Your manager asked you to price a 6-month European call option on a stock index with a strike price of 800. The index is currently 810 and has a volatility of 20% per annum and a dividend yield of 2% per annum. The risk-free rate is 6% per annum. All rates expressed in percentages are continuously compounded.

Therefore, to obtain the price estimates, you used the Black-Scholes-Merton formula and you obtained 56.2761 which you multiplied by $100 to obtain the price estimate of $5627.61 for one option.

However, your manager said that the Black-Scholes-Merton formula relies on very strong assumptions and insisted that you should implement a Monte Carlo simulation based on a Geometric Brownian Motion for the index. Your manager said this will look much more realistic and thus more convincing.

Consequently, you have spent the last two weeks programming a simple Monte Carlo approach as requested by your manager. You have just finished debugging your codes earlier today. With the same input parameters as above, you have run your routines with 100 time steps 5 times and you have obtained:

{79.884, 53.2573, 49.7413, 59.2395, 46.4742}

You have then repeated your 5 simulations but now increasing the number of time steps to 10000 and you have obtained the following estimates:

{55.7547, 56.5384, 56.8841, 56.1191, 55.4183}

Describe how these outcomes are different and why? Are they more realistic as opposed to the result produced by the Black-Scholes-Merton formula? Will you consider increasing the number of time steps to 1 million? Briefly discuss.