Hull-White Calibration /Hypothetical Cap Pricing

Question

I have a question regarding calibrating Hull-White (Extended Vasicek) Model to bond data. As you know, and stated in Mercurio (2005), zero coupon bond price in the Hull and White (1994);

$P(t,T)=A(t,T)e^{-B(t,T)r(t)}$

where

$B(t,T)=\frac{1}{a}\left[1-e^{-a(T-t)}\right]$

and

$A(t,T)=\frac{P(0,T)}{P(0,t)}e^{\left(B(t,T)f(0,t)-\frac{\sigma^{2}}{4a}(1-e^{-2at})B(t,T)^{2}\right)}$

My question is what happens if t=0? What is $f(0,0)$ in this case ? What $A(0,T)$ leads to ?

I wonder if it is possible to fit bond market data (zero and coupon bearing) to get parameters. Most of the works calibrating data using caplets/floorlets/inflation indexed swaps etc. But there is none of them in the market I'd like to work. So all I have is nominal and inflation indexed bonds.

Jarrow Yildirim has 7 parameters to price inflation indexed derivatives. By fitting these for nominal and inflation indexed bonds I'll have 4 of them. What is left is $\sigma_I$ and correlations between nominal, real and inflation. Any idea how to get those?

To give you the broad picture I need to explain that: My main goal is to price hypothetical caps by using only nominal bonds and inflation indexed bonds price data.

Any lead on this matter will be much appreciated.

Answer 1

You actually have more than one question here.

First Question: Price of bond in Hull White model where t=0

The $t$ refers to start date of the bond and $T$ is the maturity of the bond. If $t=0$ you are valuing a bond that starts now (or spot) and the forward will $f(0,0)=r(0)$ and $P(0,0)=1$

(Thanks to LePiddu for pointing that out)

Second Question:

Unfortunately, I don't think you will be able to price "hypothetical caps using only nominal bonds and inflation indexed bonds".

The reason the Jarrow Yildirim model has all those parameters is because it models inflation and nominal rates with a foreign-currency analogy, where you have 3 processes:

• Process for the nominal rates (domestic rate in FX world)
• Process for the real rates (foreign rates in FX world)
• Process for the the CPI (exchange rates in FX world)

Think of it as a HullWhite 1F for the nominal rate, a Hull White 1F for the real rate and a geometric brownian for the CPI.

To get the idea of why you can't price caps just with prices of bonds, consider the case of a HW1F for a single currency. If you only have bonds, you don't have any instruments to calibrate the volatility and the mean reversion. So although your model could fit the initial term structure, you wouldn't be able to price optionality (ie, caps and swaptions).

Imagine a Monte Carlo simulation where the average of your paths would give you your initial bond prices (rates) but depending on the other parameters (reversion and vols) it would have different dispersions and therefore price options differently.

Answer 2

First question

I downvoted David answer because $f(0,0) \neq 0$ (generally speaking). And that's because it's the instantaneous forward rate at time $t=0$, that is $f(0,0) = f(0, 0, \Delta t)= r(0)$ so it's the starting value of the short rate process.

In practice, you can set $\Delta t$ as one day (in years) and compute the forward rate (continuously compounded) from your yield curve.

Second question

As shown in various paper or as in Rebonato (2018) [1], you can price options on zero coupon bonds even calibrating your two parameters $a, \sigma$ on the yield curve only but you are not guaranteed it will contain enough information to account for the future costs of hedging of your option. I refer you back to Rebonato (2002) [2]: there is no derivative pricing without hedging (dynamic as in Black-Scholes or static). Traded option prices incorporates the future costs of hedging your option, so if you calibrate your model on traded prices and hedge following your model prescriptions everything will make sense (i.e. there are no arbitrages).

In this case you don't have option prices to calibrate your model to. You can calibrate to other quantities. Do they contain the correct informations? You cannot be sure.

References

[1] Rebonato, Riccardo. Bond Pricing and Yield Curve Modeling: A Structural Approach. Cambridge University Press, 2018.

[2] Rebonato, Riccardo. Volatility and correlation: the perfect hedger and the fox. John Wiley & Sons, 2005.