How to Calculate Options Prices and Their Greeks E-Book

Pierino Ursone

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Series
• Sprache: Englisch

Seitenzahl: 317

Veröffentlichungsjahr: 2015

Beschreibung

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.

Leseprobe

Table of Contents

Title Page

Copyright

Preface

Chapter 1: Introduction

Chapter 2: The Normal Probability Distribution

STANDARD DEVIATION IN A FINANCIAL MARKET

THE IMPACT OF VOLATILITY AND TIME ON THE STANDARD DEVIATION

Chapter 3: Volatility

THE PROBABILITY DISTRIBUTION OF THE VALUE OF A FUTURE AFTER ONE YEAR OF TRADING

NORMAL DISTRIBUTION VERSUS LOG-NORMAL DISTRIBUTION

CALCULATING THE ANNUALISED VOLATILITY TRADITIONALLY

CALCULATING THE ANNUALISED VOLATILITY WITHOUT

μ

CALCULATING THE ANNUALISED VOLATILITY APPLYING THE 16% RULE

VARIATION IN TRADING DAYS

APPROACH TOWARDS INTRADAY VOLATILITY

HISTORICAL VERSUS IMPLIED VOLATILITY

Chapter 4: Put Call Parity

SYNTHETICALLY CREATING A FUTURE LONG POSITION, THE REVERSAL

SYNTHETICALLY CREATING A FUTURE SHORT POSITION, THE CONVERSION

SYNTHETIC OPTIONS

COVERED CALL WRITING

SHORT NOTE ON INTEREST RATES

Chapter 5: Delta Δ

CHANGE OF OPTION VALUE THROUGH THE DELTA

DYNAMIC DELTA

DELTA AT DIFFERENT MATURITIES

DELTA AT DIFFERENT VOLATILITIES

20–80 DELTA REGION

DELTA PER STRIKE

DYNAMIC DELTA HEDGING

THE AT THE MONEY DELTA

DELTA CHANGES IN TIME

Chapter 6: Pricing

CALCULATING THE AT THE MONEY STRADDLE USING BLACK AND SCHOLES FORMULA

DETERMINING THE VALUE OF AN AT THE MONEY STRADDLE

Chapter 7: Delta II

DETERMINING THE BOUNDARIES OF THE DELTA

VALUATION OF THE AT THE MONEY DELTA

DELTA DISTRIBUTION IN RELATION TO THE AT THE MONEY STRADDLE

APPLICATION OF THE DELTA APPROACH, DETERMINING THE DELTA OF A CALL SPREAD

Chapter 8: Gamma

THE AGGREGATE GAMMA FOR A PORTFOLIO OF OPTIONS

THE DELTA CHANGE OF AN OPTION

THE GAMMA IS NOT A CONSTANT

LONG TERM GAMMA EXAMPLE

SHORT TERM GAMMA EXAMPLE

VERY SHORT TERM GAMMA EXAMPLE

DETERMINING THE BOUNDARIES OF GAMMA

