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Currently I am doing a research on the plain vanilla multi-curve framework Libor Market Model meaning that no stochastic volatility is involved. I had the idea to calibrate to the swaption market. In the volatility calibration procedure I minimize the error between the market quoted swaption volatilities and the volatilities obtained by Rebonato's swaption volatility approximation formula. Which minimization method would be convenient for this particular problem?

I have the following parametrizations for the instantaneous correlation and instantaneous volatility, respectively: $\rho_{i,j} = e^{-\beta |T_i - T_j |}$

and

$\sigma_i(t) = (a+b(T_i - t))e^{-c(T_i-t)}+d$

I would like to include the correlation parameter $\beta$ in the calibration.

Does anyone have a suggestion on how I can calibrate the correlation? I suppose this has to be done before calibrating the volatility parameters $a,b,c,d$ since the Rebonato's approximation formula requires the instataneous correlation as input. To which target value do I have to compare it to since no instantaneous correlation can directly be deduced from the market quoted volatilities and which error do I have to minimize?

Furthermore, after having calibrated the model I can use the calibrated parameters in a Monte Carlo routing where the forward Libor rates are simulated under the spot measure to price a swaption. Can anyone advise me on how I can test the accuracy of the calibrated parameters by comparing the swaption prices obtained by Monte Carlo, by Rebonato's approximation formula and the market quoted volatilities? (side question: Do I retrieve the price of swaption obtained throught Monte Carlo in terms of volatility by taking the volatility inserted in the simulation?)

Would it make sense to compare these results in a graph where different strikes are plotted (ATM + x) against the three different types of volatilities described above for a given set of calibrated parameters? Or is this only relevant when the volatility in the model is stochastic?

Thank you in advance.

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Market practitioners do the following: Correlation is calibrated most often by looking at historical correlations between liquid par swap rate pairs. One could look at implied correlations within options on the yield curve (eg 10 yr minus 2yr) also.
Swaption calibration should be done by comparing straddle prices in the market to prices produced by the simulation model. Use at least 10 liquid points covering short end and long end of the curve, and a few different expirations. The last question : the market skew if very hard to match in a simple Monte Carlo rates model. unless you're using stochastic vol you're probably not going to produce as much smile as the market. In addition, the market dynamics tend to be different for short dated forwards versus long dates , and you can't capture that.

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