# 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.

• Your English looks fine to me but I added some $\LaTeX$ to your question. can you verify the formulas? – Bob Jansen Apr 30 '16 at 21:04