## Essential Maths for Quantitative Finance

Quantitative finance uses lots of formulae. The following are the most important ones. Whether you’re studying for an exam, coding them up, or just curious about the strange things that quants do, these formulae should help:

- Pi = 3.145927: This first one isn’t a formula, but you use the famous maths constant pi not because you need to work out the area of a circle but because it’s used in the formula for the normal distribution.
- e = 2.718282: Exponentials occur all the time in quantitative finance because of the present value of money formula and the normal distribution. ‘e’ is the base of the natural logarithm, ln.
- ln(
*ab*) = ln(*a*) + ln(*b*): Use this formula when taking the logarithm of the likelihood in applying the maximum likelihood method to estimate the parameters of your model. - If
*y*=*e*, then:^{x}*x*= ln*y*. Use this formula for calculations with continuously compounded interest rates. - If
*y*=*e*, then:^{x}

The derivative of the exponential function is the same exponential function. You use this when you’re solving the Black-Scholes equation.

## Formulae for Futures and Option Payouts

How much will jam be worth tomorrow? In quantitative finance, you often want to explore the relationship between an amount of money promised in the future and the value of that money today. These formulae can help:

- In working out today’s value of option payoffs, use the present value,
*P,*for a payoff,*C,*in the future. You can use exactly the same formula to work out the present value for the coupon*C*of a bond to be paid at time*T.*The continuously compounded interest rate is r and the time in the future in years is*T:*

- To calculate the price of a European put from the price of a European call or vice versa, calculate put call parity for European options with strike price
*K.*The current stock price is*S,*call price*C,*put price*P,*the time to expiry*T*and the continuously compounded interest rate is*r:*

## Calculating Option Prices Using Black-Scholes Formulae

The Black Scholes equation, perhaps the most famous in quantitative finance, expresses how the price of an option depends upon the price of the underlying asset and its volatility. The interest rate also gets a look-in. The equation is complicated, but thankfully, mathematicians have solved it for some useful cases such as for European options.

Use the cumulative normal distribution to express the solution to the Black-Scholes equation for the price of options:

The Black-Scholes solution for the price, *C,* of a European call option on a non-dividend-paying stock is given by the following formula. The volatility is a lower-case sigma, the risk-free interest rate is *r,* the expiry time is *T, *the current time is *t*, and the underlying stock price is *S.* The strike price is *K.* The formula is complicated: using the variables *d*_{1} and *d*_{2 }makes the final formula easier to digest.

Use the Black-Scholes solution for the price, *P,* of a European put option on a non-dividend-paying stock. The parameter names are the same as for a call option in the preceding formula: