# Tag Info

1

No, the simulation is not exact in general, precisely for the reason you mentioned. By "exact", it is meant that there is no discretization error in time. Of course, there will always be a Monte-Carlo sampling error. For the Black-Scholes model, the simulation is exact if you simulate the log asset, as it is a standard arithmetic Brownian motion, and then ...

3

In stochastic calculus, expressions of the type: $$dX_t = a(t, X_t)dt + b(t, X_t) dW_t$$ are called stochastic differential equations. What the one above means for example is that $X_t$ has the following expression: $$X_t = X_0 + \int_0^t a(u, X_u)du + \int_0^t b(u, X_u) dW_u$$ The first integral is a regular one, and the second is called a stochastic ...

0

When you call ql.FlatForward it simply means you are constructing a rate curve that will lead to flat forward rates. The constructor of this curve takes the forward rate as an input. If you want to change the input (say, because the market moved and forward value changed), then you can change the quote value with the new value like this. First, keep a ...

3

You know the bond price formula takes this form: $P \left( t, T \right)= A \left( t, T \right) e^{ -r_{t} B \left(t, T \right) }$ Now apply Ito's lemma, so you will get after some manipulation: \$\frac{dP}{P}= \left(\frac{1}{A} \frac {\partial A}{\partial t} -r \frac {\partial B}{\partial t} - \kappa \theta B + \kappa r B+ \frac{1}{2} B^2 {\sigma}^2\right)...

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There are a few issues that need to be separated here. Issue "zero" is whether your MC is able to correctly represent the dynamics you've chosen for your assets. If you implement your MC properly, by construction it should converge in distribution to the postulated dynamics. No bias there. Variance yes potentially, because of discretisation, but no ...

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Mostly these assumptions are being made for mathematical simplicity and tractability. Non-Markovian processes are very difficult to work with. Gaussian processes are easy and convenient. Continuous time is a powerful assumption, although empirical data is usually in discrete time. Second order stationarity is appropriate because interest rates vary over ...

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There are two ways to look at this question. You can analyse the historical data to check whether it is in conformity with the stated assumptions around mean reversion and stationarity. This, by the nature of the real market prices/rates which are discrete, will be done in terms of the discrete analog of the OU process. Alternatively you can derive the ...

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