jump diffusion, liquidity or liquidity jump’s. n. al. er. io. model, all of the estimation error in the sample are less than Black-Scholes. It interprets. i Un. v. that those model can descript accurately what characteristic the underlying asset. Ch. engchi. possesses. In other words, the Dow-Jones Index not only has property of jump diffusion but also is tremendously affected by market illiquidity which exists phenomenon of jump diffusion. Furthermore, we also know that those pricing model have lower pricing error out of sample. We know those model’s pricing performance are better than Black-Scholes in Table 3. However, the inclusion of the liquidity effect in option pricing inevitably increases the number of model parameters and thus cause in-sample over fitting, which may harm the model's performance if the extra parameters do not improve out-of-sample pricing performance. Therefore, we examine our model's option pricing performance in different statement. 21.

(31) Prior literature shows that option prices are very sensitive to their moneyness and maturity levels. Therefore, this article analyzes option pricing performance across several categories. Table 5 presents the results of pricing performances for all stocks in different moneyness and maturity levels. The formula solution and Monte Carlo solution are similar in Table 6. It implies that the formula solution is correct. As expected, no matter what the long run or short run is, the pricing errors of the liquidity and stochastic volatility’s model are smaller than the pricing errors of the Black-Scholes model in the money (ITM) and at the money (ATM). It means that not only the liquidity of stock can be captured by liquidity model for ITM as well as ATM but also the. 治政 characteristic of liquidity is similar with stochastic volatility. 大 In other words, the stock’s 立 liquidity has a long effect on option price instead of short effect. What’s more, because ‧ 國. 學. the option pricing performance of jump diffusion is better than Black-Scholes out of. ‧. money (OTM), we further incorporate the jump into liquidity model to improve the. y. sit. io. al Greeks of Comparison Between Different i v Pricing Models n. 3.3.1. 3.3 Hedging Parameters. er. Nat. pricing performance for OTM.. Ch. n engchi U. The correlation between market liquidity and liquidity discount factor, seen in Figure 2, only have a little effect on option price. However, the sensitivity of market liquidity and coefficient about jump diffusion, seen in Figure 4, play an important role on option price. When the liquidity sensitivity is higher, the option price is more expensive. Lubos Pastor & Robert F. Stambaugh (2003) find aggregate liquidity series exhibits a negative association with market volatility. When the market illiquidity happen, the volatility of stock will rise and then heighten the option price.. 22.

(32) In. addition, seen as Figure 5, we find that when the market illiquidity exists, the delta is. more sensitive to maturity under OTM. There is a positive relation between delta and maturity. It means the higher the delta is, the longer the maturity is. According to Hyejin Ku & Kiseop Lee & Huaiping Zhu (2012), we know one follows the delta hedging strategy with the usual Black-Scholes prices, one would face a significant amount of loss with the presence of liquidity costs. In other words, while the liquidity is worse, the delta is higher than Black-Scholes. Therefore, it needs to increase the cost of hedging as executing the hedging strategies. Stefan Sandberg (2012) also finds that illiquid periods could have a time process of perfect liquidity, but the issuer is not able to follow every. 治政 shift, because each shift represents a very low volume.大 If the issuer should follow each 立 shift, the issuer may create a market impact that increases the hedging cost even more ‧ 國. 學. than the discontinuous delta hedge.. ‧. However, from OTM to ITM, the amount of variation for gamma is more acute. sit. y. Nat. than other pricing models as Figure 6. The thing that gamma first increase and then. io. al. er. decrease means market illiquidity will increase the difficulty of delta hedging at the. n. money. David Bakstein & Sam Howisonransaction (2003) think transaction costs are an increasing function of the gamma.C h. engchi. i Un. v. No matter what the model is, the amount of variation for rho appear to significant increment especially out of money. When the maturity is longer, the rho is larger. However, this phenomenon is not slightly the same as finding Stefan Sandberg (2012) discover. The issuers usually do not hedge rho, because the changes are too small, only if there are extreme movements in the interest rate market. So, the investment should execute the rho hedging strategies when the market liquidity becomes worse. Stefan Sandberg (2012) think issuers does not focus so much on theta because the cost of theta is quite expensive, which may not recompense at maturity. All the same, we still know, 23.

