The next stage (step 5 in the previous section) is to adjust the probabilities to build in the correlation between the OIS rate and the spread (ie, the correlation between dzr . and dzs ). The probabilities at the edge of the table are the branching probabilities at node (2,−2) of the ther tree and (2,2) of the s-tree. b) Fixed branching probabilities at the node (2,−2,2).

## 5 Valuation of a Spread Option

Table 8 shows how the price of the spread option is affected by the assumed correlation and volatility of the spread. As can be expected, the price of the spread option is highly sensitive to spread volatility.

## 6 Bermudan Swap Option

Comparing Tables 8 and 9, we see that the correlation between the OIS rate and the spread has a much larger effect on the valuation of a Bermudan swap option than on the valuation of a spread option. This is not the case for a Bermudan swap option because the profit depends on the LIBOR rate, which depends on the OIS rate as well as the spread.

## 7 Conclusions

We find that a correlation of approx. -0.1 between the one-month OIS and the 12-month LIBOR OIS spread is indicated by the data.20. In this case, even when the correlation between spread rate and OIS rate is relatively small, a stochastic spread can change the price by 5-10.

The model can then be used in the same way that two-dimensional tree models were used for LIBOR before the crisis. Images or other third-party material in this chapter are licensed under a Creative Commons Work License unless otherwise noted in the credit line; if such material is not covered by a Creative Commons license for the work and such action is not permitted by law, users will need to obtain permission from the licensee to duplicate, adapt or reproduce the material.

Derivative Pricing for a Multi-curve

Extension of the Gaussian, Exponentially Quadratic Short Rate Model

## 1 Introduction

In section 4 we discuss the pricing of linear interest rate derivatives and finally in section 5 the non-linear/optional interest rate derivatives.

## 2 Preliminaries

### Discount Curve and Collateralization

Regarding the explicit role of collateral in the pricing of interest rate derivatives, we refer to the above paper by Pallavicini and Brigo [28].

### Martingale Measures

Normally, the floating rate is given by the Libor rate, and since we assume arrears, we also assume that the rate is fixed at the beginning of the interest rate interval, here at T. It follows from this expression that the value of the fixed interest that makes the contract fair on time is given by.

## 3 Short Rate Model 3.1 The Model

### Bond Prices (OIS and Libor Bonds)

Integrating these ODEs with respect to the second variable and recalling (16), we get (for details, see the proof of statement 2.1 in [17]). Then, always by analogy with the proof of Statement 2.1 in [17], we can see that the coefficient vectors B(t,T) and B(t,¯ T) of the first-order terms satisfy the following system.

### Forward Measure

Based on this we will derive in the next section the announced adjustment factor that allows the transition from the pre-crisis quantities to the corresponding post-crisis quantities. We recall that the expectation regarding the measure QT+Δ is denoted by ET+Δ{·}. 2b2)2+8(σ2)2, and where we assumed initial deterministic values Ψ01, Ψ02 and Ψ03. For details of the above calculation, see the proof of Corollary 4.1.3.

## 4 Pricing of Linear Interest Rate Derivatives

### FRAs

*Adjustment Factor We shall show here the following*

Using the above expression for the relationship between the OIS and Libor bond rates and taking into account the definition of the short rates in relation to the factor processes, we get. The parameters in the model (10) for the factors Ψti and thus also in the model (11) for the short rates and the spread st are the coefficients bi andσi for i=1,2,3. From (14) note that for i =1,2, these coefficients enter the expressions for the OIS bond prices sp(t,T) that can be assumed to be observable since they can be bootstrapped from the market quotations of the OIS swap rates.

### Interest Rate Swaps

The expectations in (53) must be calculated under the measuresQTk, under which, analogously to (33), the factors have the dynamics. Now coming to the second expectation in the second line of (53) and using the second equation in (54), we set

## 5 Nonlinear/optional Interest Rate Derivatives

### Caps and Floors

Thus, the payoff of the caplet at time T +Δ is Δ(L(T;T,T +Δ)−R)+, assuming the notional N=1, and its time-t price PCpl(t;T +Δ ,R) is given by the following risk-neutral pricing formula under the forward measure QT+Δ. By setting R˜ :=1+ΔR, and recalling the first equality in (30), the time-0 price of the caplet can be expressed as.

