## Black-Scholes-Merton Model

**Black-Scholes-Merton Model**

The Black-Scholes-Merton model is a differential equation used to solve for option prices. The Black-Scholes-Merton model won a noble prize in economics. The standard BSM model is only used to price European options as it does not take into account American options because they can be exercised before maturity.

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**What is the Black-Scholes-Merton model?**

The Black-Scholes-Merton (BSM) model is a pricing model for financial instruments. It is used for the purpose of the valuation of stock options. The BSM model values options in continuous time and is derived from the same no-arbitrage assumption which is used to value options with the binomial model. It’s also used to determine the fair prices of stock options based on six variables and they are volatility, underlying stock price, type, time, risk-free rate, and strike price. To derive the BSM model, an instantaneously riskless portfolio is used to solve for the option price.

**Assumptions of the Black-Scholes-Merton model:**

- Stock prices are log-normally distributed whereas stock returns are normally distributed.
- The continuous risk-free rate is constant, known, and available for borrowing and lending always.
- The volatility of the underlying asset is constant, known, and option values depend on volatility.
- Trading is continuous.
- Markets are frictionless, which means, there are no taxes, no transaction costs, and no restriction on short sales or the use of short sale proceeds.
- The underlying asset has no cash flows. Example dividends and coupon payments.
- The options valued are European options. These can only be exercised at maturity. The model does not price American options correctly.

**Calculation with an example:**

The call and put formulas for BSM model are as follows.

**Call (C _{0}) = [S_{0} x N(d_{1})] – [X x e^{-RxT }x N(d_{2})]**

**Put (P _{0}) = {X x e^{-RxT }x [1- N(d_{2})]} – {S_{0 }x [1 – N(d_{1})]}**

Where:

d_{1} = ln (S_{0}\ X) + [R+ (0.5 x ^{2})] x T

—————————————-

x T

d_{2 }= d_{1} – ( x T)

T = Time to maturity

S_{0} = Asset Price

X = Exercise price

R = Risk free rate

= Volatility

If one value from call and put is provided then you can always use the put-call parity formula to calculate the other.

**P _{0 }+ S_{0} = (X e^{-RT}) + C_{0}**

The values of N(d_{1}) and N(d_{2}) are found in a table of probability values (i.e z-table).

Example: The stock of Vola is trading at 60 and there is a call option with an exercise price of 58 that expires in 6 months. The continuously compounded risk-free rate is 3.50% and the standard deviation is 20%. Calculate the value of call and put options using the BSM model.

The value of N(d_{1}) and N(d_{2}) using the formula above and the z-table is **0.6517** and **0.5981** respectively.

Using N(d_{1}) and N(d_{2}) and the formula for the call, we get the value of the call option

Call (C_{0}) = [S_{0} x N(d_{1})] – [X x e^{-RxT }x N(d_{2})]

**Call (C _{0}) = 4.77**

Then using put call parity we can calculate the value of put

P_{0 }+ S_{0} = (X e^{-RT}) + C_{0}

**P _{0 }= 2.14**

Hence, the value of call and put is 4.77 and 2.14 respectively.

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**Limitations of the Black-Scholes-Merton model:**

The Black-Scholes-Merton model is only used to price European options and not American options as they could be exercised before the expiration date. It also assumes that dividends and risk-free rates are constant, but this may not be true in reality. The model also assumes volatility remains constant over the option’s life but volatility fluctuates with the level of supply and demand.

As of other assumptions like there are no transaction costs or taxes; that the risk-free interest rate is constant for all maturities; that short selling of securities with use of proceeds is permitted; and that there are no risk-less arbitrage opportunities which can lead to prices that deviate from the real world where these factors are present.

**Final Thoughts:**

The Black-Scholes-Merton model was a turning point for the options world who had a mathematical foundation to build their options portfolios. It also gave rise to a number of option hedging strategies that are still being implemented.

**Author** **– Saachi Lodha**

**About the Author** – A passionate professional with knowledge of Accounting and Finance and currently exploring Financial Risk Management (FRM) to gain knowledge and exposure. As a part of the FRM course also writing blogs to explore the field more and deep dive into the content.

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