(33) seen in Figure 8, the theta will increase accompanied with increment of maturity for OTM. This is mainly difference compared with Black-Scholes. According to Figure 9, no matter what models are, it positive relation that correlation between maturity and vega exist. Especially in liquidity model, the amount of variation is stronger than others. It means that as the market illiquidity exist, it is more difficult to hedge the volatility risk in the longer maturity. Stefan Sandberg (2012) think stock market comparing to the options market is much more liquid. In addition, supposing that the stock market average volume has dropped or more, then of course the options market that is depending on the trade volume of stock, will also be affected in. Thus, we need to pay attention to vega hedging.. 學. ‧ 國. 治政 similar scale, and can therefore be consider as unavailable. 大 In other words, as long as 立 the market liquidity is lower in stock market, the option liquidity will also be lower. ‧. Comparing delta of different models, as Figure 10, the delta of liquidity model is. sit. y. Nat. higher than stochastic volatility, jump diffusion and GBM’s model for OTM but is lower. io. er. than others for ITM. However, the delta will appears to opposite result after adding the. al. jump diffusion into the liquidity model. It interprets the jump of liquidity will increase. n. iv n C the cost of hedging risk which is change price for ITM. And it decreases the h e nofgstock chi U cost of hedging for OTM. From the aspect of gamma, regardless of ITM or OTM, the gamma of liquidity model is lower than others models in Figure 11. When the options approach to ATM, the gamma is higher than others. As a result of this, the difficulty of delta hedging will increase. However, if considering effects of liquidity jump, the difficulty of delta hedging will decrease in any situation. Although the rho of liquidity model is higher than other models for OTM as Figure 12, the value is lower than others for ITM. In contrast, liquidity jump make rho lower 24.

(34) for ITM, ATM or OTM. It means that if the liquidity jump happens, the investor probably hedge actively the interest risk. In Figure 13, we know no matter what ITM, ATM or OTM is, the liquidity risk make the option is more sensitive to maturity. However, liquidity jump factor let theta of ITM and OTM’s options decrease. In other words, if considering the liquidity jump, the investor actually can neglect the effect of maturity declining. Liquidity make it difficult to hedge volatility risk for ITM and OTM as Figure 14. For ATM, because the liquidity increase the variation of volatility, the hedging cost of volatility will be higher. There is an extremely difference in Figure 14 between liquidity. 治政 model and liquidity jump model. It reduce the sensitivity 大of volatility for ATM. Hence, 立 investor holding options do not need to evade volatility risk for ATM. ‧ 國. 學. 3.3.2. Effect of Liquidity and Liquidity-Jump Parameters on Greeks. ‧. According to Table 7, Table 8, Table 9, Table 10 as well as Table 11, we know that. sit. y. Nat. market liquidity has a little effect on delta, gamma, rho, theta and vega. However, in. io. er. addition to vega, effect of market liquidity is almost inconsistent. Condition on the same. al. variation of volatility, the market liquidity make the vega declines.. n. iv n C According to Figure 15, we know delta is affected by the sensitivity of market h ethat ngchi U. illiquidity. When the option is gradually moving from OTM to ITM, the amount of change will gradually increase. In addition, delta and the sensitivity of market illiquidity is a positive correlation between each other in the money. That is, when sensitivity is gradually increased, delta gradually increasing. Nevertheless, the relation is negatively correlated with each other, that is, when the sensitivity of increasing, delta gradually decreasing. As Figure 16, gamma is affected by the sensitivity of market illiquidity. Gamma will slightly increases for OTM. That is, as market liquidity becomes bad, delta hedging 25.

(35) of OTM option will be more difficult. Except OTM, the higher sensitivity does not possess large impact on gamma. When it is OTM, rho and sensitivity of market liquidity exist a positive correlation with each other, that is, if the sensitivity is gradually increased, rho also gradually increase. On the contrary, there is a negative correlation between each other for ITM. It means as result of increasing of sensitivity, rho gradually decreases instead. This phenomenon can be observed in Figure 17. Observed as Figure 18, it shows that there is a positive correlation between theta and market liquidity. That is, when the sensitivity increase accompanied with increment of theta. Vega, as Figure 19, exists a positive relation with each other for ITM or OTM.. 治政 That is, the higher the sensitivity, the higher the vega. 大 Nevertheless, sensitivity of 立 liquidity possess a negative impact on vega. In other words, if the sensitivity is ‧ 國. 學. increases, the vega hedging will become more difficult. Prior literatures suggest. ‧. aggregate liquidity also tends to be low when market volatility is high. Suggesting that. sit. y. Nat. stocks with lower liquidity tend to be more exposed to aggregate liquidity fluctuations.. io. er. In addition, seen in Table 12 and Table 13, the correlation between liquidity. al. discount and market liquidity has a little impact on delta as well as rho. In Figure 20, we. n. iv n C find that as long as the stock marketh is in surplus or short e n g c h i U supply, delta hedging for ATM will become more difficult. Nevertheless, the stronger the positive correlation is, the. easier the delta hedging which is OTM is. In Figure 21, we also find that the stronger the positive correlation is, the more difficult the delta hedging which is OTM is. On contrary, the delta hedging which is ITM will become easier. Observed as Figure 22, we find that as long as the stock market is in surplus or short supply, delta hedging under ATM will become more difficult. However, the delta hedging which is ITM will become difficult when the positive correlation increase. The stronger the positive correlation, the easier the delta hedging which is OTM. 26.

(36) Finally, according to Figure 25, Figure 26, Figure 27 and Figure 28, the frequency of liquidity jump almost does not affect the delta, gamma, rho and vega. However, we find in Figure 23 that this factor possesses a slight impact on theta which is ITM. That is, the liquidity jump will slightly decrease the sensitivity of declining of time value. In addition, from Figure 29, Figure 30, Figure 31 and Figure 32, the correlation between liquidity jump and stock jump almost does not affect the delta, gamma, rho and vega. However, seen as Figure 24, the positive correlation compared with negative correlation significantly increase sensitivity of maturity. Not only that, the effect level in statement of positive correlation is larger than that in statement of negative correlation.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 27. i Un. v.

(37) Chapter 4. Conclusions. Market illiquidity occurs if the underlying asset is in short supply or in surplus. Precious literatures indicate market liquidity affect the option pricing and hedging strategies. Unfortunately, most traditional pricing models seldom cover the aspect of liquidity. Therefore, we adopt a new pricing model incorporating the liquidity factor to discuss what impact liquidity has on options. We find the fitting ability in liquidity is better than Black-Scholes. The pricing errors of the liquidity model are smaller than the pricing errors of traditional pricing model. Especially ITM and ATM, the option pricing. 政治大 worse. In order to cope with the disadvantage, we further add liquidity jump diffusion 立 performance get significant improvement. On the contrary, the OTM performance is. ‧ 國. 學. into pricing model. In so doing, the OTM performance actually derive a large improvement. According to empirical results, the standard deviations of the pricing. ‧. errors of the liquidity model as well as liquidity-jump model are also smaller. Thus, the. Nat. sit. y. results indicate that these model possess not only smaller also more stable pricing errors.. n. al. er. io. That is, pricing biases in traditional model especially for OTM options can be. i Un. v. substantially corrected by a model that assumes a risk-neutral distribution that is more. Ch. engchi. flexible than the log-normal distribution.. On the other hand, we also discussed options on the prices and Greeks under market illiquidity. In fact, the market illiquidity cause the option prices to be more expensive. Furthermore, the correlation between liquidity discount and market illiquidity and sensitivity of the stock to market illiquidity significantly affect all of Greeks. It indicates that if market illiquidity occurs, it is easier that investors hedge the volatility risk for ATM. In addition, delta hedging for ATM is simpler. Further, if we price the option by taking liquidity jump diffusion into account, we find the price is more expensive. The level of difficulty for delta, rho as well as Vega will decline. 28.

