摘要
现代期权定价理论提供的是基于风险中性概率密度的无套利价格,其对应的定价方法往往都要对标的资产价格过程,或者市场完备性甚至市场参与者进行模型或其它的假定,而这些预设和假定几乎都与实际的市场不相符。因此,为了定价的结果更为理性、切合真实市场的表现,不能过度地依赖模型和一些假设,而应从现实金融市场中充分获取对定价有用的信息。目前诸多非参数定价方法又往往只从标的资产市场获取相关信息,进而得到标的收益风险中性分布,为衍生产品定价。然而,期权市场也蕴含许多的有效信息,这些有效信息可以准确反映市场的各种预期,包括标的资产收益的预期分布,从而能够捕捉到与实际市场相符的风险中性分布的“形状”,比如能够考虑波动率微笑(volatility smile)和尾部行为(tail behavior)。
基于此,本论文研究并提出能够为期权理性定价的不依赖模型(或无模型,model-free)方法：除了使用标的资产价格,还从期权市场有限数量的价格数据中提取对定价有效的信息,将其数学化；并结合这些来自真实市场的信息,借鉴信息熵原理建立期权的无模型定价方法,得到更“准确”的风险中性定价测度,为期权给出符合实际市场的理性定价。论文的研究用到金融理论、随机数学、信息熵理论以及数值计算与实现技术。
论文的贡献在于将标的资产(对数)收益的风险中性矩作为约束,嵌入canonical最小二乘蒙特卡罗定价框架(Longstaff&Schwartz:2001; Stutzer,1996; Alcock&Carmichael,2008; Liu,2010),形成带矩约束的canonical最小二乘蒙特卡罗熵(MCLM)的无模型期权定价方法。MCLM从一个新的视角基于最大熵原理和最小二乘蒙特卡罗算法为欧式及美式期权定价。目前已有的方法(如Alcock and Carmichael,2008和Liu,2010,分别简记为AC和CLM),在其定价模型中,或者只使用鞅约束以及其它历史估计参数,或者需要大量的历史数据来生成风险中性分布,与之不同,MCLM使用标的资产对数收益的风险中性矩约束和少量而有效的历史数据估计出更合适的风险中性分布作为定价测度为欧式期权定价,并结合蒙特卡罗技术为美式期权定价。
在MCLM方法中,风险中性矩可以通过使用数个美式看涨期权或虚值欧式看涨看跌期权价格来估计,实现方法相当简单、易得。重要的是,这些风险中性矩在推导出风险中性分布时起着极为重要的作用,因为它们能够准确捕捉到市场中关于标的价格(收益)分布的信息,不仅能够精确估计出标的风险中性收益率,且能有效地将波动率、峰度、偏度等因素考虑进去,而不需要进行任何事先假设。相对目前已有的熵定价方法及其它一些基准方法,这是MCLM方法的优点所在。而且在理论上证明了,基于同-Black-Scholes假设下,MCLM方法得到的期权价格正好就是Black-Scholes价格。
为期权定价,MCLM方法先使用估计出的风险中性测度直接生成足够多的条标的价格样本路径,从而这些路径都是风险中性的,避免了使用大量的历史数据。我们的研究仅使用365个历史收益来为期权定价,而通常非参数方法往往要求许多的历史数据,比如Alcock和Auerswald(2010)则需要使用超过7,000个历史收益数据来计算风险中性测度。因此我们的方法更加灵活、切合实际,特别是当历史数据不可得或者过旧以至不能够准确反映当前的市场。
论文进行了两个模拟市场实验,用来评价MCLM方法并从不同角度同其它的基准方法进行了比较,在该模拟实验中使用的基准方法有Black-Scholes公式(其价格将作为无红利支付美式看涨期权的真实价格)、Crank-Nicolson Finite Difference方法(将作为美式看跌期权真实价格)、AC和CLM方法。首先,有关估计风险中性矩的结果表明,MCLM方法得到的风险中性矩估计值与其理论值非常吻合。其次,第一个实验中,MCLM得到的价格也几乎与理论价格一致；使用的MCLM方法仅对平价和深度实值看跌期权产生负的定价偏差,而AC则对看涨、看跌期权无论什么状态均产生负偏差；误差统计量分析显示出了,MCLM方法在定价时的高度无偏性以及对每一次模拟的稳定性；均方误差(MSE)和平均百分误差(MPE)的结果也表明MCLM方法优于AC方法。第三,第二个实验中,在两个收益率(漂移率)下,使用MCLM得到的美式看涨及看跌期权价格均与其对应的“真实”价格非常接近,且稍微低于其“真实”价格,而CLM则展现出一致的正向定价误差；而且,MCLM导致的误差在所有的moneyness情况下比较均匀；通过比较绝对误差发现,我们定价方法的精确度高于CLM的,特别是对于美式看跌期权更是占绝对优势。最后,有理由将AC和CLM方法看作MCLM的特例。
论文还使用IBM股票期权数据,对提出的MCLM方法进行了实证研究及与其它方法(FD、AC、CLM)的对比,所使用数据的期间覆盖2008年金融危机时期,在该时期对应的利率非常低而IBM股票又存在相对数量不小的分红,为此红利、非常数利率因素在研究中均被考虑进来。实证结果显示,对所有moneyness和：maturity下的期权,MCLM方法的定价误差均比对比的方法小很多。对IBM看涨期权,来自MCLM的误差仅仅是CLM定价误差的一半左右；对IBM看跌期权,AC方法在短期maturity时优于FD方法,而在长期maturity时FD则优于CLM,但我们的MCLM方法则比两者都占显著优势,尤其是对实值和深度实值看跌期权,MCLM能够给出非常精确的定价。无论是IBM看涨还是看跌期权,总的结果表明MCLM方法的定价比其它基准方法定价更优。
模拟和实证检验表明,从估计风险中性矩能力与降低定价误差的角度,我们的MCLM定价方法比一些基准方法,包括无模型方法AC和CLM都占有优势。原则上,MCLM方法可以应用于任何其它的虚拟市场环境和实际市场,因为其能够从期权市场有效地捕捉到标的资产收益分布的信息,从而准确估计出风险中性定价测度,而不需要对标的资产价格过程强加任何假设。
本论文的结构如下：
第一章是导论,介绍了衍生产品定价的一些金融与数学背景知识,包括文献综述和论文的组织架构。
第二章给出了一些基准定价方法,包括Black-Scholes期权定价公式、Crank-Nicolson有限差分法(Crank-Nicolson Finite Difference)、AC和CLM方法,这些方法在接下的章节中要用于作为对比方法。
第三章则提出了对数收益的风险中性矩(RNM)概念,并且建立起RNM与期权价格之间的关系式用以从期权数据中获取RNM,这一章还详细讲解了如何实现这一关系。
第四章给出了完整的定价模型以及得到风险中性分布(RND)的具体程序及其数值实现方法,同时,该章还讨论了RND的存在性与唯一性条件。
第五章则介绍了使用MCLM方法对欧式与美式期权进行定价的具体步骤,并且进行了两个模拟实验与定价结果比较。
在第六章我们使用IBM看涨、看跌期权对MCLM方法进行实证分析,并同其它方法加以比较。
第七章是总结与展望。
最后给出了本论文中一些引理与定理证明、图与表。若需,论文研究使用的程序代码也可随时提供。
论文得到的主要结果：
1.证明了基于已知的准确信息约束下,所构建的最大熵模型,能够提供最符合真实的风险中性鞅测度,可用来作为期权的风险中性定价测度；并且,在风险中性矩不相关的条件下,这样的鞅测度是唯一存在的。
2.在标的资产价格服从几何布朗运动时,论文的MCLM定价模型得到的风险中性测度恰好是Black-Scholes定价的风险中性测度(定理5.1)。
3.提取风险中性矩、估计风险中性分布的能力
在模拟环境下,检验从期权价格提取风险中性矩的实验中,我们的MCLM方法能够精确估计风险中性矩(与理论值一致),而基准方法CLM却不然(表5.4)。MCLM得到的最大熵分布是“风险中性”的(图5.1,图5.2)。
4.对于第一个模拟实验
MCLM方法得到的价格几乎均与真实价格一致(表5.5-5.6)；定价精确度基本随moneyness单调递增,在深度实值时的误差非常小(表5.7-5.8)；MCLM方法只对平值和深度实值看跌期权产生价值低估,而AC方法产生一致的负偏差(表5.9-5.10)。
我们方法得到的定价结果,高度无偏,且对每一个模拟结果均比AC方法更稳定(通过MSE统计量反映)(表5.9-5.10)。
5.对于第二个模拟实验(表5.11-5.12)
在不同的漂移率(growth rate、drift)下,MCLM价格与真实价格几乎一致但低于真实价格,无论是美式看涨或看跌期权；而CLM产生正向定价误差。
MCLM定价计算比CLM要稳定；总体上,MCLM定价比CLM定价更精确,尤其是对美式看跌期权。
6.对于IBM美式期权的实证
对IBM看涨期权,MCLM定价误差基本是CLM定价方法误差的一半(表6.2)；对IBM看跌期权,MCLM定价精确度很高,且明显优于CLM和FD方法,特别是在实值和深度实值状态(表6.2)。
除了实值和短期期权,MCLM方法基本低估期权价格,但几乎对所有类别期权其定价误差都比其它基准方法要低。
论文主要创新点体现在：
1.从期权市场提取有效信息——风险中性矩
除标的市场外,期权市场也蕴含诸多对定价有效的信启、。论文选择“标的资产对数收益的风险中性矩”作为待提取的有效信息,建立一种方法,不依赖模型地从有限的期权市场价格中获取这些风险中心矩(这些矩能够准确反映标的市场的各种预期尤其是标的资产的收益分布特征,比如可以捕捉到“波动率微笑”和“尾部现象”)。
