Preserving Martingale Constraints When Interpolating Bond Scenarios

Summary

The document asks how to fill missing annual maturities in simulated zero-coupon bond price paths. The available scenarios come from a shifted Libor Market Model and contain forward bond prices at selected maturities, while the desired grid has annual spacing. The scenarios cannot be regenerated, and the calibration swaption prices are unavailable.

The central constraint is risk neutrality: an interpolated bond price should preserve the stated deflator-weighted martingale relationship in expectation, rather than simply follow a conventional cubic curve between observed maturities. This frames the task as interpolation across both maturity and simulated paths while respecting an expectation condition. The document supplies no proposed interpolation algorithm, numerical experiment, or proof that a particular method works. It is therefore useful as a modeling problem statement and constraint, but practitioners would need to develop and validate a method against the existing scenario data and relevant bond pricing relationships.

Key ideas

• Missing maturities must be inferred from simulated zero-coupon bond prices at selected tenors.
• The scenarios use a shifted Libor Market Model and cannot be regenerated in the stated setting.
• Interpolation should preserve the deflator-weighted martingale condition under the risk-neutral measure.
• Ordinary cubic interpolation may fail to enforce that condition across trajectories.
• The document poses the problem but does not specify or test a solution.

Tags

# Interpolation of forward zeros-coupons bonds simulations for missing maturities (ESG data)


# Interpolation of forward zeros-coupons bonds simulations for missing maturities (ESG data)












I have a set of economic scenarios simulated with Barrie and Hibbert ESG. The stochastic model for interest rates used is Libor Market Model Shifted. I am facing a problem with zeros-coupons prices.

Indeed, I have for each maturity: 2,000 forward prices(in 1 year to 30 years) trajectories of zeros-coupons. I have the following maturities (1; 2; 3; 5; 8; 10; 15; 20; 30; 40; 50; 60) for each forward price but I want the maturities of 1 to 30 with an annual pace.

I cannot regenerate the scenarios: I have to work with this simulations and I don't have swaptions prices used to calibrate the model. So I have to interpolate the missing maturities throughout 2000 trajectories. Considering that I have to project in risk neutral world, zero coupon prices are martingale seen in $t = 0$: $E[B(t,T)D_t|\mathcal{F}_0]=B(0,t+T)$ with :

- $B(t,T)$: the price seen in t of zero-coupon bond with maturity T.

- $D_t$: the deflator to calculate the present value of a cash-flow in t.

- $\mathcal{F}_0$: the filtration(the information in $t=0$)

So I can not make cubic interpolation without considering the fact that the price interpolated must be martingale in all trajectories in expectation (the mean).

What can you propose to me in order to interpolate zeros-coupon prices for missing maturities so that the interpolated prices are still martingale.

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