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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Wiley Finance Editions
- Sprache: Englisch
Stress-test financial models and price credit instruments with confidence and efficiency using the perturbation approach taught in this expert volume Perturbation Methods in Credit Derivatives: Strategies for Efficient Risk Management offers an incisive examination of a new approach to pricing credit-contingent financial instruments. Author and experienced financial engineer Dr. Colin Turfus has created an approach that allows model validators to perform rapid benchmarking of risk and pricing models while making the most efficient use possible of computing resources. The book provides innumerable benefits to a wide range of quantitative financial experts attempting to comply with increasingly burdensome regulatory stress-testing requirements, including: * Replacing time-consuming Monte Carlo simulations with faster, simpler pricing algorithms for front-office quants * Allowing CVA quants to quantify the impact of counterparty risk, including wrong-way correlation risk, more efficiently * Developing more efficient algorithms for generating stress scenarios for market risk quants * Obtaining more intuitive analytic pricing formulae which offer a clearer intuition of the important relationships among market parameters, modelling assumptions and trade/portfolio characteristics for traders The methods comprehensively taught in Perturbation Methods in Credit Derivatives also apply to CVA/DVA calculations and contingent credit default swap pricing.
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Table of Contents
Cover
Title Page
Copyright
Preface
Note
Acknowledgments
Acronyms
CHAPTER 1: Why Perturbation Methods?
1.1 ANALYTIC PRICING OF DERIVATIVES
1.2 IN DEFENCE OF PERTURBATION METHODS
NOTE
CHAPTER 2: Some Representative Case Studies
2.1 QUANTO CDS PRICING
2.2 WRONG‐WAY INTEREST RATE RISK
2.3 CONTINGENT CDS PRICING AND CVA
2.4 ANALYTIC INTEREST RATE OPTION PRICING
2.5 EXPOSURE SCENARIO GENERATION
2.6 MODEL RISK
2.7 MACHINE LEARNING
2.8 INCORPORATING INTEREST RATE SKEW AND SMILE
NOTE
CHAPTER 3: The Mathematical Foundations
3.1 THE PRICING EQUATION
3.2 PRICING KERNELS
3.3 EVOLUTION OPERATORS
3.4 OBTAINING THE PRICING KERNEL
3.5 CONVOLUTIONS WITH GAUSSIAN PRICING KERNELS
3.6 PROOFS FOR CHAPTER 3
NOTES
CHAPTER 4: Hull–White Short‐Rate Model
4.1 BACKGROUND OF HULL–WHITE MODEL
4.2 THE PRICING KERNEL
4.3 APPLICATIONS
4.4 PROOF OF THEOREM 4.1
NOTES
CHAPTER 5: Black–Karasinski Short‐Rate Model
5.1 BACKGROUND OF BLACK–KARASINSKI MODEL
5.2 THE PRICING KERNEL
5.3 APPLICATIONS
5.4 COMPARISON OF RESULTS
5.5 PROOF OF THEOREM 5.1
5.6 EXACT BLACK–KARASINSKI PRICING KERNEL
NOTES
CHAPTER 6: Extension to Multi‐Factor Modelling
6.1 MULTI‐FACTOR PRICING EQUATION
6.2 DERIVATION OF PRICING KERNEL
6.3 EXACT EXPRESSION FOR HULL–WHITE MODEL
6.4 ASYMPTOTIC EXPANSION FOR BLACK–KARASINSKI MODEL
6.5 FORMAL SOLUTION FOR RATES‐CREDIT HYBRID MODEL
NOTE
CHAPTER 7: Rates‐Equity Hybrid Modelling
7.1 STATEMENT OF PROBLEM
7.2 PREVIOUS WORK
7.3 THE PRICING KERNEL
7.4 VANILLA OPTION PRICING
CHAPTER 8: Rates‐Credit Hybrid Modelling
8.1 BACKGROUND
8.2 THE PRICING KERNEL
8.3 CDS PRICING
NOTES
CHAPTER 9: Credit‐Equity Hybrid Modelling
9.1 BACKGROUND
9.2 DERIVATION OF CREDIT‐EQUITY PRICING KERNEL
9.3 CONVERTIBLE BONDS
9.4 CONTINGENT CDS ON EQUITY OPTION
NOTES
CHAPTER 10: Credit‐FX Hybrid Modelling
10.1 BACKGROUND
10.2 CREDIT‐FX PRICING KERNEL
10.3 QUANTO CDS
10.4 CONTINGENT CDS ON CROSS‐CURRENCY SWAPS
CHAPTER 11: Multi‐Currency Modelling
11.1 PREVIOUS WORK
11.2 STATEMENT OF PROBLEM
11.3 THE PRICING KERNEL
11.4 INFLATION AND FX OPTIONS
NOTE
CHAPTER 12: Rates‐Credit‐FX Hybrid Modelling
12.1 PREVIOUS WORK
12.2 DERIVATION OF RATES‐CREDIT‐FX PRICING KERNEL
12.3 QUANTO CDS REVISITED
12.4 CCDS ON CROSS‐CURRENCY SWAPS REVISITED
CHAPTER 13: Risk‐Free Rates
13.1 BACKGROUND
13.2 HULL–WHITE KERNEL EXTENSION
13.3 APPLICATIONS
13.4 BLACK–KARASINSKI KERNEL EXTENSION
13.5 APPLICATIONS
13.6 A NOTE ON TERM RATES
NOTES
CHAPTER 14: Multi‐Curve Framework
14.1 BACKGROUND
14.2 STOCHASTIC SPREADS
14.3 APPLICATIONS
CHAPTER 15: Scenario Generation
15.1 OVERVIEW
15.2 PREVIOUS WORK
15.3 PRICING EQUATION
15.4 HULL–WHITE RATES
15.5 BLACK–KARASINSKI RATES
15.6 JOINT RATES‐CREDIT SCENARIOS
NOTES
CHAPTER 16: Model Risk Management Strategies
16.1 INTRODUCTION
16.2 MODEL RISK METHODOLOGY
16.3 APPLICATIONS
16.4 CONCLUSIONS
NOTES
CHAPTER 17: Machine Learning
17.1 TRENDS IN QUANTITATIVE FINANCE RESEARCH
17.2 FROM PRICING MODELS TO MARKET GENERATORS
17.3 SYNERGIES WITH PERTURBATION METHODS
NOTES
Bibliography
Index
End User License Agreement
List of Tables
Chapter 15
TABLE 15.1 Model parameters used in USD rates evolution
List of Illustrations
Chapter 5
FIGURE 5.1 Black–Karasinski prices for
maturity cap with 6m LIBOR tenor
FIGURE 5.2 Black–Karasinski prices for caps with different maturities
FIGURE 5.3 Dependence of Black–Karasinski prices for 5y caps on volatility l...
FIGURE 5.4 Dependence of Black–Karasinski prices for 5y caps on volatility m...
FIGURE 5.5 Dependence of ATM Black–Karasinski prices for 5y caps on IR curve...
Chapter 8
FIGURE 8.1 Prices for
maturity CDS
FIGURE 8.2 Correlation impact on prices for
maturity IR swap extinguisher...
FIGURE 8.3 Correlation impact on prices for IR swap extinguisher with delaye...
FIGURE 8.4 IR volatility impact on prices for IR swap extinguisher with dela...
FIGURE 8.5 Prices for a CCDS on a 10y IR swap underlying
Chapter 10
FIGURE 10.1 Pricing of counterparty risk protection on a cross‐currency swap...
Chapter 15
FIGURE 15.1 USD rates evolution: Scenario 1
FIGURE 15.2 USD rates evolution: Scenario 2
FIGURE 15.3 USD rates evolution: Scenario 3
FIGURE 15.4 Comparison of 1st and 2nd order results for
Guide
Cover Page
Table of Contents
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Perturbation Methods in Credit Derivatives
Strategies for Efficient Risk Management
COLIN TURFUS
This edition first published 2021
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Library of Congress Cataloging‐in‐Publication Data is available
Names: Turfus, Colin, author.
Title: Perturbation methods in credit derivatives : strategies for efficient risk management / Colin Turfus.
Description: Chichester, West Sussex, United Kingdom : John Wiley & Sons, 2021. | Series: Wiley finance series | Includes bibliographical references and index.
Identifiers: LCCN 2020029878 (print) | LCCN 2020029879 (ebook) | ISBN 9781119609612 (hardback) | ISBN 9781119609629 (adobe pdf) | ISBN 9781119609599 (epub)
Subjects: LCSH: Credit derivatives. | Financial risk management.
Classification: LCC HG6024.A3 T87 2021 (print) | LCC HG6024.A3 (ebook) | DDC 332.64/57–dc23
LC record available at https://lccn.loc.gov/2020029878
LC ebook record available at https://lccn.loc.gov/2020029879
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Preface
This is a book about how to derive exact or approximate analytic expressions for semi‐exotic credit and credit hybrid derivatives prices in a systematic way. It is aimed at readers who already have some familiarity with the concept of risk‐neutral pricing and the associated stochastic calculus used to define basic models for pricing derivatives which depend on underlyings such as interest and FX rates, equity prices and/or credit default intensities, such as is provided by Hull [2018]. We shall set out models in terms of the stochastic differential equations which govern the evolution of the risk factors or market variables on which derivatives prices depend. However, we shall in the main seek to re‐express the model as a pricing equation in the form of a linear partial differential equation (PDE), more specifically a second order diffusion equation, using the well known Feynman–Kac theorem, which we shall use without proof.
Our approach will be mathematical in terms of using mathematical arguments to derive solutions to pricing equations. However, we shall not be concerned here about the details of necessary and sufficient conditions for existence, uniqueness and smoothness of solutions. In the main we shall take advantage of the fact that the equations we are addressing are already known to have well‐behaved solutions under conditions which have been well‐documented. Our concern will be to use mathematical analysis to infer analytic representation, either exact or approximate, of solutions. We shall in some cases seek to offer more rigorous justification of the methods employed. But our general approach will be to demonstrate that the results are valid either in terms of satisfying the specified pricing equation (exactly or approximately), or else replicating satisfactorily prices derived by an established method such as Monte Carlo simulation.
Our method combines operator formalism with perturbation expansion techniques in a novel way. The focus is different from much of the work in the literature insofar as:
Rather than deriving particular solutions for individual products with a specific payoff, we obtain first general solutions for pricing equations; in other words, pricing kernels. We then use these to produce prices for particular products simply by taking a convolution of the payoff function(s) with the kernel.
Rather than focussing on products whose value is contingent on spot variables such as FX or inflation rates, or equity or commodity prices, and building expansions based on the assumption of low variability of local and/or stochastic volatility, we consider mainly rates‐credit hybrid derivatives, taking the short rate and the instantaneous credit default intensity to be stochastic and building expansions based on the assumption of
low rates and/or intensities
. This latter assumption is almost always valid allowing simple expressions which are only first order, or at most second order, to be used with very high accuracy. Implementation of the derived formulae typically involve nothing more complicated than quadrature in up to two dimensions and fixed point iterative solution of one‐dimensional non‐linear equations, so are well suited to scripting languages such as Python, which was indeed used for most of the calculations presented herein.
As a consequence, we are able to derive many new approximate but highly accurate expressions for hybrid derivative prices which have not been previously available in the literature. These approximations are furthermore uniformly valid in the sense that they remain valid over any trade time-scale unlike many other popular asymptotic methods such as the SABR approximation of Hagan et al. [2015], the accuracy of which depends on an assumption of short time‐to‐maturity (low term variance). We are also able to point the reader in the direction of how to derive further results for models and products other than those considered explicitly here.
The essence of our approach is that we focus on models where the stochastic factors approximate to a good degree to being normally distributed (or lognormally, which simply means that the logarithm of the variable in question is normally distributed) and where interest rates and credit default intensities are taken to be governed by short‐rate models.1 This means that the pricing kernel can to leading order be expressed as a multivariate gaussian distribution (multiplied by a discount factor). Corrections need to be applied to this base representation to obtain a sufficiently accurate result. We show how in many cases this can be done exactly. In other cases, in particular where rates or credit intensities are lognormal rather than normal, one or two correction terms need to be added to a leading order pricing kernel formula. The prices of derivatives are then obtained by taking a convolution of the pricing kernel with the associated payoff functions, which task is typically a standard one.
