Valuation of Inflation-Linked Annuities in a Lévy Market

Abstract

We study the problem of pricing an inflation adjusted annuity in a forward rates market with jumps. Since the market will be incomplete, we use the minimal 𝑓^𝑞-martingale measure 𝑄_𝑞 which we use for computing discounted expectations. We give explicit results for 𝑄_𝑞 together with explicit results for the price of the annuity.

1. Introduction

Inflation derivatives are a new feature in the derivatives markets with active trading starting in the 1980’s. Their origin has been necessitated by the risk of inflation eroding medium to long-term maturity securities. In economies with anticipation of high inflation, it makes sense to consider the real value of future assets as opposed to the nominal value. According to the simplified Fisher’s model, the real interest rate equals the nominal interest rate minus the inflation rate. It is only in cases where inflation rate is negligible that these two rates almost give the same rate of return. Pricing annuities differ from pricing options in that an annuity makes periodic or continuous payments usually up to a certain redemption date which could be predetermined or which is contingent on the life of the policy holder. An option, when it matures and if its maturity is beneficial to its holder, will make a single payment, the redemption value of the option. An option can also expire worthless. Annuities do not have this option of expiring worthless. In elementary Actuarial analysis, the premium to be paid for an annuity is the expected present value (EPV) of cash flows. As a result, we then find striking similarities between computing option prices and insurance premiums.

Motivated by the work [1] and references therein, in this paper, we find the premium of a nonlife-based annuity in a market where bond prices are driven by jump processes. We emphasize here that the inclusion of life contingency will not bring much deviation of this result because market movements are independent of the future lifetime of a policyholder. Due to incompleteness caused by the introduction of jumps, we consider and give explicitly the minimal 𝑓^𝑞-martingale measure studied in [2].

We characterize our problem as one similar to interest rate and foreign exchange derivatives (see [3] for more). Naturally the premium of an inflation protected annuity is equal to the premium of an ordinary annuity plus some “penalty” which results from the extra benefit of the policy holder being cushioned for inflation. This paper concentrates on finding explicit results of this extra “penalty”.

The paper is organized as follows: the following section looks at the mathematical preliminaries. For this part, we use the notation in [4]. We also refer the reader to the same text for more detailed analysis of Lévy processes. Section 3 looks at the assets price dynamics. We give the Consumer Price Index (CPI) as the ratio of the nominal bond to the real bond. This approach is similar to foreign exchange derivatives where we consider the nominal bond as representing the local currency, the real bond as representing the foreign currency, and then the CPI being the exchange rate of foreign currency to local currency. A high value of exchange rate indicates a weak local currency with respect to the foreign currency. In the same vein, high inflation indicates high-interest rates which may not stimulate production due to high cost of production. Of interest in Chapter 3 is our version of Fisher’s equation (see [5]) which is an extended version that incorporates volatility coefficients and jump terms.

In Section 4, we give the main result which ends with the premium of a non-life annuity adjusted for inflation being given explicitly.

2. Mathematical Preliminaries

Consider a filtered probability space (Ω,ℱ,ℱ_𝑡,𝑃) where {ℱ}_𝑡≥0 is the filtration such that the Lévy-Itô processes discussed in this paper are adapted.

Given a Lévy process 𝑋(𝑡), the jump of 𝑋(𝑡) is Δ𝑋(𝑡)=𝑋(𝑡)−𝑋(𝑡⁻). The number of jumps, which occur before or at time 𝑡 and of a size contained in some Borel set 𝑈, can be counted by the measure 𝑁(𝑡,𝑈) called the Poisson random measure. It turns out that𝑁(𝑡,𝑈)=_{0≤𝑠≤𝑡}𝜒_𝑈(Δ𝑋(𝑠)). (2.1) For times 0≤𝑡₁≤𝑡₂<∞, we denote the differential form of 𝑁(𝑡₂,𝑈)−𝑁(𝑡₁,𝑈) by 𝑁(𝑑𝑡,𝑑𝑧). The set function 𝜈(𝑈)=𝐸[𝑁(1,𝑈)] is called the Lévy measure of 𝑋 and we require that ∫_ℝmin(1,𝑧²)𝜈(𝑑𝑧)<∞.

The measure 𝑁(𝑑𝑡,𝑑𝑧)=𝑁(𝑑𝑡,𝑑𝑧)−𝜈(𝑑𝑧)𝑑𝑡 is called the compensated Poisson random measure. In this paper, we will be considering jumps of sizes greater than −1. Therefore, we have the relationship (see [4]),𝑁()=𝑁()()|𝑧|𝑁()|𝑧|𝑑𝑡,𝑑𝑧𝑑𝑡,𝑑𝑧−𝜈𝑑𝑧𝑑𝑡,if<1,𝑑𝑡,𝑑𝑧,if≥1. (2.2) The Lévy-Itô process 𝑋(𝑡) that we will consider in this paper is of the form𝑑𝑋(𝑡)=𝛼(𝑡,𝑋(𝑡))𝑑𝑡+𝛽(𝑡,𝑋(𝑡))𝑑𝐵(𝑡)+_ℝ𝛾(𝑡,𝑋(𝑡))𝑁(𝑑𝑡,𝑑𝑧), (2.3) where 𝐵(𝑡) is a standard Brownian motion and the coefficients 𝛼(⋅,⋅),𝛽(⋅,⋅) and 𝛾(⋅,⋅) satisfy the necessary growth conditions (see [4]) for which (2.3) has a unique strong solution 𝑋(𝑡).

The unique characteristic function 𝜂_𝑋(𝑢)=𝐸[𝑒^{𝑖𝑢𝑋(𝑡)}] is given by the Lévy-Khintchine formula (see [4] or [6] for more).

Frequently we will be referring to the Lévy-Itô formula for this Lévy process. We refer the reader to ([4]) for a more general definition. We will, for the purpose of this paper, restrict ourselves to a two-dimensional version of the definition.