(38) However, it will increase the sensitivity of maturity. In other words, we need to pay attention to theta hedging if the liquidity jump happen. In sum, investors have to adopt a suitable alternative hedging strategy facing the liquidity factor. A limitation of our approach is that the equilibrium price does not derive from the optimization behavior of agents. In fact, the market-clearing condition consists of demand and supply. Rigorously, we should take supply pressure into account. That is, the dynamic process of stock prices depends on demand and supply. Hence, we have to discuss the factors affecting the supply of stocks. This is an interesting area for future research.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 29. i Un. v.

(39) References [1] Amihud, Y., 2002, Illiquidity and stock returns: cross-section and time-series effects, The Journal of Financial Markets 5, 31–56. [2] Bakstein, D., and S. Howison, 2003, Using Options on Greeks as Liquidity Protection, Mathematical Finance Group, University of Oxford, Working paper. [3] Brunetti, C., and A. Caldarera, 2006, Asset Prices and Asset Correlations in Illiquid Markets, Working Paper. [4] Cetin, H., M. Soner, N. Touzi, 2010, Option hedging for small investors under. 政治大 Cetin, U., R. A. Jarrow, P. Protter, 2004, Liquidity risk and arbitrage pricing theory, 立 liquidity costs, Finance and Stochastics 14, 317–341.. [5]. ‧ 國. 學. Finance and Stochastics 8, 311–341.. [6] Chou, R. K., S. L. Chung, Y. J. Hsiao, and Y. H. Wang, 2011, The Impact of. ‧. Liquidity Risk on Option Prices, Journal of Futures Markets 31, 1116–1141.. Nat. sit. y. [7] Cox, J. C., J. E. Ingersoll, and S. A. Ross, 1985, A Theory of the Term Structure of. n. al. er. io. Interest Rates, Econometrica 53, 385—407.. i Un. v. [8] Duffie, D., J. Pan, and K. Singleton, 2000, Transform Analysis and Asset Pricing. Ch. engchi. for Affine Jump-Diffusions, Econometrica 68, 1343–1376. [9] Eraker, B., 2004, Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices, The Journal of Finance 59, 1367–1404. [10] Eraker, B., M. Johannes, and N. Polson, 2003, The Impact of Jumps in Volatility and Returns, The Journal of Finance 58, 1269–1300. [11] Feng, S. P., M. W. Hung, and Y. H. Wang, 2014, Option pricing with stochastic liquidity risk: Theory and evidence, Journal of Financial Markets 18, 77–95. [12] Franzoni, F., E. Nowak, and L. Phalippou, 2012, Private Equity Performance and Liquidity Risk, The Journal of Finance 67, 2341–2373. 30.

