Understanding Option Price Sensitivities – European Put Options – Sales & Trading Technical Interview Guides

As part of our Sales & Trading technical interview guide series we have done a number of posts on Greeks, Delta hedging and estimation of Delta hedging Profit & Loss.

A recent client/student/interview request indicated a preference/need for a sheet dedicated to European Put Options Greek plots. Hence the Greeks Put Option suspects’ gallery. While some of these are making a second appearance, we think a Put only collection is indeed useful given our focus on European Call options.

The Analysis framework used for dissecting put option Greeks is simple. We break the contracts down by “money-ness”. The three categories are Deep In, At/Near, Deep Out of money options.

And the five option pricing variables – Spot, Strike, Vol, Interest Rates & Time. This analytical combination produces interesting results.

The options for which Greeks have been plotted below assume a volatility of 20%, a risk free rate of 5% and a time to maturity of 1 year. In addition Spot and Strike prices are:

Deep In

Spot = 50

Strike = 100



Spot = 50

Strike = 100


Deep Out

Spot = 50

Strike = 100


Greeks Against Spot – European Put Options

European Put Options – Deep In the money Options

European Put Options – At/Near Money Options

European Put Options – Deep Out of money Options


Greeks Against Spot – European Put Options

European Put Options – Deep In the money Options

European Put Options – At/Near money Options

European Put Options – Deep out of money Options

European Put Option Greeks – Against Volatility

European Put Options – At/In money Options

European Put Options – Against Changing Interest Rates

European Put Options – At/Near money Options

European Put Options – Against Changing Time to Maturity

European Put Options – At/Near money Options

Understanding Option Greeks – Relevant Sales & Trading Interview Guides prior posts

Understanding Greeks – Introduction

Understanding Greeks – Analyzing Delta & Gamma

Understanding Greeks – The Guide to delta hedging using Monte Carlo Simulation

Understanding Greeks – Quick Reference Guide to Delta, Gamma, Vega, Theta & Rho

Related posts:

  1. Sales & Trading Interview Guide – The understanding Greeks resource
  2. Sales & Trading Technical Interviews Preparation
  3. Sales & Trading Interview Guide: Understanding Greeks: Option Delta and Gamma

Greeks – Option Price Sensitivities – A cheat sheet to Delta, Gamma, Vega, Theta & Rho.

While we have done a few posts earlier about option price sensitivities, here is a quick reference guide for the truly lost and confused. For convenience the reference guide has been broken down into the following sections

  • Greeks Formula Reference
  • Greeks – Suspects Gallery – a visual review of option Greeks across 4 dimensions and money-ness

How to analyze Greeks in time for your final exam/interview/assessment/presentation tomorrow morning

While there are many ways of dissecting Greeks a framework or frame of reference helps. Here are some basic ground rules.

1. Remember the first order Greeks and separate them from second order sensitivities. Delta, Theta & Rho are first order (linear) Greeks which means that they will be different for Call Options and Put Options. Gamma and Vega are second order (non linear) Greeks which means that they will be exactly the same for Calls and Puts.

2. Remember that in most cases Greeks will behave differently depending on the “in-the-money-ness” of the option. Greeks will behave and look differently between Deep Out, At, Near and Deep In the money options.

3. Think how the Greeks will change or move as you change the following parameters:

  • Spot
  • Strike Price
  • Time to Maturity or expiry
  • Volatility of the underlying
  • Interest Rates

Rather than remember the formula try and remember behavior, shape and shifts. For example, see the following three panels that show the shift of the 5 Greek shapes across spot prices and “money-ness”. Starting off with a Deep out of money call option we plot the same curves for an At and near money option as well as a Deep in money option. Can you see the shift and the transition?

Figure 1 Delta, Gamma, Vega, Theta & Rho for a Deep out of money Call Option

Figure 2 Delta, Gamma, Vega, Theta & Rho for At and Near Money Call Option

Figure 3 Delta, Gamma, Vega, Theta & Rho for a Deep In Money Call Option


Greeks – Option Price Sensitivities – Formula Reference and one liner definition guide

The five derivative pricing and sensitivities (aka Greeks) with their equations and definition reference

Figure 4 Option Greeks: Delta & Gamma formula reference

Figure 5 Option Greeks – Vega, Theta & Rho, formula reference

Option pricing – Greeks – Sensitivities – Suspects Gallery

Greeks Against Spot Prices. Here is the short series for Deep out of Money Call Option and Deep In and Out of Money Put options.

