Stock options trading & investment : The Greeks.

17 Nov, by

Article 15 – Stock options trading & investment : The Greeks.

By Michael M Lee

  1. Introduction

In option trading risks can be managed to a greater extent than with outright buying or selling of stocks. The price of an option is contributed by several components which are affected by the changes in the stock market. The components are:

  1. Underlying stock
  2. Passage of time
  3. Implied volatility
  4. Interest rates and
  5. Dividends

Short-term interest rates and dividends are predictable. The risk/return potential of an options position can be determined by analysing the Greeks.

  1. The Greeks

The Greeks are fundamental parameters of risk assessment. They are:

  1. Delta and Gamma, which express exposure to a change in the underlying.
  2. Theta expresses exposure to the passage of time.
  3. Vega expresses exposure to a change in implied volatility.

Each Greek alphabet represents a characteristic of risk that affects option pricing and traders used them to manage their option portfolios.

The diagram below depicts how the Greeks affect the option price.

Option pricing depends on the underlying stock’s intrinsic value and extrinsic value. Intrinsic value depends on the current stock price and the strike price of the option. The strike price is the exercise price of the option at expiration. During the term of the option, the current stock price will fluctuate and at expiration, the current stock price and strike price may end up In-the-Money (ITM), At-the-Money (ATM) or Out-of-the-Money (OTM). The Greeks that reflect the changes in the stock price are Delta and Gamma.

Extrinsic value depends on the option’s time to expiry, volatility, and interest rate. And during the term of the option, the Greeks that affect them are Theta, Vega, and Rho.

Note: The tips of the ITM, ATM and OTM arrows depict  market price of the underlying, and diagram refers to a call option.

Based on an educated reading of the Greeks, the trader or investor would have an idea of where the option price is heading at expiration and can strategise towards an ITM situation, or cutting losses, if needed.

  1. Delta

Delta is the amount that an option changes with respect to a small change in the underlying. It is measured as follows:

  1. If an option is deep ITM, it is at parity with the underlying. It has a delta of 1.0 (a 100 delta) or a 100% correlation with the underlying.
  2. If the option is ATM, its delta is 0.5 or the option has a 50 delta.
  3. If the option is far OTM, it is worthless, and its delta is 0.0

Therefore, delta indicates the probability that of an option expiring ITM. An ATM option with a 0.5 delta has an even chance of expiring ITM, while a call option with 0.11 delta has a 11% probability of expiring ITM.

The delta of a call option has a range between 0 and +1, while that of a put option has a range between 0 and -1.

Option traders use delta to represent the hedge ratio to create a delta-neutral position. If a 0.40 delta call option is purchased, the trader will sell 40 shares of the underlying to be fully hedged.

The diagram below shows the impact of stock price movement on call value via Delta. Notice that as the stock moves higher, Delta increases resulting in more call value (gain more money). On the other hand, as stock moves lower, Delta decreases but less money is lost.

  1. Gamma

Delta changes continually with the changes in the underlying and so its rate of change must be assessed and analysed. This Delta rate of change is called Gamma and is the 2nd derivative of the graph of the option price relative to its stock price. Gamma, unlike Delta is not constant.

ITM and OTM options have lower Gamma, being further away from ATM options. ATM options have higher Gamma. Gamma becomes active when Theta (Days To Expiry, DTE) and Volatility set in.

Gamma indicates the amount the Delta would change when the underlying moves $1. If a call option has 0.05 Delta and 0.10 Gamma, and it moves up by $1, the Delta of the option would move by 0.01 to 0.06.

The diagram below shows the impact of Gamma on Delta (Call option).

  1. Theta

Time passage causes options to lose value through time decay, which works to the benefit of the option seller.

A longer term option will cost more than a shorter-term option because the option caller has to pay for a higher premium to enjoy the probability of the call option ending ITM. Time decay or DTE is non-linear. An option loses its value at an accelerating rate as it approaches expiration.

An OTM or ITM option enters its accelerated time decay period much earlier than the ATM option. This means that the risk/return potential also accelerates with time decay. So when a call option has a theta of 0.05, the option price would decrease by $0.05 every day, ceteris paribus.

Knowing that near-term options are less costly because DTE is short, the trader could profit hugely from an expected large move in the underlying. This could be strategized if the reading and analysis of the fundamental and technical aspects of the underlying is educated. However short the DTE, always remember that the risk of time decay remains serious and is merciless!

  1. Vega

Market conditions can change suddenly, increasing or decreasing the price of the underlying. This is where volatility of the option premium sets in and it is expressed as Vega. Vega is the amount that an option changes if the implied volatility changes by 1 percentage point. It can also be expressed in dollar amounts. Note that it is the implied volatility that is required, not the historical volatility.

As volatility is the single most important aspect of option pricing, the use of options is to identify volatility and to hedge the option’s exposure to market fluctuations. To understand volatility, we will need to get into the statistics of normal price distribution and the bell curve.

The bell curve centres around the day’s closing price of the stock and plots the next closing day’s closing price to the left or right depending on the next day’s price fluctuations downwards or upwards respectively. On a bell curve graph, the x-axis represents the magnitude of price changes and the y-axis, the frequency. A narrow bell curve indicates low volatility, while a broad one, high volatility.

Historical volatility depicts the range of price movements of an underlying over a period of 10, 20, or 30 days. It does not show price direction. It is calculated based on a one-day, one standard deviation price movements, annualised. Normally, a 20-day average is used to calculate historical volatility.

Once the historical volatility is calculated it is inserted into the Black Scholes formula together with other pricing inputs (strike price, price of underlying, DTE, interest rate, and dividends) to arrive at the option price.

