Table of Contents

Title Page

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

List of Illustrations

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

Chart 1.2 P&L distribution of the combination trade at expiry

Chart 1.3 Combination trade, long 20,000 40 puts and 60 calls, short 10,000 50 puts and 50 calls

Chart 2.1 Normal probability distribution

Chart 2.2 Chart 2.2 shows the probability distributions for the Future, currently trading at 50, at two different volatility levels and three different times to maturity.

Chart 3.1 Probability distribution at 10% volatility

Chart 3.2 Probability distribution at 20% volatility

Chart 3.3 Probability distribution at 40% volatility

Chart 3.4 Normal scale versus logarithmic scale

Chart 3.5 Volatility as a function of the daily percentage moves

Chart 3.6 Four scenarios for intraday moves

Chart 3.7 The “stretch”

Chart 4.1 Long the 50 call

Chart 4.2 Short the 50 call

Chart 4.3 Long the 50 put

Chart 4.4 Short the 50 put

Chart 4.5 Long the Future at 50

Chart 4.6 Short the Future at 50

Chart 4.7 50 Reversal

Chart 4.8 50 Conversion

Chart 4.9 Synthetic 50 call

Chart 4.10 Synthetic 50 put

Chart 5.1 Delta distribution of the 50 call at expiry

Chart 5.2 Value of the 50 call at different maturities

Chart 5.3 Delta distribution of the 50 call, volatility 20%, maturity 3 months

Chart 5.4 Delta distribution of the 50 call, volatility 20%, maturity 1 month

Chart 5.5 Delta distribution of the 50 call, volatility 20%, different maturities

Chart 5.6 Delta distribution of the 50 put, volatility 20%, different maturities

Chart 5.7 Delta distribution of the 50 call, maturity 1 year, different volatilities

Chart 5.8 Delta distribution of the 50 put, maturity 1 year, different volatilities

Chart 5.9 The 20–80% delta region

Chart 5.10 Delta distribution of the call, different strikes, volatility 20%, maturity 1 year

Chart 5.11 Delta distribution of the call, different strikes, volatility 10%, maturity 1/2 year

Chart 5.12 Velocity of the delta change

Chart 5.13 Delta distribution 50 call, volatility 20%, maturity 3 months

Chart 5.14 Delta distribution versus d1

Chart 6.1 Value of the 50 call, Future at 50, different volatilities, maturity 1 year

Chart 6.2 Value of the 50 straddle, twice the value of the call

Chart 6.3 Value of the 50 straddle, Future at 50, volatility 20%, different maturities

Chart 7.1 The gamma boundaries for 2 different maturities with different volatilities

Chart 7.2 The delta and gamma boundaries for calls, Future at 50, volatility 20%, maturity 1 year

Chart 7.3 The lineair shape of the delta with d1 between 0 and 0.5

Chart 7.4 The 30% rule for one straddle in or out of the money

Chart 7.5 The 15% rule for an additional straddle in or out of the money

Chart 7.6 50-54 call spread at expiry

Chart 8.1 Displaying the change in delta from a 50 call, relative to a change in the Future from 50 towards 48 and 54.

Chart 8.2 The delta and gamma distribution over a range of different option strikes

Chart 8.3 The gamma distribution of a 50 strike option

Chart 8.4 The boundaries for the delta in comparison with the boundaries for the gamma

Chart 8.5 The gamma distribution with a stable volatility and alternating maturities

Chart 8.6 The gamma distribution with a stable maturity and alternating volatilities

Chart 8.7 The gamma distribution with a stable volatility, a stable maturity and alternating Future levels (which are at the money).

Chart 8.8 The velocity of delta change for a range of call option strikes.

Chart 8.9 Linear approach for the gamma distribution

Chart 9.1 Vega distribution

Chart 9.2 The linearity of the price of an option compared to the volatility level

Chart 9.3 Vega distribution resembling gamma distribution. However, at the money vega is stable with regards to volatility

Chart 9.4 Distribution of gamma, Future at 50, maturity 1 year

Chart 9.5 Vega distribution with changing time to maturity, Future at 50, volatility at 20%

Chart 9.6 Vega distribution at different Future/at the money levels with stable volatility of 10% and maturity of 1 year.

Chart 9.7 Vega distribution over time with stable volatility of 10% and Future at 50

Chart 9.8 Vega distribution compared to volatility level, maturity 1 year, Future at 50

Chart 9.9 Vega distribution compared to volatility level, maturity ¼ year, Future at 50

Chart 9.10 Vega boundaries compared to delta boundaries, volatility 10%, maturity 1 year, Future/at the money level at 50.00

Chart 9.11 Boundaries for gamma, compared to boundaries for delta

Chart 9.12 Comparing vega distribution with gamma distribution

Chart 9.13 Vega is Gamma, when multiplying gamma by 2.5 the values will be identical

Chart 9.14 Linearisation of the vega distribution

Chart 10.1 Theta distribution call/put, Future at 50, volatility at 10%, maturity 1 year

Chart 10.2 linearity of theta compared to changes in volatility for at the money options

Chart 10.3 Distribution of theta different volatilities, maturity 1 year, Future at 50

Chart 10.4 Cost of theta of a portfolio of 10,000 ATM options, Future at 50, maturity 3 months

Chart 10.5 Theta distribution, Future at 50, volatility 20%, different maturities

Chart 10.6 The evolution of theta in time for a portfolio of 10,000 calls/puts, volatility 16%, Future at 50

Chart 10.7 Linear relationship between theta and the Future/ATM level for a portfolio of 10,000 calls/puts at 25% vol and 1 year maturity.

