How to Calculate Options Prices and Their Greeks

Exploring the Black Scholes Model from Delta to Vega

Pierino Ursone

• Publisher: Wiley
• Year: 2015
• Print ISBN:
  9781119011620
• Edition: 1
• eBook ISBN:
  9781119011637
• Topic: Business
• Subtopic: Finance
• Index: Business

A unique, in-depth guide to options pricing and valuing their greeks, along with a four dimensional approach towards the impact of changing market circumstances on options

How to Calculate Options Prices and Their Greeks is the only book of its kind, showing you how to value options and the greeks according to the Black Scholes model but also how to do this without consulting a model. You'll build a solid understanding of options and hedging strategies as you explore the concepts of probability, volatility, and put call parity, then move into more advanced topics in combination with a four-dimensional approach of the change of the P&L of an option portfolio in relation to strike, underlying, volatility, and time to maturity. This informative guide fully explains the distribution of first and second order Greeks along the whole range wherein an option has optionality, and delves into trading strategies, including spreads, straddles, strangles, butterflies, kurtosis, vega-convexity , and more. Charts and tables illustrate how specific positions in a Greek evolve in relation to its parameters, and digital ancillaries allow you to see 3D representations using your own parameters and volumes.

 The Black and Scholes model is the most widely used option model, appreciated for its simplicity and ability to generate a fair value for options pricing in all kinds of markets. This book shows you the ins and outs of the model, giving you the practical understanding you need for setting up and managing an option strategy.

•              Understand the Greeks, and how they make or break a strategy

•              See how the Greeks change with time, volatility, and underlying

•              Explore various trading strategies

•              Implement options positions, and more

 
Representations of option payoffs are too often based on a simple two-dimensional approach consisting of P&L versus underlying at expiry. This is misleading, as the Greeks can make a world of difference over the lifetime of a strategy. How to Calculate Options Prices and Their Greeks is a comprehensive, in-depth guide to a thorough and more effective understanding of options, their Greeks, and (hedging) option strategies.

Chapter 1
Introduction

The most widely used option model is the Black and Scholes model. Although there are some shortcomings, the model is appreciated by many professional option traders and investors because of its simplicity, but also because, in many circumstances, it does generate a fair value for option prices in all kinds of markets.

The main shortcomings, most of which will be discussed later, are: the model assumes a geometric Brownian motion where the market might deviate from that assumption (jumps); it assumes a normal distribution of daily (logarithmic) returns of an asset or Future while quite often there is a tendency towards a distribution with high peaks around the mean and fat tails; it assumes stable volatility while the market is characterised by changing (stochastic) volatility regimes; it also assumes all strikes of the options have the same volatility; it doesn't apply skew (adjustment of option prices) in the volatility smile/surface, and so on.

So in principle there may be a lot of caveats on the Black and Scholes model. However, because of its use by many market participants (with adjustments to make up for the shortcomings) in combination with its accuracy on many occasions, it may remain the basis option model for pricing options for quite some time.

This book aims to explore and explain the ins and outs of the Black and Scholes model (to be precise, the Black '76 model on Futures, minimising the impact of interest rates and leaving out dividends). It has been written for any person active in buying or selling options, involved in options from a business perspective or just interested in learning the background of options pricing, which is quite often seen as a black box. Although this is not an academic work, it could be worthwhile for academics to understand how options and their derivatives perform in practice, rather than in theory.

The book has a very practical approach and an emphasis on the distribution of the Greeks; these measure the sensitivity of the value of an option with regards to changes in parameters such as the strike, the underlying (Future), volatility (a measurement of the variation of the underlying), time to expiry or maturity, and the interest rate. It further emphasises the implications of the Greeks and understanding them with regards to the impact they will/might have on the P&L of an options portfolio. The aim is to give the reader a full understanding of the multidimensional aspects of trading options.

When measuring the sensitivity of the value of an option with regards to changes in the parameters one can discern many Greeks, but the most important ones are:

1. Delta: the price change of an option in relation to the change of the underlying;
2. Vega: the price change of an option in relation to volatility;
3. Theta (time decay): the price change of an option in relation to time;
4. Rho: the price change of an option in relation to interest rate.

