# Multi objective optimization Swaption/Caplets joint Calibration

People suppose that we have a two asset type portfolio optimization (as Intrument Type 1 and 2). In the each portfolio refered by the instrument type there are 2 asset so we have four asset in total.

How to express the objective function optimization in case of several asset classes with an appropriate weight?

If I suppose that each instrument in each portfolio have the same weight and that the two asset type or portfolio have the same weight I have expressed several objective functions using the relative form

[1] $X = \sqrt{sum_{i=0}^n ( InstrType1_{i}^M - InstrType1_{i}^N)^2} + \sqrt{sum_{j=0}^n ( InstrType2_{j}^M - InstrType2_{j}^N)^2}$

[2] $X = \sqrt{sum_{i=0}^n ( InstrType1_{i}^M - InstrType1_{i}^N)^2*wi} + \sqrt{sum_{j=0}^n ( InstrType2_{j}^M - InstrType2_{j}^N)^2*wj}$

[3] $X = p*\sqrt{sum_{i=0}^n ( InstrType1_{i}^M - InstrType1_{i}^N)^2*wi} + (1-p)*\sqrt{sum_{j=0}^n ( InstrType2_{j}^M - InstrType2_{j}^N)^2*wj}$

One can see that the weighted forms are exponentially lower than the non weighted optimization, the solver do not efficiently handle the objective due to that rescalled problem.

Does anybody already deal with multiple objectives?

Using matlab least square https://fr.mathworks.com/help/optim/ug/lsqnonlin.html how to express p in the [3] objective form as its seems to be a weighted least square optimization? as example in [2] one can weight the objective directly within the function while for the [3] its seems to be outside?

• Why use a sum of square root sum of square differences ? why not simply a sum of square differences $\sum_i p w_i (\text{InstrType}1^M_i - \text{InstrType}1^N_i)^2 + \sum_j (1-p) w_j (\text{InstrType}2^M_j - \text{InstrType}2^N_j)^2$ – Antoine Conze Jun 14 '17 at 11:10
• Yes indeed that is a notation I have found out in a book that was new to me but anyway using a weight for the asset classes and a weight for instrument contribution is reducing the level of the objective from % to 1/10000 with a loss of accuracy – Bond007 Jun 14 '17 at 11:27
• Can you clarify what it is you are calibrating ? from the title of your OP and the table you included I am assuming that you are trying to calibrate a model (which model ?) to both caps and swaptions. Also the absolute level of your objective function does not matter, what matters is that you have achieved its minima. The calibrated parameters will remain the same even if you rescale the entire objective function with an arbitrary number. – Antoine Conze Jun 14 '17 at 11:39
• I've successfully calibrated the 2 factor linear Gaussian model to both caps and swaptions in the past, however the fit was never very good but in my optinion that is a model limitation. For this type of calibration I use square difference of implied vols (market implied vol minus recomputed implied vol from model price), so as to deal with sum square error of comparable quantities (that's especially useful when using both ATM and OTM options) – Antoine Conze Jun 14 '17 at 12:44
• If I want to add constraints to the model parameters, $l_k < x_k < u_k$, then I do the change of variable $x_k(z_k) = b_k + a_k \arctan(z_k/a_k)$ with $a_k=(u_k-l_k)/\pi$ and $b_k=(u_k+l_k)/2$ and apply my Levernberg-Marcquardt to $\min_{z_1, ..., z_p}...$ instead of $\min_{x_1, ..., x_p}...$. – Antoine Conze Jun 19 '17 at 8:57