Definition 2.1. Let 𝑋(𝑡)∈ℝ² be an Itô-Lévy process of the form 𝑑𝑋(𝑡)=𝛼(𝑡,𝜔)𝑑𝑡+𝜎(𝑡,𝜔)𝑑𝐵(𝑡)+_ℝ𝛾(𝑡,𝑧,𝜔)𝑁(𝑑𝑡,𝑑𝑧), (2.4) where α:[0,𝑡]×Ω→ℝ², 𝜎∶[0,𝑇]×Ω→ℝ^2×1, and 𝛾∶[0,𝑇]×ℝ×Ω→ℝ^2×1 are adapted processes such that the integrals exist. Let 𝑓∈𝐶^1,2([0,𝑇]×ℝ²;ℝ) and put 𝑌(𝑡)=𝑓(𝑡,𝑋(𝑡)). Then 𝑑𝑌(𝑡)=𝜕𝑓𝜕𝑡𝑑𝑡+²_𝑖=1𝜕𝑓𝜕𝑥_𝑖𝛼_𝑖𝑑𝑡+𝜎_𝑖(𝑡)+1𝑑𝐵2^2𝑖,𝑗=1𝜎𝜎^∗_𝑖𝑗𝜕²𝑓𝜕𝑥_𝑖𝜕𝑥_𝑗𝑑𝑡+²_𝑘=1ℝ𝑓𝑡,𝑋(𝑡⁻)+𝛾(𝑡,𝑧)−𝑓(𝑡,𝑋(𝑡⁻))−²_𝑖=1𝛾_𝑖(𝑡,𝑧)𝜕𝑓𝜕𝑥_𝑖(𝑋(𝑡⁻))𝜈(𝑑𝑧)𝑑𝑡+²_𝑘=1ℝ𝑓𝑡,𝑋(𝑡⁻)+𝛾(𝑡,𝑧)−𝑓(𝑡,𝑋(𝑡⁻))𝑁(𝑑𝑡,𝑑𝑧), (2.5) where 𝛾∈ℝ^2×1 is column vector 𝛾=(^𝛾1𝛾2)^∗. Here the ∗ represents transposition of a matrix.

The Itô-Lévy formula above shall be useful in the next chapters.

3. Asset Dynamic Models

3.1. The Bond Price Process

Suppose that the real(𝑅) and nominal(𝑁) forward rates are given jointly by𝑓_𝑘(𝑡,𝑇)=𝑓_𝑘(0,𝑇)+^𝑡₀𝛼_𝑘(𝑣,𝑇)𝑑𝑣+^𝑡₀𝜎_𝑘()(𝑣)+𝑣,𝑇𝑑𝐵_𝑇0𝛾_𝑘(𝑣,𝑇,𝑧)𝑁_𝑘(𝑑𝑣,𝑑𝑧),𝑘∈{𝑁,𝑅}. (3.1)

We know that the price of a zero coupon nominal (real) bond is given by𝑃_𝑘(𝑡,𝑇)=exp−^𝑇_𝑡𝑓_𝑘(𝑡,𝑠)𝑑𝑠,∀𝑡∈[0,𝑇]. (3.2)

Proposition 3.1. Let 𝑎_𝑘(𝑡,𝑇)=−∫^𝑇_𝑡𝜎_𝑘(𝑡,𝑠)𝑑𝑡 and 𝑐_𝑘(𝑡,𝑇,𝑧)=−∫^𝑇_𝑡𝛾_𝑘(𝑡,𝑠,𝑧)𝑑𝑡. Define also 𝑏_𝑘(𝑡,𝑇)=−∫^𝑇_𝑡𝛼_𝑘(𝑡,𝑠)𝑑𝑡+(1/2)𝑎_𝑘(𝑡,𝑇)−∫_ℝ𝑐_𝑘(𝑡,𝑇,𝑧)𝜈(𝑑𝑧). Assume that 𝑎_𝑘(⋅),𝑏_𝑘(⋅,⋅), and 𝑐_𝑘(⋅) saisfy the necessary regularity conditions for applying Fubini’s theorem. Then 𝑃_𝑘(𝑡,𝑇) is given by 𝑑𝑃_𝑘(𝑡,𝑇)=𝑃_𝑘(𝑡,𝑇)𝑟_𝑘(𝑡)+𝑏_𝑘(𝑡,𝑇)𝑑𝑡+𝑎_𝑘(𝑡,𝑇)𝑑𝐵(𝑡)+_ℝ𝑐_𝑘(𝑡,𝑇,𝑧)𝑁(𝑑𝑡,𝑑𝑧). (3.3)

