Canonical Least-Squares Monte Carlo Valuation Of Option:Based On Moment-Constraints From Option Price Information

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. InChapter 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 experimentThe 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 experimentThe 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 optionsFor 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 measureIncorporating 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.