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


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

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 lognormal 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 rate

Chapter 5 Delta

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

Simple way to determine 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 F

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


Chapter 10 Theta

A practical example

Theta in relation to volatility

Theta in relation 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)


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 spread

Ratio spread



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/Vgamma

Vgamma in relation to time