MCBarrierEngine - 2023.2 English

Vitis Libraries

Release Date
2023.2 English

MCBarrierEngine is similar to the MCEuropeanEngine except for the path pricer.

In this engine, the path pricer samples the asset price path at the discrete time intervals \([t_j, t_j+1]\), \(j=1, ..., N\). The barrier condition is tested at each time step. When the barrier level is hit, the option becomes non-effective and the rebate is paid for the owner of option. The rebate is discounted to today for value of option. If the barrier is never hit, the payoff of the option is calculated by value of asset at maturity time \(T\) and it is discounted to today for value of option.

The detailed procedure of Monte Carlo Simulation is as follows:

  • For \(i\) = 1 to \(M\)
    • For \(j\) = 1 to \(N\)
      • generate a normal random number;
      • simulate the price of asset \(S^i_j\);
      • calculate the payoff and discount it to today.

if \(S^i_j > B\),

\[P_i = R\exp (-rt_j)\]

or if \(S^i_1, S^i_2, ..., S^i_N\) never cross the barrier,

\[P_i = max(S^i_N - K, 0)\exp (-rT).\]

So, the estimated value of option is the average of all the samples.

\[c = \frac{1}{M}\sum_{i=1}^{M} P_i\]

The \(c\) is a biased estimate of the barrier option value, because the sample path may have exceeded the barrier level, knocking out the option, during the time interval.