Approximating the optimal boundary (IMAGE)
Caption
Our method for call options is summarised in the following algorithm:
(1) We device the time interval into the grid 0≡t_0<t_1<...<t_n≡T and let D_0,D_1...,D_n be some values for the underlying asset at these nodes. We shall approximate the optimal boundary maximizing the holder’s financial result at this grid deriving the appropriate values of D’s.
(2) We use the closed-form formula for the price of a European-style option that expires when the underlying asset reaches an exponent of the piecewise linear function that relates the values D_0,D_1...,D_nat the grid–see Formula (3.21). We denote this price by V(x;{t_0,...,t_n};{D_0,...,D_n}).
(3) The optimal boundary at the maturity D_n is given by Formula (2.14).
(4) Suppose that we know the values of the optimal boundary at the moments t_m,t_(m+1)…,t_n – we denote them by D_m,D_(m+1)…,D_n – for some m<n. For a fixed value x of the asset price at the previous moment t_(m-1), we defined d(x) as the value for the boundary that maximizes the holder’s financial result:
d(x)=argmax{D:V(x;{0,t_m-t_(m-1)...,t_nt_(m-1)};{D,D_m,...,D_n })}. (2.18)
(5) We approximate the value of the optimal boundary at the previous time-node t_(m-1) as D_(m-1)=min{x:D(x)=x}. In fact, this is the lower value for which the immediate exercise is optimal.
Credit
Tsvetelin Zaevski.
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