By: Sonam Yadav
Investment means allocating money to financial assets with the goal of generating returns over time.
Mathematics involve various strategy and techniques to minimise loss in investment and financial decisions. Here we’ll discuss some important topics of mathematics used in investment.
Concept of covariance and correlation:-
Covariance shows that how the prices of two assets move together.
Positive covariance:-
prices of both assets increases or decreases together.
Negative covariance:-
prices of one asset increase then other one decreases.
Zero covariance:-
No relation between prices of assets .
For example:-
we have two shares A and B , we can calculate the covariance of A and B by using formula.
Firstly we calculate the mean of last year monthly returns of both the shares A and B separately and then subtract it from current returns and multiply the value get from both the shares .Now , divide the value by (period-1) we get some value and if we compare the value from defination of covariance we can determine the movement of prices of both shares.
Correlation also measure the relation between two variables or assets providing value between 1 and -1.
If value is equals to 1 then the asset is moving in same direction
If value equal to -1 then assets are moving in different directions
If value equal to zero then no correlation
For example :-
In above formula numerator is covariance and denominator is standard deviation, If we have two shares A and B then by calculating mean by previous data and standard deviation of both shares we can determine correlation and that value tells us about the returns of share A and B according to above definition.
Normal distribution:-
Normal distribution is a key element to modern portfolio theory.
Any investment has two aspects: risk and return. Investors look for the lowest possible risk for the highest possible return. The normal distribution quantifies these two aspects by the mean for returns and standard deviation for risk.
If we have mean and standard deviation as 10% and 3% respectively. Then we can calculate probability that expected return will be between 7% to 13%.
Here, (7-10)/3 = -1 and (13-10)/3 = 1 Now, by using Z table area between -1 and 1 represents the probability between 7% and 13% which is 68% Blue part of the graph . 3 We conclude that there are 68% chances that from previous data , stock return falls between 7% and 13%.
Limitations:- All assets doesn’t follow normal distribution .Normal distribution assumes that each return is independent of the others, but in reality, returns can be correlated across time .
Black sholes Partial differential equation :-
V(S, t) is the value of the option as a function of stock price ( S ) and time to maturity ( t ) , (r )is risk free interest rate, Sigma is volatility.
Black sholes PDE provides insights into how the price of an option evolves over time, considering various market factors like the stock price, volatility, and risk-free interest r. European call option and European put option can be determined by using formula:-
And , c = European call option , S0 = stock price
Here the values of c and p tells about the price of european call and put option per share . For eg.:- we find that the price of the European put option is approximately $7.89. This means that the current market
Conditions and the Black-Scholes assumptions, the fair price for this put option would be $7.89 per share.
Conclusion:-
Mathematical finance is a bridge of theoretical mathematics and mathematics in real world. Here, we use theoretical mathematics in real world decision and this provides us an idea of risk management and financial decisions . And using probability and statistics we can determine the chances of loss and gain by using previous year data, this gives us an idea of where to invest and where not...
Thank you!
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