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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 ...


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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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You can use this article Probability distribution of returns in the Heston model with stochastic volatility Let $$\begin{align} & d{{S}_{t}}=r{{S}_{t}}dt+\sqrt{{{\nu }_{t}}}\left( \rho dW_{1}^{Q}(t)+\sqrt{1-{{\rho }^{2}}}dW_{2}^{Q}(t) \right) \\ & d{{v}_{t}}=\kappa (\theta -{{v}_{t}}){{d}{t}}+{{\sigma }_{v}}\sqrt{{{\nu }_{t}}}dW_{1}^{Q}(t) \\...


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Have you solved it yet? For example in the drift parameter, the dt needs to be vector of time from 0 to 1 by dt. My code is: GBM<-apply(BM,2,function(x) 100*exp((cumsum((r-0.5*sigma*sigma)*time)+sigma*x))) where I'm using GBM on already cumsummed Brownian Motion (x).


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The classical and naïve procedure for generating Poisson Hypersphere samples is by acceptance rejection, which has complexity over $O(N^2)$ and is thus unfeasible for most practical usage with on-the-fly generation. This cost could be improved by space partitioning techniques at low dimensions, but at high ones afaik they become useless again with uniform ...


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for example we restrict our discuss to a European call option denoted by $C(t,S_t)$ with exercise price $K$ and expiry date $T$. The final condition at time $t=T$ can be derived from the definition of a call option. If at expiration $S>K$ the call option will be worth $S-K$ because the buyer of the option can buy the stock for $K$ and immediately sell it ...


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The only problem I see with this approach, which remains completely valid from a theoretical perspective, is the embedded (and probably not accounted for) calibration risk: what if your LV surface does not allow you to correctly reproduce the observed vanilla option prices in the first place? In that case, you'll have lost information in the process and ...


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It is valid to do that, but if your local volatility surface is calibrated to the same OTM options, then your price will converge to the same answer. A local volatility surface is mainly a way of treating path-dependent options consistently with the option volatility surface. Variance swaps are path dependent on the face of it, but as you note the math ...


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No. Antithetic variable method is usually for generating smaller standard error than your non-antithetic method, which is a direct result of the negative correlation between original variable and the antithetic variable. For OTM option, there definitely will be a lot of path ending up with value 0. What may be a choice is to use importance sampling. Write ...


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No, you can have $$ \frac{1}{2n}\sum_{i=1}^{2n} C(S^i_T,K,T) = 0 $$ First off, there's the obvious case where $n=1$ and $u_1 = 0.5$ More generally, for options way out of the money it is common to have $$ \frac{1}{n}\sum_{i=1}^{n} C(S^i_T,K,T) = 0 $$ even for very large $n$. Antithetic sampling does not change that.


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Besides the code's problem, I highly recommend the Brownian Bridge correction method which can compensate the pricing error resulting from discretization of the continuous path.


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There are many things wrong with your code. I'll leave aside the manner in which it is implemented, but note that it is: (1) not Matlab friendly with all the for loops (you should vectorise), (2) the fact that you have splitted the case j==0 in the main loop is a poor coding practice. for i=1:n I=1; for j = 0:(m-1); Z(j+1)= randn (1 ,1); dW=sqrt (T/...


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This looks to me like a range accrual. Let $t_1, \ldots, t_n$, where $0 < t_1 < \cdots < t_n$ be business days that are being considered. We compute \begin{align*} E\left(\sum_{i=1}^n \pmb{1}_{b_1 < S_{t_i} < b_2} \right) &=\sum_{i=1}^n E\left(\pmb{1}_{b_1 < S_{t_i} < b_2} \right)\\ &=\sum_{i=1}^n \left[E\big(\pmb{1}_{S_{t_i} >...


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"Monte Carlo convergence" means that you've sampled enough individuals to represent (and understand) a general population. If the probability models behind your Monte Carlo simulation are accurate, then your results will match reality as you increase your sampling size. Monte Carlo convergence becomes difficult when you try to study a low-probability sub-...


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Here is a good paper which can help you. https://www.rocq.inria.fr/mathfi/Premia/free-version/doc/premia-doc/pdf_html/mc_jourdainlelong_doc.pdf



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