measuring the market risk of illiquid positions alexander shklyarevsky, senior vice president senior...

Measuring the market risk of illiquid positions

Alexander Shklyarevsky, Senior Vice PresidentSenior Quantitative Finance AnalystGlobal Markets Risk Management

BANK OF AMERICA

Content

• Market risk overview• Illiquid risk• Quantitative approaches to managing market

risk in a volatile environment• Valuation models for derivative products and

their application in managing market risk• Single-trade level and portfolio level market

risk methodologies

Market risk overview

• Under a Federal Reserve Bank definition, a Market risk is “the potential change in a bank's earnings or value due to adverse movements in market rates or prices, such as interest rates, foreign exchange rates, equity prices or commodity prices”

• Because of the current financial crisis and its credit nature, we would like to add to the list above “credit (CDS) spreads, default correlations, volatilities, option-adjusted spreads (OAS), credit indices”

• A major measure of market risk at both a single-trade level and a portfolio level is a Value-at-Risk (VaR)

• It is defined as such an amount of money that the probability of a potential loss over a certain Time Period (say, 1 day) being less than VaR equals to a Confidence Level (say, 99%, the Confidence Level is, generally speaking, different for different financial institutions)

• VaR is reported at a desk level, book level, geographic region level, business unit level, company level, etc.


• There are several VaR methodologies:• Variance-Covariance VaR• Delta VaR• Delta-Gamma VaR and • Full Valuation VaR methodologies:

– Historical Simulation VaR– Monte Carlo Simulation VaR– Semi-Analytical VaR– Analytical VaR

• Factor Analysis plays an important role in measuring market risk as well• Besides VaR, other classic market risk measures are the Greeks including the

below ones that are defined as follows:• Delta – a change in a derivative (portfolio) price P w.r.t. a change in the

value(s) of the underlying(s) (Delta = P’(S), S – an underlying)• Gamma – a change in Delta w.r.t. a change in the value(s) of the underlying(s)

(Gamma = P”(S))


• Vega – a change in a derivative (portfolio) price w.r.t. a change in the value(s) of the implied volatilitie(s) σ (Vega = P’(σ))

• Theta – a change in a derivative (portfolio) price w.r.t. a change in the passage of time t (Theta = P’(t))

• Rho – a change in a derivative (portfolio) price w.r.t. a change in the value(s) of the (domestic) risk-free interest rate(s) r (Rho = P’(r))

• Phi – a change in a derivative (portfolio) price w.r.t. a change in the value(s) of the dividend yield(s) d (foreign risk-free interest rate(s)) (Phi = P’(d))

• Delvega (Vanna) - a change in a derivative (portfolio) price w.r.t. a change in the value(s) of the underlying(s) and the implied volatilitie(s) (Vanna = d²P / (dS dσ))

• Volga – a change in vega w.r.t. a change in the value(s) of the implied volatilitie(s) (Volga = P”(σ))


For the yield/spread curve-dependent derivatives:• Duration (Modified) – minus relative change in a derivative

(portfolio) price w.r.t. a change in the value(s) of the entire yield/spread curve(s) (if Y is a yield/spread and B(Y) is such a derivative’s price, then Modified Duration = - B’(Y) / B(Y))

• Duration (Macaulay) – Modified Duration * (1 +Yield/Spread) (Macaulay Duration = - B’(Y) / B(Y) * (1 + Y))

• Convexity – a relative change of change in a derivative (portfolio) price w.r.t. a change in the value(s) of the entire yield/spread curve(s) (Convexity = B”(y) / B(y))

• Key Rate Duration – - ∂B(Y1,…,Yi,…,Yn)/∂Yi / B(Y1,…,Yn)• Key Rate Convexity – ∂²B(Y1,…,Yi,…,Yn)/(∂Yi ∂Yj) / B(Y1,…,Yn)• Effective Duration – the derivative in the Modified Duration

definition should be replaced by a central finite difference• Effective Convexity – the second derivative in the Convexity

definition should be replaced by a second central finite difference


• One of the major drawbacks of the VaR market risk measure is that it gives an upper bound of the potential losses with the probability that equals to a Confidence Level (say, 99%)

• It means that with the probability that equals to 1 – Confidence Level (say, 1%) the potential loss will be greater than VaR, so our risk number is not adequately reflecting such events

• In this case, such a measure as Expected Shortfall should be used, especially when the loss distribution is exhibiting fat tails

• Expected Shortfall (coherent semi-additive market risk measure) is a conditional expectation of all the potential losses that are greater or equal to VaR; this measure is capturing the extreme scenarios above

