In this project, I priced the European call option on S&P 500 index by a 5000-path Monte Carlo simulation. Assuming the current index level (S0) as 1,544, annualized volatility of the index 20%, 1-year LIBOR rate 1.25%, trading days 250 in a year, and strike price 1,550, I worked out the call option's intrinsic value as $129.
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In the above project, I assumed a constant volatility. How would we price the option with randomly changing volatility. Next, I would introduce how to simulate volatility (Cholesky Factorization is applied) and see its impact on the S&P 500 index and the option pricing. In this case, the spot index is 1,065, strike price equals to 1,100, annualized long term volatility is 20%, and the 1-year LIBOR rate is 1.25%.
In addition, we could also apply GARCH (generalized auto-regressive conditional heteroskedastic) and EWMA (exponentially weighted moving average) approaches to generate auto-correlated volatility, Both approaches calculate "t" period volatility by "t-1" period return and variance, however, GARCH involves three parameters, whereas EWMA only has one decay factor.
In addition, we could also apply GARCH (generalized auto-regressive conditional heteroskedastic) and EWMA (exponentially weighted moving average) approaches to generate auto-correlated volatility, Both approaches calculate "t" period volatility by "t-1" period return and variance, however, GARCH involves three parameters, whereas EWMA only has one decay factor.
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