DETERMINING THE GAMMA VALUE OF AN AT THE MONEY STRADDLE

GAMMA IN RELATION TO TIME TO MATURITY, VOLATILITY AND THE UNDERLYING LEVEL

PRACTICAL EXAMPLE

HEDGING THE GAMMA

DETERMINING THE GAMMA OF OUT OF THE MONEY OPTIONS

DERIVATIVES OF THE GAMMA

Chapter 9: Vega

DIFFERENT MATURITIES WILL DISPLAY DIFFERENT VOLATILITY REGIME CHANGES

DETERMINING THE VEGA VALUE OF AT THE MONEY OPTIONS

VEGA OF AT THE MONEY OPTIONS COMPARED TO VOLATILITY

VEGA OF AT THE MONEY OPTIONS COMPARED TO TIME TO MATURITY

VEGA OF AT THE MONEY OPTIONS COMPARED TO THE UNDERLYING LEVEL

VEGA ON A 3-DIMENSIONAL SCALE, VEGA VS MATURITY AND VEGA VS VOLATILITY

DETERMINING THE BOUNDARIES OF VEGA

COMPARING THE BOUNDARIES OF VEGA WITH THE BOUNDARIES OF GAMMA

DETERMINING VEGA VALUES OF OUT OF THE MONEY OPTIONS

DERIVATIVES OF THE VEGA

VOMMA

Chapter 10: Theta

A PRACTICAL EXAMPLE

THETA IN RELATION TO VOLATILITY

THETA IN RELATION TO TIME TO MATURITY

THETA OF AT THE MONEY OPTIONS IN RELATION TO THE UNDERLYING LEVEL

DETERMINING THE BOUNDARIES OF THETA

THE GAMMA THETA RELATIONSHIP α

THETA ON A 3-DIMENSIONAL SCALE, THETA VS MATURITY AND THETA VS VOLATILITY

DETERMINING THE THETA VALUE OF AN AT THE MONEY STRADDLE

DETERMINING THETA VALUES OF OUT OF THE MONEY OPTIONS

Chapter 11: Skew

VOLATILITY SMILES WITH DIFFERENT TIMES TO MATURITY

STICKY AT THE MONEY VOLATILITY

Chapter 12: Spreads

CALL SPREAD (HORIZONTAL)

PUT SPREAD (HORIZONTAL)

BOXES

APPLYING BOXES IN THE REAL MARKET

THE GREEKS FOR HORIZONTAL SPREADS

TIME SPREAD

APPROXIMATION OF THE VALUE OF AT THE MONEY SPREADS

RATIO SPREAD

Chapter 13: Butterfly

PUT CALL PARITY

DISTRIBUTION OF THE BUTTERFLY

BOUNDARIES OF THE BUTTERFLY

METHOD FOR ESTIMATING AT THE MONEY BUTTERFLY VALUES

ESTIMATING OUT OF THE MONEY BUTTERFLY VALUES

BUTTERFLY IN RELATION TO VOLATILITY

BUTTERFLY IN RELATION TO TIME TO MATURITY

BUTTERFLY AS A STRATEGIC PLAY

THE GREEKS OF A BUTTERFLY

STRADDLE–STRANGLE OR THE “IRON FLY”

Chapter 14: Strategies

CALL

PUT

CALL SPREAD

RATIO SPREAD

STRADDLE

STRANGLE

COLLAR (RISK REVERSAL, FENCE)