Proof. Simplifying and using the Fubini's theorem, we have ln𝑃_𝑘(𝑡,𝑇)=−^𝑇_𝑡𝑓_𝑘(𝑡,𝑠)𝑑𝑠=−^𝑇_𝑡𝑓_𝑘(0,𝑠)𝑑𝑠+^𝑇_𝑡^𝑡₀𝛼_𝑘(𝑣,𝑠)𝑑𝑣𝑑𝑠+^𝑇_𝑡^𝑡₀𝜎_𝑘()(𝑣)+𝑣,𝑠𝑑𝐵𝑑𝑠_{𝑡𝑡𝑡0ℝ}𝛾_𝑘(𝑣,𝑠,𝑧)𝑁(𝑑𝑣,𝑑𝑧)𝑑𝑣𝑑𝑠=−^𝑡_𝑡𝑓_𝑘(0,𝑠)𝑑𝑠−^𝑡₀^𝑇_𝑡𝛼_𝑘(𝑣,𝑠)𝑑𝑠𝑑𝑣−^𝑡₀^𝑇_𝑡𝜎_𝑘()(𝑣)−𝑣,𝑠𝑑𝑠𝑑B_{𝑡0𝑇𝑡ℝ}𝛾_𝑘(𝑣,𝑠,𝑧)𝑁(𝑑𝑠,𝑑𝑧)𝑑𝑠𝑑𝑣=−^𝑇₀𝑓_𝑘(0,𝑠)𝑑𝑠+^𝑡₀𝑓_𝑘(0,𝑠)𝑑𝑠−^𝑡₀^𝑇_𝑣𝛼_𝑘(𝑣,𝑠)𝑑𝑠𝑑𝑣+^𝑡₀^𝑡_𝑣𝛼_𝑘()−𝑣,𝑠𝑑𝑠𝑑𝑣_{𝑡0𝑇𝑣}𝜎_𝑘(𝑣,𝑠)𝑑𝑠𝑑𝐵(𝑣)+^𝑡₀^𝑡_𝑣𝜎_𝑘(𝑣,𝑠)𝑑𝑠𝑑𝐵(𝑣)−^𝑡₀^𝑇_𝑣ℝ𝛾_𝑘(𝑣,𝑠,𝑧)𝑁(𝑑𝑠,𝑑𝑧)𝑑𝑠𝑑𝑣+^𝑡₀^𝑡_𝑣ℝ𝛾_𝑘(𝑣,𝑠,𝑧)𝑁(𝑑𝑠,𝑑𝑧)𝑑𝑠𝑑𝑣=ln𝑃_𝑘(0,𝑇)+^𝑡₀𝑟_𝑘(𝑠)𝑑𝑠−^𝑡₀^𝑇_𝑣𝛼_𝑘(𝑣,𝑠)𝑑𝑣𝑑𝑠−^𝑡₀^𝑇_𝑣𝜎_𝑘()(𝑣)−𝑣,𝑠𝑑𝑠𝑑𝐵_{𝑡0𝑇𝑣ℝ}𝛾_𝑘(𝑣,𝑠,𝑧)𝑑𝑣𝑑𝑠𝑁(𝑑𝑠,𝑑𝑧). (3.4) Therefore, we have ln𝑃_𝑘(𝑡,𝑇)=ln𝑃_𝑘()+0,𝑇_𝑡0𝑟_𝑘(𝑠)−^𝑇_𝑣𝛼_𝑘()1𝑣,𝑠𝑑𝑣−2^𝑇_𝑣𝜎²_𝑘(𝑣,𝑠)𝑑𝑣−^𝑇_𝑣ℝ𝛾_𝑘()(−𝑣,𝑠,𝑧𝑑𝑣𝜈𝑑𝑧)𝑑𝑠_{𝑡0𝑇𝑣}𝜎_𝑘()(𝑠)+1𝑣,𝑠𝑑𝑣𝑑𝐵2^𝑡₀^𝑇_𝑣𝜎²_𝑘()−𝑣,𝑠𝑑𝑣𝑑𝑠_{𝑡0𝑇𝑣ℝ}𝛾_𝑘(𝑣,𝑠,𝑧)𝑑𝑣𝑑𝑠𝑁(𝑑𝑠,𝑑𝑧)+^𝑡₀^𝑇_𝑣ℝ(𝑣,𝑠,𝑧)𝑑𝑣𝑑𝑠𝜈(𝑑𝑧). (3.5) Let 𝑎_𝑘(𝑡,𝑇)=−∫^𝑇_𝑡𝜎_𝑘(𝑡,𝑠)𝑑𝑡 and 𝑐_𝑘(𝑡,𝑇,𝑧)=−∫^𝑇_𝑡𝛾_𝑘(𝑡,𝑠,𝑧)𝑑𝑡. Define also 𝑏_𝑘(𝑡,𝑇)=−^𝑇_𝑡𝛼_𝑘()1𝑡,𝑠𝑑𝑡+2𝑎_𝑘(𝑡,𝑇)−_ℝ𝑐_𝑘(𝑡,𝑇,𝑧)𝜈(𝑑𝑧) (3.6) and assume that 𝑐_𝑅(𝑡,𝑇,𝑧)≠−1.
Then from the previous expression we have, ln𝑃_𝑘(𝑡,𝑇)=ln𝑃_𝑘(0,𝑇)+^𝑡₀𝑟_𝑘(𝑠)+𝑏_𝑘(𝑠,𝑇)𝑑𝑠+^𝑡₀𝑎_𝑘()(𝑠)−1𝑣,𝑠𝑑𝐵2^𝑡₀𝑎²_𝑘(𝑣,𝑠)𝑑𝑠+^𝑡_0ℝ𝑐_𝑘(𝑣,𝑠,𝑧)𝑁(𝑑𝑠,𝑑𝑧)𝑑𝑠−^𝑡_0ℝ𝑐_𝑘(𝑣,𝑠,𝑧)𝑑𝑠𝜈(𝑑𝑧),𝑑𝑃_𝑘(𝑡,𝑇)=𝑃_𝑘(𝑡,𝑇)𝑟_𝑘(𝑡)+𝑏_𝑘(𝑡,𝑇)𝑑𝑡+𝑎_𝑘(𝑡,𝑇)𝑑𝐵(𝑡)+_ℝ𝑐_𝑘(𝑡,𝑇,𝑧)𝑁(𝑑𝑡,𝑑𝑧). (3.7)

3.2. The CPI Process

As we said before, the Consumer Price Index (CPI), 𝐶(𝑡), is taken as the ratio of the nominal bond price to the real bond price. By Lévy-Itô formula on 𝐶(𝑡)=𝑃_𝑁(𝑡,𝑇)/𝑃_𝑅(𝑡,𝑇), we have𝑑𝐶(𝑡)=𝐶(𝑡)𝑎(𝑡)𝑑𝑡+𝜎(𝑡)𝑑𝐵(𝑡)+_ℝ𝛾(𝑡,𝑧)𝑁(𝑑𝑡,𝑑𝑧),𝐶(0)=𝑐, (3.8) where𝑎(𝑡)=𝑟_𝑁(𝑡)−𝑟_𝑅(𝑡)+𝑏_𝑁(𝑡,𝑇)−𝑏_𝑅(𝑡,𝑇)−𝑎_𝑅(𝑡,𝑇)𝑎_𝑁(𝑡,𝑇)−𝑎_𝑅()+𝑡,𝑇_|𝑧|<11+𝑐_𝑁(𝑡,𝑇,𝑧)−𝑐_𝑁(𝑡,𝑇,𝑧)−𝑐_𝑅(𝑡,𝑇,𝑧)+11+𝑐_𝑅(𝑡,𝑇,𝑧)1+𝑐_𝑅()𝜈(),𝑡,𝑇,𝑧𝑑𝑧 (3.9)𝜎(𝑡)=𝑎_𝑁(𝑡,𝑇)−𝑎_𝑅()(𝑡)=𝑐𝑡,𝑇,𝛾^𝑁(𝑡,𝑇,𝑧)−𝑐_𝑅(𝑡,𝑇,𝑧)1+𝑐_𝑅(𝑡,𝑇,𝑧). (3.10) Note that the solution of (3.8) is𝐶(𝑡)=𝐶(0)exp^𝑡₀𝑎(𝑠)−12𝜎²(𝑠)𝑑𝑠+^𝑡₀𝜎(𝑠)(𝑠)+𝑑𝐵_{𝑡0|𝑧|<1}()(𝜈()+ln1+𝛾𝑠,𝑧−𝛾𝑠,𝑧)𝑑𝑧𝑑𝑠_𝑡0ℝln1+𝛾(𝑠,𝑧)𝑁(𝑑𝑠,𝑑𝑧),𝐶(0)=𝑐 (3.11) or alternatively𝐶(𝑡)=𝐶(0)exp^𝑡₀𝑎^∗(𝑠)𝑑𝑠+^𝑡₀𝜎(𝑠)(𝑠)+𝑑𝐵_{𝑡0|𝑧|<1}()(𝜈()+ln1+𝛾𝑠,𝑧−𝛾𝑠,𝑧)𝑑𝑧𝑑𝑠_𝑡0ℝln1+𝛾(𝑠,𝑧)𝑁(𝑑𝑠,𝑑𝑧),𝐶(0)=𝑐, (3.12) where 𝑎^∗(𝑡)=𝑎(𝑠)−(1/2)(𝜎^𝑁(𝑡))².

