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In SV model, it is well-known that the integrand for the call price can sometimes show high oscillation, can dampen very slowly along the integration axis, and can show discontinuities. Remedy The ‘‘Little Trap’’ formulation of Albrecher et al. Also , you can use Fourier transforms Bakshi and Madan (2000) Lewis,(2001). Gatheral (2006) Carr and ...

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It is hard to imagine why a trader would want to buy a forward start option to express a market view, unless there is a one-off event like an election which they don't want to have as part of the live period for the volatility. A forward start option is mainly exposed to the volatility relating to the period after the strike is set. Forward start options ...

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They would only have been equal (up to the usual MC accuracy and bias) should the black-box model had assumed a GBM dynamics as in the classic Black-Scholes framework. $D_{MC}$ and $D_{BS}$ will indeed differ in general because digital options are sensitive to the implied volatility skew, which is inexistent in a Black-Scholes world where $\sigma (K,T)=\... 3 That's impossible. Since neither the vanilla options nor the underlyings have any exposure to the correlation, no portfolio of these instruments can either. 2 You are trying to write a program which solves the following pricing PDE (Black-Scholes assumed) $$\frac{\partial V}{\partial t}(t,S) + (r-q)S\frac{\partial V}{\partial S}(t,S) + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}(t,S) - rV(t,S) = 0$$ where$V_0:=V(0,S_0)$is the target option premium. The terminal condition is that, at$t=T$(... 2 Assuming$\theta>0$(take$\tilde{X}=\mu-X$if it is not the case) Let us denote$\text{erfi}(x)$the imaginary error function Let us denote$\tau_L$,resp.$\tau_U$the hitting time of$L$resp.$U$where$L<U$1) Using Ito's lemma, prove that : $$Y_t = \text{erfi}\left(\sqrt{\frac{\theta}{\sigma^2}}\left(X_t-\mu\right)\right) \text{ is a martingale}$$ ... 2 There has been a huge amount of work on this. Generally a Fourier transform approach is used. First, be careful to use the form of the characteristic function that does not wind about zero in order to avoid having to count the normal of windings. Second, using contour shifts can make the integral much better behaved. eg integrate along the line with$0.5$... 2 I'd use FFT or similar rather than direct integration. Here is an old paper with Heston example: Option pricing using fractional FFT 1 There's no such engine at this time. If you want to code it, you can clone and rename the MCEuropeanBasketEngine and the EuropeanMultiPathPricer classes. The new path-pricer class must be modified so that its operator() returns the payoff of your option as calculated on a given path; the new engine will be mostly unmodified, except for the pathPricer method ... 1 Judging from the oscillations near$S=0$, it looks like the payoff function is causing these problems. Your payoff should go towards -1 as$S$goes towards zero, but your computer might just evaluate it at$S=0$, producing nonsense as a result. Depending on the exact implementation, this will then spread through the neighborhood of that point, causing ... 1 Call put parity is : $$C(T,K) - P(T,K) = ( F_{t,T} - K ) B(t,T)$$ and with your notation : $$C(T,K) - P(T,K) = ( F_{t,T} - K ) + \text{TimeValueCall}(T,K) - \text{TimeValuePut}(T,K)$$ Reference: https://en.wikipedia.org/wiki/Put%E2%80%93call_parity 1 You don't need any assumption about the distributional properties of$S_t$. What matters for the FTAP is the drift only. By definition, the risk neutral measure$Q$is the measure, equivalent to the natural measure$P$(*), under which the local rate of return (i.e. the instanteneous drift of the SDE of$S_t$per unit of$S_t$) of "any" traded asset$S_t\$ (...

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