GAMMA PORTFOLIO

GAMMA HEDGING STRATEGIES BASED ON MONTE CARLO SCENARIOS

SETTING UP A GAMMA POSITION ON THE BACK OF PREVAILING KURTOSIS IN THE MARKET

EXCESS KURTOSIS

BENEFITTING FROM A PLATYKURTIC ENVIRONMENT

THE MESOKURTIC MARKET

THE LEPTOKURTIC MARKET

TRANSITION FROM A PLATYKURTIC ENVIRONMENT TOWARDS A LEPTOKURTIC ENVIRONMENT

WRONG HEDGING STRATEGY: KILLERGAMMA

VEGA CONVEXITY/VOMMA

VEGA CONVEXITY IN RELATION TO TIME/ VETA

Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Begin Reading

List of Illustrations

Chart 1.1

Chart 1.2

Chart 1.3

Chart 2.1

Chart 2.2

Chart 3.1

Chart 3.2

Chart 3.3

Chart 3.4

Chart 3.5

Chart 3.7

Chart 4.1

Chart 4.2

Chart 4.3

Chart 4.4

Chart 4.5

Chart 4.6

Chart 4.9

Chart 4.10

Chart 5.1

Chart 5.2

Chart 5.3

Chart 5.4

Chart 5.5

Chart 5.6

Chart 5.7

Chart 5.9

Chart 5.10

Chart 5.11

Chart 5.12

Chart 5.13

Chart 5.14

Chart 6.1

Chart 6.2

Chart 6.3

Chart 7.1

Chart 7.2

Chart 7.3

Chart 7.4

Chart 7.5

Chart 7.6

Chart 8.1

Chart 8.2

Chart 8.3

Chart 8.4

Chart 8.5

Chart 8.6

Chart 8.7

Chart 8.8

Chart 8.9

Chart 9.1

Chart 9.2

Chart 9.3

Chart 9.4

Chart 9.5

Chart 9.6

Chart 9.7

Chart 9.8

Chart 9.9

Chart 9.10

Chart 9.11

Chart 9.12

Chart 9.13

Chart 9.14

Chart 10.1

Chart 10.2

Chart 10.3

Chart 10.4

Chart 10.5

Chart 10.8

Chart 10.9

Chart 10.10

Chart 10.11

Chart 10.12

Chart 10.13

Chart 10.14

Chart 10.15

Chart 10.16

Chart 10.18

Chart 10.19

Chart 11.1

Chart 11.2

Chart 11.3

Chart 11.4

Chart 11.5

Chart 12.2

Chart 12.3

Chart 12.4

Chart 12.5

Chart 12.6

Chart 12.7

Chart 12.8

Chart 13.1

Chart 13.3

Chart 13.4

Chart 13.5

Chart 13.6

Chart 13.7

Chart 13.8

Chart 13.9

Chart 13.10

Chart 13.11

Chart 13.12

Chart 13.13

Chart 13.14

Chart 13.15

Chart 14.1

Chart 14.2

Chart 14.3

Chart 14.4

Chart 14.5

Chart 14.6

Chart 14.7

Chart 14.8

Chart 14.9

Chart 14.10

Chart 14.11

Chart 14.12

Chart 14.13

Chart 14.14

Chart 14.15

Chart 14.16

Chart 14.17

Chart 14.18

Chart 14.19

Chart 14.20

Chart 14.21

Chart 14.22

List of Tables

Table 3.1

Table 3.2

Table 3.3

Table 3.4

Table 3.5

Table 4.1

Table 4.2

Table 4.3

Table 5.1

Table 5.2

Table 5.3

Table 5.4

Table 5.5

Table 5.6

Table 6.1

Table 7.1

Table 7.2

Table 7.3

Table 7.4

Table 7.5

Table 8.1

Table 8.2

Table 8.3

Table 8.4

Table 8.5

Table 8.6

Table 8.7

Table 8.8

Table 8.9

Table 8.10

Table 8.11

Table 8.12

Table 8.13

Table 8.14

Table 8.15

Table 8.16

Table 8.17

Table 8.18

Table 9.1

Table 9.2

Table 9.3

Table 9.4

Table 9.5

Table 9.6

Table 9.7

Table 9.8

Table 9.9

Table 9.10

Table 9.11

Table 9.12

Table 10.1

Table 10.2

Table 10.3

Table 10.4

Table 10.5

Table 10.6

Table 10.7

Table 10.8

Table 10.9

Table 11.1

Table 12.1

Table 12.2

Table 12.3

Table 12.4

Table 12.5

Table 12.6

Table 12.7

Table 12.8

Table 12.9

Table 12.10

Table 12.11

Table 12.13

Table 13.1

Table 13.2

Table 13.3

Table 13.4

Table 13.5

Table 13.6

Table 14.1

Table 14.2

Table 14.3

Table 14.4

Table 14.5

Table 14.7

Table 14.8

How to Calculate Options Prices and Their Greeks: Exploring the Black Scholes Model from Delta to Vega

PIERINO URSONE

This edition first published 2015

© 2015 Pierino Ursone

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John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

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Preface

In September 1992 I joined a renowned and highly successful market-making ­company at the Amsterdam Options Exchange. The company early recognised the need for hiring option traders having had an academic education and being very strong in mental calculation. Option trading those days more and more professionalised and shifted away from “survival of the loudest and toughest guy” towards a more intellectual approach. Trading was a matter of speed, being the first in a deal. Strength in mental arithmetic gave one an edge. For instance, when trading option combinations, adding prices and subtracting prices – one at the bid price, the other for instance at the asking price – being the quickest brought high rewards.