Chart 10.8 Theta distribution at different at the money levels

Chart 10.9 Theta has the same boundary levels as delta, gamma and vega

Chart 10.10 Gamma at different volatilities

Chart 10.11 Gamma distribution for different Future/ATM levels

Chart 10.12 Gamma at different maturities.

Chart 10.13 Gamma Theta relationship

Chart 10.14 Gamma Theta relationship, theta multiplied by 29.2

Chart 10.15 Gamma Theta relationship, theta multiplied by - 29.2

Chart 10.16 Theta distribution compared to time to maturity, volatility 10%, Future/ATM level at 50.00

Chart 10.17 Theta distribution compared to volatility, Maturity 1 year, Future at 50

Chart 10.18

Chart 10.19 Linearisation of the Theta distribution

Chart 11.1 Different types of skew shapes

Chart 11.2 Smiles at different times to maturity

Chart 11.3

Chart 11.4 Sticky at the money volatility, Future decreasing

Chart 11.5 Sticky at the money volatility, Future increasing

Chart 12.1 50–55 Call spread at expiry

Chart 12.2 50–55 Call spread at expiry minus investment of $1.50

Chart 12.3 50–45 Put spread at expiry

Chart 12.4 Delta of spreads, longer time to maturity

Chart 12.5 Delta of spreads, short time to maturity

Chart 12.6 Gamma distribution of the 45–50 call spread

Chart 12.7 Vega distribution of the 45–50 call spread

Chart 12.8 Theta distribution of the 45-50 call spread

Chart 13.1 Profit Loss distribution of a 46–50–54 butterfly

Chart 13.2 Profit Loss distribution of a 46–50–54 butterfly minus initial investment

Chart 13.3 A butterfly is the combination of two spreads

Chart 13.4 All the butterflies from 30 to 70

Chart 13.5 All the butterflies from 0 to infinity

Chart 13.6 Butterflies with $2 strike differential

Chart 13.7 Butterfly versus vega

Chart 13.8 Linearisation of the butterfly distribution

Chart 13.9 Distribution of butterflies at different volatility levels

Chart 13.10 Price evolution of a butterfly

Chart 13.11 Price evolution of a butterfly

Chart 13.12 Trendline of a delta distribution

Chart 13.13 Gamma distribution of the 40–50–60 butterfly

Chart 13.14 Vega distribution of a 40–50–60 butterfly

Chart 13.15 Theta distribution of a 40–50–60 butterfly

Chart 14.1 50 Call at expiry

Chart 14.2 50 Put at expiry

Chart 14.3 50–55 Call spread at expiry

Chart 14.4 1 by 2 50-45 ratio put spread at expiry

Chart 14.5 50 Straddle at expiry

Chart 14.6 40–60 Strangle at expiry

Chart 14.7 40-60 Collar at expiry

Chart 14.8 Trending Future market

Chart 14.9 Retracing Future market

Chart 14.10 Sideways Future market

Chart 14.11 Different types of distribution

Chart 14.12 Platykurtic distribution

Chart 14.13 Leptokurtic distribution

Chart 14.14 Transition from a platykurtic environment towards a leptokurtic environment

Chart 14.15 Vega distribution at 20% volatility

Chart 14.16 Vega distribution, values for 40, 50 and 64 strike

Chart 14.17 Vega distribution at 20% and 25% volatility

Chart 14.18 Vega distribution at 20% and 30% volatility

Chart 14.19 Vega distribution at 20% and 10% volatility

Chart 14.20 Vega distribution, maturity 1 year and 9 months

Chart 14.21 Vega distribution, maturity 1 year and 6 months

Chart 14.22 Vega distribution, maturity 1 year and 3 months

List of Tables

Table 1.1

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 6.2

Table 6.3

Table 6.4

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 The vega distribution in values from the 36 to 72 strike, Future at 50.00, Volatility 10% and Maturity 1 year

Table 9.2

Table 9.3

Table 9.4

Table 9.5

Table 9.6

Table 9.7 Vega values, Black Scholes model, maturity 1 year, volatility 10% Future at 50

Table 9.8 Print from Black Scholes model, Future/at the money at 50, volatility at 10%, maturity 1 year

Table 9.9

Table 9.10

Table 9.11

Table 9.12

Table 10.1

Table 10.2

Table 10.3 Theta values (for calls), Black Scholes model, maturity 1 year, volatility 10%, Future/ATM level @ 50.00

Table 10.4

Table 10.5

Table 10.6

Table 10.7

Table 10.8

Table 10.9

Table 11.1

Table 12.1 In other words a formula could be formed (in this case 30 and 35 strike with the same maturity):

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 12.12

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 14.6

Table 14.7

Table 14.8 Long 20,000 40 puts, short 20,000 50 calls/puts, long 20,000 64 calls, volatility at 20% at inception, Future at 50 at inception, maturity 1 year

This edition first published 2015

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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