### Swaptions

As for the caps, here too we consider the Gaussian joint distributionf(ΨT1. 0,ΨT20,ΨT30)(x,y,z) of the factors under the measurement T0−before QT0 and we have. Therefore, the square root of the last expression in the various proposition statement formulas is well defined.

Multi-curve Construction

## Definition, Calibration, Implementation and Application of Rate Curves

Although this approach is generally not supported by an economic concept, it also introduces several (self-made) problems, e.g. interpolation of (so-called) overlapping instruments, see section 5.3. While forward linear interpolation is a common scheme,1 forward value interpolation seems to be a new approach.

## 2 Foundations, Assumptions, Notation

In this case, the linear interpolation of the discount curve and the forward value curve would not introduce an arbitrage violation. The symbol V refers to the value of the product in question, while N indicates the numerical, for example the collateral account built up by OIS.

## 3 Discount Curves

Since the valuation formulas are identical to the case of a "special" collateral account (corresponding to the funding account), we will consider an uninterrupted product as a product with a different collateralization. For example X will indicate the fixed rate in an exchange, I will indicate the floating rate index in an exchange, Indicates the currency of the two legs, N will be used to define the discount factor and the value of the exchange.

## 4 Forward Curves

### Performance Index of a Discount Curve (or “Self-Discounting”)

LetIC(Ti) := 1−PPU,CU,C(T(iT+i+d;dT;iT)i), where PU,C(Ti+d;Ti) is the discount factor for maturity Ti+das, visible in timeTi. Conversely, we can define the discount curve from the forward curve "implicitly" so that relation (7) holds.

## 5 Interpolation of Curves

*Implementing the Interpolation of a Curve: Interpolation Method and Interpolation Entities**Interpolation Time**Interpolation of Forward Curves**The Classical Approach**Alternative Interpolation Schemes for Forward Curves**Assessment of the Interpolation Method*

Given 0

## 6 Implementation of the Calibration of Curves

*Generalized Definition of a Swap**Calibration of Discount Curve to Swap Paying the Collateral Rate (aka. Self-Discounted Swaps)**Calibration of Forward Curves**Calibration of Discount Curves When Payment and Collateral Currency Differ**Fixed Payment in Other Currency**Float Payment in Other Currency**Lack of Calibration Instruments (for Difference in Collateralization)**Implementation*

The value of the swap leg can be expressed in terms and discount factors as. The value of the notional exchange swap leg can be expressed in terms of forward and discount factors as.

## 7 Redefining Forward Rate Market Models

Another option is to start with a stochastic model for forward rates, where now the forward curve defines the initial value of the SDE model, and then define the discount curve (numéraire) via deterministic or stochastic spreads. An implementation of the standard LMM with deterministic adjustment for the discount curve is provided by the author at [9].

## 8 Some Numerical Results

### Impact of the Interpolation Entity of a Forward Curve on the Delta Hedge

This model is given by joint modeling of processesLi(t):= FU,CT(i+1Ti,−TTi+i1;t), e.g. as log-normal processes under the QNC measure and the additional assumption that the processPC( Ti;t) is deterministic in its short period t∈(Ti−1,Ti). We have described how to use the standard LIBOR market model as a term structure model for the account NC insurance (eg OIS curve).

### Impact of the Lack of Calibration Instruments for the Case of a Foreign Swap Collateralized

1 Forward curve (USD-3M) calibrated from swaps with different collateralization (USD-OIS and EUR-OIS) assuming that the market rates are independent of the type of collateralization.

Impact of the Interpolation Scheme on the Hedge Efficiency

## 9 Conclusion

Open Access This chapter is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, so as long as you give appropriate credit to the original author(s) and the source, a link to the Creative Commons license is provided and any changes noted. Fries, C.P.: Funded Replication: Fund Exchange Process and the Valuation with Different Funding Accounts (Cross Currency Analogy to Funding Revisited).

## Impact of Multiple-Curve Dynamics in Credit Valuation Adjustments

This general prescription is what guides us in this paper, where we attempt to adapt valuation interest rate models to the current environment. Here we focus our updated valuation framework on the following key points: (i) focus on interest rate derivatives;. ii) understand how an updated valuation framework can be useful in defining the key market rates underlying the multiple yield curves that characterize current interest rate markets; (iii) definition of secured valuation criteria; (iv) form consistent realistic dynamics for the rates resulting from the above analysis; (v) show how the framework can be applied to the valuation of particularly sensitive products such as underlying credit default swaps and collateral postings; (vi) highlight limitations in some current market practices, such as interpreting multiple curves with deterministic factors or shifts. where the option built into the Credit Valuation Adjustment (CVA) calculation would be priced without any volatility.