Definition 3.2 (Fisher’s inflation model). The rate of inflation 𝐼(𝑡) at time 𝑡 is the drift process of the CPI, so that 𝐼(𝑡)=𝑟_𝑁(𝑡)−𝑟_𝑅(𝑡)+𝑏_𝑁(𝑡,𝑇)−𝑏_𝑅(𝑡,T)−𝑎_𝑅(𝑡,𝑇)𝑎_𝑁(𝑡,𝑇)−𝑎_𝑅()+𝑡,𝑇_|𝑧|<11+𝑐_𝑁(𝑡,𝑇,𝑧)−𝑐_𝑁(𝑡,𝑇,𝑧)−𝑐_𝑅(𝑡,𝑇,𝑧)+11+𝑐_𝑅(𝑡,𝑇,𝑧)1+𝑐_𝑅()𝜈(),𝑡,𝑇,𝑧𝑑𝑧 (3.13) which is essentially our version of the Fisher equation.

In particular, if the volatility coefficients are zero and there are no jumps in the bond price dynamics, then the Fisher’s equation says that the real interest rate is approximately equal to the nominal rate less inflation rate. High inflation coupled with low nominal rates may result in negative returns and no incentives for depositors to keep their money in the bank. This will also affect pensioners whose future payments in the form of annuities will be eroded by inflation. Keeping inflation low is one of the main tasks of the central bank of any country since hyperinflation brings with itself also economic and social chaos.

In our case, we see that volatility and jumps can further reduce the real rate to far lower than the original rate given by Fisher. It is thus crucial to offer annuities that are inflation adjusted in that case.

Theorem 3.3 (Girsanov Theorem; see [4]). Suppose that there exists a process 𝑢(𝑡) and 𝜃(𝑡,𝑧) which are ℱ_𝑡-adapted and such that 𝑎_𝑘(𝑡,𝑇)𝑢(𝑡)+_ℝ𝑐_𝑘(𝑡,𝑇,𝑧)𝜃(𝑡,𝑧)𝜈(𝑑𝑧)=𝑏_𝑘(𝑡,𝑇), (3.14) and such that the process 𝑍(𝑡)∶=exp−^𝑡₀𝑢(𝑠)(𝑠)−1𝑑𝐵2^𝑡₀𝑢²(𝑠)𝑑𝑠+^𝑡_0ℝ((𝑁()+ln1−𝜃𝑠,𝑧))𝑑𝑠,𝑑𝑧_𝑡0ℝ{ln(1−𝜃(𝑠,𝑧))+𝜃(𝑠,𝑧)}𝜈(𝑑𝑧)𝑑𝑠,0≤𝑡≤𝑇 (3.15) is well defined and satisfies 𝐸[𝑍(𝑇)]=1 where expectation is taken with respect to the original probability measure 𝑃. Define the probability measure 𝑄=𝑄^{(𝑢(𝑡),𝜃(𝑡,𝑧))} on ℱ_𝑇 by 𝑑𝑄(𝜔)=𝑍(𝑇)𝑑𝑃(𝜔). Then (1)𝑃_𝑘(𝑡,𝑇) is a local martingale with respect to 𝑄, (2)the process 𝐵(𝑡) given by 𝑑𝐵(𝑡)=𝑢(𝑡)𝑑𝑡+𝑑𝐵(𝑡) is a 𝑄-Brownian motion, (3)the random measure measure 𝑁^𝑄(𝑑𝑡,𝑑𝑧) given by 𝑁^𝑄(𝑑𝑡,𝑑𝑧)=𝜃(𝑡,𝑧)𝜈(𝑑𝑧)𝑑𝑡+𝑁(𝑑𝑡,𝑑𝑧) is a 𝑄 compensated Poisson random measure in the sense that 𝑀(𝑡)=∫^𝑡₀∫_ℝ𝛾_𝑘(𝑠,𝑧)𝑁^𝑄(𝑑𝑠,𝑑𝑧);0≤𝑡≤𝑇 is a local 𝑄-martingale provided the condition ∫^𝑇₀∫_ℝ𝛾_𝑘(𝑠,𝑧)𝜃(𝑠,𝑧)𝜈(𝑑𝑧)𝑑𝑠<∞ℎ𝑜𝑙𝑑𝑠𝑎.𝑠.For a proof, we just consider the Itô-Lévy formula on 𝑌_𝑘(𝑡)=𝑍(𝑡)𝑃_𝑘(𝑡) and proceed as in ([4]).

We have to observe here that for 𝑘∈{𝑅,𝑁}, (3.14) will result in infinitely many solutions (𝑢(𝑡),𝜃(𝑡,𝑧)) and as such the market of nominal (real) bonds is incomplete. We do have an infinite number of equivalent martingale measures 𝑄. Let ℳ^𝑒(𝑃_𝑘) denote the set of all equivalent martingale measure 𝑄 for 𝑃_𝑘(𝑡).

