Valuation of Standard Call Options Using the Euler–Maruyama Method with Strong Approximation

The objective of this work is to valuate a European standard call option with different types of volatilities and a European exchange option based on the price of underlying assets through numerical experiments using the Euler–Maruyama stochastic numerical method with strong approximation. Different trajectories of Brownian motion are simulated with different time steps, different risk-free interest rates, different levels of volatility, different strikes or exercise price and different expiration times, assuming constant initial values for the different subjacent assets. The results obtained in the valuation of the options considered show that the proposed method presents very low mean squared errors compared to the valuation obtained from the reference methods: The Black–Scholes formula for an asset, Margrabe for two assets and the Euler–Maruyama scheme with weak approximation are analyzed for all the scenarios proposed. The strong Euler–Maruyama method becomes an attractive method for future research in terms of options valuation where there is no explicit formula. The results show that the proposed method can also be considered to value options, over one or more assets, since it produces a low mean square error in the analyzed scenarios.