首先引入随机数学中的特征函数,使用随机积分方法建立起风险中性矩与期权价格之间的理论关系；其次采用一种稳定的数值计算技术,实现这一理论关系,使得能够直接从数量有限的期权市场价格中准确提取出富含有效信息的各阶风险中性矩。
2.理性定价测度的确立
将以上提取到的富含信息的风险中性矩,作为约束条件嵌入到最大熵框架,根据信息熵理论求解出更加“理性”的风险中性定价测度,进而可以为期权给出符合实际市场表现的理性定价。
这一定价测度与所提取的市场有效信息相吻合,且是风险中性的、唯一的,从而有望突破以往不完备市场中等价鞅测度不唯一的难题。
在标的资产价格服从几何布朗运动时,论文的MCLM定价模型得到的风险中性测度恰好是Black-Scholes定价的风险中性测度。
3.无需大量市场数据、很容易考虑红利因素,可以为多类衍生品定价
论文提出的MCLM方法,从其具体定价过程可以看出,该方法能够应用到美式期权以及其他路径依赖期权定价；且很容易将红利情况考虑进来。
不需要大量的标的以及期权历史价格数据(同目前已有的非参数定价方法比较,此为MCLM方法的一大优势)。
Modern option pricing theory provides arbitrage-free price that is ensured by employing risk-neutral pricing densities, but the valuations usually rely on a series of assumptions such as assuming a specific process for the underlying price or presupposing the completeness of market which are almost not consistent with the realistic markets. This thesis aims to recover some efficient information from the available option prices and incorporate those information into our pricing framework to generate superior estimate of the risk-neutral pricing measure for option pricing.
For obtaining a rational price consistent with the market, one therefore couldn't rely on any unreasonable assumption but make the most of information from financial market. Currently, however, some nonparametric methods just utilize related information from the underlying market, and ignore those from option market which reflect market expectations about future asset market events and returns distributions so that we can learn the shape of RND by taking into account volatility smile and tail behavior.
In view of mentioned above, this thesis aims to propose a model-free valuation approach to offering a rational value for options:retrieving some efficient information (RNMs) from the available option prices and incorporate those information into entropy pricing framework to generate superior estimate of the risk-neutral pricing measure for option pricing. This study involves Finance Theory, Stochastic Mathematics, Information Theory, Numerical Computation and Implementation Technique.
This thesis contributes to the improvement of option pricing accuracy by incorporating risk-neutral moments as constraints into the canonical least-squares Monte Carlo valuation framework (Longstaff&Schwartz,2001; Stutzer.1996; Alcock&Carmichael.2008; Liu.2010). An ideal moment-based canonical least-squares Monte Carlo valuation is introduced here and this method can be taken as an improved nonparametric valuation technique to price European and American options based on the principle of maximum entropy and least-squares Monte Carlo approach from a new perspective. Unlike the current methods in the literature (e.g. Alcock and Carmichael,2008and Liu,2010, abbreviated as AC and CLM, respectively) either relying on only martingale constraint or employing a large set of historical data to generate a risk neutral distribution, our valuation approach uses risk-neutral moments of underlying asset return as constraints and a much smaller set of historical data to generate a better estimate of risk-neutral distribution as the pricing measure and then utilizes it to price European options or into the tractable Monte Carlo techniques to price American options.
The risk-neutral moments in our valuation approach can be estimated using several American call options or out-of-the-money European call and put options, and the implementation technique is relatively simple and tractable. More importantly, these risk-neutral moments play a significant role for deriving risk-neutral distribution, owning to their ability of capturing market information, not only can they accurately estimate the risk-neutral growth rate for underlying asset, but also effectively take volatility, skewness, and kurtosis into consideration without imposing any structural assumption. Compared with other existing entropic valuation methods and some benchmark methods, this is an outstanding feature for our valuation approach. Further, it is theoretically proved that, given the Black-Scholes assumptions, the evaluated price from MCLM is just that one from Black-Scholes formula.
MCLM valuation approach uses the estimated risk-neutral measure to directly generate a large number of risk-neutral underlying price paths for valuing options. This procedure avoids requirement of a large set of historical data which is a common issue in a nonparametric valuation method. In this study, we use365historical returns to derive a risk-neutral measure for all options to be priced, while Alcock and Auerswald (2010) needs to calculate the risk-neutral measure for each option which requires more than7,000historical return observations. Hence our approach is more practical, especially when a large set of historical data is not available.
We conduct two simulation experiments to evaluate the usefulness of our method and compare its performance from several aspects with that of Black-Scholes formula (as the "true" price of a call option), Crank-Nicolson Finite Difference (FD, hereafter, as the "true" price of a put), AC and CLM. First, the results on extracting risk-neutral moments suggest that our moment estimates match quite well with their corresponding theoretical values in both simulation experiments. Second, in the first experiment, the resultant prices from MCLM are almost same as the true prices. Our approach underprices only at-the-money and deep-in-the-money put options whereas AC exhibit negative bias for all levels of moneyness regardless of calls or puts. Error metrics analysis suggests that the price estimates using our approach are largely unbiased and stable for every simulated price. The mean-square error (MSE) and (mean percentage error) MPE results suggest that our approach outperforms significantly over AC. Third, in the second experiment, the estimated prices using MCLM are fairly close to the "true" prices for American call and put options in both cases of growth rates and all price estimates are less than the "true" prices for both call and put options, whereas CLM persistently exhibits positive bias. Furthermore, the price bias of MCLM is more stable for two cases of growth rates. By comparing the absolute difference between estimated and "true" prices, the overall accuracy of our approach is higher than that from CLM, and particularly is dominant in pricing American puts. Finally, it is not unreasonable to imagine that MCLM nests CLM or AC method as a special case.
We also test our valuation approach and compare its performance with AC for call options and CLM and FD for puts using the IBM option data with a period covering the2008US financial turmoil while the interest rate had been quite low. The results show that the pricing bias by our approach is much lower than other compared methods for almost levels of moneyness and time to maturity. For IBM calls, the pricing errors from MCLM nearly equals to a half of that from CLM method. With regard to IBM put option valuation, CLM performs better than FD when the time to maturity is short, whereas FD outperforms CLM across the moneyness with long maturity, but our method is significantly dominant over CLM and FD, especially for in-the-money and deep in-the-money, MCLM can price put options very well with a quite high accuracy. In brief, all the results suggest again that our approach performs well and much better than other benchmark methods.