Observe that, with respect to 𝑄, we have𝑑𝑃_𝑘(𝑡,𝑇)=𝑃_𝑘(𝑡,𝑇)𝑟_𝑘(𝑡)𝑑𝑡+𝑎_𝑘(𝑡,𝑇)𝑑𝐵(𝑡)+_ℝ𝑐_𝑘(𝑡,𝑇,𝑧)𝑁^𝑄(𝑑𝑡,𝑑𝑧). (3.16)

Then with respect to 𝑄, we have 𝐶(𝑡) given by𝑑𝐶(𝑡)=𝐶(𝑡)𝛼(𝑡)𝑑𝑡+𝜎(𝑡)𝑑𝐵(𝑡)+_ℝ𝛾(𝑡,𝑧)𝑁^𝑄(𝑑𝑡,𝑑𝑧),𝐶(0)=𝑐, (3.17) where𝛼(𝑡)=𝑟_𝑁(𝑡)−𝑟_𝑅(𝑡)−𝑎_𝑅(𝑡,𝑇)𝑎_𝑁(𝑡,𝑇)−𝑎_𝑅()+𝑡,𝑇_|𝑧|<11+𝑐_𝑁(𝑡,𝑇,𝑧)−𝑐_𝑁(𝑡,𝑇,𝑧)−𝑐_𝑅(𝑡,𝑇,𝑧)+11+𝑐_𝑅(𝑡,𝑇,𝑧)1+𝑐_𝑅()𝜈()𝑡,𝑇,𝑧𝑑𝑧 (3.18) and as before 𝜎(𝑡)=𝑎_𝑁(𝑡,𝑇)−𝑎_𝑅(𝑡,𝑇) and 𝛾(𝑡)=𝑐_𝑁(𝑡,𝑇,𝑧)−𝑐_𝑅(𝑡,𝑇,𝑧)1+𝑐_𝑅(𝑡,𝑇,𝑧). (3.19) Consequently, the solution of (3.17) is𝐶(𝑡)=𝐶(0)exp^𝑡₀𝛼(𝑠)−12𝜎²(𝑠)𝑑𝑠+^𝑡₀𝜎(𝑠)𝑑𝐵(𝑠)+^𝑡_0|𝑧|<1()(𝜈()+ln1+𝛾𝑠,𝑧−𝛾𝑠,𝑧)𝑑𝑧𝑑𝑠_𝑡0ℝln1+𝛾(𝑠,𝑧)𝑁^𝑄(𝑑𝑠,𝑑𝑧),𝐶(0)=𝑐 (3.20) or alternatively𝐶(𝑡)=𝐶(0)exp^𝑡₀𝛼^∗(𝑠)𝑑𝑠+^𝑡₀𝜎(𝑠)𝑑𝐵(𝑠)+^𝑡_0|𝑧|<1()(𝜈()+ln1+𝛾𝑠,𝑧−𝛾𝑠,𝑧)𝑑𝑧𝑑𝑠_𝑡0ℝln1+𝛾(𝑠,𝑧)𝑁^𝑄(𝑑𝑠,𝑑𝑧),𝐶(0)=𝑐, (3.21) where 𝛼^∗(𝑡)=𝛼(𝑠)−(1/2)(𝜎^𝑁(𝑡))².

4. Pricing an Inflation Adjusted Annuity

Pricing in incomplete markets requires one to choose one of the equivalent martingale measures in ℳ^𝑒(𝑃_𝑘). The most popular measures are minimal martingale measure which reduces the squared variance between the payoff of a contingent claim and the terminal value of a portfolio of stocks and bond chosen to hedge the claim. Alternatively the minimal relative entropy measure has been widely suggested as one of the best measures “closest” in some sense to the original probability 𝑃. In this paper, we will consider the minimal 𝑓^𝑞-martingale measure.

Definition 4.1 ((Minimal 𝑓^𝑞-martingale measure) (see [2])). Let 𝐼=(−∞,0)⋃(1,∞) and let 𝑞∈𝐼 be arbitrary but fixed. For 𝑓^𝑞(𝑧)=𝑧^𝑞, the minimal 𝑓^𝑞-martingale measure (qMMM) 𝑄_𝑞 is the equivalent martingale measure which minimizes the 𝑓^𝑞-divergence 𝑓^𝑞(𝑄⧵𝑃)∶=𝐸_𝑃𝑓^𝑞(𝑍)=𝐸_𝑃𝑍(𝑇)^𝑞. (4.1)

Corollary 4.2 (see [2]). Let ℳ^𝑞∶={𝑄^(𝑢,𝜃)∈ℳ^𝑒(P_𝑘)∣𝑓^𝑞(𝑄⧵𝑃)<∞} and let ℳ={𝑄^{(𝑢(𝑡),𝜃(𝑡,𝑧))}∈ℳ^𝑒(𝑃_𝑘)∣(𝑢(𝑡),𝜃(𝑡,𝑧))𝑡𝑖𝑚𝑒𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡a𝑛𝑑d𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑖𝑠𝑡𝑖𝑐}, then 𝑓^𝑞𝑄_𝑞⧵𝑃=inf_𝑄∈ℳ𝑓^𝑞(𝑄⧵𝑃). (4.2)

The above corollary tells us that 𝑄_𝑞^{(𝑢(𝑡),𝜃(𝑡,𝑧))} consists only of deterministic and time independent pair (𝑢,𝜃) which minimizes the 𝑓^𝑞-divergence.

Theorem 4.3. Let 𝜔₁=∫_ℝ𝜈(𝑑𝑧) and 𝜔₂=∫_ℝ𝑐_𝑘(𝑡,𝑇,𝑧)𝜈(𝑑𝑧). Assume that the Girsanov parameters of Theorem 3.3 are time-independent and constant, that is, assume that 𝑢(𝑡)=𝑢 and 𝜃(𝑡,𝑧)=𝜃. Then the minimal 𝑓^𝑞-martingale measure 𝑄_𝑞 is given by the optimal Girsanov pair (𝑢,𝜃) where 𝜃 is a solution of the equation 𝜔₁𝑎²_𝑘(𝑡,𝑇)(1−𝜃)^𝑞−1−𝜔²₂𝑞−1𝜃+𝑞−1𝑏_𝑘𝜔₂+𝜔₁𝑎²_𝑘(𝑡,𝑇)=0, (4.3) and 𝑢 and 𝜃 are related by 𝑎_𝑘(𝑡,𝑇)𝑢+𝜃_ℝ𝑐_𝑘(𝑡,𝑇,𝑧)𝜈(𝑑𝑧)=𝑏_𝑘(𝑡,𝑇). (4.4)

