In spot exchange rate modeling with Geometric Brownian Motion (GBM), the drift term is interest rate differential...Many practitioners while computing PFE for a portfolio with interest rate and currency swap, model exchange rate with constant drift (rd-rf), but uses stochastic interest rate model like HW for interest rates swap!!!What we have done is, simulate interest rate using HW and plug it in GBM equation ..Process is described below....Let me know if the following approach is correct....1.Exchange rate follows the dynamicsd S(t)= (rd (t) -rf(t) )S(t) dt+ sigma S(t) dW(t) (1)2. Simulate S(t) as per Euler schemeLn {S(ti+1)/S(ti)} = (rd (ti) -rf(ti) ?0.5sigma^2) (ti+1-ti) + sigma(x) sqrt (ti+1-ti) Zx i+1 (2)3. Here rd (t) & rf(t) are short rates as per Vasicek which are generated as per Euler schemerk(ti+1)= rk (ti)+k(theta - rk (ti))(ti+1-ti)+ sigma(k) sqrt (ti+1-ti) Zk i+1 (3) k= domestic, foreign.Sigma (x) refers to spot exchange rate volatility. Sigma(k) refers to interest rate volatility4. Z i+1 s follows correlated standard normal distribution distribution.5. From historical over night interest rate and spot exchange rate compute Zi series for domestic and foreign interest rate and exchange rate using relation (2) & (3).6. Compute correlation matrix betwen Zi s. Monte carlo simulation is done using copula approach...7. Above approach can be generalized to multiple currencies and exchange rates.

The basic idea is just fine, and quite standard. Amin and Jarrow (1991) is a classic reference for the underlying theory, and I am aware of commercial implementations of the model going back to 1993. The model is fully analytical, and you should not need to use any Euler schemes. Also, since you are working with a multivariate Gaussian distribution to begin with, there is no reason to introduce copulas.

Do you get analytical solution to spot exchange rate dynamics with stochastic interest rates? Please provide me reference... standard textbooks have provided analytical solution to GBM spot exchange rate dynamics under constant drift..........

Hi, can anybody answer my question?

brigo and mercurio Interest rate options book - they deal with Longdated equity ( ie single interest rate)

QuoteOriginally posted by: sandipan2011In spot exchange rate modeling with Geometric Brownian Motion (GBM), the drift term is interest rate differential...Many practitioners while computing PFE for a portfolio with interest rate and currency swap, model exchange rate with constant drift (rd-rf), but uses stochastic interest rate model like HW for interest rates swap!!!What we have done is, simulate interest rate using HW and plug it in GBM equation ..Process is described below....Let me know if the following approach is correct....1.Exchange rate follows the dynamicsd S(t)= (rd (t) -rf(t) )S(t) dt+ sigma S(t) dW(t) (1)Equation (1) can not be applied for a FX model. If you put base and counter currencies be equal we get on the left hand dS = 0 as S = 1 on the right hand side we get a constant sigma* dW. Therefore at least the diffusion sigma can not be a constant in the model. The other problem which relates to (1) is that this equation should be consistent with IRP and it actually follows from it at least in deterministic case. Assumption of the type (1) implies that actual interest rates of the bonds domestic and foreign should be of the 'white' noise type which in turn implies higher values than its face value for each t during the lifetime of the bonds. If these remarks were stated explicitly it might be that it could not popular in these days.

Nonsense!

QuoteOriginally posted by: bearishNonsense!Let exchange rate d S(t)= (rd (t) - rf(t) )S(t) dt+ sigma S(t) dW(t) (1)Apply exchange a currency to itself. Then S ( t ) =1, dS = 0, and r_d = r_f. From (1) it follows then 0 = sigma * dW.Diffusion must contain additional factor [ rd (t) - rf(t) ] or a convenient power of it. Correct me please if I am wrong here.

OK - I'm going out on a limb here. I think "Apply exchange a currency to itself" means that we look at something like the value of USD in terms of USD. In that case, rd(t) = rf(t) for all t, S(t) = 1, dS(t) = 0, and ... to have it all make sense ... sigma = 0. Trivial in the extreme, but it still makes sense.

QuoteOriginally posted by: bearishOK - I'm going out on a limb here. I think "Apply exchange a currency to itself" means that we look at something like the value of USD in terms of USD. In that case, rd(t) = rf(t) for all t, S(t) = 1, dS(t) = 0, and ... to have it all make sense ... sigma = 0. Trivial in the extreme, but it still makes sense.If one use the equation (1) and say that sigma would depend on the difference r_d - r_f not for example on its ratio or other its function it would be too general even from math point of view. If sigma will be replaced by an expression g ( r_d - r_f ) where g ( x ) is a smooth function such that g ( 0 ) = 0 the adjusted equation will look sufficiently good. The simple class of appropriate g are functions of the type sigma*[ r_d - r_f ]. Of course my remark refines the equation but does not reject it. For a particular pair of currency we can hope for more accurate forecast. If we think that this adjustment does not bring anything then we can apply such argument to the drift and state that the factor ( r_d - r_f ) can be replaced by a constant.

QuoteOriginally posted by: list QuoteOriginally posted by: bearishNonsense!Let exchange rate d S(t)= (rd (t) - rf(t) )S(t) dt+ sigma S(t) dW(t) (1)Apply exchange a currency to itself. Then S ( t ) =1, dS = 0, and r_d = r_f. From (1) it follows then 0 = sigma * dW.Diffusion must contain additional factor [ rd (t) - rf(t) ] or a convenient power of it. Correct me please if I am wrong here.You're wrong. What is the volatility of dollars against dollars?

'the volatility of dollars against dollar' is 0 as far as the volatility term should contain the factor[ r_d - r_f ] as it was suggested.

It doesn't need any additional term, it is zero already.

QuoteOriginally posted by: TinManIt doesn't need any additional term, it is zero already.Then replace drift factor (rd (t) -rf(t) ) in (1) d S(t)= (rd (t) -rf(t) )S(t) dt+ sigma S(t) dW(t) (1)for mu and mu will be already what you expect it to be.

I know how to use it, you said it isn't applicable when it clearly is.Look at FX options in any elementary textbook.