Weighting Multiple Objectives in Swaption and Caplet Calibration

Summary

The document asks how to combine calibration errors for two instrument groups, each containing multiple assets. It compares an unweighted sum of group-level Euclidean errors with versions that apply weights within each group and a mixing parameter between groups. The author is concerned that weighting makes the objective much smaller and may impair solver performance, and asks how to express the mixed objective with MATLAB’s least-squares optimizer.

The material frames a practical calibration design question rather than presenting a resolved method. It gives no empirical comparison, recommended normalization, solver settings, or explanation of how the within-group weights should be chosen. In particular, the question remains open about representing an outer group weight in a least-squares residual formulation, where residual scaling affects the objective. Researchers should treat the formulas as candidate constructions and consider scale normalization and the relative importance of each instrument class when designing a calibration loss.

Key ideas

• The document compares unweighted and weighted Euclidean calibration errors across two instrument groups.
• A mixing parameter can control the relative contribution of each group to the total objective.
• The author reports concern that weighting reduces objective scale and may hinder solver performance.
• The document asks how to represent group-level weighting in a least-squares solver but supplies no answer or test evidence.

Tags

# Multi objective optimization Swaption/Caplets joint Calibration


# 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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.