Proof. First observe that (4.4) is actually (3.14) with 𝑢(𝑡) replaced with 𝑢 and 𝜃(𝑡,𝑧) replaced with 𝜃. Next, we need also the following proposition which can be found as an exercise in [4]:

Proposition 4.4. Suppose that 𝛾(𝑠,𝑧) is deterministic and satisfies some regularity conditions. Then 𝐸_𝑃exp^𝑡_0ℝ𝛾(𝑠,𝑧)𝑁(𝑑𝑠,𝑑𝑧)=exp^𝑡_0ℝ𝑒^{𝛾(𝑠,𝑧)}−1−𝛾(𝑠,𝑧)𝜈(𝑑𝑧)𝑑𝑠. (4.5) The proof is trivial. We now go back to the proof of Theorem 4.3.
Assume that 𝑢(𝑡)=𝑢 and 𝜃(𝑡,𝑧)=𝜃, then due to independence of the Brownian motion and the poisson random measure, one has 𝐸_𝑃𝑍^𝑞(𝑇)=𝐸_𝑃𝑒^{−𝑞𝑢𝑘𝐵(𝑇)}⋅𝐸_𝑃exp^𝑇_0ℝ()𝑁(),−1𝑞ln1−𝜃𝑑𝑠,𝑑𝑧exp2𝑞𝑢²𝑇+^𝑇_0ℝ𝑞{()}𝜈()𝑇ln1−𝜃+𝜃𝑑𝑧𝑑𝑠=exp2𝑢²𝑞𝑞−1+𝑇_ℝ(1−𝜃)^𝑞−1−𝑞𝜃𝜈(𝑑𝑧). (4.6) Therefore 𝑄_𝑞 is found by solving the following deterministic optimization problem, which is a particular case of Problem 𝒫_𝑞 in [2]: minimize𝑘_𝑞()=𝑞𝑢,𝜃𝑞−12𝑢²+_ℝ(1−𝜃)^𝑞−1−𝑞𝜃𝜈(𝑑𝑧),subjectto𝑎_𝑘(𝑡,𝑇)𝑢+𝜃_ℝ𝑐_𝑘(𝑡,𝑇,𝑧)𝜈(𝑑𝑧)=𝑏_𝑘(𝑡,𝑇). (4.7) It was proved in [2] that a general solution exists under certain conditions. We conjecture here that the necessary conditions for a solution exist. We the strive to find the optimal solution explicitly.
Let 𝜔₁=∫_ℝ𝜈(𝑑𝑧) and 𝜔₂=∫_ℝ𝑐_𝑘(𝑡,𝑇,𝑧)𝜈(𝑑𝑧). Thus (𝑢,𝜃) is a solution of minimize𝑒_(𝑢,𝜃)𝑞𝑞−12𝑢²+𝜔₁(1−𝜃)^𝑞−1−𝑞𝜃subjectto𝑎_𝑘(𝑡,𝑇)𝑢+𝜃𝜔₂=𝑏_𝑘(𝑡,𝑇). (4.8) Define the Lagrangian ℒ()=𝑞𝜆,𝑢,𝜃𝑞−12𝑢²+𝜔₁(1−𝜃)^𝑞−1−𝑞𝜃+𝜆𝑎_𝑘(𝑡,𝑇)𝑢+𝜃𝜔₂−𝑏_𝑘(𝑡,𝑇), (4.9) then one gets the following: 𝑎_𝑘(𝑡,𝑇)𝑢+𝜃𝜔₂=𝑏_𝑘(),𝑞𝑡,𝑇𝑞−1𝑢+𝜆𝑎_𝑘()𝜔𝑡,𝑇=0,₁𝑞(1−𝜃)^𝑞−1+𝑞𝜔₁=𝜆𝜔₂. (4.10)

The result comes from solving these equations.

As a special case, we now let 𝐵^𝑞(𝑡) given by 𝑑𝐵^𝑞(𝑡)=𝑢𝑑𝑡+𝑑𝐵(𝑡) denote the 𝑄_𝑞-Brownian motion while 𝑁^𝑄𝑞(𝑑𝑡,𝑑𝑧) given by 𝑁^𝑄𝑞(𝑑𝑡,𝑑𝑧)=𝜃𝜈(𝑑𝑧)𝑑𝑡+𝑁(𝑑𝑡,𝑑𝑧) denote the 𝑄_𝑞 compensated Poisson random measure.

In that case, with respect to 𝑄_𝑞, we have𝑑𝐶(𝑡)=𝐶(𝑡)𝛼(𝑡)𝑑𝐵^𝑞(𝑡)+_ℝ𝛾(𝑡,𝑧)𝑁^𝑄𝑄(𝑑𝑡,𝑑𝑧),𝐶(0)=𝑐. (4.11) An inflation protected annuity pays if inflation (as measured by the CPI) exceeds a certain threshold 𝐾. The payout at any time 𝑡 will be given by a payout function 𝜌(𝑡) given by𝜌(𝑡)𝐶(𝑡)=𝑓𝑁1+max𝐶(0)−1−𝐾,0, (4.12) where 𝑓 is the nominal annuity and 𝑁 is the notional. Assume that this payment is for a fixed period [0,𝑇] where 𝑇 is a predetermined limiting age for the policy holder. In most cases 𝑇 is a random variable with a given distribution. For nonlife policies, this may not be necessary. Assume further that the nominal interest rate 𝑟_𝑁 is constant over time. Then, the premium to be paid for such an annuity at any time 𝑡 is given by𝜋₀(𝑐)=𝐸_𝑄𝑞^𝑇₀𝜌(𝑡)𝑒^{−𝑟𝑁(𝑇−𝑡)}𝑑𝑡. (4.13) In order to simplify this value, we may want to consider the fact that 𝑓 and 𝑁 are constant (or at best time dependent) and as such we are interested in the value 𝜋^∗₀(𝑐) given by𝜋^∗₀(𝑐)=𝐸_𝑄𝑞^𝑇₀𝜌^∗(𝑡)𝑒^{−𝑟𝑁(𝑇−𝑡)}𝑑𝑡, (4.14) where 𝜌^∗(𝑡)=max{(𝐶(𝑡)/𝐶(0)−1)−𝐾,0}. The following result follows