These Greeks are called the first order Greeks. Next to that there are also second order Greeks, which are derivatives of the first order Greeks – gamma, vanna, vomma, etcetera – and third order Greeks, being derivatives of the second order Greeks – colour, speed, etcetera.

The most important of the higher order Greeks is gamma which measures the change of delta.

Table 1.1

Parameters	First order Greeks	Second order Greeks	Third order Greeks
Strike	Delta	Gamma	Colour
Underlying	Vega	Vanna	Speed
Volatility	Theta	Vomma	Ultima
Time to maturity	Rho	Charm	Zomma
Interest rate		Veta
		Vera

When Greeks are mentioned throughout the book, the term usually relates to delta, vega, theta and gamma, for they are the most important ones.

The book will also teach how to value at the money options, their surrounding strikes and their main Greeks, without applying the option model. Although much is based on rules of thumb and approximation, valuations without the model can be very accurate. Being able to value/approximate option prices and their Greeks off the top of the head is not the main objective; however, being able to do so must imply that one fully understands how pricing works and how the Greeks are distributed. This will enable the reader to consider and calculate how an option strategy might develop in a four dimensional way. The reader will learn about the consequences of options pricing with regards to changes in time, volatility, underlying and strike, all at the same time.

People on the verge of entering into an option strategy quite often prepare themselves by checking books or the internet. Too often they find explanations of a certain strategy which is only based on the payoff of an option at time of maturity – a two-dimensional interpretation (underlying price versus profit loss). This can be quite misleading since there is so much to say about options during their lifetime, something some people might already have experienced when confronted with adverse market moves while running an option strategy with associated losses. A change in any one of the aforementioned parameters will result in a change in the value of an option. In a two-dimensional approach (i.e. looking at P&L distribution at expiry) most of the Greeks are disregarded, while during the lifetime of the option they can make or break the strategy.

Throughout the book, any actor who is active in buying or selling options – i.e. a private investor, trader, hedger, portfolio manager, etcetera – will be called a trader.

Many people understand losses deriving from bad investment decisions when buying options or the potentially unlimited losses of short options. However, quite often they fail to see the potentially devastating effects of misinterpretation of the Greeks.

For example, as shown in Chart 1.1, a trader who sold a 40 put at $1.50 when the Future was trading at 50 (volatility at 28%, maturity 1 year), had the right view. During the lifetime of the option, the market never came below 40, the put expired worthless, and the trader consequently ended up with a profit of $1.50.

Chart 1.1 P&L distribution of a short 40 put position at expiry

The problem the trader may have experienced, however, is that shortly after inception of the trade, the market came off rapidly towards the 42 level. As a result of the sharp drop in the underlying, the volatility may have jumped from 28% to 40%. The 40 put he sold at $1.50 suddenly had a value of $5.50, an unrealised loss of $4. It would have at least made the trader nervous, but most probably he would have bought back the option because it hit his stop loss level or he was forced by his broker, bank or clearing institution to deposit more margin; or even worse, the trade was stopped out by one of these institutions (at a bad price) when not adhering to the margin call.

So an adverse market move could have caused the trader to end up with a loss while being right in his strategy/view of the market. If he had anticipated the possibility of such a market move he might have sold less options or kept some cash for additional margin calls. Consequently, at expiry, he would have ended up with the $1.50 profit. Anticipation obviously can only be applied when understanding the consequences of changing option parameters with regards to the price of an option.

A far more complex strategy, with a striking difference in P&L distribution at expiry compared to the P&L distribution during its lifetime, is a combination where the trader is short the 50 call once (10,000 lots) and long the 60 call twice (20,000 lots) and at the same time short the 50 put once (10,000 lots) and long the 40 put twice (20,000 lots). He received around $45,000 when entering into the strategy. The...

Table of contents

• Cover
• Title Page
• Copyright
• Table of Contents
• Preface
• Chapter 1: Introduction
• Chapter 2: The Normal Probability Distribution
• Chapter 3: Volatility
• Chapter 4: Put Call Parity
• Chapter 5: Delta Δ
• Chapter 6: Pricing
• Chapter 7: Delta II
• Chapter 8: Gamma
• Chapter 9: Vega
• Chapter 10: Theta
• Chapter 11: Skew
• Chapter 12: Spreads
• Chapter 13: Butterfly
• Chapter 14: Strategies
• Index
• End User License Agreement