We start off in Chapter 1 by discussing why perturbation methods are not currently seen as “mainstream” quantitative finance, concluding that some of the reasons are seen on closer inspection to be invalid, while others, despite having some validity, do not apply to the methods set out in this book, which seeks to pioneer a new approach with wider applicability. We seek to justify this claim in the remainder of the book, starting with Chapter 2, which is dedicated to case studies illustrating how the approach we propose allows flexible response to evolving needs in a risk management context. In Chapter 3, we set out the mathematical approach and core tools which we will make use of throughout. We apply these in Chapters 4 and 5 to the construction of pricing kernels for the popular Hull–White and Black–Karasinski short‐rate models, respectively, using these kernels to derive important derivative pricing formulae; as exact expressions in the former case and as perturbation expansions in the latter.
We then turn our attention to hybrid and multi‐factor models, devoting Chapter 6 to setting out a generic framework for handling models with multiple factors following the Ornstein–Uhlenbeck processes, the detailed calculation associated with which method turns out to depend only on the (stochastic) discounting model employed. We set out the details for both Hull–White and Black–Karasinski discounting models. The next four chapters deal with two‐factor hybrid models: rates‐equity; rates‐credit; credit‐equity; and credit‐FX. Kernels are deduced, either exact or as perturbation expansions, and used to infer the prices of a number of semi‐exotic derivatives in each case. Some evidence is provided of the favourable performance of approximate results against calculation performed by numerical schemes capable of delivering arbitrarily high precision.
Chapter 11 expands the envelope one step further, looking at a three‐factor model incorporating an FX rate and two interest rates, deducing an exact pricing kernel and using this to infer option prices. It is noted that the model considered is of Jarrow–Yildirim type so is applicable also to the pricing of inflation derivatives. A further turn of the handle in Chapter 12 also brings credit risk into the mix, resulting in a four‐factor model. A pricing kernel expansion is deduced and used to price a number of semi‐exotic credit derivatives. Most notably we revisit quanto CDS pricing (covered in the first instance in Chapter 10), now allowing interest rates to be stochastic as well as credit and FX rates.
The next two chapters of the book take us off in slightly different directions. First we look forward to the new risk‐free LIBOR replacement rates which are set in arrears on the basis of compounding daily (or overnight) rates (Chapter 13). This approach is intended to supplant the currently used multi‐curve frameworks where LIBOR rates embed a tenor‐dependent stochastic spread, the modelling of which is the subject of Chapter 14. In each of these cases we consider in the first instance how the pricing kernel for the short‐rate model is affected then look at how the integration with a Black–Karasinski credit model impacts the resulting hybrid kernel and assess the consequent impact on credit derivatives formulae.
The remaining chapters are devoted to applications of the methods and results herein expounded in various areas of contemporary interest in a risk management context. Chapter 15 looks at scenario generation where interest rate and credit curves need to be evolved alongside spot processes to allow risk measures such as market risk, counterparty exposure and CVA, depending on a projected distribution of future prices, to be calculated. In Chapter 16 we look at model risk, noting that our methods have utility here too, both in providing useful, easily implemented benchmarks for model validation purposes and for making quantitative assessments of the influence of model parameters and modelling assumptions on portfolio evaluations. Finally the newly evolving application of machine learning to problems in quantitative finance and the question of how asymptotic methods could complement this approach in practice are addressed in Chapter 17.
C. Turfus
London, 2020
Note
1
We exclude for the former reason rates (interest or credit) which are governed by a model of the CIR type defined by Cox et al. [
1991
) (where the underlying stochastic factor follows a
distribution), and for the latter reason rates which are governed by either a HJM model of the type defined by Heath et al. [
1992
) or a LIBOR market model. Most of the standard models for spot underlyings are encompassed within the framework, the main exceptions being Lévy models and rough volatility models.