Theorem 4.5. Let the CPI be given by (3.8) and 𝜌 given by (4.12).
Let Λ(𝑥)=max{(𝑥/𝑐−1)−𝐾,0} for some constant 𝑐.(a)Suppose that𝛾(𝑡,𝑧)=0. (4.15) Then the price of an inflation protected annuity is 𝜋₀(𝑐)=𝑓𝑁𝑟_𝑁1−𝑒^{−𝑟𝑁𝑇}+𝑓𝑁⋅𝑣(𝑡,𝐶(𝑡)), (4.16) where 𝑣()=𝑥𝑡,𝑥𝑐exp−𝑟_𝑁(𝑇−𝑡)+^𝑇_𝑡𝛼^∗(𝑠)+12𝜎²(𝑠)𝑑𝑠𝑁𝑑₁−(1+𝐾)𝑒^{−𝑟𝑁(𝑇−𝑡)}𝑁𝑑₂,𝑑₁=ln(𝑥/𝑐(1+𝐾))+∫^𝑇_𝑡𝛼^∗(𝑠)+𝜎²(𝑠)𝑑𝑠∫^𝑇_𝑡𝜎²(𝑠)𝑑𝑠^1/2,𝑑₂=𝑑₁−^𝑇_𝑡𝜎²(𝑠)𝑑𝑠^1/2=ln(𝑥/𝑐(1+𝐾))+∫^𝑇_𝑡𝛼^∗(𝑠)𝑑𝑠∫^𝑇_𝑡𝜎²(𝑠)𝑑𝑠^1/2. (4.17)(b)If on the other hand𝛾()𝛼(𝑡)>1𝑡,𝑧≠0,2𝜎²(𝑡)+1𝑇ln𝑥 (4.18) then 𝜋₀(𝑐)=𝑓𝑁𝑟_𝑁1−𝑒^{−𝑟𝑁𝑇}+𝑓𝑁^𝑇₀𝐴𝑥^𝑛∗𝑒^{−𝑡𝜀∗(𝑛∗)}𝑑𝑡, (4.19) where, 𝐴 is a constant, 𝜀^∗(𝑛)(𝑡)+1=𝑛𝛼2𝜎²(𝑡)𝑛(𝑛−1)−𝑟_𝑁+_|𝑧|<1^𝑛_𝑘=2𝑛𝑘𝛾^𝑘(𝑡,𝑧)𝜈(𝑑𝑧), (4.20) and 𝑛^∗ is a solution of the equation 12𝜎²(𝑡)𝑛²(𝑡)1+𝑛𝑇𝛼−ln𝑥−𝑇2𝜎²(𝑡)+𝑇_|𝑧|<1^𝑛_𝑘=2𝑛𝑘𝛾^𝑘()𝜈()Λ(𝑥)𝑡,𝑧𝑑𝑧+ln𝑓(0)−𝑟_𝑁=0. (4.21)

Note that the second expression of (4.16) and (4.19) represents the “penalty” charged to the investor for protection against inflation. We also assume that payments are not linked to an inflation index with a time lag, which is a common scenario in real life. We also should point out here that the assumption on part (a) of the theorem that 𝛾(𝑡,𝑧)=0 will not result in a unique equivalent martingale measure because (3.14) depends on 𝑐_𝑘(𝑡,𝑇,𝑧) and 𝛾(𝑡,𝑧)=0⇏𝑐_𝑘(𝑡,𝑇,𝑧). However, 𝐶(𝑡) given by (4.11) will be a 𝑄_𝑞-geometric Brownian motion.

