Black Scholes Model

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Black Scholes Model Theory and Methodology Calculations Pros and Cons Conclusion References

BLACK SCHOLES MODEL

By ProbabilityofProfit.com

The Black Scholes Model (BSM) is the most frequently used option-pricing model used to calculate the theoretical price of financial options, taking into account valuation factors including the expected volatility, term, exercise price, current price, dividend yield, and the risk-free interest rate.

Originally published in 1973 by Fischer Black, Myron Scholes, and Robert Merton with the purpose to value European-style exchanged-traded short-term options that are exercisable only upon expiration for non-dividend paying stocks.

THEORY & METHODOLOGY

THEORY

The Black-Scholes Model incorporates various valuation factors, such as expected volatility, time to expiration, strike price, current stock price, dividend yield, and the risk-free interest rate, to calculate an option's price. Its simplicity and analytical tractability have made it a foundational tool in financial economics.

Introduced in 1973 by Fischer Black, Myron Scholes, and Robert Merton, the BSM was designed to value European-style, exchange-traded short-term options that can only be exercised at expiration, specifically for non-dividend-paying stocks. The groundbreaking work earned Scholes and Merton the Nobel Prize in Economic Sciences in 1997, highlighting its significance in the field.
- Since then, it has been widely adopted for evaluating expenses for U.S. employee stock options by the Financial Accounting Standards Board (FASB) under ASC 718.
- Further adaptations have been applied to the model to price American-style options and other financial instruments like dividend paying stocks, commodity futures, and FX forwards.

METHODOLOGY

The key idea behind the Black-Scholes model is that the specific manner in which the underlying asset evolves over time isn't crucial for valuation—how the underlying reached that stage doesn't impact the option's worth.

For European-style options, which are exercisable only at contract expiration, what truly counts is the range of potential prices at that moment.
For American-style options, which are exercisable at any point before the contract’s expiration, the core principle remains similar to European options
The model's primary limitation persists in its ability to accurately capture the dispersion of future prices rather than the complexity of the dispersion process itself—this means it may not accurately capture the true complexity of the dispersion of future prices, especially in cases of extreme market events or changes in volatility over time.

Based on financial market assumptions:

Arbitrage Free Markets
Frictionless, Continuous Markets
Risk Free Rates is time dependent
Log-normally Distributed Price Movements under the standard geometric Brownian motion
Volatility is constant

CALCULATIONS

BLACK-SCHOLES ASSUMPTIONS

In the most fundamental form of the Black-Scholes model these assumptions are to be expected:

Arbitrage Free Markets: The Black Scholes formulas work under the idea that traders always aim to make the most profit for themselves and won't let any risk-free profit opportunities last for long.
Frictionless, Continuous Markets: This idea assumes that markets operate smoothly without any obstacles, meaning you can buy or sell as much as you want of something at any time without paying extra fees.
Risk Free Rates: There's a way to borrow and lend money at an interest rate without any risk.
Log-normally Distributed Price Movements: Prices tend to follow a certain pattern called Geometric Brownian Motion, where they're distributed in a specific mathematical way that looks like a log-normal curve.
Constant Volatility: The Black Scholes-style formulas for options assume that the level of volatility in the market remains the same throughout the life of the option contract.

INPUT VALUES

The Black-Scholes formula require these input values for its calculation:

Underlying Price (F or S): The current price of the asset you're dealing with. Use S for the spot price (current price) and F for a future price.
Strike Price (X): The price at which you can buy or sell the asset if you choose to exercise the option.
Time to Expiration (T): The time left until the option expires, measured in years. You can calculate this by finding the difference in days between the expiration date (t_1) and the valuation date (t_0), then dividing by 365.
Valuation Date (t_0): The date you're valuing the option. For example, if you're evaluating the option's worth today, then today's date would be t_0.
Expiration Date (t_1): The date by which the option must be exercised or it expires worthless.
Volatility (V): How much the price of the asset is expected to fluctuate over time. It's typically calculated from past option prices.
Continuous Yield (q): This accounts for any dividends or payments the underlying asset makes over time. It affects the option's value.
Risk-Free Rate (r): The expected return on an investment with no risk. Usually approximated by government bond yields or LIBOR rates.
Foreign Risk-Free Rate (rf): The risk-free rate of the foreign currency if you're dealing with international assets. Each currency has its own risk-free rate.

