In real-world, volatility is changing every moment in an uncertain manner which suggests, that the asset price can change at any moment and may take any value (i.e., continuous-time and continuous value stochastic behavior). Any variable which changes in an uncertain manner is said to be following a stochastic process. An option pricing model has to specify the nature of the stochastic process followed by the underlying asset.

Unlike Binomial Model, the BSM Model correctly retains the continuous stochastic process behavior of the price of an underlying. A continuous stochastic process means a variable that changes in continuous time is said to follow a continuous-time stochastic process and which takes any value is said to follow a continuous value stochastic process.

### Assumptions in BSM Option Pricing Model

It is better to thoroughly understand the assumptions of the BSM Option Pricing Model-

The underlying asset price follows a Geometric Brownian Motion (GBM), which means, the underlying price follows Log-Normal Distribution. The concept of log-normal distribution is very closely related to the Normal Distribution. A log-normal distribution is commonly used to describe the distribution of an asset such as share price and in this case, log-normal is more suitable because as like share price, the data points in the log-normal distribution cannot go negative. To identify the same, if the continuously compounded returns of a share follow a normal distribution, then the share price follows a log-normal distribution. In case even if the continuously compounded returns do not follow a normal distribution, it is logical to conclude that since the share price is always positive, and therefore, it follows a log-normal distribution. Hence, the log-normal property is invoked as it prevents the underlying asset price from becoming negative.

As mentioned, the Geometric Brownian Motion (GBM) implies continuous value i.e., there are no jumps in the share price. However, in the emerging countries or companies that are going into some major corporate events like mergers, etc. exhibits jumps in the share price, and therefore, BSM should not be applicable in such cases.

Volatility in the underlying asset is known and constant and is not changing (throughout the option period) but in real life, the implied volatility calculated by reverse engineering using BSM Model is generally greater than the expected volatility derives from the historical data of the underlying and is changing over time.

The annualized continuously compounded dividend yield on the underlying is known and constant throughout the option period.

Since BSM Model is an analytical closed-form solution to price an option, and therefore, it can be only applied to European options as it can be exercised only on the expiration date.

Markets are frictionless (i.e., there are no transaction cost, no taxes, and no regulatory constraints such as no restrictions on short selling with full use of proceeds and continuous trading is available).

Markets are efficient (i.e., no-arbitrage opportunity available and the underlying asset is liquid).

The annualized continuously compounded risk-free interest rate is constant throughout the option period the same is applicable for both- unlimited lending and borrowing.

### Factors Determining Option's Price

The Black-Scholes-Merton option pricing model is based on six inputs: underlying price, strike price, volatility, time to expiration, interest rate, and dividend yield. These inputs or factors are so important that a successful trader has to be a master at identifying the key factors and their dimensions and understanding why and how they directly impact the price of an option. The favorable or unfavorable aspect depends on the type of the trader's position.

Current Underlying Price: The price of an option is affected by changes in the underlying asset price. If the underlying asset price goes up, the call option should gain in value because the buyer of a call option is now able to buy the underlying asset at a lower price (strike price). In other words, as the option gets further in the money (ITM), the probability that the option will end up in the money also increases, and therefore, the call option price gains more value.

Strike Price: It is also called the exercise price of an option (determined initially at the time of entering into a contract) at which an option trader transacts if the option is ITM on the expiration date. This is the price at which a call option buyer has the right to transact in order to purchase the underlying asset, and a put option buyer has the right to transact in order to sell the underlying asset.

Volatility: The BSM option pricing model requires the trader to identify the future volatility prevailing during the life of the option. Obviously, the correct future volatility is not known, and therefore, the trader has to predict based on historical data and calculate the volatility. The volatility engine could be the Standard Normal Volatility, EWMA, or GARCH(1,1), depending on the precision required. However, in order to identify the other factors, one can also use the implied volatility calculated by reverse engineering using the BSM Model, which allows the trader to determine the market consensus on the future volatility likely to be.

Time to Expiration: All option contracts come with an expiration date. The more time left until expiration, the greater the chances of making profitable moves. Likewise, the less time left until expiration, the fewer the chances. This holds true for both call and put options, and therefore, time decay always hurts the price of an option.

Interest Rate: Like most other financial instruments, the option price is also impacted by changes in interest rates. Logically, if a trader chooses a call option over the stock, then the extra cash the trader has left should earn some interest income. As the interest rate rises, the price of a call option also rises, and that of a put option falls. Therefore, interest rates have a positive relationship with call options and a negative relationship with put options.

### Black-Scholes-Merton Option Pricing Formula

The basic idea behind the BSM Model is that- it considers infinite possibilities that the underlying price can take on the expiration data and compare it with the exercise price after considering the probabilities of its being exercised/lapse.

[ the infinite possibilities can be achieved using this Simulator ]

E^[ Max ( S(T) - K, 0 ) ]

The option price is the present value of the expected payoff calculated above discounted at the continuous compounded risk-free interest rate.

C(0) = e^-rt * E^[ Max ( S(T) - K, 0 ) ]

Where,

S(T) = Underlying Stock Price at Expiration

K = Strike Price of an Option

r = Continuously Compounded Risk-free Interest Rate

T = Time to Expiration of an Option

Black-Scholes-Merton formula is a differential equation divided into two parts-

N(d1) is the probability that determines the expected payoff in case the option is getting exercised in the risk-neutral world, and therefore, the expression [ S(0) * N(d1) * e^rt ] in the equation is the expected stock price at expiration date (S(0)*e^rt) calculated using the probability (N(d1)) in a risk-neutral world.

On the other hand, the exercise/lapse is dependent on the moneyness of an option, and therefore, the strike price (K) is only transacted if the option is ITM standing on the expiration date whose probability is N(d2)- represented by [ K * N(d2) ].

Therefore, the expected payoff in a risk-neutral world is- [ S(0) * N(d1) * e^rt ] - [ K * N(d2) ]

The option price is the present value of the expected payoff calculated above discounted at the continuous compounded risk-free interest rate gives us the equation for BSM Model.

Call Price = S(0) * N(d1) - K*e^-rt * N(d2)

Where,

d1 = ( LN ( S(0) / K ) + ( r - q + σ^2 / 2 ) * T ) / σ * √T

d2 = d1 - σ * √T

S(0) = Underlying Stock Price at T0

K = Strike Price of an Option

r = Continuously Compounded Risk-free Interest Rate

q = Continuously Compounded Dividend Yield

T = Time to Expiration of an Option

σ = Underlying Stock Price Volatility

Let us take an example to understand the maths here-

Asset Price $500.00

Strike Price $450.00

Maturity 1 Year

Volatility 22.315%

Risk-free Interest Rate 6.00%

d1 = ( LN ( S(0) / K ) + ( r - q + σ^2 / 2 ) * T ) / σ * √T

= ( LN ( $500 / $450 ) + ( 0.06 - 0 + 0.22315^2 / 2 ) * 1 ) / 0.22315 * √1

= 0.8526

d2 = d1 - σ * √T

= 0.8526 - 0.22315 * √1

= 0.6295

Call Price = S(0) * N(d1) - K*e^-rt * N(d2)

= $500 * NORM.S.DIST(0.8526,TRUE) - $450*e^-0.06*1 * 0.6295

= $89.84

This was easy and simple, right? -- not actually!

### Limitations of Black-Scholes-Merton Model

Although the BSM formula is a one-liner easy and simple to put the inputs and derive the option price but to work this formula, it has already sacrificed few things which are highly important to be considered in reality.

The underlying share price assumed to be following Geometric Brownian Motion (GBM) implies random walk, and therefore, large jumps are not incorporated in the model.

Volatility in the underlying asset is assumed to be constant throughout the option period, but the volatility is dynamic in real life.

The BSM formula nowhere considers dividends distributed during the option period, and thus, not correctly pricing the option.

Frictionless markets do not exist in reality. Therefore, transaction costs, taxes, and regulatory constraints come into play.