Black-Scholes Model Explained: How Options Are Valued
In India, options are used widely for hedging, earning and directional plays. However, option premium is not merely a function of whether the market is bullish or bearish. Factors like the price of the underlying asset, strike price, volatility, expiry and interest rates affect the theoretical value of the premium. Here comes the importance of Black Scholes Model . The Black Scholes Model is one of the best-known models to estimate the theoretical value of European style options. Also, the Black Scholes Model is applicable to the Indian derivatives market as NSE indicates that theoretical values of the option for a number of index and individual stock options are based on the Black-Scholes approach . Explore option pricing with confidence and use the best app for trading to analyze market opportunities, premiums and key option factors.
What Is the Black Scholes Model?
The Black Scholes Model, otherwise known as the Black-Scholes-Merton Model (BSM), is a mathematical equation which determines the theoretical price of an option . This model was derived through the works of Fischer Black, Myron Scholes, and Robert Merton and has been very crucial in forming the basis of contemporary derivative pricing models. Originally, the model was created to cater to European options, which means the option is exercised only at the expiration date and not before it . Simplified, the question that the Black-Scholes Option Pricing model tries to answer is:
What should a European Call or Put option be worth under prevailing market circumstances?
The model factors in the price of the underlying asset, strike price, time to maturity, volatility and risk-free interest rate. Where the underlying asset pays dividends, the expected dividend yield may also be factored in.
Why Is the Black Scholes Model Important in India?
Black Scholes model formula is not just some academic formula for Indian derivatives traders. In NSE documents, they use Black Scholes model based theoretical valuation for option contracts. This is evident from NSE derivatives specification on Nifty 50 where it is stated that theoretically valued options introduced are using the Black Scholes model . Also, NSE mentions that their theoretical valuations for option contracts consider the relevant interest rate where they mention MIBOR in their specification . In this way, knowing the Black Scholes formula becomes important for those who are analyzing Nifty, Bank Nifty, and stock options.
Black Scholes Formula
The Black Scholes formula for a European call is as follows:
C = S N(d1) − K e??? N(d2)
A European put can be priced by:
P = K e??? N(−d2) − S N(−d1)
The NSE uses the same basic formulas for their options contract specifications.
The two intermediate quantities are given by:
d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T)
d2 = d1 − σ√T
Where:
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C = theoretical call price
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P = theoretical put price
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S = underlying asset price
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K = strike price of option
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T = time to expiry in years
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r = risk free interest rate
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σ = annualised volatility
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N(d) = standard normal cumulative distribution function
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ln = natural logarithm
This is according to the BSM formula formulated by the NSE.
How the Black Scholes Option Valuation Model Works
The Black Scholes option pricing formula model transforms five primary inputs from the market into the theoretical price of an option.
1. Underlying Price
A high underlying price increases the value of a call and decreases the value of a put, ceteris paribus . For instance, a high value of Nifty increases the value of a call, assuming that the price of a call strike is constant.
2. Strike Price
It refers to the price at which the option can be exercised at expiry . Low strike prices favour call buyers while high strike prices Favor put buyers.
3. Time to Expiry
High time to expiry increases the value of an option since the possibility of favourable changes in the underlying is higher over a longer period of time . As the expiry date nears, the value of time to expiry drops. It is considered through theta, one of the main option Greeks.
4. Volatility
Volatility is one of the most crucial factors in the black Scholes option pricing formula . High expected volatility will result in higher call and put prices since more movement in the price implies that there are higher chances of ending up being in-the-money for the option holder . This explains why option premiums tend to increase significantly despite little change in the underlying asset's price due to expected volatility.
5. Interest Rate
The interest rate is a factor in the time value of money aspect of the strike price. The correct interest rate that should be used in India can depend on the methodology of the stock exchange and market rates.
Black Scholes Calculation Example
Example of a theoretical Nifty style Index Option:
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Price of underlying (S): Rs.25,000
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Strike price (K): Rs.25,000
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Times to maturity: 30 days
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Volatility: 20% per year
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Risk free rate of interest: 6%
With T = 30/365, the result would be:
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d1 = 0.115
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d2 = 0.057
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Value of Call option ≈ Rs.634
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Value of Put option ≈ Rs.511
The above example has been created for understanding how Black Scholes works. The figure cannot be taken as the actual fair value of Nifty options . In practical scenarios, market premium can be different since actual values depend on prices based on market factors.
Black Scholes Formula for Put Option
The Black Scholes formula for put option is:
P = K e??? N(−d2) − S N(−d1)
Explanation of how the factors of the underlying, volatility, interest rates, and time affect the theoretical price of a put is provided in the put pricing formula.
For instance, an increase in the anticipated level of volatility will cause an increase in the theoretical prices of both calls and puts. This is also one of the factors affecting call and put prices in the same direction.