FORMULAS

Black-Scholes Call Price Formula:

$C = Fe^{(b-r)T} N(D_1) - Xe^{-rT} N(D_2)$

Black-Scholes Put Price Formula:

$P = Xe^{-rT} N(-D_2) - Fe^{(b-r)T} N(-D_1)$

Supplemental Calculations:

D1 Parameter Formula:

$D_1 = \frac{ln\frac{F}{X} + (b+\frac{V^2}{2})T}{V*\sqrt{T}}$

D2 Parameter Formula:

$D_2 = D_1 - V\sqrt{T}$

BLACK-SCHOLES CALL PRICE EXAMPLE

Input Values:

Underlying Price (S): The current price of the stock you're interested in, let's say it's $100.
Strike Price (X): The price at which you can buy the stock if you choose to exercise the option, let's set it at $110.
Time to Expiration (T): Let's assume there are 6 months (0.5 years) left until the option expires.
Valuation Date (t_0): Today's date.
Expiration Date (t_1): 6 months from today.
Volatility (V): The volatility of the stock, let's assume it's 20%.
Continuous Yield (q): This stock doesn't pay dividends, so q is 0.
Risk-Free Rate (r): The current risk-free rate, let's assume it's 2%.

Now, let's calculate:

Now, you would use a standard normal distribution table or a calculator that can compute the cumulative standard normal distribution function (N) to find the values of (-0.3913) and N(-0.5327). After substituting these values, with 0.34827 and 0.29806 respectively, you'll get the Call Price.

BLACK-SCHOLES PUT PRICE EXAMPLE

Input Values:

Underlying Price (S): $100 (current price of the stock).
Strike Price (X): $110 (the price at which you can sell the stock if you choose to exercise the option).
Time to Expiration (T): 6 months (0.5 years) until the option expires.
Valuation Date (t_0): Today's date.
Expiration Date (t_1): 6 months from today.
Volatility (V): 20% (historical volatility of the stock).
Continuous Yield (q): 0 (the stock doesn't pay dividends).
Risk-Free Rate (r): 2%.

Now, let's calculate:

Now, you would use a standard normal distribution table or a calculator that can compute the cumulative standard normal distribution function (N) to find the values of (0.5327) and N(0.3913). After substituting these values, with 0.70194 and 0.65173 respectively, you'll get the Put Price.

PROS & CONS

PROS	CONS
Widely Used: The Black-Scholes model is the most frequently used option-pricing model in finance, making it a standard tool for pricing options and other derivative securities.	Assumptions: The model relies on several assumptions that may not always hold true in real-world market conditions, such as constant volatility, continuous and frictionless markets, and log-normal distribution of price movements.
Theoretical Pricing Objective: The model provides a theoretical framework for calculating the fair value of financial options, taking into account valuation factors such as expected volatility, time to expiration, exercise price, and risk-free interest rate.	Limited to European Options: The original Black-Scholes model is designed for European-style options, which can only be exercised at expiration.
Arbitrage-Free Pricing: The model is based on the assumption of arbitrage-free markets, which ensures that the calculated option prices are consistent with market conditions and do not allow for risk-free profits.	Inability to Capture Extreme Events: The model's limitation in accurately capturing the true price options during periods of extreme market volatility or uncertainty.
Applicability: While originally designed for European-style options on non-dividend-paying stocks, the Black-Scholes model has been adapted to price a wide range of financial instruments, including American-style options, dividend-paying stocks, commodity futures, and FX forwards.

