Option Pricing Engine Mathematical Framework

For $S_0,r,\sigma >0$ the Black-Scholes stock price process is defined as

$$S_t=S_0\exp \left(\left(r-\frac{1}{2}{\sigma }^2\right)t+\sigma W_t\right)$$

where $\sigma$ is the volatility, $W_t$ is a Brownian Motion and $r$ is the interest rate. Then for a $g:{\mathbb{R}}_{+}\to \mathbb{R}$ for $g\in \mathcal{B}({\mathbb{R}}_{+},\mathbb{R})$, $\mathbb{E}[|g(S_T)|]<\infty$ we have for the price at time $t\in [0,T]$ of a European call option with payoff $g(S_T)$ at maturity $T$

$$V_t=e^{-r(T-t)}\mathbb{E}[g(S_T)|{\mathcal{F}}_t],t\in [0,T]$$

where $\mathbb{F}=({\mathcal{F}}_t{)}_{t\in [0,T]}$ is a filtration. From this follows the following possible Monte Carlo pricing engine:

Input: S0, r, σ, T, payoff function g, number of simulations N  
  
for i in 1,...,N:  
    Sample $Z_{i}$ ~ N(0,1)  
  
    Compute terminal stock price
    
    $S_T^{(i)} = S0 * \exp((r - 1/2 σ^2)T + σ√T Z_i)$
  
    Compute payoff  
    
	$P_i = g(S_T^{(i)})$
  
Compute sample mean of payoffs  
    $\overline{P}= \sum_{i=0}^{N} P_{i}$   
  
Discount expected payoff  
    $\widehat{V}_{0 }= \exp(-rT) * \overline{P}$ 
  
Return $\widehat{V}_{0}$ 

This algorithm approximates the conditional expectation appearing in the pricing formula by the empirical average of simulated payoffs.