Numerical Option Pricing Project
The file S&P500.csv contains the daily S&P500 index values in the last 10 years, downloaded from Yahoo Finance. Use the most recent index value in the table as the current index. Suppose that the continuously-compounded risk-free rate is 0.75% per annum, and the continuously compounded dividend yield on the index portfolio is 2.30% per annum.
CBOE S&P 500 index option has a multiplier of 100, and it is of the European style. This means that one S&P500 call option with a strike price K has a payoff of 100*Max(ST-K, 0) at maturity, where STthe spot index value at maturity.
1) Please estimate the annual volatility of the S&P 500 index using the given data (hint, you should estimate the standard deviation of the continuously compounded price return. A shortcut for estimating volatility is to use the excel function STDEV).
2) Consider a CBOE S&P index call option that has a strike price of 2000 and a time to maturity of 6 months. Calculate the value of this option using the Black-Scholes-Merton formula.
3) Consider the option described in 2). Estimate the value of this option using a binomial tree with 100 time steps.
4) Consider again the option described in 2). Estimate the value of this option using the Monte Carlo simulation approach with 1000 trials. Please report both your best estimate and the 95% confidence interval of your estimate.
5) Asian option. Consider a 6-month European-style Asian call option on the S&P 500 index. Assume the multiplier for the Asian option is also 100, and the strike price is 2000. For each unit of the option, the payoff at maturity is given by
CT = 100 * max(SAVE – K, 0),
where SAVE is defined as the arithmetic average of the index values observed at the monthly intervals during the option life (i.e., the index value is measured once per month, with the first observation taken in one month, and the last one in 6 months). Please estimate the value of this option using the Monte Carlo simulation approach with 1000 trials, and report both your best estimate and the 95% confidence interval of your estimate.
6) Compare your results in 2), 3) and 4), and briefly point out the differences.
7) Compare the values of the Asian option with the value of the traditional CBOE S&P 500 option you estimate in 2), and briefly explain why they are different.