The Black-Scholes model also called the Black-Scholes-Merton model is a mathematical equation that evaluates the theoretical value of pricing of bonds, stocks etc, based on six main variables. It provides a mathematical model for the derivatives of the financial market. The Black-Scholes formula gives an estimate of the price according to the European style option.
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The equation and the model are proposed by the economists Fischer Black and Myron Scholes. The idea was first put forward by Robert C. Merton, which is why he is also credited for this model. The main idea behind the model is to hedge the buying and selling alternatives of the invested assets with minimum risk of loss or maximise the gains.
Some Important Terms Related to Black-Scholes Model
Underlying are basic inputs of the Black-Scholes model. The model gives us a geometric Brownian motion with constant drift and volatility which appears just like a smirk with constant drift and volatility. By adjusting the value of these variables in constant proportion a riskless hedge portfolio can be created by minimising the possible market risks. Let us discuss these variables in brief.
- Strike price: It is the price at which the holder with the option has the right to buy or sell an owned security, depending on the alternative they have for a call option or put option. It is generally denoted by K.
- Spot price/ Stock price: It is simply the current market price of the available asset. For assets that are not liquid, it becomes quite difficult to find the exact price but in some situations, the closing market price is taken into consideration. It is denoted by S(t) or St or simply S.
- Time until expiration: By the term itself it is quite clear, that it is the time span (in years) until the available option expires. It is denoted by t.
- Risk-free interest rate: It is the constant rate of return on an asset particularly, riskless-asset. It is often denoted by r.
- Volatility: The most pivotal input of the model. It is the degree of variation of price of trade over time, calculated by taking the standard deviation of the returns for that span of time. The volatility of the option pricing model can be measured in different ways:
- Historical volatility: The variation of the market price is calculated by taking into consideration the price over more than five years. Historical volatility can be sometimes flawed as it is assumed that what happened in the past the same pattern will be repeated which is not always the case.
- Implied volatility: It is measured using the volatility surface which is the function of both strike price and maturity time. It is a curved surface of three-dimensional space whose x-axis represents the time to maturity, the y-axis represents the strike price of the stock and the z-axis represents the resultant current market implied volatility. Thus, implied volatility is implied by the market price of the trading option. To determine the volatility here we need to reverse the pricing model with the known price.
Basic Hypothesis of the Black-Scholes Pricing Model
The model gives us a theoretical value of the stock option for both the call and put option. Before proposing this model Black and Scholes took the following assumptions which are quite necessary for the pricing model. Let us discuss them also in brief:
- Riskless rate: This is nothing but the risk-free interest rate. The amount of returns on an asset is considered to be constant, hence risk-free.
- Constant Volatility: The volatility is the measure of the variance of a stock over time. It is assumed for the price model, the variation for the price of the option is constant for the time period. However, in a real-world scenario, this never happens.
- No dividends: The underlying stock has no dividends over the life span of the option which is again not possible in the actual case, companies do have to pay dividends to their shareholders. This anomaly can be adjusted by subtracting the discounted value of a future dividend from the current stock price.
- Frictionless market: Black and Scholes considered the market to be ideal and almost efficient in the following ways:
- Transactions happen without any processing charges.
- All parties have equal opportunities and access to information at the same time.
- Liquid market
- Random walk: The instantaneous logarithmic interest on stock price is a very minute random walk with drift, in other words, at any instant the price of the underlying stock option can deviate in a proportionate manner and with the same probability. Again market does not happen to be this much efficient for a very long time span.
- The interest rates in the market are constant and known.
- Log-normal distribution of returns: Interest returns are normally distributed which is quite relative to the actual scenario.
- European-style option: They considered the European-style option call which works only at the time of expiration whereas in the American style it can be exercised anytime bere the expiration date.
Black-Scholes Differential Equation
It is a partial differential equation that depicts the pricing of an option for a given time period. The equation is given below-
\(\begin{array}{l}\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0 \end{array} \)
Where t is time in years (if t = 0, then it represents the present year)
r is the annual risk-free interest rate.
S is the function of time t represents the price of the underlying asset at t (sometimes also denoted as St).
σ is the standard deviation of the returns on the stock.
V is the function of S and t represents the price of the option.
The equation suggests the hedging of the option either by buying or selling the underlying asset in a way which eliminates the risk factor.
Black-Scholes Formula for the Model
The Black-Scholes formula is obtained by solving the above partial differential equation by the terminal and boundary conditions:
C(0, t) = 0 ∀ t
C(S, t) → S – K as S → ∞
C(S, T) = max{S – K, 0}, where
C(S,t) is the price of the European-style call option.