• The present financial crisis situation is an extreme event scenario as well, and a pretty severe one, so we suggest to measure the market risk using an Expected Shortfall measure in addition to other measures

• As this financial crisis is the liquidity one, we are suggesting to incorporate liquidity considerations into pricing, hedging and risking of derivative securities and their portfolios

Illiquid risk

• Under a Federal Reserve Bank definition, a Liquidity risk is “the inability to sell assets or obtain adequate funding on reasonable terms. Very large players may also be exposed to the inability to unwind or offset exposures without significantly influencing market prices”

• As we suggest to factor the liquidity into pricing, hedging and risk estimation of derivative securities and their portfolios, the terms “Illiquid Risk” and “Liquidity Risk” should be equivalent

• Same is true for the terms “Illiquid Positions” and “Liquidity Risky Positions”, so both pairs of terms should be used interchangeably

• Let’s look at the quantitative approaches to managing market risk in a volatile environment (such as the current extremely turbulent financial crisis environment) including incorporation of liquidity into pricing, hedging and risk estimation models for derivative securities and their portfolios

• After this analysis, we will see how these quantitative approaches result in corresponding valuation models and market risk methodologies

Quantitative approaches to managing market risk in a volatile environment• Our general ideas for the quantitative approaches to managing market

risk in a volatile environment like the current financial crisis in terms of valuation, calibration, risk estimation and data processing for derivatives and their portfolios include the following:– These approaches should include Valuation, Risk Estimation, Hedging,

Volatility, Correlation, Calibration, Liquidity & Transaction Cost Models– These models should be preferably closed-form ones or analytical

approximations – for sure in risk management and potentially on the desk – we at Risk Management Analytics run exotic structures valuations for up to 1 second per trade vs. 2-5 minutes and more on the desk

– This could be accomplished via application of integral, differential and other operator transformations to solutions of valuation partial differential equations (PDEs) and partial integro-differential equations (PIDEs) and computations of 1-dimensional and multi-dimensional integrals (expectations), as derivative’s (risk-neutral) MTM is a discounted expected payoff, and removing a risk-neutrality assumption (e.g., in conditional pricing) brings another layer of integration to the valuation

Quantitative approaches to managing market risk in a volatile environment• As analytical modeling (closed-form solutions and analytical

approximations) was always a nontrivial task for exotic derivative products, iterative processes – binomial and trinomial trees, finite differences and various types of Monte Carlo simulation – are used for a majority of these products not only in terms of their risk estimation, but in terms of their valuation as well

• The iterative processes approach to valuation and risk estimation of these derivative products pose significant challenges for both practitioners and academics including the following problems:

• The speed of calculation is either slow or very slow (except enhanced Monte Carlo simulation methodologies, such as Monte Carlo with importance sampling / principal components analysis, variance reduction techniques, etc.)

• The process is not converging, unless a significant number of iterations is used (for regular Monte Carlo, it should be from 10,000 to 100,000 iterations)

Quantitative approaches to managing market risk in a volatile environment

• Continuous dynamics is approximated by a discrete one that makes these iterative processes approximate as well and might produce imperfect valuation and risk estimation of multi-asset and hybrid derivative products

• A further strain for risk estimation is when valuation is using a regular Monte Carlo simulation methodology with more than 10,000 iterations; for MBS, in particular, a current Risk Management practice is to use a regular Monte Carlo for valuation purposes and a duration-convexity approximation VaR for Risk Management purposes, as a full revaluation methodology (Monte Carlo or Historical simulation) would take too long (sometimes days for a typical portfolio!) to complete (as it would be Monte Carlo on Monte Carlo)

• Perspective directions for iterative processes and their usage to value and risk exotic derivatives should include the following:

– To use and further develop enhanced Monte Carlo simulation methodologies, such as Monte Carlo with importance sampling / principal components analysis, variance reduction techniques

– To try to derive formulaic expressions from binomial trees, trinomial trees and finite differences; a successful experience in that for single-asset products has been encouraging

Quantitative approaches to managing market risk in a volatile environment• Perspective directions for analytical methodologies and their usage to value and risk

exotic derivatives should include the following:– Analytical multi-dimensional PDE, Integral Equation and PIDE valuation

methodologies should get further developed including the ones that incorporate liquidity and transaction costs into the differential equations’ dynamics

– Stochastic volatility and correlation models should get incorporated into such models increasing their dimensionality, but simultaneously easing up an analytical solution development compared to a general case volatility/variance smile and correlation/covariance smile models

– Generalize specific volatility smile models that lead to closed-form solutions and analytical approximations for valuation models

– Further develop stochastic Prepayment Models for MBS, ABS and options on them in conjunction with OAS Stochastic models, Interest Rate Stochastic models and CDS Spread Stochastic models (accounting for their volatilities, PCAs, correlations and mean reversion parameters)

– Mean Reversion and its nonlinear generalizations should be further explored in PDE, PIDE and SDE context

– Move in the direction of structural models for multi-asset Credit Derivatives (including bespoke and index CDOs) should be further developed


– Along with numerical procedures for path-dependent exotic derivatives, analytical methodologies should further get developed

– Further develop analytical models for variance, volatility and correlation derivatives– Further develop models for Stochastic Volatility, Stochastic Variance, Stochastic

Correlation and Stochastic Covariance– Further develop analytical models for other exotic derivatives, existing and to be

developed– Further use operator methods in exotic derivatives valuation and develop new

operator method for this and other purposes– Further use Fourier and other integral transforms in exotic derivatives valuation and

develop new integral transform approaches for this an other purposes– Further use non-integral transforms in exotic derivatives valuation and develop new

non-integral transform for this and other purposes– Further development of the best approximation orthonormal bases usage in exotic

derivatives valuation and develop new such approaches for this and other purposes– Broaden the range of exotic derivatives that could be valued analytically to include


– Convertible Bonds and Convertible Preferreds (Fixed Rate and Floating Rate ones)– SCONes (Structured Convertible Bond Option Notes)– Quanto Options (certain types)– Spread Options on any asset classes– Performance Options an any asset classes– Basket Options on any asset classes– Variance, Volatility and Correlation Derivatives– FX Derivatives with Stochastic Interest Rates– Mortgage-Backed and Asset-Backed Securities (MBS and ABS) and Options on them

including Reverse MBS new generation of models– CDS on ABS, MBS and CMBS– Credit Derivatives – to include

» CDOs» CDS baskets and associated options» Nth To Default Swaps and associated options» Bespoke and Index synthetic CDOs» Forward start CDOs and options on them» CDOs on CDOs (CDO squared)

– TRS (Total Rate of Return Asset Swaps) where the underlying asset is anything above– Repos (Repurchase Agreements) and Reverse Repos for anything above


– Insurance Derivatives – to include» Annuities» Insurance Policies» Premium Finance Notes (including the ones with Contingent

Interest)» STEP Options» Real Estate Options

• In such a volatile environment as we are having now, it is extremely important to incorporate liquidity into valuation of derivative securities and their portfolios

• We will show in the next section how to price liquidity into derivatives valuation analytically and other analytical approaches to derivatives valuation for certain types of exotic derivative securities and their portfolios

• These dynamic approaches represent a step forward from the past static approaches and make a tough job of managing market risk in a current volatile market crisis environment much more transparent

Valuation models for derivative products and their application in

managing market risk• Independently and because of the challenges that iterative processes

pose, analytical approaches – closed-form solutions and analytical approximations – started to be more and more popular, first in Risk Management and then (if the solution is there) on the trading desks

• We will go over analytical approaches to valuation of derivative products giving more color to some of them

• Application of analytical approaches to valuation and risking of exotic derivatives could revolutionize a data processing of these products by making processing times hundreds and thousands times faster and more reliable

• There is no convergence problem that a regular Monte Carlo poses• Both continuous and discrete dynamics could be there as modeled, and

the results accuracy is much better in the case of closed-form solutions and could be good enough in the case of analytical approximations


managing market risk• We suggest also a dynamic and analytical approach to incorporating liquidity (and

later – transaction costs) into pricing (valuation) models by generalizing PDEs (and later – PIDEs) to account for them

• These PDEs and PIDEs will become thus nonlinear in terms of nonlinearity of a differential (and later – integral) operator

• This differential operator is nonlinear in terms of nonlinearity of the functions of Gamma a weighted sum of which is added to a Black-Karasinski PDE that is a generalization of Black-Scholes-Merton PDE

• This result is obtained similarly to a Black-Karasinski PDE itself using Ito’s lemma for a continuous differentiable function of the underlying asset and time (to become a derivative’s price) and a non-arbitrage assumption, that now includes not only a hedged (and then rehedged) derivative, but liquidity as well

• From the results of Krakovsky, Bakstein and Howison, etc., as well as our own results, a single-asset derivative PDE that incorporates liquidity into the derivative’s pricing will look in the general enough form as follows:

• In this nonlinear PDE, S is an Underlying Asset Value, t is Valuation Time, V(S, t) is a Derivative’s Price, μ is a term-structured Long-Term Mean, ν is a term-structured Mean Reversion Speed, σ is a term-structured Implied Volatility, c(k) are term-structured weights that depend on liquidity (stochastic in future models), fk() are, generally speaking, nonlinear (and not necessarily differentiable) functions of Gamma, r is a term-structured risk-free interest rate

• For some partial cases of fk(), using the ideas above, we obtained closed-form solutions for European options, though corresponding PDEs, similarly to MBS PDE that will be discussed below, proved to be reduced to non-homogeneous ones, along with analytical approximations for non-European optionality

• For other cases of fk(), we obtained analytical approximations as well• Other authors suggested their own analytical approaches to solve such PDEs• For multi-dimensional (multi-asset and hybrid) derivatives and portfolios of

one-dimensional (one-factor) and multi-dimensional (multi-factor) derivatives, the nonlinear PDE above will be generalized to the following PDE:

VrS

VfSc

S

VS

S

VSS

t

Vtk

kn

k

kt

ttt

)(2

)ln(2

2

1

)(2

222

• In this nonlinear multi-dimensional PDE, S1,…, Sm are Underlying Values for m Assets (possibly, from different Asset Classes) or Factors, t is Valuation Time, V(S1,…, Sm, t) is a Derivative’s (Portfolio) Price, μ are term-structured Long-Term Means, ν are term-structured Mean Reversion Speeds, σ are term-structured Implied Volatilities, ρ are term-structured Correlation coefficients, d(i) and c(l)(k) are term-structured weights that depend on liquidity (stochastic in future models), fk() are, generally speaking, nonlinear (and not necessarily differentiable) functions of Gamma, r is a term-structured risk-free interest rate

• For some partial cases of fk(), using the ideas above and generalizing a one-dimensional case, we obtained closed-form solutions for European options, though corresponding PDEs, similarly to MBS PDE that will be discussed below, proved to be reduced to non-homogeneous ones, along with analytical approximations for non-European optionality and other cases of fk()

• Such a generalization (with possible switch for some assets from the lognormal mean-reverted dynamics above to a normal mean-reverted dynamics) could be done for all the multi-asset and hybrid products described below, as well as exotic one-dimensional ones, so let’s briefly describe them

VrS

VfSc

SS

V

S

V

S

VSSdd

SS

VSS

S

VSS

t

Vt

m

l l

kk

l

n

k

klt

jiji

ji

m

i

m

j

ijt

jt

it

jt

it

jiji

ijt

jt

m

i

m

j

it

m

l lll

lt

lt

12

2

1

))((2

2

2

2

22

1 1

)()()()()(2

)()(

1 1

)(

1

)()( )()(2

1

2

1)ln(


managing market risk• As Hybrid Derivatives is a major class of multi-asset derivatives, let’s start

with them• Hybrid Derivatives are derivative securities that have more than 1 underlying

asset class (while Multi-Asset Derivatives are derivative securities that have more than 1 underlying asset)

• A good example of Hybrid Derivatives (or, in short, hybrids) are Convertible Bonds, Convertible Preferreds and other Convertible Arbitrage (Arb) trades (structure could include CDS, Options, etc.)

• They include also SCONes – Structured Convertible Option Notes – that are American Asset Swaptions on Convertible Bonds or Convertible Preferreds; methodology in this case could be split into an American Option valuation methodology (analytical or otherwise) and a Convertible Bond valuation methodology

• They include also Quanto Options, Performance Options, Variance, Volatility and Correlation Derivatives, FX Derivatives with Stochastic Interest Rates, TRS and Repos on them, etc.


managing market risk• Convertible Bonds are bonds that have embedded an option to convert a bond into

common stock at a certain Conversion Ratio (Strike) (i.e., when Conversion Ratio * Common Stock Price Per Share becomes greater than Bond Price)

• Convertible Preferreds are bonds that have embedded an option to convert a preferred stock into common stock at a certain Conversion Ratio

• Unlike a popular belief that it is an Equity Option that has to be valued, hedged and risked as such, we think of it as an option on Equity, CDS Spread (Option Adjusted Spread – OAS), Interest Rate (and FX Spot Rate if traded in foreign currency) accounting for volatilities and correlations for these 3 (or 4) variables that should be valued, hedged and risked as such (e.g., one should not only Equity Delta-hedge the equity risk of the Convertible Bond, but also CDS Spread Delta-hedge the CDS credit spread risk and Interest Rate Duration-hedge the Interest Rate risk of the Convertible Bond, as well as FX Spot Rate if necessary)

• Considering the above reasoning, Paul Wilmott proposed a PDE approach to compute a Convertible Bond MTM, assuming a lognormal distribution for a stock and a normal distribution with mean reversion (Hull-White model) for an interest rate; the following is this PDE (not accounting for liquidity yet):

• Here V is a price of a Convertible Bond, S is a Stock Price Per Share, r is a Credit-Risky Interest Rate, d is a Dividend Yield, a and b are mean reversion parameters, σ is a Stock Implied Volatility, w is an Interest Rate Implied Volatility and ρ is a correlation coefficient between the Stock and the Interest Rate

• Paul Wilmott’s PDE could be solved using a Crank-Nicholson (finite differences) method for either European, Bermudan or American type of optionality that is assumed to be occurring in a Convertible Bond

• In 2002, when CDS spreads of Convertible Bonds skyrocketed sometimes to over 1,000 bps (and their volatilities sometimes to over 200%!), the CDS spread component of a Credit–Risky Interest Rate became a risk factor of the highest volatility, so we suggested to generalize Wilmott’s model to 3 dimensions – CDS Credit Spread, Equity (Stock) Price Per Share and Credit Risk-Free Interest Rate (with possible further inclusion of FX Spot Rate if stocks and bonds are traded not in domestic currency); CDS Credit Spread could follow the following process: dC = (A – B*C)dt + Wc dZ + dP (here C is a CDS spread, A and B are mean reversion coefficients, Wc is Volatility of CDS spreads, dZ is a Wiener process & dP is a Poissonian jump-diffusion process); today it’s important too

• Floating Rate Convertible Bonds and Convertible Preferreds should be valued using the Paul Wilmott PDE above or its 3- and 4-dimensional generalizations; we used Fourier and other integral transforms, appropriate changes of variables and non-integral transforms to accomplish that to get a closed-form solution for European Optionality and an Analytical Approximation or otherwise for American and Bermuda Optionality

• Fixed Rate Convertible Bonds and Convertible Preferreds could be valued under certain assumptions using the following PDE (also not accounting for liquidity yet):

rVr

Vrba

S

VSdr

r

Vw

rS

VSw

S

VS

t

Vttt

tttt

t

)()(22 2

222

2

222

• This PDE differs from the previous one only by the 2 terms where a Floating Interest Rate r is replaced by a Fixed Interest (Coupon) Rate c (please note that a subscript t here and for other parameters means a term structure; as in a Black-Scholes PDE, this does not prevent the results’ analytical tractability)

• This equation could be solved in a closed-form fashion for European optionality and analytically or otherwise for American and Bermuda optionality using Fourier transforms and an appropriate change of variable

• My colleague Michael Pykhtin obtained the same result for Fixed Rate Convertible Bonds and Convertible Preferreds independently by using a risk-neutral valuation argument

• The approaches for valuation of Quanto Options (quantos) and Performance Options should be similar to the approaches for valuation of Fixed Rate Convertible Derivatives described above

• The approaches for valuation of FX Derivatives with Stochastic Interest Rates (TARNs, PDRCs, etc.) should be similar to the approaches for valuation of Floating Rate Convertible Derivatives described above

• As for any other multi-asset or hybrid product, TRS and Repos on them will just use their corresponding valuation models described above inside TRS and Repos valuation models

Vcr

Vrba

S

VSdc

r

Vw

rS

VSw

S

VS

t

Vttttt

tttt

t

)()(22 2

222

2

222


managing market risk• Variance, Volatility and Correlation Swaps are becoming more and more

liquid products lately (Variance Swaps market has even commoditized)• Variance and Volatility Swaptions market is evolving rapidly, as Variance and

Volatility Swaps represent natural hedges for Variance and Volatility Swaptions correspondingly, profit margins for these products could be higher and model development provided a good basis for their valuation, calibration and risk estimation

• The first approach that led to a practical solution for valuation of variance swaptions (as well as variance swaps) appeared in a 1998 article by Alex Lipton and Dmitry Pugachevsky “Pricing of volatility-sensitive products in the Heston model framework”

• In this article, the authors assume that a stock and its volatility follow the following system of stochastic differential equations (SDEs):

dS = r S dt + S√ v dZs dv = a (b – v) dt + w √ v dZv

corr(Zs,Wv) = ρ


managing market risk• Here S is a stock Price Per Share, v is the stock variance (volatility squared), r is a

risk-free rate (if we generalize to a non-zero dividend, r should be replaced by r – d where d is a dividend yield), a and b are mean reversion parameters, Zs and Zv are corresponding Wiener processes and ρ is a correlation coefficient between them

• All parameters are assumed to be constants, though this assumption could be generalized to all the parameters to be term-structured

• Then the approach for Variance Swaps and Swaptions was developed by both using a risk-neutral valuation argument (here the authors assumed a non-central χ 2 distribution with n degrees of freedom for a variance v (as a stock S is lognormally distributed) and construct a risk-neutral measure using this distribution and a Green function approach) and a PDE solution approach

• We suggest to further improve the approach by using an Enhanced Heston framework, writing a probability density function of a non-central χ 2 distribution with n degrees of freedom in a closed-form fashion and obtain an analytical solution for variance swaptions by using this probability density function and a probability distribution function obtained from it via a risk-neutral argument OR to use the PDE solution techniques described for Convertible Bonds and Preferreds to solve a Lipton-Pugachevsky PDE


managing market risk• There are several other approaches to the valuation of Variance and Volatility Swaptions

including a Carr-Lee approach that suggests to price Variance and Volatility Swaptions using a Variance and Volatility Swaps delta-hedging argument and a lognormal distribution of realized variance and our approach that generalizes it to an empirically proven lognormal distribution with mean reversion for a realized variance; both approaches yield a closed-form solution for a European exercise and an analytical approximation or otherwise for American and Bermuda exercise

• Correlation Derivatives is an evolving market and a great tool for hedging a correlation risk across asset classes; it also poses great modeling, valuation, risk estimation, calibration and IT challenges

• The development of Variance, Volatility and Correlation Derivatives markets and models started with an appearance of Variance Swaps on the market after they were designed and modeled by Emanuel Derman and other researchers

• Closed-form solutions and analytical approximations for Basket Options is a subject of our current research (a basket distribution is being derived)

• Same is true for Spread Options if each asset in the spread is considered separately; if the Spread is considered as an asset with a normal mean-reverted (say, Hull-White) distribution, there is a closed-form solution for valuation of European options and an analytical approximation and otherwise for American and Bermuda options


managing market risk• For multi-asset Credit Derivatives (especially CDOs, CDS baskets and

associated options), we have observed a tremendous shift from Monte Carlo – based methodologies to the analytical ones; no wonder, as to risk, say, a CDO, one needs a triple Monte Carlo – to compute a copula, to compute MTM and then all the risk numbers

• A closed-form result should be implemented to compute a copula (at least for a Gaussian copula)

• An analytical approximation should be used to compute a CDO MTM for risk purposes with further attempts to obtain a closed-form solution

• Same is true for Nth To Default Swaps and CDOs on CDOs (CDO squared), but valuation and especially risk estimation remains challenging, especially for CDO squared

• Other approaches like Random Factor Loading should be applied, especially after a Correlation Crunch in May 2005


managing market risk• CDS on ABS (including CDOs), MBS and CMBS should use a closed-form CDS

valuation model and closed-form or analytical approximation models for ABS, MBS and CMBS, existing or to be developed including a correlation structure of an underlying instrument

• Associated options for Credit Derivatives should be treated similarly to Credit Default Swaptions (CDSOs) where CDS spreads follow the same SDE with Poissonian jump-diffusion process as we showed for Convertible Bonds and Preferreds, so instead of solving a 1-dimensional partial integro-differential equation (PIDE) for CDSOs, multi-dimensional PIDEs need to be solved for multi-asset Credit Derivatives with optionality

• These PIDEs solution process should use similar techniques to the solution of PDEs described above, as an additional element there is a convolution of a solution and a Poissonian jump-diffusion distribution density function, and its Fourier-image is a product of both Fourier-image of a solution and a Fourier-image of the Poissonian jump-diffusion distribution density function

• Another approach to the valuation of Credit Derivative Options is extending a Merton-76 Jump-Diffusion model to a multi-dimensional case to derive analytical approximations and closed-form solutions in partial cases; we continue this research after successful results for a one-dimensional case


managing market risk• MBS, ABS and options on them remain a big challenge in terms of modeling,

valuation, calibration, risk estimation and data processing almost as much as 20-30 years ago being (together with CDOs) the most complex derivative securities traded, especially during current financial crisis

• These trades are usually valued using Monte Carlo simulation, regular or enhanced, and risked using a duration-convexity approximation (otherwise, it would be Monte Carlo on Monte Carlo and take days to complete a computation for a typical portfolio)

• Calibration and data processing also pose a challenge, though some third party vendors (including Intex Solutions) are successfully overcoming it

• Having the data problem resolved (these trades could have 100-200 transaction details!), one could start developing analytical valuation approaches for these extremely complex derivative securities

• The first approach is a natural development of numerous attempts to enhance Monte Carlo simulation in valuation routines for MBS, ABS and options on them via importance sampling, principal components, variance reduction, stratified sampling, etc.


managing market risk• These enhanced Monte Carlo methodologies brought up a question of whether

it is feasible to compute multi-dimensional integrals that are used to value these derivatives analytically including the methodology of lowering a dimensionality of integration

• We generalized the results obtained by Brodie Johnson in his 1992 paper “Conditional Stratified Sampling: A New Variance Reduction Technique” and computed certain integrals in a closed-form fashion and some – as analytical approximations; we conduct further research on how to compute analytically prices of various types of MBS, ABS and options on them using integral transforms, operator approach and orthonormal bases approach

• A second approach is to construct a PDE or a system of PDEs for such a derivative security and solve it using techniques similar to the ones used for valuation of Convertible Derivatives

• These PDEs will be similar to a Paul Wilmott PDE for Convertible Bonds with also similar condition at maturity (for European optionality)

• It means that we will have a closed-form solution for European optionality and analytical approximation or otherwise for American and Bermuda optionality


managing market risk• The risk factors (with SDEs for them) are Price(s) of the Underlying Asset(s)

and Loan (Mortgage) Rate; Rental Rate could be an additional parameter• The third approach is also PDE-based, but is different from the second one:

this time, we consider an MBS/ABS as a function of Prepayment Speed(s) (lognormally distributed with mean reversion), Option-Adjusted Spread(s) (OAS) and/or CDS Spread(s) for ABS (normally distributed with mean reversion and possibly volatility smile and jump-diffusion Poissonian process) and Interest Rate(s) (normally distributed with mean reversion) in a correlated fashion

• The model will depend also on Weighted-Average Coupon (WAC), Weighted-Average Maturity (WAM), Original Principal, Fixed Coupon Rate, Pass-Through Rate, etc.

• This PDE will be forward-looking, as we do not know a payoff function at maturity or any exercise date (and it may be many maturities of the mortgages in the pool, so it’s reasonable to talk about WAM), though we know the MBS/ABS value at inception, and it equals to 0

• This PDE is non-homogeneous, unlike all linear PDEs we considered before


managing market risk• Nevertheless, all the approaches that we used before are relevant – Fourier and other

integral transforms, one-dimensional and multi-dimensional integration techniques including lowering of dimensionality of integration, appropriate changes of variable, non-integral transforms, operator approach, orthonormal bases approach, etc.

• We are conducting further research in both the application of all these techniques to valuation and risk estimation of all the exotic products and further development of all these mathematical approaches themselves

• And the last example: a new market has evolved lately – multi-asset Commodity and Energy Derivatives and hybrid products that include commodity underlying instruments

• These underlyings are modeled as lognormally distributed with mean reversion, term-structured coefficients and jumps, and their spreads are modeled as normally distributed with mean reversion, term-structured coefficients and jumps

• These Commodity and Energy Derivatives could be modeled then using multi-dimensional integration and/or solving PIDEs similarly to the described above approaches to analytical valuation of Convertible Bonds, Credit Derivatives Options and MBS/ABS and options on them

• We are continuing our research by enhancing the approach suggested by Dr. Deng in the paper “Stochastic Models of Energy Commodity Prices and Their Applications: Mean Reversion with Jumps and Spikes”; for a case that was thought to have only a numerical solution we already found an analytical one

Single-trade level and portfolio level market risk methodologies

• Computation of VaR and Expected Shortfall for multi-asset and hybrid derivative products remains a challenge both computationally and in terms of data integration, so mostly these risk numbers are obtained using Monte Carlo or Historical Simulation

• The development is being done, though, using enhanced Monte Carlo techniques and developing analytical methodologies based on some successes on one-dimensional derivatives

• Analytical valuation models for multi-asset derivatives are very helpful for these computation, as well as analytical risk estimation methodologies

• We suggest to compute VaR, Expected and other market risk measures by using, when possible, closed-form solutions, analytical approximations or enhanced Monte Carlo simulations together with definitions of these risk measures and portfolio valuation (pricing) functions obtained as solutions of the PDEs of the type presented on Page 19 where liquidity of this derivatives portfolio (or a multi-asset derivative) to be risked is priced into this equation

• Though today Market Risk (and Counterparty Credit Risk) of portfolios of multi-asset and hybrid derivatives is computed primarily using Monte Carlo

Single-trade level and portfolio level market risk methodologies

Simulation, Historical Simulation, Delta (Duration) Approximation or Delta-Gamma (Duration-Convexity) Approximation, a lot of research is being conducted to derive analytical methodologies of computing Portfolio Market and (Counterparty) Credit Risk for multi-asset and hybrid products

• We participate in this research trying to expand our analytical results and approaches from one-dimensional products to multi-asset and hybrid (multi-dimensional) products and their portfolios and incorporate liquidity there too

• Multi-dimensional integration techniques, lowering of integration dimensionality, operator theory and its application to integrals computation and solution of Integral Equations, PDEs and PIDEs, orthonormal bases, their properties of integration and differentiation, integral and non-integral transformations & appropriate changes of variables are among the approaches to be taken for the purpose of analytical modeling, valuation and risk estimation for both one-dimensional and multi-dimensional (multi-asset and hybrid) derivative products and their portfolios, accounting for liquidity too

• For that purpose, it is also crucial to have proper trade data, market data and calibration parameters, especially volatility and correlation structures

• These analytical approaches to valuation & measuring the market risk of derivatives (portfolios) with liquidity dynamics are crucial in today’s market

References

1. Lipton, A. and D. Pugachevsky, Pricing of volatility-sensitive products in the Heston model framework, July 29, 1998.

2. Carr, P. and R. Lee, Realised volatility and variance: options via swaps, Risk Magazine, May 2007.

3. Laurent, J.-P. and J. Gregory, Basket Default Swaps, CDO’s and Factor Copulas, September 2003.

4. B. Johnson, Conditional Stratified Sampling: A New Variance Reduction Technique, November 1992.

5. Kau, J. B. and D. C. Keenan, An Overview of the Option-Theoretic Pricing of Mortgages, Journal of Housing Research, Volume 6, Issue 2, Fannie Mae 1995.

6. Papi, M. and M. Briani, A PDE-Based Approach For Pricing Mortgage-Backed Securities.

7. Shklyarevsky, A. and A. Khazan, Volatility smile using real market distribution, trading risk management and euro issues, Euromarket 2000, London, 6/1999, p.141.

References

8. Shklyarevsky, A. and A. Khazan, Volatility models, trading risk management and the ESFM, European Single Financial Market, Euromoney Publications PLC, London, 9/98, pp 119-125.

9. Shklyarevsky, A., A. Khazan, S. Shuster and L. Gorbonosov, Advances in Derivatives, Economic and Monetary Union: Perspectives for business and finance, Euromoney Publications PLC, London, 7/97, pp. 89-104.

10. Nagar, D., B.Potter and A. Shklyarevsky, A Brief Chronicle of Option Valuation Models, Euromoney Derivatives Handbook 1997, 1/97, pp 27-37.

11. Derman, E., M. Kamal, J. Zou and K. Demeterfi, A Guide to Volatility and Variance Swaps, The Journal of Derivatives, Summer 1999.

12. E. Derman, Outperformance Options, Goldman Sachs & Co, January 1992.13. Bardhan, E., A. Bergier, E. Derman, C. Dosembet and I. Kani, Valuing Convertible Bonds as

Derivatives, Goldman Sachs & Co, Quantitative Strategies Research Notes, November 1994.

14. P. Wilmott, Derivatives: The Theory and Practice of Financial Engineering, pp. 468-471, University Edition, John Wiley & Sons 1998.

References

15. M. Davis, Discounting in Convertible Bonds.16. S. Deng, Stochastic Models of Energy Commodity Prices and Their

Applications: Mean Reversion with Jumps and Spikes, Industrial and Systems Engineering, Georgia Institute of Technology, October 1999.

17. C. F. Lo et al, Pricing multi-asset financial derivatives with time-dependent parameters – Lie algebraic approach, Hong Kong 2002.

18. Derman, E. and I. Kani, The Volatility Smile and its Implied Tree, Goldman Sachs & Co, Quantitative Strategies Research Notes, January 1994.

19. A. Krakovsky, Pricing Liquidity Into Derivatives, Risk, December 1999, Volume 12.

20. Bakstein, D. and S. Howison, A Non-Arbitrage Liquidity Model with Observable Parameters for Derivatives, Oxford Mathematical Institute.

21. Federal Reserve System, Market and Liquidity Risk Assessment Overview.

measuring the market risk of illiquid positions alexander shklyarevsky, senior vice president senior...

Documents

risk var

measuring market risk

potential change

relative change

market risk overview

market rates

market risk singletrade

major measure of market