## 2 Valuation Equation with Credit and Collateral

### Valuation Framework

Cuis the collateral process, and we use the convention that Cu>0 whileI is the collateral receiver and Cu<0 whenI is the collateral poster.(ru−cu)Keep the collateral margin cost and the collateral rate is defined asct :=c+ t 1{Ct >0}+ c−t 1{Ct<0} withc± defined in the CSA contract. Note that the above valuation equation (2) is not suitable for explicit numerical evaluations, as the right-hand side still depends on the derived price via the indicators within the collateral rates and possibly via the closing term, leading to recursive/non-linear characteristics .

### The Master Equation Under Change of Filtration

In order to derive an explicit valuation formula, we further assume that the gap risk is not present, namely V˜τ−= ˜Vτ, and we consider a special form of collateral and closing prices, namely that we model the closing value as. This means that the close is the risk-free mark to market at the time of first default and the collateral is a fraction of the closing value.

## 3 Valuing Collateralized Interest-Rate Derivatives

### Overnight Rates and OIS

Such contracts exchange a fixed payment portion with a variable portion, paying a discrete compound interest based on the same overnight interest rate used for their collateral. Furthermore, we can introduce the (nominal) fixed rates K = Et(T,x;e) that make the OIS contract for one period fair, namely priced at time 0.

### LIBOR Rates, IRS and Basis Swaps

Thus, for any payoff φT, perfectly hedged at the overnight rate et, we can express prices as expectations under the hedged T-forward target, and in particular we can write LIBOR forward rates as. Once covered zero-coupon bonds are derived, we can start up forward yield curves from such rates.

### Modeling Constraints

In this sense, we can define the instantaneous term interest rate(T;e) starting from (collateralized) zero-coupon bonds, as given by. We can derive the instantaneous forward interest rate dynamics from Itô's lemma, and we obtain the following dynamics under the QT measure.

## 4 Interest-Rate Modeling

### Multiple-Curve Collateralized HJM Framework

In the generalized version of the HJM framework proposed by [23], we have an explicit expression for both the secured zero-coupon bonds Pt(T;e) and the LIBOR forward rates Ft(T,x;e). In particular, we can obtain a variant of the Ch model ([10]), considering a common square root process for all entries of h, as in [29].

### Numerical Results

To better appreciate the difference between the Ch model and the MP model, the quantity can be calculated. In fact we can see that the Ch model and the MP model are almost indistinguishable while the results of the HW model are different from those of stochastic volatility.

## A Generalized Intensity-Based Framework for Single-Name Credit Risk

Our goal is to start with even weaker assumptions about the standard time and allow for jumps in the standard time compensator at deterministic times. The structure of this article is as follows: In Sect.2, we introduce the general framework and study conditions in an extended HJM framework, which guarantees the absence of arbitrage in the bond market.

## 2 A General Account on Credit Risky Bond Markets

### The Generalized Intensity-Based Framework

To minimize the technical difficulties that arise, we assume that there is an increasing process. However, one can use the right-continuous extension and we refer to [15] for a precise treatment and for a guide to the related literature.

### An Extension of the HJM Approach

In a stylized form, the Merton model can be represented by a Brownian motion W indicating the normalized logarithm of the firm's assets, a constant K >0 and the standard time. The reduction of the fixed value at U corresponds to considering a standard limit with an upward jump at that time.

## 3 Affine Models in the Generalized Intensity-Based Framework

The idea is to consider an affine processX and study arbitrage-free double random term structure models where the compensator Λof the standard indicator processH=1{ ≤τ} is given by . Then the continuous unique strong solution of the stochastic differential equation d Xt=μ(Xt)dt+σ(Xt)d Wt, X0=x, (22) is an affine process X on the state space X, see Chap.

## 4 Conclusion

ISDA: Credit Derivatives Designation Committee ISDA Americas: Credit Default Event in the Argentine Republic. -pay-credit-event(2014).