Acknowledgments
The author is grateful to co‐researcher Alexander Shubert for his important contribution in implementing in Python the asymptotic formulae presented in Chapter 15 and in preparing the associated graphs.
Acronyms
ATM
at the money
CCDS
contingent credit default swap
CDS
credit default swap
CDF
cumulative distribution function
CMS
constant maturity swap
CVA
counterparty value adjustment
ITM
in the money
OTM
out of the money
PDE
partial differential equation
probability density function
FRTB
Fundamental Review of the Trading Book
VaR
value at risk
CCR
counterparty credit risk
HJM
Heath Jarrow Morton
LMM
LIBOR Market Model
CHAPTER 1Why Perturbation Methods?
1.1 ANALYTIC PRICING OF DERIVATIVES
How important are analytic formulae in the pricing of financial derivatives? The way you feel about this matter will probably determine to a large degree whether this book will be of interest to you. Current opinion is undoubtedly divided and perhaps for good reasons. On the one hand, presented with the challenge of some new financial calculation, financial engineers these days are likely to spend considerably less time looking for analytic solutions or approximations than, say, twenty years ago, citing the ever‐increasing power and speed of computational resources at their disposal. On the other hand, where known analytic solutions exist, those same financial engineers are unlikely to eschew them and to persist doggedly in replicating the known solution using a Monte Carlo engine or a finite difference method.
So, it might be suggested, the resistance to analytic solutions that we observe is not to their use as such when they are already available, but to making the effort to find (and implement) them. One of the reasons for this is a perception that, given the huge amount of research effort that has been invested into finding solutions over the past few decades, most of the interesting and useful solutions have been found and published. It is the experience of the author that the reaction to the announcement of discovery of a new and interesting analytic solution tends to be indifference or scepticism rather than interest. At the same time, it is often assumed (correctly?) that such effort as is being invested into finding analytic solutions is these days directed mainly towards approximate solutions, most particularly using perturbation methods, which area continues to be a reasonably fertile ground for research effort, at least in academic institutions. We shall look more closely at the areas which are attracting attention below.
It is of interest to ask then why, despite the continuing effort being invested on the theoretical side into the development of analytic approximations, the take‐up in practice appears to be relatively limited, certainly compared to the heyday of options pricing theory when the choice of models made by practitioners was significantly influenced by the availability of analytic solutions, even of analytic approximations such as SABR [Hagan et al., 2002]. For example Brigo and Mercurio [2006] observed of the short‐rate model of Black and Karasinski [1991] that
the rather good fitting quality of the model to market data, and especially to the swaption volatility surface, has made the model quite popular among practitioners and financial engineers. However,…the Black–Karasinski (1991) model is not analytically tractable. This renders the model calibration to market data more burdensome than in the Hull and White (1990) Gaussian model, since no analytic formulae for bonds are available.
It is undoubtedly true that the relative tractability of the Hull–White model has been an important factor resulting in its much wider adoption as an industry standard.
No single reason can be cited to account for the relatively limited use to which analytic approximations are put. Practitioners' views vary greatly depending on the types of models they are looking at and what they are using them for. A number of factors can be pointed to, as we shall elaborate in the following section. For the moment we make the following observations, specifically comparing analytic pricing with a Monte Carlo approach.
There is a general distrust by financial engineers of methods involving any kind of approximation. The fact that, if results involve power series‐like constructions, it may not be possible to guarantee arbitrage‐free prices in 100% of cases is often cited as a reason to avoid use of such approximations in pricing models intended for production purposes. Furthermore, it can be more work to assess the error implicit in a given approximation than it is to compute prices in the first place.
While analytic methods are computationally more efficient, they appear to be intrinsically less scalable than Monte Carlo methods from a development and implementation standpoint. Whereas the Monte Carlo implementation of a model mainly involves the simulation of the underlying variables, with different products merely requiring different payoffs to be applied, each product variant tends to have a different analytic formula with limited scope for reuse with reference to other products. Also, if an additional stochastic factor is included in a Monte Carlo method, this can often be handled as an incremental change, while in the case of analytic methods, they will often break down completely when an additional risk factor is added.
