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3

A butterfly in general has a payoff of the form \begin{align*} (X_T-K_c)^+ + (K_p-X_T)^+-(X_T-K_{atm})^+-(K_{atm}-X_T)^+, \end{align*} where $X_T$ is the asset value at maturity $T$, while $K_c$, $K_p$, and $K_{atm}$ are strike levels.


0

This study seems to be on point: http://christian-fries.de/finmath/foresightbias/Fries_ForesightBias.pdf


3

you just add in any auxiliary variables accumulated along the path that determine the pay-off to the regression variables. So path-dependence is not a problem. If you have previous decisions, you may need to do different regressions based on their possible values or make them into a continuous variables that can be used for regression.


2

As your code works for the short maturity case, I assume that it is correct. The volatility of $80 \%$ is simply huge. Thus the area covered by the paths is huge too. As you can read e.g. here the sampling error is proportional to the variance of the process, which is huge in your case. As a brute force solution you can just enlarge the number of samples. ...


0

Increase the number of paths in your simulation for the getting the terminal prices, and at some point your monte carlo option price will finally converge to Black scholes option price as you are using a very longer maturity call option i.e. 10 year call option.


0

I can share how a pricing application (eg: QuantLib) calculates the VaR with Monte-Carlo. Generate a vector of independent Gaussian random numbers. A typical (and simple) implementation is Box-Muller. I prefer the inverse transform method, and I think this is also the default for QuantLib. Now, we will need to generate correlated returns. We will need a ...


3

Yes you can! Any SDE that has an analytic solution can be simulated exactly. The vasicek model has dynamics $dr=a(b-r)dt+\sigma dW_t$. By Ito's lemma, $$d\left(e^{at}r\right)=e^{at}\left(a(b-r)dt+\sigma dW_t\right) +a e^{at} r dt$$ Simplifying, $$d\left(e^{at}r\right)=e^{at} ab +e^{at}\sigma dW_t$$ Integrating, $$e^{aT} r_T=r_0+b(e^{aT}-1)+\sigma \int_0 ...



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