The affine inflation market models

ABSTRACT

Interest rate market models, such as the LIBOR market model, have the advantage that the basic model quantities are directly observable in financial markets. Inflation market models extend this approach to inflation markets, where two types of swaps, zero-coupon and year-on-year inflation-indexed swaps, are the basic observable products. For inflation market models considered so far, closed formulas exist for only one type of swap, but not for both. The model in this paper uses affine processes in such a way that prices for both types of swaps can be calculated explicitly. Furthermore, call and put options on both types of swap rates can be calculated using one-dimensional Fourier inversion formulas. Using the derived formulas, we present an example calibration to market data.

KEYWORDS:

• Market models
• inflation options
• affine processes

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Here, we mean price in terms of money which has to be paid at a transaction. In fact it is essentially this number that is quoted on trading screens. However, in case such a bond is traded, the cash-flow is then the quoted number multiplied by the appropriate index ratio.

2. One could interpret $I(t)$ as a numeraire, but one has to be careful not to use this as a mathematical numeraire, since $I(t)$ is not actually traded.

3. $R(t,T)$ is also referred to as the real bond yield.

4. That is, the fixed rate which an investor would be willing to exchange at time $T$ for the payoff in (5).

5. We consider a market consisting of nominal and inflation-linked zero coupon bonds for a finite number of maturities.

6. One can extend bond price processes to $[0,T]$ by setting $P(t,T_k):=\frac{P(t,T)}{P(T_k,T)}$ for $t>T_k$, so that $P(\cdot ,T_k)/P(\cdot ,T)$ is a martingale on $[0,T]$ if and only if it is a martingale on $[0,T_k]$. Economically this can be interpreted as immediately investing the payoff of a zero coupon bond into the longest-running zero coupon bond.

7. The assumption that for each zero coupon maturity there is a ILB with the same maturity is used only for notional convenience.

8. The inflation index is only described through the bond prices $P_{\mathrm{I}\mathrm{L}\mathrm{B}}$. That is, the distribution of $I(t)$ is only given at times $T_k$, where it coincides with the distribution of $P_{\mathrm{I}\mathrm{L}\mathrm{B}}(T_k,T_k)$.

9. Although interest rates are currently negative in certain countries, interest rates are still bounded below by the costs of physically keeping money. We can incorporate bounds different from $0$ by setting $\frac{P(t,T_k)}{P(t,T)}:=c_k{M}_t^{u_k}$. Alternatively one can drop this assumption altogether and allow for negative interest rates.

10. This also shows that $X$ is a time-inhomogeneous affine process under ${\mathbb{Q}}^{T_k}$.

11. Up to some technical conditions the variance of a one-dimensional affine process is

$\mathbb{V}\mathrm{a}\mathrm{r}\left[X_t^i\right]=\frac{{\mathrm{\partial }}^2}{{\mathrm{\partial }}^{2}u}{|}_{u=0}({\varphi }_t^i(u)+{\psi }_t^i(u)X_0^i).$

12. We actually require only $N$ parameters, since the odd rows in Table 2 do not contribute for the considered annual inflation rates. For notional simplicity we nevertheless consider $2N$ parameters.

13. Although only stated for even forward rates for notional simplicity this is true for all forward rates.

14. Negative correlations in this setup are only possible if the sequence $({\tilde{u}}_k)$ is not decreasing which means that forward interest rates can become negative.

15. We decided to introduce correlation for the inflation part only via the common factor process. We could also have changed the parameters ${\bar{u}}_k$ in order to introduce correlation. In this case even more complicated correlation patterns could be created.

16. We only used parameter sets where of the two choices for ${\bar{v}}_k$, one choice was smaller than zero and the other one larger than zero, which we then picked.

17. $\mathcal{V}$ can be described as the (convex) set, where the extended moment generating function of $X_t$ is defined for all times $t$ and all starting values $x$. By Lemma 4.2 in Keller-Ressel and Mayerhofer (2015), the set $\mathcal{V}$ is in fact equal to the seemingly smaller set $\left\{u\in {\mathbb{C}}^d:\mathrm{\exists }x\in \mathrm{i}\mathrm{n}\mathrm{t}(D):{\mathbb{E}}^x[{\mathrm{e}}^{\mathrm{R}\mathrm{e}(u)\cdot X_T}]<\mathrm{\infty }\right\}$.

18. This was also shown for affine processes with general state spaces in Keller-Ressel, Schachermayer, and Teichmann (2013b) and Cuchiero and Teichmann (2013).