Another argument that is not infrequently heard against the introduction of new analytic results is that it is just too much trouble to integrate them into pricing libraries which are already quite mature. An accompanying argument may be that, since the libraries of financial institutions are already written in highly optimised C++ code, any gains that might be made are only likely to be marginal.
There is also a suspicion concerning the utility of perturbation methods insofar as, while the most interesting and challenging problems in derivatives pricing occur where stochastic effects have a significant impact on the pricing, most perturbation approaches have some kind of reliance on the smallness of a volatility parameter, usually a term variance.
1
But, for this parameter to have a significant impact on pricing it cannot be too “small”, so we are led to the expectation that we will need a large number of terms in any approximating series to secure adequate convergence in many cases of importance.
A more recent argument which the author has encountered in a number of conversations with fellow researchers is that, insofar as more efficient ways are sought to carry our repetitive execution of pricing algorithms, the strategy adopted in the future will increasingly be to replace the time-consuming solution of SDEs and PDEs not with analytic formulae but with machine-learned algorithms which can execute orders of magnitude faster (see for example Horvath et al. [2019]). The cost of adopting this approach is a large amount of up-front computational effort in the training phase where the full numerical algorithm is run multiple times over many market data configurations and product specifications to allow the machine-learning algorithm to learn what the “right answer” looks like so that it might replicate it. There will also be a concomitant loss of accuracy. But if, as is often the case, the requirement is to calculate prices for a given portfolio or the CVA associated with a given “netting set” of trades with a given counterparty over multiple scenarios for risk management or other regulatory purposes, the upfront cost can be amortised against a huge amount of subsequent usage of the machine-learned algorithm. Since machine-learning approaches are a fairly blunt instrument, there is not the need to customise the approach to the particular problem addressed, as would be necessary if perturbation methods were used instead as a speed-up strategy wherein some accuracy is traded for speed.
Finally, there is not uncommonly a perception that, unlike with earlier analytic options pricing formulae which were deduced using suitable application of the Girsanov theorem, with which financial engineers tend to be familiar, perturbation‐based methods are by comparison something of a dark art. Many of the results are derived using Malliavin calculus or Lie theory, with which relatively few financial engineers are familiar, and often presented in published research papers in notation which is relatively opaque and quite closely tied in to the method of derivation. Other derivations are performed using methodologies and notations borrowed from quantum mechanics or other areas of theoretical physics, areas with which a contemporary financial engineer is unlikely to be familiar. There is, furthermore, not a clearly defined body of theory which the practitioners of perturbation analysis seek to rely on; books which offer a unified approach to perturbation methods applicable to a range of problems in derivatives pricing such as Fouque et al. [
2000
], Fouque et al. [
2011
] and Antonov et al. [
2019
] are few and far between.
1.2 IN DEFENCE OF PERTURBATION METHODS
Although the arguments presented above challenging the merit of attempts to extend the range of analytic formulae available for derivatives pricing by means of perturbation expansion techniques may appear compelling, we suggest that, when they are unpicked a little, their apparent validity starts to unravel. More specifically they are seen to be premised on a view of what is possible with perturbation methods which is challengeable in the light of recent theoretical developments, in particular those set out in this book. They, furthermore, depend on a view of what practical purposes option pricing methods need to address in the industry and the consequent constraints they must satisfy which is likewise challengeable and not altogether up to date.
While the development of derivatives pricing methods was based on the concept of risk‐neutral pricing to guarantee the absence of arbitrage opportunities through which market makers could systematically lose money, the use of pricing models is increasingly in practice for risk management purposes, rather than the calculation of prices for market‐making purposes. So, even if it is the case that an approximation method might technically give rise to arbitrage opportunities in a small number of extreme cases, provided no trading takes place at these prices this is not necessarily a problem. Indeed we are often in a risk management context more interested in real‐world probabilities than in their risk‐neutral counterparts, on account of the fact it is extreme real‐world events and their frequency of occurrence in practice which can lead to the destabilisation or demise of a financial institution. For example, a report by Fintegral and IACPM [
2015
] surveying 37 global and regional financial institutions concludes that calculation of counterparty credit risk (CCR) tends to operate under “real‐world” assumptions using historical volatilities to calibrate the Monte Carlo simulation.