CONCLUSION

The Black Scholes Model continues to stand the test of time as a fundamental tool in the world of finance, helping investors and analysts price options with a systematic approach that factors in volatility, time to expiration, exercise price, and the risk-free interest rate. While it relies on certain assumptions, like constant volatility and frictionless markets, its versatility and broad application across various financial instruments highlight its lasting utility.

As you venture into options trading, remember that understanding the underlying assumptions and methodologies of any pricing model is crucial. The Black Scholes Model is just one piece of the puzzle. Its value lies in how well you can apply its principles while keeping an eye on the bigger market picture and your own risk comfort levels.

We trust this guide has shed light on the intricacies of option pricing and provided you with useful insights for your trading journey. Good luck with your trading, and thank you for exploring this guide with us!

REFERENCES

At ProbabilityofProfit.com, we are dedicated to delivering accurate and trustworthy content to our readers. Our team meticulously researches each topic, ensuring that the information we provide is both comprehensive and reliable. We reference high-quality sources, including academic journals, industry reports, and market analysis, to support our content. Additionally, we draw upon original studies and data from respected publishers to provide a well-rounded perspective.

Our commitment to accuracy means we constantly review and update our articles to reflect the latest developments and trends in options trading. We strive to present unbiased information, allowing our readers to make informed decisions based on solid evidence. Discover more about our rigorous standards and our dedication to excellence on our website.

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Table of Contents

BLACK SCHOLES MODEL

The Black Scholes Model (BSM) is the most frequently used option-pricing model used to calculate the theoretical price of financial options, taking into account valuation factors including the expected volatility, term, exercise price, current price, dividend yield, and the risk-free interest rate.

Originally published in 1973 by Fischer Black, Myron Scholes, and Robert Merton with the purpose to value European-style exchanged-traded short-term options that are exercisable only upon expiration for non-dividend paying stocks.

THEORY & METHODOLOGY

THEORY

Since then, it has been widely adopted for evaluating expenses for U.S. employee stock options by the Financial Accounting Standards Board (FASB) under ASC 718.

Further adaptations have been applied to the model to price American-style options and other financial instruments like dividend paying stocks, commodity futures, and FX forwards.

METHODOLOGY

The key idea behind the Black-Scholes model is that the specific manner in which the underlying asset evolves over time isn't crucial for valuation—how the underlying reached that stage doesn't impact the option's worth.

For European-style options, which are exercisable only at contract expiration, what truly counts is the range of potential prices at that moment.

For American-style options, which are exercisable at any point before the contract’s expiration, the core principle remains similar to European options

Based on financial market assumptions:

Arbitrage Free Markets

Frictionless, Continuous Markets

Risk Free Rates is time dependent

Log-normally Distributed Price Movements under the standard geometric Brownian motion

Volatility is constant

CALCULATIONS

BLACK-SCHOLES ASSUMPTIONS

In the most fundamental form of the Black-Scholes model these assumptions are to be expected:

Arbitrage Free Markets: The Black Scholes formulas work under the idea that traders always aim to make the most profit for themselves and won't let any risk-free profit opportunities last for long.

Frictionless, Continuous Markets: This idea assumes that markets operate smoothly without any obstacles, meaning you can buy or sell as much as you want of something at any time without paying extra fees.

Risk Free Rates: There's a way to borrow and lend money at an interest rate without any risk.

Log-normally Distributed Price Movements: Prices tend to follow a certain pattern called Geometric Brownian Motion, where they're distributed in a specific mathematical way that looks like a log-normal curve.

Constant Volatility: The Black Scholes-style formulas for options assume that the level of volatility in the market remains the same throughout the life of the option contract.

INPUT VALUES

The Black-Scholes formula require these input values for its calculation:

Underlying Price (F or S): The current price of the asset you're dealing with. Use S for the spot price (current price) and F for a future price.

Strike Price (X): The price at which you can buy or sell the asset if you choose to exercise the option.

Time to Expiration (T): The time left until the option expires, measured in years. You can calculate this by finding the difference in days between the expiration date (t_1) and the valuation date (t_0), then dividing by 365.