(40) [13] Hameed, A., W. Kang, and S. Viswanathan, 2010, Stock market declines and liquidity, The Journal of Finance 65, 257–293. [14] Heston, S. L., 1993, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, The Review of Financial Studies 6, 327-343. [15] Hu, G. X., J. Pan, and J. Wang., 2013, Noise as Information for Illiquidity, The Journal of Finance 68, 2341–2382. [16] Kou, S. G., 2008, Jump-Diffusion Models for Asset Pricing in Financial Engineering, Handbooks in Operations Research and Management Science 15, 73–116.. 立. 政治大. [17] Ku, H., K. Lee, and H. Zhu, 2012, Discrete time hedging with liquidity risk,. ‧ 國. 學. Finance Research Letters 9, 135–143.. ‧. [18] Leland, H., 1985, Option pricing and replication with transactions costs, The. sit. y. Nat. Journal of Finance 40, 1283—1301.. io. al. n. Studies 13, 127– 153.. er. [19] Mello, A. S., and J. E. Parsons, 2000, Hedging and liquidity, Review of Financial. [20] Merton, R. C., 1976,. iv n C Optionh Pricing When U e n g c h i Underlying. Stock Returns Are. Discontinuous, Journal of Financial Economics 3, 125-144. [21] Pastor, L., and R. F. Stambaugh, 2003, Liquidity Risk and Expected Stock Returns, Journal of Political Economy 111, 642–85. [22] Rouah, F. D., 2013, The Heston Model and Its Extensions in Matlab and C#, Wiley & Sons, Inc., Hoboken, New Jersey. [23] Zhu, J., 209, Applications of Fourier Transform to Smile Modeling: Theory and Implementation, Springer, New York.. 31.

(41) Appendix Appendix I. . . f1    E Q e  r  X T  X t i X T.  1i   12  2  at2dt    2 1  2   2 at2 dWtS ,Q    at dWta ,Q    e E e       1 2 1 2 2 2 2 a ,Q i  r  X t    1 i i  1i 1    1i     at dt 1i    at dWt  2 e EQ  e2    1 i  r  X t    2 1 i  2. Q. (2.65). At the same time, the dynamic process of at2 is denoted by Ito’s Lemma as. .  政治大. dat2  2 at  2 at2   dt  2 at dWt a ,Q   2 J a at  J a2  dNt. 立. We can integrate at2 and obtain. ‧ 國. 學. aT2  at2  2  at dt  2  at2 dt    2  at dWt a ,Q. (2.66). (2.67). Rearranging this equation yields. y. 2. (2.68).  a dW ainto (2.65) and then we have iv l C n hengchi U . er. a ,Q. t. t. n. . . io. By inserting. t. aT2  at2  2  at dt  2  at2 dt   2. sit. t. ‧. Nat.  a dW. a ,Q. f1    E Q e r  X T  X t i XT e.  2 1 i   at2 1 i   1 i  r  X t    2 1 i i   2 2 2.  1 1i  1  2   2 1i   2  2    at2 dt  2 1i   at dt  aT  1i     2 2 Q  E  e2      2. (2.69). 32.

(42) Appendix II By Ito’s Lemma, X T is written as (2.70) 2   S   S  2  1 2 e   ln ST  ln St    rt     2 at2      1  dt    2  1   2   2 at2 dWt S ,Q    at dWt a ,Q   J S dNt 2   1   J a     . (2.70) According to method of Appendix I, we derive the dynamic process of at2 as (71). . . dat2  2 at  2 at2   dt  2 at dWt a ,Q   2 J a at  J a2  dNt. (71). By integrating at2 and then rearranging this equation yields.  a dW. t. . a. 政治大  a  2  a dt  2  a dt       2 J a  J dN 立 2 t. 2 t. t. 2. a t. 2 a. t. (72). 2. 學. ‧ 國. t. a ,Q. 2 T. Finally, we can derive the characteristic function of X T as (73). n.   2  S  2S  e 1   i  r  X t    1 i  1    2 1 i i  1  J a  2    . e. Ch. eE n eg c h i. ni U. v. 1 1 i  1  2  2 1 i    2 2. Q. . y. . .  . .  2  1  2  2 at2 dWt S ,Q   at dWta ,Q  J S dNt . sit. io. al.  1i   12  2  at2 dt    EQ  e  . er. Nat. e.   2  S  2S  e 1  i  r  X t    1 i  1   2 1 i   1  J a  2    . . ‧. . f1    E Q1  ei ln ST   E Q e  r  X T  X t i X T.   a dt 1i    a dW 2 t. t. t. a ,Q.    . .  1 i  J S dNt. .   .      2 1 i   at2 1 i   1 i  r  X t    1 i  1   2 1 i i    1  J a  2 2 2     2 S  S 2 e. e.  1 1i  1  2   2 1i   2  2    at2 dt  2 1i   at dt  aT  1i  1i   J S dNt  1i    J a2dNt  1i   J a  at dNt   2 2 2   E  e2   2. Q. (73). 33.    .

