Formula Sheet

1 Binomial model

1.1 Parameters

• Price of underlying security at time \(0\): \(S\)
• Simple interest rate over one period: \(r\)
• Possible stock price multipliers over one period: \(u\), \(d\)
• Possible prices of the stock after one period: \(\left\{S\cdot u, S\cdot d\right\}\).

1.2 Risk-neutral probabilities

\begin{eqnarray*}&& p_*=\frac{1+r-d}{u-d}\quad\quad q_*=\frac{u-1-r}{u-d}. \end{eqnarray*}

1.3 Price of derived security over one period

\begin{eqnarray*}\mbox{Price}&=& \frac1{1+r}\mathbb E_*\left[\mbox{Payoff}\right]. \end{eqnarray*}

2 Continuous Black-Scholes Model

2.1 Parameters

• Price of underlying security at time \(0\). \(S_0\)
• Rate of continuously compounded interest. \(r\)
• Expiration time. \(t\)
• Volatility of the stock. \(\sigma\)

2.1 Black-Scholes model for the price of underlying security under risk-neutral probability

\begin{eqnarray*} S(t)&=&S_0e^{\left(r-\frac{\sigma^2}2\right)t+\sigma W_t}. \end{eqnarray*}

2.2 Price of derived security

\begin{eqnarray*}\mbox{Price}&=& e^{-rt}\mathbb E_*\left[\mbox{Payoff}\right]. \end{eqnarray*}

2.3 Prices of European call and European put

\begin{eqnarray*}C&=&S_0\Phi\left(d_+\right)-Ke^{-rt}\Phi\left(d_-\right)\\ P&=&Ke^{-rt}\Phi\left(-d_-\right)-S_0\Phi\left(-d_+\right)\\ d_{\pm}&=&\frac{\ln\frac{S_0}K+\left(r\pm\frac{\sigma^2}2\right)t}{\sigma \sqrt t}. \end{eqnarray*}

3 CPP Tables

3.1 Types of variables

In C++ real numbers are of type double, integers are of type long. If you have to declare a variable whose name is pStar and is supposed to be a real number, then you need to write double pStar;

3.2 Conversion

Assume that the variable x is of type double and contains a real number and that y is of type long. If you want to convert the real value from x to integer and store the result in y, then you need to write y=static_cast<long>(x);

3.3 Math and probability functions

• If x and y are two variables of type double and you want to assign the value \(e^x\) into the variable y, then you need to write this instructions y=std::exp(x);
• If x and y are two variables of type double and you want to assign the value \(\ln(x)\) into the variable y, then you need to write y=std::log(x);
• If x, y, and z are three variables of type double and you want to assign the value \(x^y\) into the variable z, then you need to write z=std::pow(x,y); or z=std::exp(y*std::log(x));
• To make assignment \(y=\sqrt{x}\), write y=std::sqrt(x);
• To make assignment \(y=\pi\), write y=MATH::PI;
• To make assignment \(y=\frac1{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-\frac{u^2}2}\,du\), write y=MATH::standardNormalCDF(x);