Option Pricing Techiniques

INDEX

1. INTRODUCTION

2. METHODS OF ESTIMATION AND MODEL SELECTION

2.1 Volatility and distribution analysis of my data

2.2 Naïve (Geometric Brownian Motion)

2.3 LASSO

2.4 QMLE

2.5 GARCH

3. OPTION PRICING

3.1 Comparison between different methods for option pricing:

- European Options: Black & Scholes, Monte Carlo, Levy process

- American Options: Implicit and Explicit Finite-Difference methods, Broadie & Glasserman, Longstaff and Schwartz Least Squares Method, Black & Scholes and Barone-Adesi Wiley approximations.

3.2 Option pricing for volatility trading strategies

4. CONCLUSIONS

5. APPENDIX: R codes

1. INTRODUCTION

The paper I will develop and deepen will be based on Amazon.com Inc, eBay Inc., Best Buy Co. Inc. and Barnes and Noble Inc. stocks (from now on called with their symbols: AMZN, EBAY, BBY, BKS), whose historical data is taken from Finance.Yahoo.com, through the software called 'R'.

Being these companies the four biggest and most important online retailers in the US, I will analyze in general how the E-commerce market, one of the most young markets that it is still in development, behave in a one year time horizon (data are mostly taken from 23-05-2011 to 23-05-2012).

In the first part of the paper I will focus my attention on different models through which I can do inference on my data, taking into account the Naïve method (for the Geometric Brownian Motion, that will be proved to be the best model for doing inference), the so called LASSO estimation, the Quasi - Maximum Likelihood Estimation (QMLE) and finally the GARCH estimation.

In the second part I will analyze different methods for both European option pricing (Black & Scholes, Monte Carlo and Levy process) and American option pricing (Implicit and Explicit finite methods, Broadie & Glasserman, Black & Scholes and Barone-Adesi Wiley approximations).

I will then try to build contracts for volatility trading strategies and give suggestions on which of the four stocks can be considered the most profitable and less costly, focusing on strategies such as straddles and strangles, built with combinations of two different options.

In the last part I will provide a conclusion related to the results given by the various methods of option pricing and a comparison between them in order to demonstrate which is the most reliable.

2. METHODS OF ESTIMATION AND MODEL SELECTION

2.1 VOLATILITY AND DISTRIBUTION ANALYSIS OF MY DATA

First of all, I will go through the analysis of the volatility and distribution of AMZN's, EBAY's, BKS's and BBY's log-returns, plotting the paths for the log-returns (volatility of the stocks), the density and the Normal Q-Q Plot (needed to show how much my data can be compared with the normal Gaussian), together within a single chart.

As it can be noticed from the figure above, the distribution of AMZN's log-returns fits pretty well the one of a standard normal distribution N(0,1) given by the blue dashed line in the middle plot. This statement is moreover highlighted in the last plot, the Normal Q-Q Plot, where most of the dots are positioned on the red line, which represents the normal standard distribution. As a matter of fact the Normal Q-Q Plot shows that the more the circles are placed on the red line, the more the data fits the standard normal distribution.

From the first of the three plots, it is possible to infer that AMZN's stock is quite volatile but within a (little) range (-0,05;0,05), despite the data shows 2 peaks right before November 2011 (downward peak) and May 2012 (upward peak).

Hereinafter I continue with the same analysis of the data as I did for AMZN, reiterating the same sequence of commands in R for each of AMZN's competitors:

1. eBay Inc.

The first plot shows a higher volatility of the stock with many more peaks than in AMZN's. The data distribution fits quite well the one of a standard normal distribution and the normal Q-Q plot chart confirms it, because most of the dots are placed on the red line.

2. Best Buy and Co.

Even for this company the chart above shows a high level of volatility, and a very good distribution of data with the blue dashed line almost overlapping the density distribution. In addition, the normal Q-Q plot has most of the dots placed on the red line, even in the right tail.

3. Barnes & Noble Inc.

The volatility of this company is still high as in its competitors but this is the company that most of all shows a data distribution quite far from the normal standard distribution as it can be noticed by the blue dashed line which is a lot smaller than the real density distribution.

All in all the E-commerce market, well represented by these four companies, shows a high degree of volatility and has data which is distributed, with the only exception of BKS, as a normal standard distribution.

This will lead to the conclusion that option pricing models based on the Geometric Brownian Motion, whose basic assumption is the log-normal distribution of stock returns, are the most appropriate to replicate the market.

From now on I will analyze and compare different methods of estimation and model selection starting from the Geometric Brownian Motion.

2.2 NAÏVE - GEOMETRIC BROWNIAN MOTION PARAMETERS

I will now compute the parameters for the Geometric Brownian Motion model, whose stochastic differential equation is:

This model is a particular case of the more general class of Stochastic Differential Equations and in order to be precise I will use the log-normal version of gBm because the classical Brownian Motion (Wiener process) has the peculiarity to be nowhere differentiable with respect to time. This can be demonstrated, as we saw during lectures, via a graphical view. The graph above shows an explosive behavior of the increments of the Brownian motion because the increments W(t+Δt) -W(t) behave like √Δt instead of Δt. The following limit demonstrates this trend:

At this point, computing the parameters for the gBm

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