Simulation and empirical testing results demonstrate that our valuation approach outperforms the methods of some benchmark valuations including several model-free valuation approaches in terms of reducing pricing errors and capability of recovering risk-neutral moments. In principle, MCLM can be applied in any other artificial circumstances and real markets due to their ability in effectively capturing information in option market for generating a better estimate of risk-neutral measure without imposing any structural assumption of underlying asset price.
This dissertation is organized as follows.
Chapter One briefly presents some financial and mathematical background and the literature review, the structure of this thesis as well.
In Chapter Two, some benchmark valuations to be compared in following chapters including Black-Scholes price formula, Crank-Nicolson Finite Difference, AC and CLM are given.
Chapter Three provides the risk-neutral log-return moments (RNMs) and bridges the relationship between RNMs and option prices so that RNMs can be extracted using option prices, also the implementation technique is specified.
Chapter Four presents our valuation framework with detailed numerical procedures for obtaining the risk-neutral distribution (RND), meanwhile the existence and uniqueness of RND are discussed.
Chapter Five then exhibits the detailed steps of MCLM for pricing European and American call and put options, two simulation experiments for comparing our method with other benchmark approaches are also conducted in this chapter. In
Chapter Six, we conduct the empirical analysis on valuation of IBM call and put options and make several comparisons with other methods.
Conclusions and remarks are given in Chapter Seven.
The main results are:
1. With all the known information, our entropic model can produce market-oriented risk-neutral measure and this measure would be used as the pricing measure; This measure solution is unique if the risk-neutral moments are independent.
2. As the underlying price obeys a GBM, the risk-neutral measure from MCLM is just the same as that from Blask-Scholes.
3. Ability in extracting risk-neutral moments and estimating risk-neutral distribution:
In that simulation circumstance, MCLM method can accurately estimate the risk-neutral moments, but not for CLM method. The resultant entropy distribution from MCLM is risk-neutral.
4. For the first experiment
The resultant prices from MCLM are almost same as the true prices (Table5.5-5.6); The estimate accuracy increases monotonically with moneyness and pricing error is small when option is deeply in the money; MCLM underprices only at-the-money and deep-in-the-money put options, whereas AC exhibit negative bias for all levels of moneyness regardless of calls or puts.
The price estimates using our approach are largely unbiased and more stable (by MSE) than AC method for every simulated price.
5. For the second experiment
The estimated prices using MCLM are fairly close to the "true" prices for American call and put options in both cases of growth rates; All price estimates are less than the "true" prices for both call and put options, whereas CLM persistently exhibits positive bias.
The price bias of MCLM is more stable for two cases of growth rates than CLM; The overall accuracy of our approach is higher than that from CLM, and particularly is dominant in pricing American puts.
6. Empirically investigation using IBM options
For IBM calls, the pricing errors from MCLM nearly equals to a half of that from CLM method; With regard to IBM put option, MCLM can price put options very well with a quite high accuracy and is significantly dominant over CLM and FD, especially for in-the-money and deep in-the-money.
MCLM underprices calls as well as puts except for ITM and short term to maturity, but our approach outperforms other methods for both calls or puts. The pricing bias of our approach is much lower than other methods for almost all levels of moneyness and time to maturity.
Three main highlights are as follows:
1. Extracting efficient information directly from option market--RNMs
In addition to the underlying market, option matket also contains much efficient information for option pricing. We choose the RNMs of underlying log-return as the information to be recovered from option prices in that these RNMs can correctly reflect the market expectations about future asset market events and returns distributions such as volatility smile and tail behaviour.
The characteristic function is first introduced and the stochastic mathematics is used to bridge the RNMs with option prices. One stable numerical technique is then employed to implement the RNMs extracting from option market.
2. Determination of the rational pricing measure
Incorporating the above risk-neutral moments as constraints into the canonical least-squares Monte Carlo valuation framework provides the rational pricing measure to price option.
This pricing measure matchs the information recovered from option market and is risk-neutral and unique so that one might bypass the non-uniqueness of equivalent martingale measure in an incomplete market.
