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19 Nov 0

Black-Scholes-Merton (BSM) Model

Summary

The Black-Scholes-Merton Model is one of the most famous and widely used models for pricing European Options. It was developed by economists Fischer Black and Myron Scholes in 1973, with contributions from Robert Merton. It revolutionized the field of financial markets by providing a way to calculate the theoretical price of options. The model is based on the assumption that financial markets behave in a specific way and that asset prices follow a stochastic (random) process.

The Black-Scholes model provides a theoretical framework for pricing options based on several key variables. The model assumes that the underlying asset price follows a geometric Brownian motion, which incorporates both a drift (average return) and a random component (volatility). The most widely known formula from this model is used to calculate the price of a European call option (the right to buy an asset at a predetermined price) and the price of a European put option (the right to sell an asset at a predetermined price).

The Black-Scholes Model is based on the principle of arbitrage-free pricing. In an efficient market, there must be no opportunity for riskless profit. The model assumes that the underlying asset follows a log-normal distribution, meaning that the price of the asset over time evolves in a random manner, with a certain expected drift (average return) and volatility.
Delta-Hedging: One of the key insights of the Black-Scholes model is that the option price can be replicated by holding a portfolio of the underlying asset and a risk-free bond. This portfolio must be continuously rebalanced to remain “delta-neutral,” which means that changes in the price of the underlying asset do not affect the portfolio’s value. The delta of an option, which is the rate of change of the option price with respect to the price of the underlying asset, is a critical component of this rebalancing strategy.

The Black-Scholes model is derived using stochastic calculus and assumptions about stock price behavior. The key assumptions of the model are:

Lognormal Distribution of Prices: The model assumes that stock prices follow a lognormal distribution, meaning their logarithms are normally distributed. This means stock prices cannot become negative and typically grow exponentially over time.
No Arbitrage: The model assumes that markets are efficient and free of arbitrage (i.e., there are no opportunities to make riskless profit).
Constant Volatility: Volatility is assumed to remain constant over the life of the option, although in reality, it may change over time (this is often accounted for with models like the Implied Volatility Surface).
European Options: The model is designed for European options, which can only be exercised at expiration (as opposed to American options, which can be exercised anytime before expiration).
No Dividends: The basic Black-Scholes model assumes that the underlying asset does not pay dividends. However, there are variations of the model that account for dividends.
Continuous Trading: The model assumes continuous trading of the underlying asset and the ability to continuously adjust portfolios, including borrowing and lending at the risk-free rate.

Variables

S = Current stock price
K = Strike price of the option
r = Risk-free interest rate (annualised)
σ = Volatility of the stock (annualised)
T = Time to expiration (in years)
d₁ and d₂ = Intermediate variables
N = Cumulative Distribution Function (CDF) of the standard normal distribution

1. Current Stock Price (S)

The current stock price (𝑆), is the price of the underlying asset today. This is a critical factor in determining the value of the option, as the option’s price is directly related to the current price of the asset. If the stock price is higher than the strike price, the call option becomes more valuable (in-the-money).

2. Strike Price (K)

The strike price (K), is the price at which the option holder can buy the underlying asset. It is the predetermined price set in the option contract. The relationship between the stock price and strike price determines whether the option is “in the money” (profitable) or “out of the money” (not profitable).

3. Risk-Free Interest Rate (𝑟)

The risk-free interest rate (𝑟) is typically based on the yield of government bonds, often considered a “safe” investment with minimal risk. It is used to calculate the time value of money — essentially, the present value of future cash flows.
The term 𝑒^−𝑟𝑇 in the formula represents the discounting factor, which adjusts the strike price for the time value of money over the life of the option.

4. Volatility (𝜎)

Volatility (𝜎) represents the annualized standard deviation of the asset’s returns. It is a measure of how much the price of the underlying asset fluctuates over time. Higher volatility increases the likelihood of the asset’s price moving favorably for the option holder (e.g., moving above the strike price for a call option).
In the Black-Scholes model, volatility is assumed to be constant over the life of the option.