(43) Appendix III Table 1 Model’s Parameter Estimation LM. SVC. GBM. JM. LCJC. κ. 0.0046. θ. 0.1595. β. 0.0046. α. 0.2460. 6.2556. 1.4670. ρ. 0.3620. 0.6522. 0.2154. ζ. 0.2108. 0.2654. η. 2.0372. 0.2054. 0.1011 0.0196. 0.6523. σ. 0.3515. σ0. 0.0495. λ. 立. μ. 0.0319 0.0101 0.0287. y. Nat. νS. 18.4751. 0.0292. ‧. μS. 11.0793. 0.0020 0.0104. io. sit. νa. 0.0430. 政治大. ‧ 國. μa. 0.1495. 學. ν. 0.2327. n. al. Average Error Average Error (Unit Stock Price). er. Table 2 Estimation Error In the Sample. LM SVC GBM iv n C h e n g20.7388 24.4803 c h i U 29.8578 0.0021. 0.0018. 0.0338. JM. LCJC. 29.3572. 26.4183. 0.0024. 0.0022. Table 3 Pricing Error Out of the Sample LM. SVC. GBM. JM. LCJC. Average Error. 68.9133. 56.6133. 91.8611. 82.7756. 50.4544. Average Error (Unit Stock Price). 0.0022. 0.0021. 0.0033. 0.0035. 0.0019. The average errors in the sample mean the value of the squared 2-norm of the residual at parameter. On the contrary, the average errors out of sample mean the forecasting errors of option prices.. 34.

(44) Table 4 Pricing Error Out of Sample With Different Situation In the Money. At the Money. Out of the Money. JM. LM. SVC. LCJC. GBM. JM. LM. SVC. LCJC. GBM. JM. LM. SVC. LCJC. GBM. 72.98. 51.21. 42.51. 45.43. 94.43. 65.78. 43.21. 28.86. 37.59. 61.96. 12.94. 9.58. 7.78. 7.71. 8.37. 51.69. 43.59. 36.25. 26.27. 58.40. 54.25. 44.55. 30.12. 41.95. 54.85. 27.92. 19.51. 15.70. 3.89. 20.15. Media. 134.52. 100.89. 72.53. 57.59. 111.66. 34.35. 22.64. 17.62. 14.16. 24.57. Term. 45.06. 50.66. 44.82. 31.11. 54.73. 43.40. 34.15. 26.40. 8.97. 37.11. 145.55. 123.38. 111.59. 71.79. 128.85. 55.12. 70.50. 71.04. 24.43. 55.05. 50.42. 50.08. 50.90. 19.51. 47.26. 55.06. 61.51. 16.32. 47.80. 47.64. 50.28. 50.99. 53.04. 學. Long Term. 165.46 106.57政 82.67 治 55.84 79.64 大 44.18 52.15 51.02 39.65 53.52 立 176.40 117.17 116.14 101.75 115.75. ‧ 國. Short Term. 68.69. 51.70. For a given model, compute the price of each option using previous day’s implied parameters. The reported pricing error is the sample average of the market price minus the. Nat. At the Money. al. er. io. In the Money. sit. Table 5 Improvement Rate of Pricing. LM. SVC. LCJC. Short Term. 0.23. 0.46. 0.55. 0.31. Media Term. 0.19. 0.39. 0.56. 0.36. iv -0.06 0.53 n C h 0.3 g c h i U0.5 0.05 e n0.26. Long Term. 0.17. 0.3. 0.37. 0.16. 0.09. n. JM. JM. LM. 0.1. SVC. 0.21. y. ‧. model price. The corresponding standard errors are recorded in this table. The sample period is from January 1 to December 31, 2011.. Out of the Money. LCJC. JM. LM. SVC. LCJC. 0.39. 0.55. -0.14. 0.07. 0.08. 0.29. 0.4. 0.08. 0.28. 0.42. 0.1. 0. -0.28. -0.29. 0.56. The improvement rate is defined as the average of the differences between the pricing errors from the Black-Sholes and other models over the pricing errors from the other model across all stocks. The definition is formulated as.  pricing error of Black-Scholes - pricing error of other model  / pricing error of other model .. 35.

