LEAST -SQUARE MONTE CARLO BASED OPTION PRICING OF EUROPEAN AND BERMUDAN STOCK INDEX OPTIONS

Abstract

On the financial markets, there are a large number of financial instruments. Two of these instruments is the European and Bermudan option, where the Bermudan option can be seen as a discrete version of the American option. Meaning, if one can price the Bermudan option one can also estimate the price of an American option. A method used to estimate the Bermudan option price is the Least-square Monte Carlo approach. It is a numerical approach that uses simulated values of the underlying asset and fits a polynomial for each date exercise is possible.
The function is used to estimate the holding value of the option, by which one can determine whether to exercise the option. Using four different price movement models to simulate the value of the underlying asset, European option prices were estimated using the standard Monte Carlo method and Bermudan option prices were estimated using the Least-square Monte Carlo approach. The results show that the pricing of the European options frequently results in options prices outside the ASK/BID-spread. It also shows tendencies towards better estimations using price movement models containing more parameters, but that these models do not always show better results. Probably, it is because of external problems such as parameter fitting. The results also show that the Least-square Monte Carlo approach works sufficiently well when pricing the Bermudan option, but that in some cases incorrect estimations are made stemming
from the fitted polynomials. To conclude, the Monte Carlo based option pricing methods are considered to work and result largely in satisfactory estimations, but contain problems such as the choice and fitting of polynomials and parameter calibration.

Popular Abstract

In the financial markets, there is a never-ending development of asset prices. Thus,for traders to make a profit from them,there exist a need to develop superiormethods to determine these instruments’value. One asset that is in particular com-plex to price is options.Options are contracts that give the holder theright to either buy or sell its underlying assetfor a predetermined price at one or more futuredates. In the former case, the contract is calleda call option, while it is called a put option whenthe latter terms apply. There are several ap-proaches to price these contracts, one of whichis through Monte Carlo-based methods.Theidea is to simulate trajectories of the underlyingasset price and estimate the option price basedon these paths. At first, the method was ap-plicable for only European options, which areoptions with one date of possible exercise, but intime the methods developed to include contractswith several dates of possible exercise as well.One of which is Bermudan options, a contractthat allows the owner to exercise the option ata finite number of dates. However, the methodsare not flawless. The aim of our thesis was toexamine the accuracy and robustness of some ofthese pricing methods when pricing Europeanand Bermudan options. In addition, attemptshave been made to develop them further. Ourresearch is important for several reasons, for in-stance for traders that want to set a proper pricewhen issuing an option for which there is no cur-rent market. Also when option prices already areavailable, there is a need to check whether theyare considered reasonable or not.Through our research, we found that the meth-ods’ accuracy varied a lot depending on howthe underlying asset price was simulated. TheMonte Carlo-based European prices were morealigned with the real ones when using modelswith time-varying volatility. However, when in-cluding random jumps in the model to considerfor news that affect the underlying asset price,the resulting prices got worse.The pricing of Bermudan options is much morecomplex. The general idea is to decide whichdate the holder receives the optimal payoff ineach price trajectory, and use these payoffs whenestimating the option price. However, due tolack of data on the real option prices, the onlyway of checking whether our prices were reason-able, was to compare them to the correspondingEuropean option price. From basic financial the-ory is it known that the Bermudan prices shouldbe equally as large or larger. This was the casefor almost all put options, but almost none ofthe call options. The reason is that it is difficultto approximate the correct size of the payoffsand when they should occur for each trajectory.The first method implemented was the LSMapproach. In short, it is about regressing a func-tion at each point in time the holder can exercisethe option, and using it as a criterion to decidewhether the holder should exercise the optionor not. To improve the results, a modificationof the standard LSM was made to form the so-called exercise boundary and use it as a decisioncriteria for whether the holder should exercisethe option. Basically, the exercise boundary is agraph from which one can identify for which as-set prices and times left to the options expirationdate, it is worth exercising the option. However,the result got even worse with the method dueto amplified errors that already existed with thestandard LSM approach.

• author: Brink Bolin, Oscar and Ahnvik, Joel
• supervisor: Magnus Wiktorsson
• organization: Mathematical Statistics
• alternative title: LEAST-SQUARE MONTE CARLO BASERAD OPTIONSPRISSÄTTNING AV EUROPEISKA OCH BERMDUA AKTIEINDEXOPTIONER
• course: FMSM01 20212
• year: 2022
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: Option, Monte Carlo *, Least-square *, Black-Scholes, Merton, Heston, Bates
• publication/series: Master's Theses in Mathematical Sciences
• report number: LUTFMS-3436-2022
• ISSN: 1404-6342
• other publication id: 2022:E4
• language: English
• id: 9072444
• date added to LUP: 2022-01-21 10:20:53
• date last changed: 2022-01-21 10:20:53