Also, since risk management is generally about portfolio aggregates rather than individual trades, and typically involves computing prices under hypothetical future scenarios, it is not so important to be able to estimate the size of errors associated with the pricing of individual trades as the expected aggregate error, which can often be estimated to a sufficient degree of accuracy by fairly heuristic methods. This is recognised in the Basel IV (FRTB) regulatory framework which has been proposed to replace VaR: internal models used for risk management purposes don't have to be validated in terms of their ability to price individual trades accurately, but rather the aggregate risk numbers produced need to be sufficiently close to those obtained using end‐of‐day pricing models in a back‐testing exercise.
Another factor is that, whereas the main criterion pricing models have to satisfy is accurate calculation of the first moment of a distribution, risk models are much more focussed on the distribution of prices, typically in the extreme quantiles where the greatest risk is usually deemed to lie, so their ability to give an accurate assessment of second and higher moments tends to be at least as important, if not more so. While, in the market, prices of a large number of traded financial securities can be considered known to a reasonable degree of accuracy (the bid–offer spread), this is not the case if one is asking about the distribution of those prices in the future, market information about which is typically much scarcer and the uncertainty about which is correspondingly much greater.
One of the issues with the way in which perturbation expansions were derived and presented historically is that they were deduced as particular solutions associated with a specific payoff structure: often this was a vanilla European‐style payoff. Such results could not therefore be used for related problems like, say, forward‐starting options. Likewise, restrictive assumptions were often made about market data, such as volatilities being constant, without clarification of how results could be generalised. We shall refer to such approaches to perturbation analysis as
first generation
. The last five years or so have seen focus shift more and more to deriving instead pricing kernels; in other words, general solutions to the pricing equation which can be used relatively straightforwardly to derive solutions for multiple payoff configurations. This approach to perturbation analysis we shall refer to as
second generation
. A good introduction to this subject is provided by Pagliarani and Pascucci [
2012
], where pricing kernels are referred to as transition densities.
The approach we shall take below is robustly second generation, seeking from the outset a pricing kernel associated with a given model before applying it to calculate derivative prices or risk measures. As we shall see, our approach has the further advantage that it provides a way not only to derive a pricing kernel (approximate or exact) systematically from a pricing equation, but also to extend it to include additional risk factors so that pricing kernels derived for simpler problems can be recycled in producing kernels for new and more complex problems. Indeed we will see (in Chapter 6) that, where risk factors do not impact on the discounting applied, they can often be included in a generic way using vectorised notation so that a single generic pricing kernel can present a unified solution to many different pricing equations. Also it will be seen that the resulting formulae all tend to have very similar structures so the coding effort involved in implementing them in a pricing library tends to be fairly light with significant code reuse possible.
In this way we will seek to go beyond existing second generation approaches and create the tools or frameworks from which pricing kernels can be constructed, rather than focussing on the pricing kernels themselves. In effect we are advocating for a Generation 2.5 approach.
Although the requirements of investment banks have historically been the main driver of innovation in pricing methodologies, banks' pricing libraries are, in general, in a fairly mature and stable state: there is relatively little happening in the way of new product or model development. New implementation of pricing routines is much less likely to be happening in the front office of investment banks than in small investment houses, hedge funds, consultancies or other service providers. Rather than developing high‐powered generic Monte Carlo engines in C++ code, they are more likely these days to be developing or buying‐in more bespoke risk and pricing models, increasingly in more flexible scripting languages such as Python rather than in C++, since rapid development, easy maintenance and transparency are more likely to be at a premium. Further, the availability for more than a decade now of the Boost.Python C++ library allows routines and software objects written in C++ to be seamlessly accessed from or integrated into Python scripts, allowing the advantages of both worlds potentially to be enjoyed. It is in relation to such usage that we see the type of easily implemented perturbation solutions presented in this book as being most relevant.
