two factor modelling of credit derivatives: a numerical
TRANSCRIPT
Two Factor Modelling of Credit Derivatives:
A Numerical Implementation
MARIO PETRUCCI
21 April 2004
Introduction
A credit derivative may be defined as a financial product whose value is driven by the likelihood of default of an underlying asset.
Total rate of return swapsCredit spread productsCredit default products
Introduction
A Credit Default Swap is a bilateral contract where one counterparty (the credit protection buyer) pays a periodic fee to the other (the credit protection seller) in order to receive an agreed amount if a credit default event occurs. If the fee is paid upfront, this agreement is usually called a Credit Default Option.
IntroductionReference entity Reference asset (Delivery option)Credit event Materiality Publicly available information
Default payment : in general, 3 different types of payouts are negotiable:
(initial price minus post-default price) (par minus post-default price) digital cash payment
IntroductionThere are two main approaches to default modelling: the structural approach and the reduced form approach.The first attempt to model default risk with a structural approach was by Merton (1974). In this model a firm has got a liability in the form of discount bond and the asset value follows a (geometric) Brownian motion. At the time of servicing the debt, the firm defaults if the asset value is below the face value of the bond. This approach is very intuitive but do not take into account observable market prices, in particular the current credit spread. Therefore its applicability for pricing traded instruments is limited. Jarrow and Turnbul (1995) proposed a different type of approach. It is called reduced form approach because there is not attempt to model assets or liabilities dynamics : default is a surprise event which is modelled as the first jump time of a Poisson process. A risk adjusted hazard rate is derived from market data. Duffie and Singleton (1999), Lando (1998) extended this work: default is a stopping time in a process with (risk adjusted) stochastic intensity.Tavella and Randall (2000) introduced a second risk factor (the short term rate) and presented a finite difference solution of this problem.
Arbitrage, Martingales and PDE
Theorem (Harrison-Pliska) : A market consisting of financial instruments and a numeraire instrument is arbitrage free if and only if there exists a measure with respect to which the prices of all (tradable) financial instruments normalised with thenumeraire instrument are martingales. This measure is unique if and only if the market is complete.
Ε=
)()(
)()(
TYTV
tYtV P
Arbitrage, Martingales and PDEFor a general discussion on this topic see Rebonato(1998 and 2002) and Deutsch (2001). A numeraireY(t) is a generic financial instrument, tradable in the market , with strictly positive pay-out. V(t) is a generic tradable asset. Arbitrage free market means that it is not possible to set up a portfolio at zero cost today, that will exhibit in the future only positive (or zero) values with probability 1. Measuremeans a probability distribution: probability distributions can be interpreted as measures since the expectation of a function F(X) of a random variable X, distributed with density p, can be interpreted as an integral with respect to a certain integral measure.
Arbitrage, Martingales and PDE
Absence of arbitrage opportunities in the economy implies that normalised prices of alltradable instruments are martingales with respect to the same probability measure.
A market is complete if and only if any traded asset can be replicated (hedged) with a self-financed portfolio of other assets. Heuristically speaking we can say that an arbitrage free market is complete (or equivalently that the martingale measure is uniquely determined) if and only if the number of traded assets, whose prices are exogenously given, is equal to the number of random sources.
The Risk Factors Driving the Market
The market for credit derivatives is modeled as follows : V(t) is the price at time t of a generic credit derivative. V(t) is driven by two risk factors:Factor 1: Poisson process with stochastic intensity p Factor 2 : Diffusion process of short term rate r A Poisson process x(t) models the event of default of the Obligor of the reference asset: default is the first jump time. The intensity of the process p represents the instantaneous probability of default of the Obligor. The intensity p is stochastic and driven by a Brownian motion.
The Risk Factors Driving the Market
Poisson process: [p=0.3 ,dt= 0.01]
-1-0.5
00.5
11.5
22.5
33.5
0 0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
5.4
6.1
6.7
7.3
7.9
8.5
9.1
9.7
time
x(t)M(t)
The Risk Factors Driving the Market
1dWdtdp δγ +=
2wdWudtdr +=dtdWdW ρ=Ε )( 21
)),(()),(( −−− ttxVttxV
If default occurs, the value of V(t) jumps instantaneously by an amount:
Martingale Pricing MethodWe will now focus on the simplest credit derivative product: a discount bond subject to default risk. If we assume, for the moment, that V vanishes instantaneously if default occurs (this assumption will be relaxed later to incorporate a fixed recovery rate), we get:
)()),(()),(( tVttxVttxV −=− −−
The price of an asset today can be derived as an expectation (both globally and locally) of the relative price in the future,taken over the martingale measure associated to a chosen numeraire. We will choose as numeraire for these calculations a money market account rolling at the risk free rate r. With this choice of numeraire the associated probability measure is referred to as the Risk Neutral Measure (or Q-Measure).
Martingale Pricing MethodThe price of an asset today can be derived as an expectation (both globally and locally) of the relative price in the future, taken over the martingale measure associated to a chosen numeraire. In switching from one equivalent measure to the other the drift of the various financial quantities (in our case the drifts: will in general vary, but their volatilities will not (Girsanov Theorem).
dtrVwdt
rpVwdt
pVdr
rVdp
pVdt
tVtdV 2
22
21
2
2
22
21)(
∂∂+
∂∂∂+
∂∂+
∂∂+
∂∂+
∂∂= δρδ
VdMpdtttxVttxV +−+ −− )]),(()),(([
Martingale Pricing Method
pdtttxVttxVttxVttxVtdM V )]),(()),(([)),(()),(()( −−−− −−−=
212
22
2
2
2
22
2)( wdWr
dWp
dtr
wrp
wpr
upt
tdV∂∂+
∂∂+
∂∂+
∂∂∂+
∂∂+
∂∂+
∂∂+
∂∂= δδρδγ 11 VVVVVVVV
VdMpdtttxVttxV +−+ −− )]),(()),(([
Martingale Pricing MethodUnder the Q-Measure (locally) with the money market account as numeraire, the risk neutral expectation of dV(t) must be equal to rV(t)dt, as V(t) is tradable.
The new drifts can be thought as the original drifts under the initial probability measure adjusted by two parameters known as market price of jump risk and market price of interest rate risk. The role of these parameters is to allow the switch into the Q-measure and they are uniquely determined if the market is complete.
0)()()( 2
22
21
2
2
22
21 =+−
∂∂+
∂∂∂+
∂∂+
∂∂−+
∂∂−+
∂∂ Vpr
rVw
rpVw
pV
rVwu
pV
tV
rp δρδλδλγ
PDE ApproachWe will assume the existence of two traded assets: a zero couponbond not affected by default risk : and a zero coupon bond affected by default risk:
),( trZZ =
),,( tprZZ=
),,(),(),,( 21 tprZtrZtprV ∆−∆−=Π
ZddZdVd 21 ∆−∆−=Π
rZ
rZ
rV
pZpV
∂∂
∂∂∆−
∂∂
=∆
∂∂∂∂
=∆
2
1
2
PDE Approachsetting :
We can easily allow (in both derivations) for a (instantaneous) fixed recovery value for the bond in default: if a quantity R is recovered in default, we get an additional term pR into the PDE; if we further assume we get:
This is the fundamental equation that a generic bond traded in the market will satisfy if we assume absence of arbitrage. The particular functional form chosen for the (arbitrary) risk neutral drifts will determine, in conjunction with the exogenous market prices of the two benchmarks, the arbitrage free evolution of the (instantaneous) probability of default and short term rate.
dtrd Π=Π
VR α=
))1(()()(2
22
21
2
2
22
21 −+−
∂∂−+
∂∂−+
∂∂+
∂∂∂+
∂∂+
∂∂ Vpr
pV
rVwu
pV
prVw
rVw
tV
pr αδλγλδδρ 0=
Numerical ImplementationWe will now assign a functional form under the Q measure to the stochastic processes for p and r. As mentioned, the risk neutral drifts will be different from the initial drifts under the physical measure but their volatilities will remain invariant.
As we are working under the assumption that there are only two benchmark bonds, exogenously given, we can use the two (arbitrary) constant levels of mean reversion to calibrate the model.
)()( 11 pkp −=− θδλγ
)()( 22 rkwu r −=− θλ
11111)( dWppkdp βσθ +−=
22222)( dWrrkdr βσθ +−=
Numerical Implementation Implementation 1: Martingale Pricing Method
Use system of SDEs with risk neutral drifts.Draw random variable Normal (0,1) using the Polar Marsaglia method.Simulate correlation using two-dimensional Cholesky’s decomposition.Given initial values for r and p , check if jump occurs using condition:
Uniform<p(1-alpha)dtIf jump occurs (during the time interval) set r =infinity, otherwise draw new value for p and r using dW1 and dW2.Repeat recursively the process for each time interval.Calculate simulated path for :
Finally, generate n paths and take average value. This is our numerical value for the risky zero coupon bond.
∑−T
t
ssr
eδ)(
VBA Code (Fragment)dt = expiry / samplesFor I = 1 To paths ' outer monte carlo loop'simulate a single pathsum = 0rate1 = hazardrate2 = spotFor j = 1 To samplesdriftterm1 = (kappa1 * (theta1 - rate1)) * dtranscale1 = vol1 * (rate1 ^ beta1) * Sqr(dt)driftterm2 = (kappa2 * (theta2 - rate2)) * dtranscale2 = vol2 * (rate2 ^ beta2) * Sqr(dt)temp = Marsaglia()rate1 = rate1 + driftterm1 + ranscale1 * temp
VBA Code (Fragment)rate1 = rate1 + driftterm1 + ranscale1 * tempIf Rnd() < rate1 * (1 - alpha) * dt Thenrate2 = 1000000Elserate2 = rate2 + driftterm2 + ranscale2 * (temp * rho + Marsaglia() * (1 - rho * rho) ^ 0.5)End Ifsum = sum + rate2 * dtNext jpath = Exp(-sum)runsum = runsum + pathNext Imc_risky2factorh = runsum / paths
Numerical ImplementationImplementation 2: Solving the PDEUse two-dimensional system of SDEs for dr and dp (neglecting dx) with risk neutral drifts.Draw random variable Normal (0,1) using the Polar Marsaglia method.Generate correlated random walks for p and rCalculate simulated path for
Finally, generate n paths and take average value: this is our numerical value for the risky zero coupon bond.
∑ +−−T
t
ssrsp
eδα )]()1)(([
Zero Coupons BondsGiven the fact that the BPE admits exact formulae for the price of a zero coupon bond, once the risk-adjusted form of the parameters is specified, it is interesting to compare our numerical output with some closed form solutions, as a reality check. In particular if we chose the combination:p = constant , We should expect to retrieve the exact CIR (Cox Ingersoll Ross) solution for a zero coupon bond (adjusted by a factor ) as limit value for our numerical solution.
2222 )( dWrrkdr σθ +−=
))(1( tTpe −−− α
FACTOR 1 FACTOR 2
Hazard Rate : 0 .010 Re c ove ry Rate : 0 .500 Inte re s t Rate : 0 .030Kappa1: 0 .000 Corre lation: 0 .000 Kappa2: 0 .250The ta1: 0 .000 Expiry: 5 The ta2: 0 .035Be ta1: 1 .000 S am ple s : 100 Be ta2: 0 .500S igm a1: 0 .250 Paths : 900 S igm a2: 0 .030
M1: 83 .15% Us e s Ma rtinga le P ricing Me thod a nd P ois s on P roce s s
M2: 83 .09% Us e s Fe ynm a n-Ka c Form ula To S olve P DE
Exac t: 83 .07% For Cons ta n t p (CIR) be ta =0 .5 83 .07% For Cons ta n t p (Adjus te d Va s ice k) be ta =0
Ris k Ne utra l1 : dp = [ka ppa 1*(the ta 1 - p )]d t+s igm a 1*p^be ta 1*dW1
Ris k Ne utra l2 : dr = [ka ppa 2*(the ta 2 - r)]d t+s igm a 2*r^be ta 2*dW2
0.01
0.035
0.05
0.070.01
0.030.06
0.5500.6000.6500.7000.7500.8000.8500.900
spot rate
hazard rate
Tw o factor valuation - defaultable zero coupon
Valuing a Credit Default OptionWhen pricing a credit default option we need to specify the behavior of the reference asset when an event of default occurs.The value of the reference asset under default is equal to a fraction of its face value (100%)We therefore have two different uses of the recovery rate. A rateα is used to price a zero coupon bond issued by the reference obligor and to determine the risk neutral values for the parameters that regulate the dynamics of the (instantaneous) probability ofdefault and the short term rate. A second rate α* is used to model the reference asset under default and to determine the pay-out of the default option.
Valuing a Credit Default OptionRecovery rates are a function of the credit quality of the issuer and of the seniority and security of the particular bond issued (Senior Secured, Senior Unsecured, Subordinated).
In our numerical implementation we consider the recovery rate as a constant. It represents the average recovery value that a reference asset of a given credit quality is expected to experience under default. Under this assumption the jump component of the model has got a fixed amplitude. An hedging portfolio composed of two traded asset can be constructed to cope with the model
incompleteness induced by two non-tradable risk factors (r and p).
Valuing a Credit Default OptionMaking use of the Harrison-Pliska theorem we can determine the value of a credit default option in the following way:
Using the money market account as numeraire (equal to 1 at time t) and choosing the time of default of the reference credit as the
observation point in the future, we get:
Ε=
)()(
)()(
TYTV
tYtV Q
∫Ε=
Dt
t
dssr
DQ
e
tVtV)(
)()(
Valuing a Credit Default OptionUnder the assumption that the reference asset in default has got a recovery value (expressed as percentage of its face value) equal toα* this pay-out can be expressed as :
at time , or zero if no defaultDt
This expression can be easily modified to incorporate a digital payment or any pre-determined amount (e.g initial price of reference asset minus recovery value). In our numerical calculation we will stick to the above pay-out which is the standard market practice.
)1( *α−
Valuing a Credit Default OptionUse risk neutral SDE for p.Simulate path using Normal(0,1) and the Polar Marsaglia method.Check if jump occurs using condition : Uniform < pdt.Exit loop and determine td when jump occurs. Use risk neutral SDE for r and simulate path using N(0,1) and Polar Marsaglia method. Stop at tD (if a jump occurred) or set tD=T (if no jump occurred).Calculate numeraire as :
∑Dt
t
ssr
eδ)(
Valuing a Credit Default OptionCalculate default option path as:
,or zero− teα * )1(
Generate n paths and take average value. This is the numerical value of a credit default option.
∑−Dt
ssr δ)(
FACTOR 1 FACTOR 2
Hazard Rate : 0 .060 a lpha* (re f. a s s e t):0 .25 0 Inte re s t Rate : 0 .0 30Kappa1: 0 .300 ZCB re c ov. ra te : 0 .30 0 Kappa2: 0 .2 50The ta 1 : 0 .060 Corre la tion: 0 .00 0 The ta 2 : 0 .0 35Be ta 1 : 1 .000 Ex piry: 6 Be ta 2 : 0 .5 00S igm a1: 0 .290 S am ple s : 100 S igm a2: 0 .0 30
Paths : 900 0
C a lib ra tion to 6 YR Dis coun t Bon d Be nchm a rks :
R is ky Ze ro : 64 .18 % YTM: 7 .39 %R is k Fre e Ze ro : 82 .34 % YTM: 3 .24 %
De fa ult Option: 20 .4 8%
p / r 0 .01 0 .03 0 .060 .0 1 13 .97 % 13 .19% 1 2 .64 %0 .0 6 21 .21 % 20 .39% 1 9 .41 %0 .0 9 24 .52 % 24 .51% 2 2 .62 %
0.010.03
0.060.01
0.060.09
10.00%12.00%14.00%16.00%
18.00%20.00%
22.00%
24.00%
26.00%
spot rate
hazard
rate
Default Option
An Upper Bound Value The expression:
can be interpreted as the probability that a jump will NOT occur during the time interval (t, T).This suggests that an upper bound value can be calculated if we ignore the normalizing effect of the numeraire.
∫−T
t
dssp
eE)(
An Upper Bound Value
The expected value of the pay-out of the option is simply given by :
∫−−
−T
t
dssp
eE)(
* 1)1( α
VBA Code (Fragment)
For i = 1 To paths ' outer monte carlo loop simulate a single pathsum = 0rate1 = hazardFor j = 1 To samplesdriftterm1 = (kappa1 * (theta1 - rate1)) * dtranscale1 = vol1 * (rate1 ^ beta1) * Sqr(dt)rate1 = rate1 + driftterm1 + ranscale1 * Marsaglia()sum = sum + rate1 * dtNext jpath = Exp(-sum)runsum = runsum + path Next imc_upperbound = (1 - alpha) * (1 - runsum / paths)
F AC T O R 1 F AC T O R 2
Ha z a rd R a te : 0 .0 6 0 a lp h a * (re f . a s s e t ) : 0 .2 5 0 In te re s t R a te : 0 .0 3 0Ka p p a 1 : 0 .3 0 0 ZC B re c o v . ra te : 0 .3 0 0 Ka p p a 2 : 0 .2 5 0Th e ta 1 : 0 .0 6 0 C o rre la t io n : 0 .0 0 0 Th e ta 2 : 0 .0 3 5B e ta 1 : 1 .0 0 0 Ex p iry : 6 B e ta 2 : 0 .5 0 0S ig m a 1 : 0 .2 9 0 S a m p le s : 1 0 0 S ig m a 2 : 0 .0 3 0
P a th s : 9 0 0 0
C a lib ra tio n to 6 YR D is c o u n t B o n d B e n c h m a rk s :
R is k y Z e ro : 6 4 .1 5 % YT M: 7 .4 0 %R is k F re e Z e ro : 8 2 .3 4 % YT M: 3 .2 4 %
D e fa u lt O p t io n : 2 0 .5 4 %
U p p e r B o u n d : 2 2 .4 8 %
M a tu rity D e fa u lt O p tio n U p p e r B o u n d1 3 .9 8 % 4 .3 7 %2 8 .0 2 % 8 .4 5 %3 1 1 .7 8 % 1 2 .3 0 %4 1 5 .1 0 % 1 5 .9 3 %5 1 7 .6 1 % 1 9 .3 2 %6 2 0 .5 1 % 2 2 .4 6 %7 2 2 .4 3 % 2 5 .4 0 %8 2 5 .1 5 % 2 8 .2 9 %9 2 7 .2 4 % 3 0 .9 5 %
1 0 2 9 .0 6 % 3 3 .4 1 %
D e fa u lt O p t io n V a lu a t io n
0 .0 0 %5 .0 0 %
1 0 .0 0 %1 5 .0 0 %2 0 .0 0 %2 5 .0 0 %3 0 .0 0 %3 5 .0 0 %4 0 .0 0 %
1 2 3 4 5 6 7 8 9 1 0
M a t u r it y
D e fa u l t O p tio nU p p e r B o u n d
Conclusion• Given the fact that the two driving risk factors in our model (hazard rate and
short term rate) are not tradable, the functional form of the two associated risk neutral SDEs can be introduced by the user without violating any no arbitrage principle. Our numerical calculations converge correctly to some analytic solutions (e.g. adjusted CIR bond formula) once the appropriate functional form is chosen.
• The presence of credit risk in the form of (discontinuous) jump risk in addition to (continuous) interest rate risk does not represent an obstacle when performing risk neutral valuation of a generic derivative product. A unique arbitrage free price for the instrument can be determined without making any assumption about market agents attitudes toward jump risk as long as the model is completed by the market prices of two exogenously given traded assets.
Conclusion• Two equivalent pricing approaches can be employed to determine
the arbitrage free price of a derivative product : PDE approach that builds a replicating portfolio and Martingale Pricing Method that uses a risk neutral expected value normalised by a numeraire. We have shown as these two approaches lead to the same numerical
valuation of zero coupon bonds. • A credit default option can be finally priced using risk neutral SDEs
calibrated to observed zero coupon prices to evolve the driving risk factors and the money market account as numeraire. We have also presented an upper bound valuation for the option price based solely on risk adjusted cumulative probability of default as a reality check.