# Monte Carlo path for rate option price
# (problem 6)
def P_MC(a, b, rho, v0, F0, T, K, call_put, numsteps, numsims):
    # running sum of rate option price for each MC path
    P = 0.

    # MC iterations
    for i in range(numsims):
        F = F0
        v = v0
        dt = 1.*T / numsteps

        # path simulation for this iteration
        for _ in range(numsteps):
            # correlated 2D normal random variables
            x, y = corr_norm(rho)

            # F(i+1) = max(F(i) + sigma(i)*F(i)^beta * sqrt(dt) * x, 0)
            F = max(F + v*F**b*sqrt(dt)*x, 0.)

            # sigma(i+1) = sigma(i) * e^(alpha * sqrt(dt) * y - alpha^2 * dt / 2)
            v = v * exp(a*sqrt(dt)*y - a*a*dt/2.)

        # call(1) or put(-1) multiplier
        opt_type = 1. if 'c' in call_put else -1.

        # add rate option value to running sum
        P += max(opt_type*(F-K), 0.) * 1  # is this correct rate option price scheme??

    # rate option average value
    P /= numsims

    return P