Black Scholes Formula With Dividends
The original model does not consider any dividends. In case of securities that pay continuous dividend yield, Black-Scholes pricing formula for dividends adjusts the stock price portion.
The formulas with dividends are:
C = S e??? N(d1) − K e??? N(d2)
P = K e??? N(−d2) − S e??? N(−d1)
In this case, q is the continuous dividend yield rate.
The modified d1 would become:
d1 = [ln(S/K) + (r − q + σ²/2)T] / (σ√T)
and:
d2 = d1 − σ√T
A dividend adjustment is especially necessary for equity or index options. It describes how a known continuous dividend yield can be accounted for using the adjustment factor of S0e??? . Parents exploring early investing options can open minor demat account.
What the Black Scholes Model Does Not Capture Perfectly
Despite the value of its application, black-Scholes is not a crystal ball . Among the traditional models' key assumptions is constant volatility, constant interest rates, frictionless markets and lognormal distribution of the underlying asset's prices . Markets are much more complex . Volatility is not constant, there can be wide price spreads, transaction costs are present, option prices may exhibit volatility smiles or skewness. Implied volatility might be non-constant across strikes and maturities . That is why professional option traders do not just compare the price of an option to one calculated from BSM model to determine whether it is overpriced or underpriced.
Black Scholes Model and Implied Volatility
One of the best uses of the black Scholes model is the computation of implied volatility.
Instead of putting in the volatility and computing the option price, traders can use the market premium and solve for the implied volatility from the black Scholes model.
For instance:
Market option price -> BSM -> Implied Volatility
Implied volatility is useful for comparisons between options premiums since implied volatility is the volatility at which the value of an option-pricing model equals market price.
Black Scholes Model vs Actual Option Premium
A significant consideration for an investor is that the theoretical price does not always match the traded premium . For example, consider the scenario where a theoretical computation gives the call premium to be Rs.634 and the trading premium of the option is Rs.670. This discrepancy doesn't necessarily imply that the call option is overvalued . This is because the market premium takes into account the expectation of future volatility, liquidity, demand and supply among others . At the moment, NSE has a structure that illustrates this significant difference as well.
Advantages and Limitations
Merits
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Gives a structured method of valuing options.
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Explains the relationship between volatility and premiums.
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Can serve as a basis to study option Greeks.
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Can be used for implied volatility estimation.
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A well-known concept in derivatives market.
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Used in NSE’s theoretical pricing model for related contracts.
Demerits
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Primarily intended for European type exercise.
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Basic assumptions don’t fit the actual market.
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Assumption of constant volatility is unreal.
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Sudden changes in the market can cause pricing discrepancies.
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Consideration of dividend and other underlying features needs to be considered.
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Should not be considered alone while trading.
How Indian Traders Can Use Itz
Black Scholes model should be used by the Indian options trader only as a reference model and not as a predictive model.
An ideal process would be:
Underlying asset price -> Strike -> Expiry -> Volatility -> Interest rate -> Theoretical value -> Comparison with market premium -> Analysis of Implied Volatility and Greeks
For instance, an investor studying a Nifty option can go through the option chain, compare the premium in the market with theoretical figures and then analyze implied volatility and Greeks like delta and theta. This option-chain facility is available online with NSE.
Final Takeaway
Even today, the Black Scholes equation continues to be one of the important models to explain the valuation of European Options. The importance is not only in the equation but also lies in the way volatility, time value of options and option Greeks and Implied Volatility can be understood using the equation . While it is important for the Indian investor to understand the Black Scholes Model and the assumptions associated with it, it is also essential that he takes into account other factors along with the theoretical value before making a trade decision . Understand option pricing better with the Black-Scholes Model, and take the next step toward informed trading with a seamless MCX account opening process.
Frequently Asked Questions
What is the Black Scholes Model?
The Black-Scholes Model is a mathematical framework that estimates the theoretical value of European call and put options.
Which variables determine the Black-Scholes option value?
The underlying price, strike price, time to maturity, volatility, interest rate, and dividend yield (wherever applicable) impact the Black-Scholes option value.
Can we apply Black Scholes Model on Indian options?
Yes, the Black Scholes Model is applied in theory to calculate the price of certain Indian options in the derivative market.
Does the Black Scholes Model give the actual premium on the options?
No, the Black Scholes Model gives the theoretical option price while actual premium may vary due to various other considerations like market demand, liquidity etc.
What is the connection between implied volatility and the Black Scholes Model?
Implied Volatility is calculated based on the market premium in Black Scholes Model.
Disclaimer: This blog is dedicated exclusively for educational purposes. Please note that the securities and investments mentioned here are provided for informative purposes only and should not be construed as recommendations. Kindly ensure thorough research prior to making any investment decisions. Participation in the securities market carries inherent risks, and it's important to carefully review all associated documents before committing to investments. Please be aware that the attainment of investment objectives is not guaranteed. It's important to note that the past performance of securities and instruments does not reliably predict future performance.
