R: Doubt in Montecarlo generation of paths

[Berardi Luca]

Ciao Luigi,

it would indeed be useful to simulate the log of the asset if we are interested in equity prices. However what if I'd like to simulate a generic diffusion process? For example I could be interested in simulating the instantaneous interest rate for pricing complex fixed income derivatives...

In this case I wonder whether it is necessary (or useful) to simulate the series of log-returns. In such a case, I should:

1) define a model for the risk-free rate r(t) (let's say CIR for example) and write down the corresponding SDE (stoch. diff. equation);

2) define Y(t) = log (r(t)/r(0)) (log return of the asset) and by means of an Ito differentiation write down the SDE for Y(t) separating the drift and diffusion components;

3) use this last drift and diffusion coefficients for simulating Y(t) and then
4) exponentiate Y(t) to get r(t)/r(0), which is what I need to put in the path that should be evaluated in the payoff function

Wouldn't it be simpler to directly simulate r(t), instead (given that we have already the drift and diffusion coefficients already available in the SquareRootProcess class)?

Am I missing something?

Luca

-----Messaggio originale-----
Da: Luigi Ballabio [[hidden email]]
Inviato: mercoledì 7 aprile 2004 12:18
A: Berardi Luca
Cc: [hidden email]
Oggetto: Re: [Quantlib-users] Doubt in Montecarlo generation of paths

Ciao Luca,

On 2004.04.07 11:38, Berardi Luca wrote:
> I was looking at the code for setting up a generic Montecarlo
> simulation, in particular the PathGenerator class.
> There is something I could not really understand in the method next. 
> Looking at the section of code (where the brownian bridge is not
> used):
>
>             for (Size i=0; i<next_.value.size(); i++) {
>                 t = timeGrid_[i+1];
>                 dt = timeGrid_.dt(i);
>                 next_.value.drift()[i] = dt *
>                     diffProcess_->drift(t, asset_);
>                 next_.value.diffusion()[i] = sequence_.value[i] *
>                     QL_SQRT(diffProcess_->variance(t, asset_, dt));
>                 asset_ *= QL_EXP(next_.value.drift()[i] +
>                                  next_.value.diffusion()[i]);
>             }
>
> It not clear to me why asset_ is re-assigned as: asset_ *=
> QL_EXP(...).
> We are numerically integrating a diffusion process given by:
>
> d asset_ = drift(t, asset_)*dt + diffusion(t,asset_)*dW

No. We're integrating

d log(asset_) =  drift(t, asset_)*dt + diffusion(t,asset_)*dW

hence the exponential

> asset_ *=
>     QL_EXP(next_.value.drift()[i] + next_.value.diffusion()[i]);

(which by the way, can be written more concisely as

> asset_ *= QL_EXP(next_.value[i]);

if you're not concerned in separating the two components.)

Bye,
        Luigi

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