The great majority of work on derivatives pricing using perturbation expansions from the SABR model of Hagan et al. [
2002
] has been in the context of local and/or stochastic volatility (mainly the latter). Furthermore, relatively little attention has been paid to short‐rate models, either for interest rates or for credit. One might ask why this is the case. This is a surprisingly difficult question to answer definitively because no one tends to report the reasons why they did
not
do research in a given area. The bias may in part be a consequence of the fact that short‐rate models are considered to have been surpassed by other more flexible frameworks such as that of Heath et al. [
1992
] (HJM) and the LIBOR market model (LMM). There appears to be a sense that short‐rate models are “harder” than models of spot processes on account of the non‐linear interaction between the short rate appearing both in the payoff specification
and
in the discount factor. While this may limit considerably the scope for
exact
analytic solutions, it ought not to be considered overly problematic in relation to perturbation expansion approaches: the impact on the discount factor is invariably quite weak, which plays into a perturbation strategy. Much of the analysis in the present volume seeks to exploit precisely this fact, building rapidly convergent perturbation series in powers of the short rate(s). It remains unclear to the author why more advantage has not been taken of this possibility by other researchers.
We shall, as indicated, focus on short‐rate models at the expense of consideration of the HJM framework and the related LIBOR market model (although some short‐rate models such as Hull–White can be derived from within a HJM framework). This is mainly because the latter do not lend themselves so well to analysis by the techniques we expound below. A word should be said in defence of this decision. In the first instance, as the book's title suggests, we are specifically interested in credit derivatives, and the HJM and LMM frameworks have far less utility in that space than do short‐rate models. Further, insofar as our focus in the interest rate modelling we do perform is mainly on hybrid derivatives pricing rather than the types of interest rate option that tend to be the main targets of HJM and LMM approaches, short‐rate modelling is what is likely already being used in practice in the contexts we address.
The main shortcoming of the argument that traditional pricing libraries are about to be replaced by machine-learned versions of the algorithms they embody is that it currently lacks any real evidential basis. The strong recent interest in machine learning in the financial engineering community can be attributed to its increasingly being used, evidently to good effect, in devising and improving trading strategies by detecting signals in market data and trading patterns which a human observer might miss; and of course in algorithmic trading where the speed of the algorithm is key to profitability and even the latency of internet connections can have a significant impact. This has led to an upsurge of interest in this area, not only from those who already have relevant domain knowledge but also from many researchers who have established credentials in other areas, including I should add perturbation methods. But the colonisation by machine learning of the space currently occupied by pricing libraries is currently more an aspiration than a defined programme of research.
While it is probably too early to call how things will pan out in this area, we would venture that the future is best viewed not as a competition between analytic formulae and machine-learned alternatives but as an opportunity for mutually beneficial collaboration. For example, it is noteworthy that Horvath et al. [2019] in their highly influential paper target not option prices but Black-Scholes implied volatility in the learning process for a rough volatility model. So they are implicitly making use of the Black-Scholes pricing formula to provide an approximation for the rough volatility model price. Perhaps more interestingly, the recent work of Antonov et al. [2020] addresses the challenging problem of how to handle efficiently the outer limits of the (high-dimensional) phase space addressed by a machine-learned representation of a pricing algorithm, effectively by substituting in an asymptotic representation of the pricing algorithm for points outside a core region of the phase space which is sampled fairly exhaustively in the learning process. The utility of such an approach hinges crucially on the availability of an asymptotically valid approximate solution to the problem at hand, which can effectively be used as a control variate in the learning process.
Finally, although it is true that there is a dearth of unified presentations of perturbation methodologies in the literature, we seek to address this in what follows by demonstrating how the particular second generation approach we advocate is applicable across a wide range of financial products, market underlyings and modelling assumptions for numerous tasks ranging from straightforward valuation to scenario generation, XVA and model risk quantification. We also seek to present results in a form which is at the same time transparent, so as to facilitate implementation, and fully general to allow real market data to be used without any modification or re‐working of results.
NOTE
1
