Error in barrier option pricing Monte Carlo

Question

I am currently trying to price an up-and-out call with Monte Carlo simulation. For an option with these parameters :

• Barrier: 65
• $K$ = 50
• $\sigma$ = 30%
• $R $ = 1%
• $T$ = 1Y
• $S_0$ = 50

With 10.000 simulations and $dt = \frac{1}{500}$ I obtain an option price close to 0.80 (95% interval confidence : [0.76 ; 0.853]) whereas a pricer gives 0.73. When rising number of simulations the price increases so I am likely doing something wrong.

Here is the python chunk of code that I use:

all_final_payoffs = np.zeros((nb_simulations,1),dtype=float)

for i in tqdm(range(nb_simulations)):
    path_generated_asset = np.zeros((1, nb_time), dtype=float)
    path_generated_asset[0, 0] = S0
    for j in range(1, nb_time):
        X = np.random.randn(1)
        path_generated_asset[0, j] = simulate_price(path_generated_asset[0, j - 1], interest_rate, volatility, dt, X)
    all_final_payoffs[i,0] = compute_barrier_call_payoff(path_generated_asset[0,:],strike,barrier)
option_price = np.mean(all_final_payoffs)*math.exp(-interest_rate*maturity)

And the two functions used :

def compute_barrier_call_payoff(asset_path,strike,barrier):
    if max(asset_path)>=barrier:
        return 0
    else:
        if asset_path[-1]-strike>0:
            return asset_path[-1]-strike
        return 0

def simulate_price(S,R,Vol,dt,X):
    return S*math.exp((R-(Vol**2)/2)*dt + Vol*math.sqrt(dt)*X)

Does someone know where my error is? Thanks a lot!

Edit: When increasing the number of simulations to 20.000 and time steps to 2000, I get the price in 3 minutes (very long) on Python. Same code in C# on the same Mac, the program gives me the result in 7 seconds. When increasing simulations to 50.000 and time steps to 4.000 it takes roughly 35 seconds to give me 0.74$.

Edit 2 : Launching the Python code with Cython and typed variables it takes 1 minute.

Answer 1

The path dependency of barrier options requires a sufficient number of steps to accurately model price evolution. For example, the stock price simulation,

for dt=1/10,

for dt=1/500,

for dt=1/1000,

It can be seen that, if you use fewer number of steps, a barrier might not be triggered which would otherwise have been triggered if more number of steps were used. That is, increasing number of steps better models the underlying's price movement, thus increasing accuracy of the simulation. This becomes very important while pricing path dependent derivatives.

Answer 2

You can also read through the answer to this related question: How are Brownian Bridges used in derivatives pricing in practice?

Please also note that the timings mentioned are terribly slow. I know speed is not Python's strong point, but still. 3m50s for 20000 simulations with 2000 time steps (dt=1/2000) gives one the wrong idea of how efficient MC can be or not. Downloading this pricer you can see that this shouldn't really take more than half a second (on a budget laptop). Using this app you can also play around with the the Brownian Bridge technique and see how if you use it you then don't need such low dt's to get accurate Monte Carlo barrier prices.