Proof. (a) By Fubini’s Theorem on expectations, it is enough to consider 𝑣(𝑡,𝐶(𝑡))=𝐸_𝑄𝑞𝑒^{−𝑟𝑁(𝑇−𝑡)}𝐶(𝑡)𝐶(0)−1−𝐾⁺∣ℱ_𝑡. (4.22) Note that 𝑣(0,𝐶(𝑡))=𝑣(𝑡,0)=0 for all 𝑡∈[0,𝑇] and 𝑣((𝑇=𝐶(𝑇)𝑇,𝐶))𝐶(0)−1−𝐾⁺. (4.23) If 𝛾(𝑡,𝑧)=0, then 𝑣𝑡,𝐶(𝑡)+𝐶(𝑡)𝛾(𝑡,𝑧)=𝑣(𝑡,𝐶(𝑡)). (4.24) Now with respect to 𝑄_𝑞, we have 𝐶(𝑇)=𝑐(𝑡)exp^𝑇_𝑡𝛼^∗𝑑𝑠+^𝑇₀𝜎(𝑠)𝑑𝐵^𝑞(𝑠), (4.25) Therefore, by direct integration or otherwise, (4.22) easily simplifies to 𝑣((𝑡=𝐶(𝑡)𝑡,𝐶))𝑐exp−𝑟_𝑁(𝑇−𝑡)+^𝑇_𝑡𝛼^∗(𝑠)+12𝜎²(𝑠)𝑑𝑠𝑁𝑑₁−(1+𝐾)𝑒^{−𝑟𝑁(𝑇−𝑡)}𝑁𝑑₂ (4.26) with 𝑑₁=ln(𝐶(𝑡)/𝐶(0)(1+𝐾))+∫^𝑇_𝑡𝛼^∗(𝑠)+𝜎²(𝑠)𝑑𝑠∫^𝑇_𝑡𝜎²(𝑠)𝑑𝑠^1/2,𝑑₂=𝑑₁−^𝑇_𝑡𝜎²(𝑠)𝑑𝑠^1/2=ln(𝐶(𝑡)/𝐶(0)(1+𝐾))+∫^𝑇_𝑡𝛼^∗(𝑠)𝑑𝑠∫^𝑇_𝑡𝜎²(𝑠)𝑑𝑠^1/2. (4.27)(b)If 𝛾(𝑡,𝑧)≠0, consider the discounted value function 𝑌(𝑡)=𝑒^{−𝑟𝑁𝑡}𝑣(𝑡,𝐶(𝑡)), then by the one-dimensional Lévy-Itô formula, we have𝑑𝑌(𝑡)=𝑒^−𝑟𝑁𝑣_𝑡(𝑡,𝐶(𝑡))+𝛼𝐶(𝑡)𝑣_𝑐((𝑡((𝑡+1𝑡,𝐶))−𝑟𝑣𝑡,𝐶))2𝜎²(𝑡)𝑐²(𝑡)𝑣_𝑐𝑐((𝑡+𝑡,𝐶))_|𝑧|<1𝑣𝑡,𝐶(𝑡)1+𝛾(𝑡,𝑧)−𝑣(𝑡,𝐶(𝑡))−𝐶(𝑡)𝛾(𝑡,𝑧)𝑣_𝑐((𝑡𝜈(+𝑡,𝐶))𝑑𝑧)𝑑𝑡_ℝ𝑣𝑡,𝐶(𝑡)1+𝛾(𝑡,𝑧)𝑁^𝑄𝑞(𝑑𝑡,𝑑𝑧)+𝜎𝐶(𝑡)𝑣_𝑐(𝑡,𝐶(𝑡))𝑑𝐵^𝑞(𝑡). (4.28) The discounted value function is a 𝑄_𝑞-martingale if the 𝑑𝑡 term equals zero, or alternatively, if 𝑣_𝑡(𝑡,𝑥)+𝛼𝑥𝑣_𝑥()+1𝑡,𝑥2𝜎²(𝑡)𝑥²𝑣_𝑥𝑥()+𝑡,𝑥_|𝑧|<1𝑣𝑡,𝑥1+𝛾(𝑡,𝑧)−𝑣(𝑡,𝑥)−𝑥𝛾(𝑡,𝑧)𝑣_𝑥(𝑡,𝑥)𝜈(𝑑𝑧)=𝑟_𝑁𝑣(𝑡,𝑥). (4.29) Then 𝑣(𝑡,𝑥) is a solution of the following boundary value problem: 𝑣_𝑡(𝑡,𝑥)+𝛼𝑥𝑣_𝑥()+1𝑡,𝑥2𝜎²(𝑡)𝑥²𝑣_𝑥𝑥()+𝑡,𝑥_|𝑧|<1𝑣𝑡,𝑥1+𝛾(𝑡,𝑧)−𝑣(𝑡,𝑥)−𝑥𝛾(𝑡,𝑧)𝑣_𝑥(𝑡,𝑥)𝜈(𝑑𝑧)=𝑟_𝑁𝑣(),𝑣()𝑣()𝑥𝑡,𝑥𝑇,0=0,𝑇,𝑥=max𝑐,𝑣()−1−𝐾,00,𝑥=0, (4.30) where 𝑥=𝐶(𝑡).
We assume a solution of the form 𝑣(𝑡,𝑥)=𝑓(𝑡)𝑥^𝑛 then by substitution into 𝑣_𝑡(𝑡,𝑥)+𝛼𝑥𝑣_𝑥()+1𝑡,𝑥2𝜎²(𝑡)𝑥²𝑣_𝑥𝑥()+𝑡,𝑥_|𝑧|<1𝑣𝑡,𝑥1+𝛾(𝑡,𝑧)−𝑣(𝑡,𝑥)−𝑥𝛾(𝑡,𝑧)𝑣_𝑥(𝑡,𝑥)𝜈(𝑑𝑧)=𝑟_𝑁𝑣(𝑡,𝑥) (4.31) we get 𝑓^′(𝑡)+𝜀^∗(𝑛)𝑓(𝑡)=0 where 𝜀^∗(𝑛)(𝑡)+1=𝑛𝛼2𝜎²(𝑡)𝑛(𝑛−1)−𝑟_𝑁+_|𝑧|<1^𝑛_𝑘=2𝑛𝑘𝛾^𝑘(𝑡,𝑧)𝜈(𝑑𝑧). (4.32) Therefore 𝑣(𝑡,𝑥)=𝑓(0)𝑥^𝑛𝑒^{−𝑡𝜀∗(𝑛)} and if we let Λ(𝑥)=𝑓(0)𝑥^𝑛𝑒^{−𝑇𝜀∗(𝑛)}, then the terminal conditions imply 12𝜎²(𝑡)𝑛²(𝑡)1+𝑛𝑇𝛼−ln𝑥−𝑇2𝜎²(𝑡)+𝑇_|𝑧|<1^𝑛_𝑘=2𝑛𝑘𝛾^𝑘()𝜈()Λ(𝑥)𝑡,𝑧𝑑𝑧+ln𝑓(0)−𝑟_𝑁=0. (4.33) Suppose that 𝑇𝑎(𝑡)−ln𝑥−𝑇(1/2)𝜎²(𝑡)>0, then there exists a unique solution 𝑛^∗ to this equation.
Then 𝑣(𝑡,𝑥)=𝑓(0)𝑥^𝑛∗𝑒^{−𝑡𝜀∗(𝑛∗)}.

Remark 4.6. Note that if 𝛼(𝑡)=𝛼 and 𝜎(𝑡)=𝜎 are constants, then one can explicitly simplify the integral of Theorem 4.5 since for any 𝑦>𝑛, then 𝜀(𝑦)>0 and ∫^𝑇₀𝑣(𝑡,𝑥)=𝐴𝑥^𝑛/𝜀(𝑛)(1−𝑒^{−𝑇𝜀(𝑛)}) which increases with term to maturity. Therefore, inflation adjusted annuities of this nature have capital losses in that the price charged at issue or purchase is more than redemption value. This applies to the case of course where we consider the price as a function of maturity time 𝑇.

5. Conclusions

Indexation helps to adjust future earnings in line with the cost of living at that time. Usually the income is indexed by reference to the values of inflation with some time lag. This is done to allow for publishing the RPI values. In that case proper adjustment must be made to the premium. In this paper we have managed to get explicit results to the price of inflation indexed annuities without time lag. We looked at non-life-based annuities and conjecture that there is not much value added by considering life based annuities.

Acknowledgments

The author would like to thank staff in the School of Management Studies at UCT for useful comments during a presentation of this paper at its draft stages. Last but not least, the author is grateful to the reviewers for their useful comments to the first draft of the paper. This paper was funded by the University of Cape Town Research Grant 461091.