Black-Scholes Model

Black-Scholes Model and Present-day Economy

Black-Scholes Model: -

The model is used to determine the fair price value for a call or a put the option is called the Black Scholes Model. The Black-Scholes pricing model is used by the option traders to buy options. This call or a put option are based on six variables as under

• Volatility
• Type of options
• Underlying stock price
• Time
• Strike price
• Risk-free rate

Introduction to the Black-Scholes formula: -

Let’s look into the Black Scholes model formula and how it works in options.

V(C)=S_o*N(d_1 )-(E*e^(-rt) )*N(d_2)

V(P)=V(C)+(〖E*e〗^(-rt))-S_0

Here,

C= price of a call option

P= price of a put option

Ln= Natural Log

So= price of the underlying asset/ current stock price
E = strike price of the option/Exercise price
r= rate of interest/risk-free interest rate based on continues compounding 

e^(-rt) =  A factor for determining the present value/PV factor

t = time to expiration/time remaining for maturity interest in years

σ= Standard deviation of the log-returns or we call volatility/volatility of the underlying/standard deviation of share price

N= Cumulative Area under Normal Curve.

N(d_2 ) = The probability that the stock price will be at the or above the strike price when the option expires.

N(d_1 ) = Expected value of stock multiplied by the probability that the stock the price will be at or above the strike price. 

Normal distribution concept in the Black-Scholes Model: -

In the Black-Scholes model, we use the Normal distribution concept to determine the value of the N(d_1)and N(d_2 ). Major part here we look into the left side of the graph from the value of d_1 and d_2 as shown below.

Standard deviation is essentially a distribution occurrence around the means. Here in the below normal distribution which shows the majority of the moves pretty much be closer or it supposed to be near to where the stock price is. However, the large risk moves are the tail risk moves. These are the upside and the downside shown in the normal distribution. It is to be pretty much symmetrical across both the side. When we analyze the distribution, we can breakdown some percentage of stock moves were around 68% of the probability that a stock price is in a particular range. high implied volatility is going to widen the graph which means it has the larger S.D and low implied volatility is going to narrow the graph or a smaller SD.  

• Options are more valuable when we are dealing with the stocks which have more volatility
•    Black-Scholes Formula European call option: - European call the option is mathematically simpler than an American call option.

Alright, let’s see the procedure of arriving at the value of call the option and value of a put option with an example

Example: -

The present market value of the share $530, 4 months of a call an option is available at an exercise price of $500, risk – free rate of interest is 5% and S.D of the share price is 20%.

Solution: -

Points to be noted -

·        Time for maturity will be 4/12 or 0.30%.

·        S.D will be 0.20 or 20%.

·        Risk-Free Interest Rate will be based on continuous compounding.

Step – 1 

Calculation of d_1

d_(1=) (L_n [S_o/E]+[r+(0.5*σ^2 )].t)/(σ√t)

d_(1=) (L_n [530/500]+[0.05+(0.5*〖(0.20)〗^2 )].0.30)/(0.20√0.30)

d_(1=) (L_n [1.06]+[0.05+0.02].0.30)/(0.20*0.55)

d_(1=) (0.05826+0.021)/0.11

d_(1=) 0.72

Step – 2

Calculation of d_2 

d_2=d_1-σ√t d_2=0.72-0.11 

d_2=0.61

Step – 3

Calculation of N(d_1 )

Z table value of d_1 (0.73) is (0.7673 -0.5 =0.2673)

The area under Z= 0 to negative (left in the graph) will be 0.5

So, the value of N(d_1 )= 0.5 +0.2673 = 0.7673

Step – 4

Calculation of N(d_2 )

Z table value of d_2 (0.61) is (0.7291 - 0.5 = 0.2291)

Area under Z = 0 to negative (left in the graph) will be 0.5

So, the value of N(d_2 ) = 0.5 +0.2673 = 0.7291

Step – 5

Calculate the value of Call option V(C)

V(C)=S_0*N(d_1 )-(E*e^(-rt) )*N(d_2)

V(C)=530*0.7673-(500*0.9851)*0.7291

V(C)=406.669-359.119

V(C)=47.55

Step – 6

Calculation of value of put option V(P)

V(P)=(47.55+492.55)-530 

V(P)=540.1-530 

V(P)=10.1

Current Economy and Options price model: -

Usually, factors such as interest rates, inflation, unemployment, and economic growth often move stock markets. Stock markets are always rooting for more economic growth, however, in options, it is not the same one who chooses options trading will be able to succeed in any economic conditions but the major thing is the percentage of right prediction. The model is determining the value of call and put option of in any market conditions.

In the present day, the COVID crisis has created such uncertainty around the world that the market has been completely crashed and the major economies of the world have shut for the months. For Option traders, there was a call opportunity at the time over the period of time it will definitely put option in the money.

If rightly getting to the Black Scholes Model will really help to an options trader in this current economic situation because of the volatility it creates.

Some of the economic factors to be considered while trading in options:

Volatility: - Implied volatility is one of the major things to execute the call or put options that will make the option trade more effectively.

Right opportunity: -Current market situation will definitely test the patience of a trader but after following any type of a predictive model for the upcoming stock prices, it is very much necessary that waiting for the right opportunity.

Learnings: - The major effort for the options traders in the current economic condition is to be an active learner about the volatility factors, options trading strategy so that one can create the right opportunity for himself.

Up to date: - An options trader should always be updated about the current issues or the ongoing and upcoming news. Try to analyze it and find out the reality of it.

Conclusion: -

After considering the six major factors the Model which is used to determine the fair price value for a call or a put option is more effective.

Options are one of the important financial derivatives. Following the Black-Scholes option pricing model. The Black-Scholes model uses the spot price of the underlying asset, strike price, the risk-free rate, termination of option and, volatility. Among these, volatility is the only input that can be predicted.