Black-Scholes-Merton
This section illustrates the Black-Scholes (BS) model [BS73] and the pricing form under it by using a European call option that based on the assumption of deterministic interest rates in the market for a particular stock. Such an assumption the stochastic interest rates have a stronger influence on the pricing option for a long-maturity, however, deterministic feature of interest rates is harmless in most situation.
Suppose we are studying this on a complete filtered probability space (Ω, f, F = (ft)_{t≥0},P), consists of a probability space (Ω, f, P)and a filtration F = (ft)_{t≥0}contained in f.
Assume that the asset-price S_{t} follows a geometric Brownian motion GBM if it satisfies the following stochastic differential equation (SDE):
dS_{t} =rS_{t}dt + σS_{t} dW_{t}^{1}
where r is the riskless rate, the standard deviation parameter σ determines its volatility as well W_{t}^{1} is a standard Brownian motion. The BS model requires that both the riskless rate and the volatility of the asset-price remain constant over the period of analysis. Moreover, the asset-price considers being a right-continuous with left limit. In addition, there is no way to make a riskless profit (i.e., There is no arbitrage opportunity) as well as the transaction does not incur any fees or costs in the market. However, it has been observed that riskless rate and volatility should be stochastic [HW93] and [Hes93] respectively.