Figure 6 Deep out of money call options – Greeks plot

Figure 7 Deep In money put options – Greeks plot

Figure 8 Deep out of money put option – Greek plot

The way to read the above graphical set is to take one Greek at a time. So starting with Delta you will see that while the shape is the same, the sign is different between the Call and the Put. For illustration we have also produced the Greek plot for a Deep out of money Put option and while there are some similarities between the Deep out of money Call and the Deep in money Put, they disappear completely when we look at the Deep out of money Put contract.

Option Price Sensitivities – Plotting Greeks against changing volatility

Figure 9 At money Call option – Greek Plot against changing volatilities

Figure 10 At money put option – Greek plot against changing volatilities

However the difference really crops up between Calls and Puts when you switch the frame of reference from changing spot prices to changing volatilities. With this new point of view Calls and Put are clearly different animals. Why is that? Or is that really the case? If you look closely you will see that as far as Vega, Delta and Rho are concerned the basic shape and shift is similar, it looks different because the LHS axis has shifted. Still Delta is different because of the sign change. But its Gamma and Theta that are really different when it comes to dissecting the behavior of Greeks across Calls and Puts. But would these differences stay if you plot the 5 Greeks across money-ness?

Option Pricing Sensitivities – Greeks – An alternate dimension

Figure 11 Plotting N(d1), N(d2) and Price against volatility

What do you think is the most common question most students have when they see figure 9 above? Do you see a contradiction? Need a hint? Take a look at Delta. Then think about how we calculate Delta for a European call option. We look at N(d1) as a conditional probability? Intuitively speaking what should we expect N(d1) to do as volatility rises? Rise or Fall? What is N(d1) doing in Figure 9 above?

Now take a look at figure 11 above? What are N(d1) and N(d2) doing as volatility rises? Is that intuitive or counter intuitive? Need a hint? Two words – volatility drag.

Think about the above question and tell us about your answers through the comment sections below in this post. Would love to hear more from you.

Related posts:

  1. Sales & Trading Interview Guide: Understanding Greeks: Option Delta and Gamma
  2. The Sales and Trading Interview Guide Series – Understanding Greeks and Delta Hedging – Coming soon to an iPad near you…
  3. Online Finance – Option Terminology Glossary – Greeks, exotics and volatility

Preparing for the quantitative portion of a sales and trading interview for a main street bank is a nightmare. Specially if the bank is an active derivative trader and wants it intake class of interns and full time analysts and associates to hit the Sales and Trading desk running.

While basic option concepts generally get covered quite well in the MBA curriculum, when it comes to option price sensitivities and Greeks, our understanding remains rudimentary and superficial. One reason is the focus on formulas and calculation rather than intuition and understanding. Most courses have run out of time when it comes to delta, gamma, vega, theta and rho and stop after a basic rudimentary coverage of the material.

There is a lot of good material available on basic quantitative and numerical techniques tested in a Sales and Trading interview. But when it comes to option price sensitivities or Greeks, available material generally looks like this.

As part of the work we do with customers and students, our Apple iPad iBooks team is working on two very interesting and exciting titles.

The first is the Sales and Trading Interview Guide – Understanding Greeks for Dummies. Using the interactive iBook template we will help you master your Greeks to such a level where future mention of delta, gamma, vega, theta and rho would no longer break you out in cold sweat and palpitations. The iBook will cover Greeks behavior across time, volatility, spot and strike prices using easy to understand language, graphs and self assessment quizes.

But it’s the second iPad iBook title that we are really excited about. Sales and Trading Interview Guide – Delta Hedging and other higher dimensions, will help you build your own delta hedging sheet in excel using Monte Carlo Simulation. Both iPads books will have options for purchasing supporting excel spread sheets that extend the concepts covered in the iBook.

Planned for release in mid September, the two books will increase our inventory of interactive iBook for iPad titles to 5. Please feel free to drop me a note if you would like to learn more about the release dates and table of content for both titles.

Related posts:

  1. Asset Liability Management (ALM) Study Guide for Board Members – coming soon to an iPad near you.
  2. Understanding ALM and Calculating Value at Risk iBooks featured on the Apple iTunes store
  3. Accounting Crash Course: Small Business Accounting Training – Sales Journal and Sales Ledgers