price_option_trapezoid_method module¶
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price_option_trapezoid_method.nu_hat(omega_m, x_0, r, t_discount, sigma, alpha)[source]¶ Calculate the fourier transform of the normalized European option value.
The transform is available in closed form as:
\[\hat{\nu}_c(w) = \frac{e^{-r(T-t)}}{(\alpha - i \omega)(\alpha - i \omega + 1)} \hat{q}(\omega + (\alpha + 1)i)\]With q written as:
\[\hat{q}(\omega') = e^{-i (x_0 + (r - \delta - \sigma^2 / 2) (T - t_0) ) \omega' - \frac{\sigma^2 (T - t_0)}{2} \omega' ^ 2}\]Parameters: omega_m : Numpy array
The discretized domain the calculate the transform over.
x_0 : double
The log of the initial stock price.
r : double
The risk free interest rate to discount at.
t_discount : double
The difference of t_T - t_0. This is the discount rate.
sigma : double
The volatility of the stock.
alpha : double
The damping parameter. If positive, it prices a call. If negative, a put.
Returns: transform : Numpy array
The transform calculated at each omega_m.
See also
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price_option_trapezoid_method.price_option_trapezoid_method(s_0, k, r, t_0, t_T, sigma, n, h, alpha)[source]¶ Price a European option by the trapezoid method.
This method calculates the pricing integral obtained from an inverse fourier transform involving the fourier transform of the normalized option price.
The approximation that it calculates is
\[V_k = Re \Big\{ \frac{e^{-\alpha k}}{\pi} \sum_{m = 0}^{N} e^{i \omega_m k} \hat{\nu}(w_m) \Delta \omega_m \Big\}\]Parameters: s_0 : double
The initial price of the asset.
k : double
The stike price for the option.
r : double
The risk free interest rate to discount at.
t_0 : double
The initial time.
t_T : double
The terminal time.
sigma : double
The volatility of the stock.
n : int
The number of points to discretize over. Together with h, this determines the frequency domain endpoint on [0, B]. B = n * h.
h : double
The size of the discretization steps.
alpha : double
The damping parameter. If positive, it prices a call. If negative, a put.
Returns: price: double
The price of the option.