# Consistent Valuation Across Curves Using Pricing Kernels

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## Abstract

**:**

## 1. Introduction

## 2. Across-Curve Pricing Formula

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

**Corollary**

**1.**

**Proof.**

**Remark**

**2.**

## 3. Pricing Kernel Approach to Multi-Curve Systems

#### 3.1. Discounting Systems in Emerging Markets

#### 3.2. Discounting Systems in Developed Markets

**Proposition**

**2.**

**Proof.**

**Lemma**

**1.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Definition**

**2.**

**Corollary**

**2.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

**Corollary**

**3.**

**Proof.**

**Remark**

**5.**

- (i)
- The quantities ${\overline{L}}_{t}^{xy}({T}_{i-1},{T}_{i})$ and ${\overline{K}}^{x}$ that determine the FRA’s floating and fixed cash flows are derived from the curve-converted quantities ${v}_{t}^{xy}({T}_{i-1},{T}_{i})$ and ${v}_{K}^{x}$, respectively. This is in contrast with ${L}_{t}^{xy}({T}_{i-1},{T}_{i})$ and ${K}^{x}$, the directly comparable curve-converted quantities used in the xy-framework. Therefore, these derived quantities are no longer consistent with a currency modelling analogy, with each differing from the correctly converted quantities by an additive factor of $({Q}_{t{T}_{i}}^{xy}-1)/{\delta}_{i}$.
- (ii)
- Observation (i) is further supported by Equation (41) which shows that the conversion factor process effectively models the spread between the multi-curve y-tenored FCF and the corresponding x-tenored FCF, as opposed to the classical forward exchange rate. Moreover, the derived y-market system has almost no relation to the developed market y-tenored interest rate system, that one seeks to model, since the model derived y-market system dominates the x-market system, i.e., ${P}_{tT}^{x}\le {P}_{tT}^{y}$ for $0\le t\le T$.

**Remark**

**6.**

**Remark**

**7.**

**Theorem**

**2.**

**Proof.**

**Remark**

**8.**

**Corollary**

**4.**

**Proof.**

#### 3.3. Consistent Multi-Curve Discounting in Emerging Markets

**Remark**

**9.**

**Definition**

**3.**

**Definition**

**4.**

- (i)
- the x-curve with varying notional defined by the forward conversion factor ${Q}_{tT}^{xy}$; or
- (ii)
- the y-curve with varying notional defined by the spot conversion factor ${Q}_{tt}^{xy}$.

**Remark**

**10.**

## 4. xy-HJM Multi-Curve Models

**Definition**

**5.**

**Definition**

**6.**

**Proposition**

**3.**

**Proof.**

- (i)
- ${\overline{L}}_{t}^{xy}({T}_{i-1},{T}_{i})$ is specified in an ad hoc fashion (usually) inspired by the LIBOR market model (LMM) such that this approach is referred to as a hybrid HJM-LMM;
- (ii)
- ${\overline{L}}_{t}^{xy}({T}_{i-1},{T}_{i}):=({v}_{t}^{y}({T}_{i-1},{T}_{i})-1)/{\delta}_{i}$ where, as before, ${v}_{t}^{y}({T}_{i-1},{T}_{i}):={P}_{t{T}_{i-1}}^{y}/{P}_{t{T}_{i}}^{y}$ defines the FCF such that under certain parameter restrictions (see Proposition 4 below) ${({h}_{t}^{x}{P}_{t{T}_{i}}^{x}{v}_{t}^{y}({T}_{i-1},{T}_{i}))}_{0\le t\le {T}_{i-1}}$ is a $\mathbb{P}$-(local) martingale; and
- (iii)
- ${\overline{L}}_{t}^{xy}({T}_{i-1},{T}_{i}):=\mathbb{E}[{h}_{{T}_{i}}^{x}{L}_{{T}_{i-1}}^{y}({T}_{i-1},{T}_{i})\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\mathcal{F}}_{t}]/({h}_{t}^{x}{P}_{t{T}_{i}}^{x})$ assuming the classical HJM drift condition for the y-market ZCB system.

**Proposition**

**4.**

**Proof.**

**Remark**

**11.**

**Proposition**

**5.**

**Proof.**

**Remark**

**12.**

**Definition**

**7.**

**Remark**

**13.**

## 5. Rational Multi-Curve Models

#### 5.1. Hybrid Rational-LMM Multi-Curve Models

**Proposition**

**6.**

**Proof.**

**Remark**

**14.**

**Proposition**

**7.**

**Proof.**

**Remark**

**15.**

#### 5.2. Pure-Rational Multi-Curve Models

#### 5.3. Linear-Rational Term Structure Models

**Definition**

**8.**

**Proposition**

**8.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Remark**

**16.**

**Definition**

**9.**

**Theorem**

**4.**

- 1.
- Let $\left({X}_{t}\right)$ be the Markov process $\left({Z}_{t}\right)$ that satisfies (94).
- 2.
- $F(t,{X}_{t})={Z}_{t}$, for all $t\ge 0$.
- 3.
- $w(t,u)={\mathrm{e}}^{-\beta (t+u)}$, $\beta \in {\mathbb{R}}^{m\times m}$ invertible where $\beta \kappa =\kappa \beta $ for $\kappa \in {\mathbb{R}}^{m\times m}$ invertible.
- 4.
- The functions ${f}_{0}\left(t\right)$ and ${f}_{1}\left(t\right)$ are given by:$$\begin{array}{ccc}\hfill {f}_{0}\left(t\right)& =& {f}_{1}\left(t\right)\left[{(\beta +\kappa )}^{-1}-{\beta}^{-1}\right]{\mathrm{e}}^{-\beta t}\theta +{\mathrm{e}}^{-\alpha t}\varphi ,\hfill \end{array}$$$$\begin{array}{ccc}\hfill {f}_{1}\left(t\right)& =& {\mathrm{e}}^{-\alpha t}\psi {\mathrm{e}}^{\beta t}(\beta +\kappa ),\hfill \end{array}$$

**Proof.**

## 6. Pricing of Inflation-Linked, FX and Hybrid Securities

#### 6.1. Inflation-Linked Pricing

#### 6.2. Exchange to Foreign Currencies

**Proposition**

**9.**

**Proof.**

#### 6.3. Multi-Curve Interest Rate Foreign-Exchange Hybrid

**Proposition**

**10.**