Valuation Date (t_0): The date you're valuing the option. For example, if you're evaluating the option's worth today, then today's date would be t_0.

Expiration Date (t_1): The date by which the option must be exercised or it expires worthless.

Volatility (V): How much the price of the asset is expected to fluctuate over time. It's typically calculated from past option prices.

Continuous Yield (q): This accounts for any dividends or payments the underlying asset makes over time. It affects the option's value.

Risk-Free Rate (r): The expected return on an investment with no risk. Usually approximated by government bond yields or LIBOR rates.

Foreign Risk-Free Rate (rf): The risk-free rate of the foreign currency if you're dealing with international assets. Each currency has its own risk-free rate.

FORMULAS

Black-Scholes Call Price Formula:

Black-Scholes Put Price Formula:

Supplemental Calculations:

D1 Parameter Formula:

D2 Parameter Formula:

BLACK-SCHOLES CALL PRICE EXAMPLE

Input Values:

Underlying Price (S): The current price of the stock you're interested in, let's say it's $100.

Strike Price (X): The price at which you can buy the stock if you choose to exercise the option, let's set it at $110.

Time to Expiration (T): Let's assume there are 6 months (0.5 years) left until the option expires.

Valuation Date (t_0): Today's date.

Expiration Date (t_1): 6 months from today.

Volatility (V): The volatility of the stock, let's assume it's 20%.

Continuous Yield (q): This stock doesn't pay dividends, so q is 0.

Risk-Free Rate (r): The current risk-free rate, let's assume it's 2%.

Now, let's calculate:

Now, you would use a standard normal distribution table or a calculator that can compute the cumulative standard normal distribution function (N) to find the values of (-0.3913) and N(-0.5327). After substituting these values, with 0.34827 and 0.29806 respectively, you'll get the Call Price.

BLACK-SCHOLES PUT PRICE EXAMPLE

Input Values:

Underlying Price (S): $100 (current price of the stock).

Strike Price (X): $110 (the price at which you can sell the stock if you choose to exercise the option).

Time to Expiration (T): 6 months (0.5 years) until the option expires.

Valuation Date (t_0): Today's date.

Expiration Date (t_1): 6 months from today.

Volatility (V): 20% (historical volatility of the stock).

Continuous Yield (q): 0 (the stock doesn't pay dividends).

Risk-Free Rate (r): 2%.

Now, let's calculate:

Now, you would use a standard normal distribution table or a calculator that can compute the cumulative standard normal distribution function (N) to find the values of (0.5327) and N(0.3913). After substituting these values, with 0.70194 and 0.65173 respectively, you'll get the Put Price.

PROS & CONS

PROS

CONS

Widely Used: The Black-Scholes model is the most frequently used option-pricing model in finance, making it a standard tool for pricing options and other derivative securities.

Assumptions: The model relies on several assumptions that may not always hold true in real-world market conditions, such as constant volatility, continuous and frictionless markets, and log-normal distribution of price movements.

Theoretical Pricing Objective: The model provides a theoretical framework for calculating the fair value of financial options, taking into account valuation factors such as expected volatility, time to expiration, exercise price, and risk-free interest rate.

Limited to European Options: The original Black-Scholes model is designed for European-style options, which can only be exercised at expiration.

Arbitrage-Free Pricing: The model is based on the assumption of arbitrage-free markets, which ensures that the calculated option prices are consistent with market conditions and do not allow for risk-free profits.

Inability to Capture Extreme Events: The model's limitation in accurately capturing the true price options during periods of extreme market volatility or uncertainty.

Applicability: While originally designed for European-style options on non-dividend-paying stocks, the Black-Scholes model has been adapted to price a wide range of financial instruments, including American-style options, dividend-paying stocks, commodity futures, and FX forwards.

CONCLUSION

We trust this guide has shed light on the intricacies of option pricing and provided you with useful insights for your trading journey. Good luck with your trading, and thank you for exploring this guide with us!

REFERENCES