5. Time to Maturity (𝑇)

The time to maturity (𝑇) is the amount of time left before the option expires. It is crucial because the longer the time to expiration, the more time the option has to become profitable (i.e., the stock price may move in the favorable direction).
Time is expressed in years, so if an option has 6 months until expiration, 𝑇=0.5.

6. Intermediate Variables 𝑑₁ and 𝑑₂

d₁ and 𝑑₂ are intermediate variables that incorporate the relationship between the current stock price, strike price, time to maturity, interest rate, and volatility.
𝑑₁ represents the normalized difference between the current price and the strike price, adjusted for time and volatility. It can be interpreted as a measure of how far the stock price is expected to move, adjusted for the time value and volatility.
𝑑₂ is simply 𝑑₁ minus the volatility term 𝜎√𝑇, adjusting for the time remaining to expiration. 𝑑₂ helps estimate the probability that the option will be exercised at expiration.

7. Cumulative Distribution Function N(𝑑)

N(𝑑₁) and N(𝑑₂) represent the cumulative probabilities under a standard normal distribution. These values give us the likelihood of the option finishing in-the-money, accounting for the randomness of the stock’s price movements.
N(𝑑₁) gives the probability that the option will be exercised, and N(𝑑₂) helps adjust the strike price for the time value of money. The standard normal CDF N(𝑑) gives the probability that a standard normally distributed random variable is less than or equal to 𝑑. This is a crucial concept in the Black-Scholes model because financial markets are assumed to follow a log-normal distribution (i.e., the logarithm of the asset price follows a normal distribution).

Assumptions

European-style options: These options can only be exercised at expiration, not before.
No dividends: The model assumes that the underlying asset does not pay dividends during the life of the option.
Efficient markets: The market for the underlying asset is efficient, meaning that all information is immediately reflected in the asset’s price.
No transaction costs: There are no costs for buying or selling the asset or for trading the options.
Constant volatility: The volatility of the underlying asset is constant over the life of the option.
Constant risk-free interest rate: The risk-free rate, often represented by the rate on government bonds, remains constant over the life of the option.
Log-normal distribution: The price of the underlying asset follows a log-normal distribution, meaning the asset prices change according to a random walk but can’t fall below zero (they are strictly positive).

Limitations

While the Black-Scholes-Merton Model is widely used and important, it has several limitations:

Constant volatility assumption: The model assumes that volatility is constant over the life of the option, which is not always true in real markets. In practice, volatility can change over time.
No dividends: The model assumes that the underlying asset does not pay dividends, but many stocks do pay dividends, and this can affect the option price.
European options only: The model applies only to European-style options, which can only be exercised at expiration. It does not account for American-style options, which can be exercised at any time before expiration.
Market inefficiencies: The model assumes that markets are efficient, meaning that all information is instantly reflected in the asset’s price, but in reality, markets may be subject to inefficiencies, such as delays in information dissemination or irrational behavior by investors.

Formulas

STEP 1: Calculate Intermediate Value (𝑑₁)

$$d_1=\frac{ln(\frac{S}{K})+(r+\frac{σ^2}{2})T}{σ\sqrt{T}}$$

STEP 2: Calculate Intermediate Value (𝑑₂)

$$𝑑_2=𝑑_1−𝜎\sqrt{T}$$

STEP 3(a): Calculate Call Option Price (C)

$C = S_0 N(d_1) – X e^{-rT} N(d_2)$

$$C=SN(d_{1})−Ke^{−rT}N(d_{2})$$

- Ke^-rT = This component discounts the strike price back to today. In other words, present value of future cashflow.

STEP 3(b): Calculate Put Option Price (P)

$$P=Ke^{−rT}N(−d_{2})−SN(−d_{1})$$

- N(-d₁) = Probability of the investor exercising the option.

Conclusion

The Black-Scholes Model has become a cornerstone of modern financial theory and practice, providing a way to price European options based on certain key factors, such as the current price of the asset, the strike price, time to expiration, volatility, and the risk-free interest rate. While the model has its limitations, it is still widely used for pricing and hedging options in financial markets today, and it laid the foundation for much of the options trading strategies employed by institutions and individuals alike. The Black-Scholes model is widely used for pricing options because it provides a closed-form solution, making it easy to calculate the theoretical price of options in real-time. However, due to its assumptions (such as constant volatility and no dividends), the model may not always capture market realities perfectly, especially during periods of high volatility or when stocks pay dividends.

Important: Calculator still under construction. The option price is incorrect! 🙁

Option Type:

Option Type

Spot Price ($):

Spot Price

Strike Price ($):

Strike Price

Risk-Free Interest Rate (%):

entered as a percentage (i.e. 10)

Volatility (%):

entered as a percentage (i.e. 12)

Expiration (T):

entered as a decimal (i.e. 1 year = 1, 6 months = 0.5, 3 months = 0.25 etc.)

d1:

d2:

N(d1):

N(d2):

Price of Option$

Admin

Financial Analysis, Financial Instruments

17 Nov 0

Correlation

The correlation between multiple stock assets refers to the statistical relationship between the price movements of those assets over time. It helps investors understand how different stocks move in relation to each other. Understanding this correlation is essential for portfolio diversification, risk management, and making informed investment decisions.

What is Correlation?

Correlation is a measure of the degree to which two or more assets move in relation to each other. It is represented by a correlation coefficient, which ranges from -1 to +1:

+1 (Perfect Positive Correlation): When one stock moves in the same direction as another stock (i.e., both go up or down together in perfect sync).
0 (No Correlation): When the movements of the two stocks are completely unrelated. One stock may go up while the other goes down, or vice versa, without any predictable relationship.
-1 (Perfect Negative Correlation): When one stock moves in the opposite direction of another stock (i.e., when one stock goes up, the other goes down in perfect inverse relation).
Between 0 and ±1: A correlation coefficient between 0 and ±1 indicates some degree of relationship between the assets, with the strength and direction of the relationship varying depending on the value.

### Types of Correlation

1. **Positive Correlation (+1):**
– If two stocks have a **positive correlation**, they tend to move in the same direction. When one stock goes up, the other tends to go up as well, and vice versa.
– Example: Stocks within the same industry, such as **Apple** and **Microsoft**, often exhibit positive correlation because they are influenced by similar market factors (e.g., technology trends, interest rates, etc.).

2. **Negative Correlation (-1):**
– If two stocks have a **negative correlation**, they tend to move in opposite directions. When one stock increases in value, the other typically decreases, and vice versa.
– Example: A **stock index (e.g., S&P 500)** and **gold** often have a negative correlation because when the stock market rises, investors may prefer riskier assets, and gold, which is considered a safe-haven asset, may decline. Conversely, during market downturns, gold might increase as investors seek safety.

3. **Zero or No Correlation (0):**
– If two stocks have **zero correlation**, their movements are independent of each other. There is no predictable relationship between their price movements.
– Example: A stock in **the airline industry** and a stock in **the pharmaceutical industry** may have a low or zero correlation because their price movements are driven by different factors (e.g., air traffic and healthcare news).

### Understanding the Correlation Between Multiple Assets

When analyzing multiple stock assets, it’s essential to look at **pairwise correlations** between each pair of assets. The correlation between multiple assets can be summarized in a **correlation matrix**, which is a table that shows the correlation coefficient for each pair of stocks.

For example, if you have three stocks, A, B, and C, the correlation matrix might look like this:

| | **A** | **B** | **C** |
|——-|——–|——–|——–|
| **A** | 1 | 0.8 | -0.2 |
| **B** | 0.8 | 1 | 0.1 |
| **C** | -0.2 | 0.1 | 1 |

– **A and B** have a **0.8 positive correlation**, meaning they tend to move in the same direction.
– **A and C** have a **-0.2 correlation**, meaning their movements have a slight inverse relationship.
– **B and C** have a **0.1 correlation**, suggesting they move independently of each other.

### Importance of Correlation in Portfolio Diversification

**Portfolio diversification** is the practice of holding a variety of assets to reduce the overall risk of an investment portfolio. The goal is to invest in assets that do not move in perfect sync with each other, thereby reducing the risk that all investments will decline at the same time. Correlation plays a key role in diversification:

– **High Positive Correlation (+1):** If stocks in a portfolio are highly correlated (i.e., they move together), diversification is limited. If one stock goes down, it’s likely that others in the portfolio will also go down.

– **Low or Negative Correlation (0 or -1):** If stocks in a portfolio are less correlated or negatively correlated, the portfolio is more diversified, which can reduce overall risk. When one stock drops in value, another may rise, helping to stabilize the portfolio’s returns.

### Practical Example: Portfolio Diversification Using Correlation

Let’s assume you have two stocks in your portfolio:

– **Stock A**: Technology company
– **Stock B**: Energy company

You find that Stock A and Stock B have a correlation of **0.3**, meaning their price movements have a weak positive relationship. By adding Stock B to your portfolio, you reduce the overall risk because the stocks are not perfectly correlated.

However, if you add a third stock, **Stock C** (say a healthcare company), which has a correlation of **-0.5** with Stock A, the portfolio’s overall risk is further reduced because Stock A and Stock C tend to move in opposite directions. In other words, when Stock A goes up, Stock C tends to go down, and vice versa.

### Key Takeaways

1. **Positive Correlation:** Assets move together in the same direction.
2. **Negative Correlation:** Assets move in opposite directions.
3. **Zero Correlation:** Assets move independently of each other.
4. **Diversification:** By combining assets with low or negative correlations, you can reduce overall portfolio risk.
5. **Risk Management:** Correlation helps in assessing the risk of a portfolio. Assets with low correlation provide better diversification benefits than assets with high correlation.

In summary, understanding the correlation between multiple stock assets is a crucial aspect of portfolio management, as it allows investors to make better decisions about risk, diversification, and asset allocation. By selecting assets with low or negative correlations, investors can minimize the overall volatility of their portfolios.

How to Calculate Correlation

The **correlation coefficient** is a statistical measure that quantifies the relationship between two variables. It tells you the strength and direction of their relationship. To calculate the correlation between two assets (or two variables), the **Pearson correlation coefficient** is most commonly used.

### Formula for Pearson’s Correlation Coefficient

The formula to calculate the **Pearson correlation coefficient (r)** between two variables **X** and **Y** is:

\[
r = \frac{\sum{(X_i – \overline{X})(Y_i – \overline{Y})}}{\sqrt{\sum{(X_i – \overline{X})^2} \sum{(Y_i – \overline{Y})^2}}}
\]

Where:

– $ X_i $ and $ Y_i $ are the individual data points of variables X and Y.
– $ \overline{X} $ and $ \overline{Y} $ are the mean (average) values of X and Y, respectively.
– $ \sum $ represents the sum of all the data points.
– The formula computes the covariance between X and Y divided by the product of their standard deviations.

### Step-by-Step Process to Calculate Correlation

Here’s a step-by-step breakdown to calculate the correlation between two sets of data (two variables or two stock assets):

#### 1. **Obtain the Data Points**
Collect the data for both variables (or stock prices). For example, you might have the monthly returns or prices of two stocks over several months. Let’s assume you have data points for two stocks over five periods:

| Period | Stock A | Stock B |
|——–|———|———|
| 1 | 10 | 12 |
| 2 | 12 | 14 |
| 3 | 14 | 16 |
| 4 | 16 | 18 |
| 5 | 18 | 20 |

#### 2. **Calculate the Means**
Find the **mean (average)** of both variables.

– Mean of Stock A ($ \overline{X} $):
\[
\overline{X} = \frac{10 + 12 + 14 + 16 + 18}{5} = 14
\]

– Mean of Stock B ($ \overline{Y} $):
\[
\overline{Y} = \frac{12 + 14 + 16 + 18 + 20}{5} = 16
\]

#### 3. **Calculate the Deviations from the Mean**
For each data point, subtract the mean of the respective variable to get the deviation from the mean:

| Period | Stock A | Stock B | $ X_i – \overline{X} $ | $ Y_i – \overline{Y} $ | Product of Deviations |
|——–|———|———|————————–|————————–|———————–|
| 1 | 10 | 12 | -4 | -4 | 16 |
| 2 | 12 | 14 | -2 | -2 | 4 |
| 3 | 14 | 16 | 0 | 0 | 0 |
| 4 | 16 | 18 | 2 | 2 | 4 |
| 5 | 18 | 20 | 4 | 4 | 16 |

#### 4. **Calculate the Sum of the Products of Deviations**
Now sum the products of the deviations from the previous column:

\[
\sum{(X_i – \overline{X})(Y_i – \overline{Y})} = 16 + 4 + 0 + 4 + 16 = 40
\]

#### 5. **Calculate the Sum of Squared Deviations**
Next, calculate the sum of squared deviations for both variables:

– For **Stock A**:
\[
\sum{(X_i – \overline{X})^2} = (-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2 = 16 + 4 + 0 + 4 + 16 = 40
\]

– For **Stock B**:
\[
\sum{(Y_i – \overline{Y})^2} = (-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2 = 16 + 4 + 0 + 4 + 16 = 40
\]

#### 6. **Calculate the Pearson Correlation Coefficient**
Now use the formula to calculate the correlation:

\[
r = \frac{40}{\sqrt{40 \times 40}} = \frac{40}{40} = 1
\]

The Pearson correlation coefficient is **1**, which indicates a **perfect positive correlation** between Stock A and Stock B. This means that for every increase in Stock A, Stock B also increases by the same proportion, in perfect synchrony.

### Interpreting the Correlation Coefficient

– **+1**: Perfect positive correlation. The two assets move together in exactly the same way.
– **0.5 to 0.8**: Strong positive correlation. The assets tend to move in the same direction, but not always perfectly.
– **0 to 0.5**: Weak positive correlation or no clear relationship.
– **-0.5 to -1**: Negative correlation. As one asset increases, the other tends to decrease.
– **-1**: Perfect negative correlation. One asset moves inversely with the other.

### Practical Use of Correlation in Finance

In finance, understanding the correlation between multiple stock assets (or asset classes) is essential for:

– **Diversification**: By selecting assets with low or negative correlations, you can reduce the overall risk of your portfolio. For example, stocks with negative correlation can help offset losses when other stocks perform poorly.
– **Risk Management**: Correlation helps you understand how stocks move relative to each other. This can help in hedging strategies, especially when you have highly correlated assets that are sensitive to the same market forces.
– **Portfolio Optimization**: Investors use correlation to construct efficient portfolios that balance risk and return. By combining assets with low correlation, you can improve the risk-return profile of the portfolio.

### Using Software for Correlation Calculations

In practice, manually calculating correlation for large datasets can be tedious. Thankfully, software like Excel, Python, or R can easily compute correlations between multiple assets:

– **Excel**: Use the `CORREL` function: `=CORREL(range1, range2)`
– **Python (Pandas)**: Use the `.corr()` method on a DataFrame.

Example in Python:
“`python
import pandas as pd

# Create a DataFrame with stock prices
data = {‘Stock_A’: [10, 12, 14, 16, 18], ‘Stock_B’: [12, 14, 16, 18, 20]}
df = pd.DataFrame(data)

# Calculate the correlation
correlation = df[‘Stock_A’].corr(df[‘Stock_B’])
print(correlation)
“`

This will give you the correlation coefficient directly without needing to calculate it manually.

### Conclusion

The **correlation coefficient** is a valuable tool in understanding the relationship between multiple stock assets. By calculating it, you can assess how assets move together, which is critical for diversification, risk management, and portfolio optimization. The closer the correlation is to +1 or -1, the stronger the relationship between the assets. In contrast, a correlation near 0 indicates little or no relationship.

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Financial Instruments