After a thorough test of my mental maths skills, I was one of only two, of the many people tested, to be employed. There I stood, in my first few days in the open outcry pit, just briefly after September 16th 1992 (Black Wednesday). On that day the UK withdrew from the European EMS system (the forerunner of the Euro), the British pound collapsed, the FX market in general became heavily volatile – all around the time the management of the company had decided to let me start trading Dollar options.

With my mentor behind me, I stood in the Dollar pit (training on the job) trying to compete with a bunch of experienced guys. My mentor jabbed my back each time when a trade, being brought to the pit by the floorbrokers, seemed interesting. In the meantime he was teaching me put–call parity, reversals and conversions, horizontal and time spreads, and whereabouts the value of at the money options should be (just a ballpark figure). There was one large distinction between us and the other traders; we were the only ones not using a computer printout with options prices. My mentor was certain that one should be able to trade off the top of the head; I was his guinea pig.

In those days, every trader on the floor was using a print of the Black Scholes model, indicating fair value for a large set of options at a specific level in the underlying asset. These printouts were produced at several levels of the underlying, so that a trader did not need to leave the pit to produce a new printout when new levels were met. Some days, however, markets could be so volatile that prices would “run off” the sheet. As a result the trader would have to leave the pit to print a new price sheet. It was exactly these moments when trading in the pit was the busiest: not having to leave the pit was an advantage as there were fewer traders to compete with. So, not having to rely on the printouts would create an edge while liquidity in trading would be booming at those times.

All the time we kept thinking of how to outsmart the others, how to value options at specific volatility levels and how, for instance, volatility spreads would behave in changing market circumstances. Soon we were able, when looking at option prices in other trading pits, to come up with fairly good estimates on the prevailing volatilities. We figured out how the delta of in the money options relate to the at the money options, how the at the moneys have to be priced and how to value butterflies on the back of the delta of spreads and more. Next to that we had our weekly company calculation and strategy sessions. There was a steady accumulation of knowledge on options pricing and valuing some of the Greeks.

After having run my own company from 1996 to 2001 at the Amsterdam exchange, I entered the energy options market, a whole different league. There was no exchange to trade on, no clearing of trades (hence counterparty risk), the volumes were much larger and it was professional against professional. As a market maker on the exchange one was in general used to earning a living on the back of the margins stemming from the differences in bid and asking prices (obviously we were running some strategies at the same time as well). Now however, with everyone knowing exactly where prices should be, all margins had evaporated. As a result, the only way to earn money was to have a proper assessment of the market and have the right position to optimise the potential profits. So I moved from an environment where superior pricing was a guarantee for success to an area where only the right strategy and the right execution of this strategy would reap rewards. It truly was a challenge how to think of the best strategy as there is a plethora of possible option combinations.

It has been the combination of these two worlds which has matured me in understanding how option trading really works. Without knowing how to price an option and its Greeks it would be onerous to find the right strategy. Without having the right market assessment it is impossible to generate profits from options trading.

In this book I have written down what I have learned in almost 20 years of options trading. It will greatly contribute to a full understanding of how to price options and their Greeks, how they are distributed and how strategies work out under changing circumstances. As mentioned before, when setting up a strategy one can choose from many possible option combinations. This book will help the reader to ponder options and strategies in such a way that one can fully understand how changes in underlying levels, in market volatility and in time impact the profitability of a strategy.

I wish to express my gratitude to my friends Bram van der Lee and Matt Daen for reviewing this book, for their support, enthusiasm and suggestions on how to further improve its quality.

Pierino Ursone

Chapter 1Introduction

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.