Affine Term-Structure Models: A Time-Changed Approach with Perfect Fit to Market Curves

Abstract

We address the so-called calibration problem which consists of fitting in a tractable way a given model to a specified term structure like, e.g., yield, prepayment or default probability curves. Time-homogeneous jump-diffusions like Vasicek or Cox-Ingersoll-Ross (possibly coupled with compound Poisson jumps, JCIR, a.k.a. SRJD), are tractable processes but have limited flexibility; they fail to replicate actual market curves. The deterministic shift extension of the latter, Hull-White or JCIR++ (a.k.a. SSRJD) is a simple but yet efficient solution that is widely used by both academics and practitioners. However, the shift approach may not be appropriate when positivity is required, a common constraint when dealing with credit spreads or default intensities. In this paper, we tackle this problem by adopting a time change approach, leading to the TC-JCIR model. On the top of providing an elegant solution to the calibration problem under positivity constraint, our model features additional interesting properties in terms of variance. It is compared to the shift extension on various credit risk applications such as credit default swap, credit default swaption and credit valuation adjustment under wrong-way risk. The TC-JCIR model is able to generate much larger implied volatilities and covariance effects than JCIR++ under positivity constraint, and therefore offers an appealing alternative to the shift extension in such cases.Model calibration is a standard problem in many areas of finance Brigo and Mercurio (2006);Joshi (2003); Veronesi (2010). It consists of tuning a model such that it "best fits" market quotes at a given time. As an example, financial markets provide a set of prices associated with liquid instruments, that openly trade on the market. Alongside with risk management (hedging), the main purpose of a model here is to act as an "interpolation/extrapolation" tool, i.e., to obtain the value of products at a given time t for which the market does not disclose prices in a transparent way. This could happen because either the product to be priced is "exotic" (i.e., is too "special", it does not quote openly on a platform, only on a bilateral basis) or because its cashflow schedule is not in line with the products that currently trade openly at t (a situation that commonly happens since products that were "standard" at inception, may have time-to-expiry or moneyness levels that are no longer "standard" afterwards).Mathematically, model calibration is nothing but an optimization problem. Starting from a set of prices quoted on the market (called "market prices") for a set of specific financial products (called "calibration instruments"), model calibration consists of computing the model parameters such that the prices generated by the model (called "model prices") best fit to the market prices, according to some error function. Model calibration is crucial in finance;it is strongly related to arbitrage opportunities. In practice, only models that are able to reproduce the market prices of "simple instruments" ...

An arbitrage-free conic martingale model with application to credit risk

“…Let us now show that S t (T ) associated with the conic martingale default model with F = Φ cannot be written as (33). It is easy to show that if ψ is regular enough for (24) to admit a unique strong solution,…”

“…It will serve as comparison for the other -more recent-models further considered. The second model is a time-changed version of the JCIR, called TC-JCIR, introduced in [33]. It is indeed an interesting alternative to the JCIR++ model.…”

“…Therefore, increasing the jump activity under the constraint that E[S t ] = G(t) for a given curve G requires to lower the shift, possibly to the negative territory. We refer to [33] for a detailed analysis of the negativity intensity issue of the JCIR++ model.…”

“…More recently, the calibration problem has been solved using a different deterministic adjustment: the shift function is replaced by a time change. These two modifications of the shift extension are refered to as the positive-shift (PS-(J)CIR) and the time-changed (TC-(J)CIR) (J)CIR [33]. We refer to [18,20] for general results within the affine term structure models and to [33] regarding the perfect fit problem.…”

“…These two modifications of the shift extension are refered to as the positive-shift (PS-(J)CIR) and the time-changed (TC-(J)CIR) (J)CIR [33]. We refer to [18,20] for general results within the affine term structure models and to [33] regarding the perfect fit problem.…”