As the underlying price obeys a GBM, the risk-neutral measure from MCLM is just the same as that from Blask-Scholes.
3. MCLM valuation approach can be used to pricing other path-dependent derivatives, considering the pricing procedures; this method can readily take the dividend into consideration.
Another outstanding feature of MCLM is that it doesn't make a requirement of used price data.
引文
8Louis Bachelier是法国伟大数学家Poincare(庞加莱,1854-1912)的博士研究生。Bachelier被誉为数理金融学的开拓者、现代期权定价之父。
91905年5月11日在Ann. Phys.,17,P549上发表的德文论文《热的分子运动论所要求的静液体中悬浮粒子的运动》。
18文献综述可参见Bahra (1997), Jarrow和Rudd (1982), Ait-Sahalia和Lo (1998), Jackwerth(1999)。方法与应用问题见Kang和Kim (2006), Jarrow和Rudd (1982), Jackwerth和Rubinstein (1996),Grundy(1991),Bates(1991),Melick和Thoraas(1997), Britten-Jones和Neuberger(2000), Jiang和Tian (2005), Bakshi和Madan (2000), Bakshi, Kapadia和Madan (2003).
19 Stutzer和Chowdhury (1999), Foster和Whiteman (1999,2006)分布对债券期货期权、CBOT上交易的大豆期货期权进行canonical方法的实证分析：Gray和Newman (2005), Alcock和Gray (2005), Gray、Edwards和Kalotay (2007)也分别对此方法进行了各方面的检验。并且所有这些研究结果都是正面的。
20与前文中的“最大交叉熵”一致,只是将交叉熵中的对数函数部分log[p(x)/q(x)]换为hg[q(x)/p(x)]、从而“最大交叉熵”变为了“最小交叉熵”。
21可参见Bossaerts(1989), Tilley (1993), Barraquand和Martineau (1995), Broadie和Glasserman (1997a,b,c),Broadie, Glasserman和Jain (1997), Raymar和Zwecher (1997), Broadie et al. (1998), Carr (1998), Carriere (1996),以及Tsitsiklis和Van Roy (1999).
30风险中性收益矩的表达式为积分形式,实际中可用数值方法求解,而且只需数个期权数据(Buchen, 2006; Rompolis,2010).
31一些实证研究,比如Dueker和Thomas (1994), Zivney (1991), Poteshman和Serbin (2002)均表明绝大多数情况下,深度虚值的美式看涨期权都不会提前执行。Broadie et al. (2000)实证研究了OEX 100指数期权,发现几乎所有提前执行的看涨期权仅发生在到期前几天,而且只有当指数价格接近执行价格时。
38该定理的传统证明方法有对偶法,比如可参见Ben-Tal(1985,第268页)。
39这里提到的数值计算方法,其计算原理与具体计算过程,可参考任何数值计算方法的书籍,在此省略。
40首先他们建议基函数中的资产价格需要用执行价格标准化,以避免数值计算的复杂化；其次,为了计算时间与精确度的平衡,Stentoft(2004)说明了2阶或3阶Legendre多项式更好(相比其它正交多项式比如Laguerre族,对计算的要求更低)。
41为方便下文中的比较(与AC和CLM方法),在此同样假设红利率为零(q=0)。但是从我们的程序实现来看,即便有存在连续或离散分红,也很容易考虑进去。
42由于使用的历史τ-期收益收据个数是H,因此要求历史数据时间窗口为[-(H+τ),0]。我们选取365个历史数据主要在于两点：首先,当时间窗口过宽时(即H太大,比如Aclock和Auerswald(2010)在得出风险中性测度时须要超过70,000个数据),从真实市场中观察到的历史收益数据可能过旧,而不能够准确反映时下的收益状态(Liu和Yu,2010),并且不一定能够收集到如此多的市场历史数据；其次是因为要考虑到计算的时间与有效性问题(Liu,2010)。
51股票价格、看涨期权价格、看跌期权价格可同时从同一个页面表格下载。
52也见"Option, Futures,and Other Derivatives"(第七版,John C. Hull)第299-300页。
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