Options Greeks

Options Valuation (Black-Scholes Model)

In this lesson, you’ll learn how institutional participants value options using the Black-Scholes model, a mathematical framework that combines multiple variables to determine fair option pricing. We’ll break down the theoretical formulas behind option pricing and explore how different factors work together to create the premium you pay or receive.

The Black-Scholes model was developed in 1973 by Fischer Black and Myron Scholes and remains one of the most important tools in quantitative finance today. This model takes into account several critical variables: the current price of the underlying asset, volatility, the risk-free interest rate, and time to expiration. The model operates under key assumptions including that the underlying price follows a normal distribution, there are no transaction costs or restrictions on short selling, no dividends are paid during the option’s lifespan, and the market is efficient with no arbitrage opportunities.

The model calculates option value by first determining the forward price using the spot price multiplied by the risk-free rate times the time minus dividends. However, for short-term maturity options, the spot price is more important than interest rates and dividends. The second critical factor is time to expiration, where time decay tends to be higher for long-term options. The third factor is implied volatility, which represents the level of uncertainty about the future price and is one of the main factors affecting option pricing.

Understanding option pricing requires breaking down the premium into two components: intrinsic value and extrinsic value (also called time value). For call options, intrinsic value equals the current price minus the strike price, while for put options it’s the strike price minus the current price. Out of the money options have zero intrinsic value. Time value represents the additional value beyond intrinsic value and reflects factors like time to expiration, implied volatility, and potential for price changes. For example, a $25 strike call on a stock trading at $27 priced at $2.50 has $2 of intrinsic value and $0.50 of time value.

Time value is composed of volatility multiplied by time. Three factors increase time value: longer maturity periods, higher implied volatility, and greater uncertainty near at-the-money strikes. In-the-money options have premiums made up of both intrinsic and time value, while out-of-the-money options have 100% time value and zero intrinsic value. As traders often say on Wall Street, “garbage in, garbage out” – the quality of your inputs directly affects the Black-Scholes model results.

To apply these concepts in your trading, focus on understanding volatility levels of the assets you trade. Volatility and premium are correlated, so market participants spend considerable time evaluating volatility to stay competitive. By understanding how the Black-Scholes model combines spot price, time, and volatility, you can better assess risks and opportunities when evaluating option prices.

Video Chapters

  1. 00:00 – Introduction to the Black-Scholes model
  2. 00:56 – Key assumptions of the Black-Scholes model
  3. 01:17 – Forward price calculation and important factors
  4. 02:24 – Time to expiration and implied volatility
  5. 03:26 – Intrinsic value explained with examples
  6. 05:17 – Time value components and out-of-the-money options
  7. 07:01 – Volatility and premium correlation

Key Takeaways

  1. The Black-Scholes model uses spot price, volatility, risk-free interest rate, and time to expiration to calculate option values
  2. Option premium consists of intrinsic value (profit if exercised immediately) and time value (volatility multiplied by time)
  3. Out-of-the-money options have zero intrinsic value and 100% time value, while in-the-money options have both components
  4. Understanding implied volatility is critical for competitive trading as volatility and premium are directly correlated
Video Transcription

[00:00:00.12] - Speaker 1
We have mentioned the valuation of an option in various parts of the course. We talked about intrinsic value and time value. In this lesson we will introduce a Black and Scores model and we will talk more theoretically about formulas and option pricing. Let's start with the Black Scores model. This standard model is particularly important because it is used to find the price of an option and its value by the majority of the institutional participants.

[00:00:22.10] - Speaker 1
The Black Scores model is a mathematical model that takes into account many variables such as the current price of the underlying asset, the the volatility, the risk free interest rate and the time to expiration. We have seen the different Greeks in the previous lessons, but we will put it all together in this section. The Black Scores model was developed in 1973 by Fischer Black and Myron Scores and is still one of the most important tools in quantitative finance today. The key assumptions of the Black Scrolls models are the following. The underlying price follows a normal distribution, meaning that its price movements are random but with a known average return and standard deviation.

[00:00:56.02] - Speaker 1
There are no transaction costs or restrictions on short selling. There are no dividends paid out by the underlying asset during the option's lifespan. The risk free interest rate is constant and non and finally the market is efficient and there are no arbitrage opportunities. This is the Black Scholes model formula. We have poor idea but we will try to simplify and extrapolate only what interests to us.

[00:01:17.25] - Speaker 1
What we look for is to understand which variables are important to create the inputs that will then be used in the Black Scholes model. The Black scores model is like a discounted cash flow model based on the parameters that we put in. We have a result. On Wall street there is a saying which is garbage inn and garbage out that describes these types of exercise. The first factor that we must take into consideration is the calculation of the forward price or the future price.

[00:01:41.01] - Speaker 1
The Black scores model in this case helps us calculate the futures value of the spot with this simple formula. What we do in this case is to use the spot price of the underlying. The spot price, as you can imagine is very important for calculating the future price of the option. In fact, the closer we get to the option strike, the more the value of the option increases. So what we do in this case is to take the spot price multiplied by the risk free rate.