K is the strike price.
N(x) denotes the standard normal cumulative distribution function defined as:
\(\begin{array}{l}N(x)=\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{x}e^{-z^{2}/2}dz\end{array} \)
And N’(x) is the standard normal probability density function.
P(S, t) is the price of the put option in the European style.
T is the time of option expiration and
𝜏 is the time until maturity, 𝜏 = T – t.
Thus, the price of the European-style call option for which no dividends are paid for the time period is given by:
C(S, t) = N(d1)S – N(d2)Ke–r(T – t), where
\(\begin{array}{l}d_{1}= \frac{1}{\sigma \sqrt{T-t}}\left [ ln\left ( \frac{S}{K} \right )+\left ( r+\frac{\sigma^{2}}{2} \right ) (T-t)\right ]\end{array} \)
\(\begin{array}{l}d_{2}= d_{1}-\sigma \sqrt{T-t}\end{array} \)
For put call option we have the formula:
P(S, t) = Ke–r(T – t) – S + C(S, t)
= N( –d2)Ke –r(T – t) – N( –d1) S.
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Solved Example on Black-Scholes Model
The stock price of the shares of a certain company A closes at ₹ 117.25 on August 1st whose strike price is ₹ 100. The stock expires on November 1st. There are no dividends that need to be paid till the expiry date and the risk-free annual interest rate is 8.5%. If the standard deviation of the volatility of the stock returns is 0.8445, calculate the price of the call option using the Black-Scholes model formula.
Solution:
The Black-Scholes formula for the call price option is given as:
C(S, t) = N(d1)S – N(d2)Ke–r(T – t), where
\(\begin{array}{l}d_{1}= \frac{1}{\sigma \sqrt{T-t}}\left [ ln\left ( \frac{S}{K} \right )+\left ( r+\frac{\sigma^{2}}{2} \right ) (T-t)\right ]\end{array} \)
\(\begin{array}{l}d_{2}= d_{1}-\sigma \sqrt{T-t}\end{array} \)
We have
S = ₹ 117.25
K = ₹ 100
T – t = 92 days = 92/365 year = 0.2520 year
r = 8.5% = 0.085
σ = 0,8445
\(\begin{array}{l}d_{1}=\frac{\frac{117.25}{100}\left (0.0085+\frac{0.8445^{2}}{2}0.2520 \right )}{0.8445\sqrt{0.2520}} =0.6369\end{array} \)
\(\begin{array}{l}d_{2}= 0.6369-0.8445\sqrt{0.2520}=0.2109\end{array} \)
N(d1) = 0.728 and N(d2) = 0.584
∴ C = 0.728 × 117.25 – 0.584 × 100e– 0.085 × 0.2520 = 29.4 (approximately)
The theoretical call price for the stock of the company is ₹ 29.4.
Practice Questions on Black-Scholes Model
1. Calculate the price of the European style call option for a non-dividend paying stock when the stock price is ₹ 52. The strice price is ₹ 50, the risk-free interest rate is 12% per annum, the volatility is 30% per annum and the time for maturation is three months.
2. Calculate the price for put option for the question given in the solved example.
Frequently Asked Questions on Black-Scholes Model – FAQs
Q1
What does the Black-Scholes model for option pricing assumes about the probability distribution of the stock price over one year?
The Black-Scholes model for option pricing assumes that the probability distribution of the stock price over one year (or any time period) is lognormal.
Q2
What does the Black-Scholes model for option pricing assumes about the continuously compounded rate of return on the stock price during a year?
The Black-Scholes model for option pricing assumes that the continuously compounded rate of return on the stock price during a year is normally distributed.
Q3
What are the assumptions of Black-Scholes model formula?
The assumptions are that stock prices follows a lognormal distribution, it cannot have negative value, no dividends are payed, frictionless market, constant volatility, riskless rate, and follows the European style option.
Q4
For what purpose th Black-Scholes model is used?
The Black-Scholes model gives mathematical formulation for the theoretical option pricing for a European-style market which eliminates the possible risk of loss and optimize gains.
Q5
Are the Black-Scholes model and the Black-Scholes-Merton model same?
Yes, Black-Scholes model is often regarded as Black-Scholes-Merton model. Although the actual formulation BSM equation is done by the economists Fischer Black and Myron Scholes, the idea of this model was first put forward by Robert C. Merton.
