The Black-Scholes model or the Black Scholes Merton is a mathematical model which estimates the theoretical value of derivatives based on other investment instruments, taking into account the effects of time and other risk factors. It is one of the leading concepts used to price options contracts.
This model came into being in 1973 and is still the best-known way of pricing the options contract. Let us dig deeper into the concept of the Black Scholes Model.
Black-Scholes Model in a nutshell
- Black-Scholes model, developed in 1973, is still a well-known model for the valuation of option contracts.
- Investors use the Black-Scholes equation to determine the exact price of European options.
- 5 Key Input Variables: Market volatility, asset price, strike price, interest rate, and expiration time.
- Limitations: Limited to European options, assumes constant volatility and may differ from real dynamics.
- The Black-Scholes model is not applicable to binary options.
The working of the Black Scholes Model
The Black Scholes Model assumes that financial instruments such as options and stocks shall have a log-normal distribution. It further believes that this log-normal distribution of prices will have constant volatility and a random walk. Investors use the Black Scholes equation to derive the European Options’ price.
Usually, an investor will need five variables to implement this model:
- Market volatility
- Underlying asset’s price
- Options’ strike price
- Interest rate
- Expiration time
This model lets the trader determine the reasonable prices for the Options held. Its prediction has its basis in the fact a heavily traded asset’s price follows a geometric Brownian motion.
When we apply this model to the stock option, it includes the price variation of the stock. It also includes other elements such as time value of money, strike price, and the options’ expiry time.
Model assumptions
Like all other models, this one assumes certain things. Let us look at the assumptions that make the Black Scholes Model:
- The holder of the Options does not receive any dividends throughout its life.
- One cannot predict the market movements, as they are random.
- No transaction costs get involved in purchasing the Options.
- The underlying assets’ returns have a log-normal distribution.
- There is a consistency in the risk-free rate and the underlying asset’s volatility.
- It is only applicable in European Options and on Options expiry.
The original Black Scholes Model did not have any provisions for the dividend effects during the lifetime of the Options. However, the assumptions of this model can get refined from time to time to suit the circumstances.
- Using the Black Scholes Model formula, you can easily determine the call option’s value.
- When this is done, you can finally compare it with the option’s current price to determine if it is worth the purchase.
Black Scholes Model formula
The mathematical formula for any model can be seen as intimidating to the trader. The beginner, especially, can feel haunted by looking at the formulas. It is because they don’t understand what to do with these equations, let alone use them to make their trading decisions.
But, don’t worry! The Black Scholes Model formula is not as intimidating as it may seem:
- C(S, t) = N(d₁)S – N(d₂)Ke-r(T-t)
- P(S, t) = N(-d₁)Ke-r(T-t) – N(-d₁)S
where:
- d₁ = (ln(S/K) + (r + σ²/2)(T-t)) / (σ√(T-t))
- d₂ = d₁ – σ√(T-t)
and where:
- C(S, t) & P(S, t) = Call and Put option prices
- S = Current stock price
- K = Option’s strike (exercise) price
- r = Risk-free interest rate
- T = Time to maturity of the option
- t = Current time
- N(x) = Cumulative distribution function of the standard normal distribution
- d₁ & d₂ = Auxiliary variables
- σ = Volatility of the underlying stock
A trader does not have to understand the intricates of this model to use the formula. You can use this formula with analysis tools and various online calculators. These days, online trading platforms offer various indicators and spreadsheets that allow the users to know the options pricing.
Example of the Black Scholes Model Formula
Let us understand how a trader can use the Black Scholes formula with the help of an example.
Suppose that a 6-month call option has an exercise price of $50. The asset currently trades at $52 and costs a trader $4.5. Let us also assume that the risk-free annual rate stands at 5%, and the stock’s return standard deviation is 12%. Considering this information, you might find yourself in a dilemma as to whether you should buy this option or not.
These are the given values:
- S = 52 (current stock price)
- K = 50 (exercise price)
- r = 0.05 (risk-free interest rate)
- T = 0.5 (time to maturity in years)
- t = 0 (current time)
- σ = 0.12 (volatility)
Firstly, we calculate d₁ and d₂:
Now we can calculate N(d₁) and N(d₂) with the cumulative standard normal distribution table:
Lastly, we put the values into the Black-Scholes formula:
Using the formula, after calculating d₁ and d₂ and applying the formula, the value of C will be $2.4601. It indicates that the option you wish to exercise has a value lower than the premium. This result leads us to an assumption that the option is overvalued. Or, we estimate the volatility lower than it is.
Thus, Black Scholes Model is one of the best methods to determine whether you are moving in the right direction in options trading. All advanced traders use this model to calculate their potential earnings with options trading.
The best part about this model is that even beginners can use the Black Scholes formula to make the right trading decisions.
Drawbacks of the Black Scholes Model
The Black Scholes Model does have its drawbacks.
Here are some drawbacks you will likely witness while using this model.
- An investor can use this model only to determine the price of European Options.
- It assumes constant volatility, which is impossible as the market always fluctuates.
- This model sometimes deviates an investor from the real-world model.
Conclusively, this model is the best to determine whether the Options you are looking to purchase are overpriced or not. It thus enhances your decision-making power.
Can you use the Black Scholes model for Binary Options?
The Black-Scholes model is not applicable to binary options, only for normal options.
Unlike traditional options, binary options are characterized by a fixed payout and expiry time, meaning that the Black-Scholes model is not suitable. Binary options use a different pricing model, such as the binomial option pricing model, which takes into account the discrete nature of binary options and their binary (yes/no) outcome.