(45) 政治大 Figure 2 Effects of Liquidity Parameter on Option Price (1) 立 ‧. ‧ 國. 學. The model parameters for stock as well as liquidity are represented in Table 1.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 3 Effects of Liquidity Parameter on Option Price (2) The model parameters for stock as well as liquidity are represented in Table 1.. 36.

(46) 政治大 Figure 4 Effects of Liquidity Jump Parameter on Option Price 立 ‧. ‧ 國. 學. The model parameters for stock as well as liquidity are represented in Table 1.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 5 Delta in Different Models The model parameters for stock as well as liquidity are represented in Table 1.. 37.

(47) 政治大 Figure 6 Gamma in Different Model 立 ‧. ‧ 國. 學. The model parameters for stock as well as liquidity are represented in Table 1.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 7 Rho in Different Model The model parameters for stock and liquidity are represented in Table 1.. 38.

(48) 政治大 Figure 8 Theta in Different Model 立 ‧. ‧ 國. 學. The model parameters for stock and liquidity are represented in Table 1.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 9 Vega in Different Model The model parameters for stock and liquidity are represented in Table 1.. 39.

(49) 政治大 Figure 10 Delta in Different Model ( 2 Dimension ) 立 ‧. ‧ 國. 學. The model parameters for stock as well as liquidity are represented in Table 1.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 11 Gamma in Different Model ( 2 Dimension ) The model parameters for stock as well as liquidity are represented in Table 1.. 40.

(50) 政治大 Figure 12 Rho in Different Model ( 2 Dimension ) 立 ‧. ‧ 國. 學. The model parameters for stock as well as liquidity are represented in Table 1.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 13 Theta in Different Model ( 2 Dimension ) The model parameters for stock as well as liquidity are represented in Table 1.. 41.

(51) 政治大 Figure 14 Vega in Different Model ( 2 Dimension ) 立 ‧. ‧ 國. 學. The model parameters for stock as well as liquidity are represented in Table 1.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 15 Effect of Sensitivity to Liquidity on Delta The model parameters for stock as well as liquidity are represented in Table 1.. 42.

(52) 政治大 Figure 16 Effect of Sensitivity to Liquidity on Gamma 立 ‧. ‧ 國. 學. The model parameters for stock as well as liquidity are represented in Table 1.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 17 Effect of Sensitivity to Liquidity on Rho The model parameters for stock as well as liquidity are represented in Table 1.. 43.

(53) 政治大 Figure 18 Effect of Sensitivity to Liquidity on Theta 立 ‧. ‧ 國. 學. The model parameters for stock as well as liquidity are represented in Table 1.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 19 Effect of Sensitivity to Liquidity on Vega The model parameters for stock as well as liquidity are represented in Table 1.. 44.

(54) 政治大 Figure 20 Effect of Correlation Between Return and Market Illiquidity on Gamma 立 ‧. ‧ 國. 學. The model parameters for stock as well as liquidity are represented in Table 1.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 21 Effect of Correlation Between Return and Market Illiquidity on Theta The model parameters for stock as well as liquidity are represented in Table 1.. 45.

(55) 政治大 Figure 22 Effect of Correlation Between Return and Market Illiquidity on Vega 立 ‧. ‧ 國. 學. The model parameters for stock as well as liquidity are represented in Table 1.. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 23 Effect of Liquidity Jump Frequency on Theta The model parameters for stock as well as liquidity are represented in Table 1.. 46.