[00:02:03.24] - Speaker 1
You can use for example a Fed fund rate or a very short term treasury rate times the time minus the dividends. It is important to know that interest rate and dividends only matter if the option has a long term maturity. In general, for option with Short term maturity, the spot price is a more important factor than the interest rate and the dividends. This then brings us to the second factor of the Black Scholes formula. Time.

[00:02:24.07] - Speaker 1
Let's talk very briefly about the time to expiration. Time decay, as we know, tends to be higher for long term options. Nothing too complicated here. We then move to the third factor. The volatility implied by the Black scores model is an estimate of the level of uncertainty about the future price of the underlying stock.

[00:02:40.26] - Speaker 1
This uncertainty is one of the main factors affecting the price of an option. So it's a very important factor to take into consideration for our valuation. Implied volatility is usually calculated using the current market price of an option and the Black Scholes formula. The model is used to generate an estimate of the theoretical option price which is then compared to the current market price. Implied volatility is used by traders to evaluate the success of option based strategies.

[00:03:04.10] - Speaker 1
In addition, implied volatility can be used as an indicator of market expectation of the future direction of the underlying stock price. All these factors put together help us calculate the price of an option. We can then derive this formula. The simple formula for calculating option prices involves understanding two intrinsic value and extrinsic value or time value. Let's start with the intrinsic value.

[00:03:26.13] - Speaker 1
This is the value that an option would have if we were to exercise immediately. For a call option, the intrinsic value is calculated as the difference between the current price of the underlying asset and the strike price. If an option is out of the money, the intrinsic value is zero. For for a put option, the intrinsic value is calculated as the difference between the strike price and the current price of the underlying asset. Also for put option out of the money option have zero intrinsic value.

[00:03:51.03] - Speaker 1
We then have the extrinsic value, also known as time value. This is the additional value of an option beyond its intrinsic value and reflect factors such as time to expiration, implied volatility and the potential for the underlying asset price to change. Time value is the difference between the option total price or premium and its intrinsic value. It represents the market's expectation of potential future price movements and the time remaining until expiration. At this point, let's look at intrinsic and time value in more details.

[00:04:19.10] - Speaker 1
Intrinsic value is the portion of the option price that can be realized if the option is exercised only in the money option have intrinsic value. In fact, until we reach the in the money strike, there is no intrinsic value. Let's take an example of a $25 strike call option on a stock trading at $27, the option is clearly in the money. Let's assume the call option is currently traded at $2.50. If this option is exercised, the option buyer would buy the stock at the price of $25.

[00:04:46.07] - Speaker 1
But the shares are currently trading at $27, which means of an instant profit of $2 if we resell the stock in the market. This $2 is what we refer to as intrinsic value or the profit if the option is exercised. So if we look at the price of this option, which is $2.50, we have $2 of intrinsic value, and the remaining $0.50 is called time value or extrinsic value. To recap, intrinsic value is the minimum value of an option at the money and out of the money option have no intrinsic value. And finally, the intrinsic value can never be negative.

[00:05:17.28] - Speaker 1
When the price of an option is higher than its intrinsic value, the difference in price is what we call time value. Time value is composed of volatility multiplied by time. In our example, we determine that the option intrinsic value is $2.00, but the option is trading at $2.50. This $0.50 of time value represent the opportunity the option still has to trade higher before expiration. More time to expiry means that there is more chance that the spot price will move in favor of the option buyer.

[00:05:44.03] - Speaker 1
And this is why the further the option expiration is, the more the premium and the time value increases. In the money options. I have premiums that are made up both by intrinsic and time value. The amount of time value will depend on the time remaining until expiration and the implied volatility of the option. Out of the money options have an intrinsic value of 0 and a time value of 100% of the premium.

[00:06:04.15] - Speaker 1
Let's do an example of a $30 strike call option on a stock trading at $27. The option is clearly out of the money. Let's assume that the call option is currently trading at $1.50, 100% of the premium. So $1.50 is represented by the time value, while the intrinsic value is 0. In previous lessons, we used this chart to help us understand the concept of time value.

[00:06:26.28] - Speaker 1
In summary, we know that an option has more value when there is more time to expiration. So the longer the maturity, the more time value. We know also that implied volatility increases time value. If you remember from the previous lesson, we compared the volatility of Procter and Gamble and Tesla. We saw that Tesla had higher implied volatility and more time value.

[00:06:45.15] - Speaker 1
And finally, uncertainty. The closer we get to add the money, the more uncertainty increases whether the option goes in the money or out of the money before expiration. This also increases the time value. Finally, let's look at volatility once again. Volatility and the premium you pay or receive are correlated.

[00:07:01.28] - Speaker 1
Market participants spend most of their time trying to understand the level of volatility of the asset they trade. This is also what you need to do if you want to be competitive and improve your trading strategy. Volatility is, as we told you before, uncertainty first of all. So when there is volatility, the market moves on the back of that uncertainty. Your role is to understand these risks and opportunities when evaluating the prices of your options.