Monte carlo methods for vanilla european options and Ito's lemma.

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Hello,

I understand that by applying Ito's lemma to the following SDE
dX = mu* X * dt + sigma * X * dW

one obtains a solution to the above SDE which is as follows:
X(t) =X (0) * exp(sigma * W(t) + ( mu- 0.5 * sigma^2)* t))

I have been told that I can use either of these equations (SDE or its solution) for applying monte carlo simulations to vanilla European options although the second one converges faster that the first one.

Can someone confirm this statement?

Furthermore, I understand that Ito's lemma allows us to obtain the differential of an SDE and that in the above case, the differential is also the solution to the SDE. Does this rule always hold for all stochastic procesesses?

Thanks in advance for your replies,

Regards,

J.

 

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Hi J,

Observation: European options are not path dependent. What I mean is it only depends on the price at time t, not before t.

For the first equation, you will have to simulate the path at every small interval of size dt. Instead, the second equation helps you to simulate the price directly at t, skipping all the small dt-intervals in between. Hence the second one converges faster.

For path dependent options like American, you need to know the prices at each of the small intervals dt since one might want to exercise before exp. For such situations the first approach works.

Ito's lemma are used to find the SDE of a process. I'm not sure how you can get the process from the SDE using Ito. The way people solve SDEs is to start with a reasonable guess for the process. Then you find the SDE of the process using Ito. If the SDE matches with the given SDE, the process is a solution. (Uniqueness can be guaranteed using boundary conditions)

I hope that helps.

Alan

 

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Hello Alan,

Thanks a lot for your reply. I understand the first part of your reply.

One point remains confuse to me: what is the difference between a stochastic differential equation and a stochastic process? In the case above what is the SDE and what is the process?

Regards,

Julien.

 

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The stochastic process is X(t) which is X (0) * exp(sigma * W(t) + ( mu- 0.5 * sigma^2)* t)) (call it (A))

If you differentiate both sides using Ito, you will have

dX(t)=d[X (0) * exp(sigma * W(t)*root(t) + ( mu- 0.5 * sigma^2)* t))] which should reduce to

dX(t) = mu* X(t) * dt + sigma * X(t) * dW(t) which is the SDE.

Start with the guess X(t)=A(t). Thus if you can show that dX(t)=dA(t) using Ito, then A(t) is a solution.

Alan

 

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thanks Alan.