A major advantage of the forward process approach is that it is invariant under the criterion change in the sense that the driving process remains a time-inhomogeneous Lévy process. Another important aspect is that in the latter model the increments of the management process translate directly into increments of the LIBOR rates.

## 2 The Lévy Forward Process Model

We begin to construct the forward process with the most distant maturity and postulate. The driving termTi∗−1 is chosen such that the forward processF(·,Ti∗,Ti∗−1) becomes a martingale under the forward measurePTi−1∗, that is.

## 3 Fourier-Based Methods for Option Pricing

The forward LIBOR ratesL(Ti∗,Ti∗) are the discretely compounded, annualized interest rates that can be earned from investment in a future interval starting at Ti∗ and ending at Ti∗−1 considered at time Ti∗. Applying Theorem 2.2 in Eberlein et al. and the torque generating functionMXT∗. Changing measures, we find MXT∗. 51) Taking into account the choice of the drift coefficient in (19), the cumulant function θs (see (9)) and the torque generating function MXT∗.

## 4 Sensitivity Analysis

### Greeks Computed by the Malliavin Approach

*Variation in the Initial Forward Price*

The following proposal provides a simpler expression for the Malliavin derivative operator Dr,0 when applied to forward process rates F(t,Ti∗,Ti∗−1) (see Di Nunno et al. In this section we give an expression for delta, the partial derivative of the expectationCplt0 (Ti∗,K) given the initial conditionF(0,Ti∗,Ti∗−1) given by.

Greeks Computed by the Fourier-Based Valuation Method

### Examples

*Variance Gamma Process (VG)**Inhomogeneous Gamma Process (IGP)*

We assume that the jump component of the driving process LT∗ is described by the inhomogeneous gamma process (IGP) introduced by Berman as follows. Assume that a function of the form A is differentiable, so we can write A(t)=A(0)+. for allt ∈R+where A˙denotes the derivation of A.

## A Appendix

*Isonormal Lévy Process (ILP)**The Derivative Operator**Integration by Parts Formula**The Chain Rule**Regularity of Solutions of SDEs Driven by Time-Inhomogeneous Lévy Processes*

Definition A.2 The stochastic derivative of a smooth random variable of the form (100) is the H-value random variable Dξ = {Dt,xξ, (t,x)∈T ×X} given by . 101) We will consider Dξ as an element of L2(T ×X×Ω)∼=L2(Ω;H); namely Dξ is a random process indexed by the spaceT ×X parameter. The expression of the derivative Dξ given in (101) does not depend on the specific representation of ξ in (100).

## Inside the EMs Risky Spreads and CDS-Sovereign Bonds Basis

This serves as a basis and gives a financial engineering intuition about the nature of the problem. It provides basic building blocks for relative value trades under the presence of the local currency yield curve which can serve as an additional pillar.

## 2 Local Currency Bonds No-Arbitrage HJM Setting

### Risky Bonds Under Marked Point Process

On the other hand, we have the conditional distribution Ft(ω;d x) of the markers X in case of a jump realization. Depending on how we specify the convention for utilization, we can get further simplification of the formulas.

### Model Formulation

*General Notes**Multi-currency Risky Bonds Model*

Namely, it is a function of: (1) the probability that the credit event will occur and the need for monetization; (2) the negative side effect of the credit event on the exchange rate through a sudden depreciation of the latter; (3) the volatility of the exchange rate; (4) the expected depreciation of the exchange rate without considering the monetization; (5) the risk aversion of market participants towards the credit event and the need for monetization, the sudden exchange rate depreciation and its magnitude; (6) the risk aversion of the market participants towards the volatility of the exchange rate. Monetization The analysis so far has considered a loss of 1−Rd,LC(T) on default of the domestic debt.

## 3 CDS-Bond Basis 3.1 General Notes

### Technical Notes

Note that, as with the EUR curve procedure, the LC curve is based on the premise that both the RMV and RP cases must share the same pLCR (t,T), which represents the probability of default on the LC debt. If we assume the limit case of zero euro-debt, they would be fully consistent with the RP in the case of euro-debt, thus providing justification for our method.

### CDS-Bond Basis Empirics

Here it is only a derived quantity, since even if we assume the same point process as a driver of default on both LC and EUR debt, we can control the compensator by changing the recoveries. This is not surprising as the result is driven by the difference in shapes between the benchmark and LC curves.

Appendix