As historical volatility does not show direction of the option price movements, and as the option price is affected by demand and supply, anticipation of future option price movement is critical. This is where we need to use implied volatility.

Implied volatility is arrived at by inserting the market price of the option into the Black Scholes formula and removing the historical volatility. It is then used to calculate market prices of options of the other strike prices within the same contract month.

Vega therefore indicates the amount an option’s price would change given a 1% change in implied volatility. A Vega of 0.10 means the option’s value is expected to change by $0.10 if the implied volatility changes by 1%.

  1. Rho

It represents the rate of change between an option’s value and a 1% change in the short-term interest rate. It measures the sensitivity of the interest rate. Suppose a call option has a rho of 0.05 and option price of $1.25. When interest rate rises by 1%, the call option would increase by $0.05 to S1.30.

Rho is greatest for ATM options with long DTE.

  1. Applying the Greeks

In an option trading or investment exercise, we must follow certain approaches.

Step 1. Consider the market outlook (political and economic events, other potential catalysts, etc)

Step 2. Select the stock (the underlying) to trade options on.

Step 3. Consider the duration of the expected market moves (when an expected event materialises) and select the contract month, as appropriate.

Step 4. Based on an educated  analysis of technical and fundamentals, estimate the probability of the occurrence of the event.

Step 5. Determine option strategy – neutral or directional, strike price)

Step 6. Analyse the Greeks and assess how each would affect option price going forward.

Step 7. Monitor and update your outlook of the market and the option position.

  1. Conclusion

Embarking on option trading and investment is not an easy endeavour. It requires knowledge of how option pricing works, how option prices behave, and the parameters which affect option prices, chief of which are the Greeks. Without the use of the Greeks to guide risk management, the trader or investor may not know what to do when option prices collapse or do not move in the direction as expected.

  • Delta is the most common Greek. It is an estimate of the probability of an option expiring in the money. Option traders use delta to represent the hedge ratio to create a delta-neutral position.
  • Gamma is the rate of change of an option given a change in the price of the underlying. ITM and OTM options have lower Gamma, being further away from ATM ATM options have higher Gamma.
  • Theta causes option to lose money through passage of time. It is non-linear and long options have +ve theta while short options have -ve theta. ATM options have the highest theta. Higher volatility increases time decay.
  • Vega indicates the amount an option’s price would change given a 1% change in implied volatility. Long options have +ve Vega; short options, -ve Vega. ATM options have highest Vega (so use DITM options to negate Vega). Volatility widens the stock price range and OTM options has therefore a better chance of ending ITM.
  • Rho is the least important of the Greeks since it requires a very large move in the interest rate to effect a change in option value (mostly in LEAPS).

Before getting into the area of options, an education on technical and fundamental analyses would be hugely helpful!





Option trading & investment is risky and may not be suitable for everyone. It involves risk and you may lose money.

This article is written for education purposes only and not an invitation to engage in option trading and investment. Such decisions are very private and personal and requires a truthful examination of the mantra “Know Thy Self, Know Thy Plan, and Know thy stocks”.


Author’s Profile

Michael Matthew Lee is a trained teacher, lecturer, C-suite coach, and facilitator, and is himself a life-long learner.

Being trained, he is skilful in explaining difficult concepts in a simple and understandable manner. With his more than 40 years of corporate working experience, which include serving as Group CFO of 3 large Main Board public-listed groups, and as CXO (MD/CEO/COO) of various companies across industries, he specialises in Finance and Strategic Marketing and is exposed to senior executive management practices, issues, and challenges. He adopts a practical approach to his training and facilitation and draws case studies from actual cases (adapted) he personally managed over his 4 decades of corporate working experience.

He authored his first Finance book entitled “The Essence of Corporate cashflow Sustainability”, launched in June 2022. His second book “Corporate Growth & Cashflow Sustainability” will be launched in early 2024.


Michael Matthew Lee’s credentials are:

Chartered Accountant (ISCA)

Chartered Marketer (CIM(UK))

Associate CVA

MBA (Finance, NUS Business School)

MBA (Strategic Marketing, University of Hull)

BAcc (NUS); DipM (UK); PDipM (APAC); ACTA; PMC, CertEd



Michael has written 14  accounting & finance articles and they can be accessed through


LinkedIn: “Michael M Lee”; and

Facebook: “Michael M Lee”

  • 8 Oct 2018: “Debit & Credit” – Upon this Rock, the House of Accountancy was built!
  • 17 Oct 2018: Financial Statement Analysis (an overview) – moving from lagging (historical) indicators to leading (predictive) indicators.
  • 26 Nov 2018: Traditional Financial Statement Analysis – using lagging indicators (Part 1)
  • 17 Dec 2018: Financial Statement Analysis – using leading indicators (Part 2)
  • 24 Jan 2019: Financial Statement Analysis – Is there an optimal capital structure? (Part 3)
  • 24 Feb 19: Financial Statement Analysis – What is V.I.S.A.? (Part 4)
  • 7 Apr 19: Financial Statement Analysis – The V.I.S.A. Approach (Part 5)
  • 26 Apr 19: Cashflow Statement – an enhanced presentation
  • 28 Jun 20: Cashflow Statement – Period & Trend Analysis
  • 3 Feb 21: The Economic Resources Approach to Accounting (ERAA)
  • 18 Apr 22: Business Valuation – A primer
  • 7 May 22: Business Valuation – Income Approach
  • 24 Sept 22: Corporate Cashflow Sustainability – The foundation for corporate growth.
  • 4 Dec 22: Stock options trading & investment : An introduction


————————————MMLee 15Nov23————————————





    Apply now

    It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout.