Download An Analytic Framework for Computing Value-at

An Analytic Framework for
Computing Value-at-Risk in
Incomplete Markets with Credit Risk
Mårten Grebäck
Abstract
This master thesis presents a framework for analytic computation of Valueat-Risk for a portfolio of arbitrary derivatives. The underlying securities are
modelled by jump-diffusion processes, where the number of jumps are Poisson
distributed and the diffusion processes are driven by Brownian motions. The
method is based on a delta-gamma approximation of the portfolio value and
Fourier inversion of the characteristic function. In the second half of the thesis
credit risk is introduced by a default event indicator variable. In case of a
default, a certain fraction of the defaulted option’s value is subtracted from
the portfolio value. Several examples are presented illustrating the effects on
Value-at-Risk with this extension.
Acknowledgements
I would like to thank my supervisor Henrik Hult, at the institution of mathematical statistics, KTH, for always giving me guidance when needed. Mr Hult’s
remarks and help has been invaluable. I would also like to thank Kaj Nyström
and Jimmy Skoglund at the financial risk control group of Swedbank/Föreningssparbanken for helping me with the subject of my thesis. I am also most
grateful for the motivating company and useful opinions of my fellow-students.
Contents
Introduction
7
1 Preliminaries
9
1.1
Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2
Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.3
Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.4
Delta-Gamma Approximation . . . . . . . . . . . . . . . . . . . .
14
2 The Framework
17
2.1
Portfolio Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
Delta-Gamma Approximation . . . . . . . . . . . . . . . . . . . .
18
2.3
Characteristic Function . . . . . . . . . . . . . . . . . . . . . . .
18
2.4
Fourier Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.5
Conditional Value-at-Risk . . . . . . . . . . . . . . . . . . . . . .
21
2.6
Delta-Gamma-Vega Approximation . . . . . . . . . . . . . . . . .
22
2.7
Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.8
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3 Credit Risk
3.1
Portfolio Value . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
29
3.2
Stochastic Intensity . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.3
Delta-Gamma Approximation . . . . . . . . . . . . . . . . . . . .
31
3.4
Default Component . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.5
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
4 Conclusions
37
A Examples: Market Risk
39
B Examples: Credit Risk
45
Bibliography
51
Introduction
Sensitivity measures such as delta, gamma and vega1 describes different aspects
of a portfolio of financial derivatives. Every night, banks (and other financial
institutions) calculate these measures, which results in an enormous amount of
data. These data are valuable for each specific trader in managing their allocated
risk. But the Greeks are not sufficient for senior management in measuring the
risk in a bigger perspective, interesting for the bank as a whole. Value-at-Risk
tries to provide such a measure. The Value-at-Risk (VaR) framework is not only
being used by financial institutions, but by central bank regulators as well. They
use it in determining how much capital a bank must keep to protect themselves,
and their customers, from future losses.
Simulations of VaR, for a large portfolio, often takes hours to run and demands
a great deal of computational power. An analytic framework for computing
VaR would make life easier for many traders and risk managers. Duffie and
Pan (2001) presents such a framework. In this master thesis we implement this
approach and investigate its pros and cons. Extensions are made for managing
vega risk and the computation of conditional Value-at-Risk.
The underlying securities are modelled by jump-diffusion processes, where the
number of jumps are Poisson distributed and the diffusion processes are driven
by Brownian motions. The size of the jumps are assumed to be Gaussian.
The portfolio value (or return) at time T , where T is the time horizon for the
VaR computation, is approximated by a second order Taylor expansion. This
approximation is called the “delta-gamma” approximation. From this form we
are able to reach the characteristic function, which after Fourier inversion gives
us the cumulative distribution function. This is carried out in Chapter 2.
1 Delta (∆), vega (ν), theta (Θ), rho (ρ), and gamma (Γ) are usually referred to as the
Greeks. They are defined as the derivative of the portfolio value w.r.t., respectively, the
underlying price, the volatility, the time, and the interest rate. Γ is defined as the second
derivative of the portfolio value w.r.t. the underlying price.
7
Market risk as well as credit risk is managed within the framework. In Chapter 3
we introduce credit risk. In case of a default the market value of the portfolio is
decreased by some exogenously given fraction of the time T value of the defaulted
contract. Individual, as well as common, default intensities are modelled with
Cox-Ingersoll-Ross processes. Basically, the same procedure as in Chapter 2 is
repeated, but with some modifications for incorporation of credit risk.
Examples can be found at the end of each chapter, and in the appendices.
8
CHAPTER
1
Preliminaries
1.1
Value-at-Risk
In the mid-nineties Value-at-Risk became a widely used measure for managing
market risk. What made VaR exceptional was that it provided an easily understood, single number, measure for the risk of a portfolio. The definition of VaR
is:
Given a time horizon T and confidence level p, Value-at-Risk is the value x such
that the probability of loosing more than x, during the time interval [0, T ], is less
than or equal to 1 − p.
In other words it gives you a lower limit of what you, with some given probability,
can expect your portfolio to be worth at some future time T . Commonly used
time horizons and confidence levels are, respectively, 1 day or 2 weeks and 1
or 5 percent. Two portfolios with exactly the same VaR may have different
potential losses beyond the quantile of interest. Artzner et al. (1999) suggests
a, by their definitions, coherent risk measure called conditional Value-at-Risk 1
to cover this shortage. Conditional Value-at-Risk (C-VaR) tells us how much
we can expect to loose, given that we are in the far end of the tail (e.g. beyond
the 5% quantile).
There are two main approaches for calculating VaR. The historical simulationand the model building approach. In the historical simulation approach past
data is used in the following manner. Suppose that we want to calculate the
one day 99%-VaR and that we have access to data for the last two years, or
500 days.2 We start by identifying market variables that affect the portfolio,
e.g. equity prices and interest rates, and note their movements. This gives
1 Other
2 One
names that are used are Tail Conditional Expectation and Tail VaR.
year has approximately 250 trading days.
9
1.1. VALUE-AT-RISK
CHAPTER 1. PRELIMINARIES
Probability density function
99%
Value−
at−Risk
0
Figure 1.1: 99% Value-at-Risk over some time horizon T .
us 500 scenarios for how the price of our portfolio may change from today till
tomorrow. For each scenario we calculate tomorrow’s portfolio value. The 1%
quantile, or the fifth worst scenario, tells us the 99%-VaR.
For an analytical model the historical approach is of no interest. The biggest
concern using the model building approach is the derivation of the probability
density function of the portfolio. For an approximation of the density function
certain assumptions have to be made about the movements of the underlying
risk factors, and how these movements will affect the value of the portfolio
(i.e. the pricing model). In most models it is assumed that the underlying
risk factors, in one way or another, are normally distributed. It is known that
the “normal” assumption often leads to under-estimation of potential losses.
To come to terms with that problem the usage of fat tailed distributions have
arisen.
Fat tailed distributions implies larger VaR at high confidence levels. Student’s-t
distribution (or just the “t-distribution”) is one example of a distribution with
fat tails. The probability density function, ft (x), of the t-distribution is defined
by
ft (x) = Tn
x2
1+
n
− n+1
2
, −∞ < x < ∞ ,
where n is the degrees of freedom (d.f.) and Tn is a constant depending on the
exogenously given variable n. As d.f. goes to infinity, student’s-t approaches
the standard normal distribution. Figure 1.2 illustrates the difference between
the probability density function of a normally distributed random variable and
a t-distributed random variable.
10
CHAPTER 1. PRELIMINARIES
1.2. CREDIT RISK
Normal probability
density function
"Fat tailed" probability
density function
0
Figure 1.2: The probability density functions of a normally distributed random variable and of a t-distributed random variable.
Another way of increasing the risk for large losses is by including jumps in the
underlying price process. In the jump-diffusion models this is done by adding a
compound Poisson process to the diffusion process with some exogenously given
intensity λ. The extreme losses in the jump-diffusion model shows up much
further out in the tail than the 1% quantile. Duffie and Pan (1997) makes a
comparison between a jump-diffusion and a pure diffusion model3 . This shows
that with an expected frequency of roughly once every 140 years, one will lose
overnight at least one quarter of the value of one’s position. In the plain vanilla
model, one would have to wait far longer than the age of the universe for this
to happen. They also bring up figures which shows that between 1986 and
1996 there was numerous daily returns, in many markets, of at least 5 standard
deviations in size. In the plain vanilla model this is expected to be the case less
than once per million days. Jaschke (2002) brings up some other aspects about
the drawbacks of VaR.
1.2
Credit Risk
While market risk means the risk of unexpected changes in prices and rates,
credit risk concerns the risk that you will suffer a financial loss because your
counterparty not is able to meet his, or hers, obligations. We will call such
an event a default event. A default event may be triggered by a number of
reasons, for example by a decline in the credit worthiness of a counterparty or
by the fact that a firm not is able to pay back its debts. The risk taker requires
compensation for this risk. In bond markets, for example, riskier issues have to
promise a higher yield than the not so risky issues.
3 Both models have IID shocks, constant mean return, and a constant annual volatility of
15%. The intensity λ is one per year and the jump standard deviation is set to 10 percent.
11
1.2. CREDIT RISK
CHAPTER 1. PRELIMINARIES
In the pricing of credit risk and credit risk derivatives the structural models
and the reduced-form models are most common. Structural models (or the
firm’s value models) are concerned with modelling and pricing credit risk that
is specific to a particular corporate obliger (a firm). The way in which a default
event occurs is by the movement of the firm’s value. Several default triggers are
possible, e.g.:
• As soon as the firm’s value hits a barrier the default event is triggered.
The barrier can be either constant or time-dependent. For example the
default can occur as soon as the firm’s value falls below the discounted
value of its outstanding debts.
• A default can only occur at the maturity of an outstanding debt. The
firm continues to operate until this happens.
The main issue in this framework is the construction of a meaningful process
describing the movements of the firm’s value. There is an extensive amount
of literature covering this field, e.g. Merton (1974), Black and Cox (1976),
Longstaff and Schwartz (1995) and Ericsson (2000).
In the reduced-form models (or the intensity-based models) a jump process,
whose first jump determines the default, defines the default process. The probability of a default in a specific time interval is given by the intensity of the jump
process. Poisson processes are common to use when modelling rare events. Two
properties of the Poisson process, which makes it an especially appropriate process to choose when modelling defaults, are:
• The Poisson process has no memory, i.e. it is not more likely for a jump
to occur only because it has been a long time since the last jump.
• The probability of two or more jumps at exactly the same time is of small
order. This may, a bit sloppy, be expressed as P (#jumps [0, t] > 1) =
O(t2 ), where t is small.
A jump in the Poisson process says that a default event has occurred, but it does
not tell the actual loss. The fractional loss, or the loss-given-default (LGD), is
often assumed to be exogenously given. In case of a default, lets say at time t,
the value of the portfolio is decreased by the LGD multiplied with the positions
value at time t. Literature concerning the intensity-based approach is, just to
name a few, Jarrow and Turnbull (1995), Duffie et al. (1996) and Schönbucher
(1996, 1998a, 1998b).
Two commercial products for quantifying credit risk are CreditMetricsTM , developed by J.P. Morgan, and CreditRisk+TM , developed by Credit Suisse First
Boston. Cossin and Pirotte (2001) gives an overview of the existing frameworks
for managing credit risk.
12
CHAPTER 1. PRELIMINARIES
1.3
1.3. INCOMPLETE MARKETS
Incomplete Markets
Imagine a market with n tradable assets whose prices are driven by some
stochastic processes S1 , . . . , Sn . Under the assumption of absence of arbitrage,
there exists an equivalent probability measure P under which the discounted
price processes S1 /B, . . . , Sn /B are martingales. When the market is complete,
all possible cash flows can be replicated by a self-financing trading strategy.4 If
the market is complete there exists an unique equivalent probability measure
under which pricing is carried out. If all possible cash flows can not be replicated
by a self-financing trading strategy the market is incomplete. The market may
still be free of arbitrage, in the sense that there exists equivalent martingale
measures,5 but there is no unique equivalent probability measure. Björk (1998)
explains completeness, and absence of arbitrage, by this meta-theorem:
Meta-theorem Let M denote the number of underlying traded assets in the
model excluding the risk free asset, and let R denote the number of random
sources. Generically we then have the following relations.
• The model is arbitrage free if and only if M ≤ R.
• The model is complete if and only if M ≥ R.
• The model is complete and arbitrage free if and only if M = R.
For an incomplete market the problem comes down to which of the existing
martingale measures that should be chosen for a valuation of the portfolio. If
the underlying asset, Sk , is following a jump-diffusion model a natural martingale measure, Q, can be derived by changing the drift of the driving Brownian
motion. This is done to obtain the drift under the new measure. Conditioning
on the number of jumps, j, gives us a log-normal distribution under Q. Hence,
to compute a price of a derivative we take expectation under Q and since the
conditional distribution is log-normal we can apply the Black-Scholes formula
(j)
to obtain the price, C(Sk ), associated with j jumps. Hence the unconditional
value of a portfolio consisting of n European options would be
EQ [C(S)] =
n ∞
(j)
C(Sk )pj
k=1 j=0
where S = (S1 , . . . , Sn ) and pj is the probability of j jumps. (Infinity is “chosen”
depending on the demands of accuracy. For a Poisson distributed number of
jumps it can be pointed out that the probability of 10 jumps in one year, with
an annual jump intensity of 2, is approximately 3.82 · 10−5 .)
4 By “self-financing” a trading strategy where no additional means are provided, or withdrawn, after t = 0 is intended.
5 A probability measure, P ∗ , and its associated expectation operator, E∗ , forms a martingale measure.
13
1.4. D-G APPROXIMATION
1.4
CHAPTER 1. PRELIMINARIES
Delta-Gamma Approximation
The delta-gamma approximation is based on a second order Taylor expansion of
the portfolio value. What makes the delta-gamma model especially appropriate
is that the characteristic function of the quadratic form is known analytically.
The value of a portfolio consisting of n different positions can be represented by
1
V = A + ∆ X + X ΓX
2
where A is a scalar, ∆ a vector of the portfolio deltas in Rn , Γ is a matrix of
the gammas for the different assets with dimension n × n and X is normally
distributed with mean 0 and covariance matrix Σ.
Solve the eigenvalue problem D ΛD = 12 (Σch ) ΓΣch , where Σch denotes the
Cholesky factorisation of the positive definite matrix Σ, Λ is a diagonal matrix
with the eigenvalues, λ1 , . . . , λn , on its diagonal and D is a matrix with the
corresponding eigenvectors as rows. Let X = Σch Y and α = ∆ Σch , where Y
is standard normal. Then V can be expressed on a quadratic form, as a linear
combination of independent random variables, as
V = A+
n
(αk Yk + λk Yk2 )
k=1
n
αk
= A+
λk Yk +
2λk
k=1
n αk2
= A+
λk Zk −
4λk
2
α2
− k
4λk
k=1
where Zk is a non-central χ2 distributed random variable with 1-degree of freeα2
dom and non-centrality parameter δk2 = 4λk2 . Now the characteristic function of
k
V , ϕ(t) = E[ eitV ], can be calculated as (see, for example, Davies (1973)),
E[ eitV ] =
n
exp it k=1 λk δk2 /(1 − 2itλk ) −
Πnk=1 (1 − 2itλk )1/2
α2k
4λk
+ itA
.
Further examining the behaviour of the delta-gamma approach we take a look
at the delta approximation. Suppose that we only got one underlying asset,
x, and that we use the Black-Scholes pricing formula to price a European call
option on that asset. Since ∆ > 0 and Γ > 0 the option price, f (x), is an
increasing, convex, function of x, where f ! C 1 (i.e. the space of functions, f ,
with continuous first derivatives). For small changes in the underlying security’s price, we know that the first order Taylor expansion is a reasonably good
approximation of the function value, i.e.
f (x + a) ≈ f (x) + ∆ · a.
The delta approximation over-estimates the loss of a long position, and underestimates the loss of a short position, when the price of the underlying asset
increases. This is illustrated in Figure 1.3.
14
CHAPTER 1. PRELIMINARIES
1.4. D-G APPROXIMATION
For the second order Taylor expansion, or the delta-gamma approximation, we
have, for f ! C 2 ,
1
f (x + a) ≈ f (x) + ∆ · a + Γ · a2 .
2
Option Price f(x)
Figure 1.4 shows that the delta-gamma approximation under-estimates the loss
of a long position, and over-estimates the loss of a short position.
f′(x)a
a
x
x+a
Figure 1.3: The delta approximation over-estimates the loss of a long position, and
Option Price f(x)
under-estimates the loss of a short position.
a
2
f′(x)a + 0.5f′′(x)a
x
x+a
Figure 1.4: The delta-gamma approximation under-estimates the loss of a long position, and over-estimates the loss of a short position.
15
1.4. D-G APPROXIMATION
CHAPTER 1. PRELIMINARIES
16
CHAPTER
2
The Framework
In this chapter we follow the model set-up by Duffie and Pan (2001).
2.1
Portfolio Dynamics
Consider a portfolio consisting of m derivatives, C1 , . . . , Cm , written on n underlying assets. Each of the price processes, S1 , . . . , Sn , for the underlying securities
are modelled by a jump-diffusion process. The total value of the portfolio then
m
becomes C(R) = k=1 Ck (R), where R = (log(S1 ), . . . , log(Sn )) denotes the
log-price process. Suppose that1
d
RT = R0 + (M − λµ) · T +
√
N (T )
T Σ X0 +
ch
(µ + Qch Xj ),
(2.1)
j=1
where M , in Rn , is the mean-return vector, Xj is standard normal in Rn , for
j = 0, 1, 2 . . ., and Xj and Xk are independent for j = k. Further, Σ and Q
are symmetric, positive semi-definite matrices in Rn×n and N (T ) is a Poisson
process, independent of X1 , X2 , . . ., with constant arrival intensity λ. Thus,
conditional on j jumps before T , we have a multi-variate normally distributed
random variable RT with mean R0 + (M − λµ)T + jµ and covariance ΣT + jQ.
The purpose of the model is to compute P (C(RT ) ≤ cq ), where C(RT ) is the
portfolio value at time T and cq is the q-th quantile portfolio value. For this we
need the cumulative distribution function (CDF) of C(RT ). We intend to reach
the CDF by Fourier inversion of the characteristic function. From a quadratic
form of independent normal random variables it is shown that the characteristic
function may be readily calculated.
1 Let
Ωch denote the Cholesky factorisation of some matrix Ω. That is Ω = Ωch (Ωch ) .
17
2.2. D-G APPROXIMATION
2.2
CHAPTER 2. THE FRAMEWORK
Delta-Gamma Approximation
A Taylor expansion around initial value R0 gives the delta-gamma approximation of the portfolio value as
C(RT ) ≈ C ∆,Γ (RT ) = C(R0 ) + ∆ (RT − R0 )
1
+ (RT − R0 ) Γ(RT − R0 )
2
where
m
∂Ci (R) ∆k =
∂Rk i=1
, Γk,l
m
∂ 2 Ci (R) =
∂Rk ∂Rl i=1
R=R0
, k, l = 1, . . . , n.
R=R0
In the case of European options delta and gamma are easily computed. Conditional on j jumps, compute the Black-Scholes delta and gamma (under the
measure derived in the way laid out in Section 1.3). Note that the underlying
assets in Black-Scholes’ framework follow a diffusion process, and that we use
the logarithm of that process. Let ∆(BS) denote ∂C
∂S . This yields
∆(R) =
and
Γ
(R)
∂C ∂S
∂C
∂eR
=
·
= ∆(BS) ·
= ∆(BS) eR
∂R
∂S ∂R
∂R
∂C ∂S
∂ 2 S ∂C
∂2C
∂
∂ 2 C ∂S
·
·
+
=
=
·
=
∂R2
∂R ∂S ∂R
∂R∂S ∂R ∂R2 ∂S
∂
∂C ∂ 2 S
∂C ∂S ∂S
=
·
+
·
= ...
∂S ∂S ∂R ∂R
∂S ∂R2
= Γ(BS) e2R + ∆(BS) eR
˜ where J˜ gives reasonable accuracy, weighted
Add up for j ranging from 0 to J,
by the probability, pj , of j jumps. Since the jump process is Poisson distributed,
the probability of N (T ) = j jumps before T is
pj = e−λT
(λT )j
.
j!
We chose “reasonable accuracy” to be J˜ = 30 jumps.2 This gives us
∆=
30
(R)
pj ∆j , Γ =
j=0
2.3
30
(R)
p j Γj .
j=0
Characteristic Function
For the computation of P (C ∆,Γ (RT ) ≤ cq ) we need the distribution function
of C ∆,Γ (RT ). One way of getting this is via the characteristic function, ϕ(t) =
∆,Γ
E[ eitCT ]. Throughout this section j is fixed and the portfolio is assumed to
consist of n contracts.
2 With an annual jump arrival intensity of 4, the probability of 30 jumps during one year
is 7.96 · 10−17 .
18
CHAPTER 2. THE FRAMEWORK
2.3. CHAR. FUNCTION
ch
We define ηj = (ηj1 , . . . , ηjn ) as the eigenvalues3 of 12 (Θch
j ) Γ Θj , i.e.
 1

ηj
1 ch 

ch
..
Dj 
 Dj = (Θj ) Γ Θj ,
.
2
ηjn
where Dj is an orthogonal, full matrix whose rows are the corresponding eigenvectors to (ηj1 , . . . , ηjn ). Denote the k-th eigenvector and eigenvalue by Djk
and ηjk , respectively. Given j jumps before T , RT has mean-value Mj =
R0 + (M − λµ)T + jµ and covariance matrix Θj = ΣT + jQ. Let Z be standard
normal, i.e. Z ∼ N (0, I) where I is the identity matrix. This enables us to
4
represent RT as Θch
j Z + Mj .
With these preparations at hand, and using the fact that Θj is symmetric, we
can express C ∆,Γ (RT ), conditional on j jumps, as
1
C ∆,Γ (RT ) = A + B RT + RT ΓRT
2
1 ch
ch
= A + B (Θch
j Z + Mj ) + (Θj Z + Mj ) Γ(Θj Z + Mj ) = . . .
2
1
= A + B Mj + Mj ΓMj + Z (Θch
j ) (B + ΓMj )
2
+ (Dj Z) Λj (Dj Z),
(2.2)
where
1
A = C(R0 ) − ∆ R0 + R0 ΓR0 ,
2
B = ∆ − ΓR0 ,
Λj = diag(ηj1 , . . . , ηjn ).
Remembering that the goal of our manipulations is to express C ∆,Γ (RT ) on
a quadratic form of independent normal variables (see Section 1.4), we define
δj = (δj1 , . . . , δjn ) by
δjk =
(Djk ) aj
2ηjk
and
1
γj = A + B Mj + Mj ΓMj −
ηjk (δjk )2 ,
2
n
k=1
where aj =
(Θch
j ) (B
C ∆,Γ (RT ) = γj +
= γj +
= γj +
+ ΓMj ). Equation (2.2) may now be re-written as
n
k=1
n
k=1
n
ηjk (δjk )2 +
n
Zk ak +
k=1
n
(Djk Zk ) ηjk (Djk Zk ) = . . .
k=1
k
Zk ak − 2ηjk Djk Zk δj +
n
ηjk (Djk Zk + δjk )2
k=1
ηjk (Djk Zk + δjk )2 .
k=1
3 All
eigenvalues are assumed to be non-zero.
ch
ch
ch
ch
=
T ] = E[Θj Z + Mj ] = Mj and E[(Θj Z)(Θj Z) ] = Θj E[ZZ ](Θj ch)
ch
ch
Θj I(Θj ) = Θj .
4 E[R
19
2.4. FOURIER INVERSION
CHAPTER 2. THE FRAMEWORK
Since Dj is an orthogonal matrix, Dj Z is standard normal, and C ∆,Γ (RT ) is
finally on a quadratic form. Davies (1973) gives us the characteristic function
by
ϕj (u) = E[exp iuC ∆,Γ (RT ) NT = j]
n
exp iu k=1 ηjk (δjk )2 /(1 − 2iuηjk ) + iuγj
=
.
Πnk=1 (1 − 2iuηjk )1/2
The unconditional characteristic function is acquired by adding up for j ranging
from 0 to a point of reasonable accuracy, weighted by the probability, pj , of j
jumps. That is
ϕ(u) =
J˜
pj ϕj (u) ,
j=0
where cutting of at J˜ jumps gives “reasonable accuracy”.
2.4
Fourier Inversion
The inversion theorem proves that the distribution function is uniquely given
by the characteristic function by (see e.g. Gut (1995))
∞
1 − e−itx
1
ϕ(t) dt.
F (x) − F (0) =
2π −∞
it
We need an explicit expression for the distribution function, hence the inversion
theorem is not satisfactory because of the constant term F (0).
Gil-Pelaez (1951) derived an explicit formula for F (x), as
1 ∞ Im[e−itx ϕ(t)]
1
dt.
F (x) = −
2 π 0
t
Applying this to our problem yields
FC ∆,Γ (cq ) = P (C
∆,Γ
T
∞
∞
1
Ij (u, cq )
1
du
(RT ) ≤ cq ) = −
pj
2 π j=0
u
0
where, for u ! R,
Ij (u, cq ) = Im[e−iucq ϕj (u)] = Φj (u)Ψj (u, cq )
where
Φj (u) =
and
Ψj (u, cq ) = sin
n
exp −2u2 k=1 (ηjk δjk )2 /(1 + 4(uηjk )2 )
Πnk=1 (1 + 4(uηjk )2 )1/4
n
n
k k 2
uη
(δ
)
1
j
j
arctan(2uηjk ) +
+ u(γj − cq ) .
2
1 + 4(uηjk )2
k=1
k=1
20
(2.3)
CHAPTER 2. THE FRAMEWORK
2.5
2.5. CONDITIONAL VAR
Conditional Value-at-Risk
Given that the time T value of the portfolio, C(RT ), is less than the q-th quantile
value, cq , conditional VaR is defined as the mean of C(RT ), i.e.
C-V aRC(RT ) (cq ) = E [C(RT ) | C(RT ) < cq ]
cq
1
=
xfC (x)dx,
FC (cq ) −∞
(2.4)
where fC (c) is the probability density function of C(RT ). The framework being used in this thesis only gives the cumulative distribution function, FC (c),
between c0 and cq .
c0
∆c
cq
Figure 2.1: The cumulative distribution function between c0 and cq . The CDF is
approximated from n data points and the step size is ∆c.
Splitting Equation (2.4) in two, yields one integral over the known section [c0 , cq ]
and one over the unknown section (−∞, c0 ],


1
FC (cq )
cq

 c0
1


xfC (x)dx =
xfC (x)dx +
xfC (x)dx .

FC (cq )  −∞

−∞
c
0 cq
I1
I2
For the calculation of I2 , let ∆c be the step size and n the number of data points
(x)
yields,
on [c0 , cq ]. The standard differential approximation f (x) ≈ F (x+∆x)−F
∆x
n−1
FC (c0 + (k + 1)∆c) − FC (c0 + k∆c)
· ∆c
I2 =
(c0 + k∆c)
∆c
k=0
=
n−1
[(c0 + k∆c) {FC (c0 + (k + 1)∆c) − FC (c0 + k∆c)}] .
k=0
21
2.6. D-G-V APPROXIMATION
CHAPTER 2. THE FRAMEWORK
All the parameters are known, so I2 may be readily calculated.
For the (−∞, c0 ] interval, some kind of approximation of FC (c) has to be done.
At least, far out in the tail FC (c) = δ · (−c)−β is a reasonable approximation,
where β is a constant and δ is given by
FC (c0 ) = P (C(RT ) ≤ c0 ) = q0
⇒ δ · (−c0 )−β = q0
⇔ δ = q0 · (−c0 )β .
Hence, given β δ is known. It should be pointed out that FC (c) is taken, almost, out of the blue. Many other approximations may work just as well. The
justification for this particular choice comes from regularly varying distributions. These models may be appropriate for modelling financial data (see e.g.
Embrechts et al. (1997)).
For the estimation of β the least square method is used. That is, let FCβ (c) =
q0 · (−c0 )β (−c)−β and solve the minimisation problem
min
β
n 2
FC (ck ) − FCβ (ck ) .
k=1
With β at hand, I1 can be calculated by
c0
I1 =
xfC (x) dx = FC (x) = fC (x) = δ · (−β)(−x)−β−1 (−1)
−∞
c0
c0
=
−x · δ(−β)(−x)−β−1 dx = −δ
β(−x)−β dx
−∞
β
(−c0 )1−β , if β > 1.
= −δ
1−β
−∞
Dividing the sum of I1 and I2 with the probability that C(RT ) is less than cq
gives us the conditional VaR.
2.6
Delta-Gamma-Vega Approximation
In the delta-gamma approximation, as well as in the Black-Scholes framework
for pricing European options, the volatility, σ, is assumed to be constant. If a
variable volatility is desirable, vega risk can be introduced to the model. Vega,
ν, is defined by ν = ∂C
∂σ , i.e. vega tells how sensitive the portfolio is against
changes in the volatility. (The Black-Scholes framework is applicable under the
assumption that σ is constant between today and the time horizon, T , for the
VaR computation and between T and the time of maturity for the options.)
A second order Taylor expansion around initial values R0 and σ0 , neglecting
the second order derivatives with respect to σ and w.r.t. σ and R, yields the
“delta-gamma-vega” approximation
C(RT , σT ) ≈ C ∆,Γ,ν (RT , σT )
= C(R0 , σ0 ) + ∆ (RT − R0 )
1
+ν (σT − σ0 ) + (RT − R0 ) Γ(RT − R0 ),
2
22
(2.5)
CHAPTER 2. THE FRAMEWORK
2.7. ERROR ANALYSIS
where ∆ and Γ are as before, and
m
∂Ci (R) νk =
∂σk i=1
, k = 1, . . . , n
R=R0
for a portfolio consisting of m derivatives written on n underlying assets. The
vega term is not affected by the fact that the logarithm of the underlying assets’
diffusion processes are being used, as the delta and the gamma terms were. (In
the same way as in Section 2.2, ν is achieved by adding up for j ranging from
˜ where J˜ gives reasonable accuracy, weighted by the probability of j
0 to J,
jumps.)
Given j jumps before T , RT has the mean-value Mj = R0 + (M − λµ)T + jµ
and covariance matrix Θj = ΣT + jQ. The diagonal of Θj is the volatility, i.e.
the mean-value of σT is Mjσ = diag(Θj ). Define R̃T in the following way
R̃T =
σT
RT
∼N
Mjσ
Mj
σ
Θj Covj
,
,
Covj Θj
where Covj = Cov(σT , RT ) < 0 and Θσ is the covarians matrix for the volatilities. Equation (2.5) may now be re-expressed as
˜ (R̃T − R̃0 ) + 1 (R̃T − R̃0 ) Γ̃(R̃T − R̃0 ),
C ∆,Γ,ν (R̃T ) = C(R̃0 ) + ∆
2
where
˜ =
∆
and
Γ̃ =
ν
∆
0n×n 0n×n
0n×n Γ
.
˜ is a vector in Rn and Γ̃ is a matrix in R2n×2n .) The approximation of the
(∆
portfolio value is now on the same form as in Section 2.2, and the remainder of
the computations may be carried out in the same way.
2.7
Error Analysis
For the evaluation of Equation (2.3) numerical integration is used in the following way,


Kj
J
1
I
((k
+
1/2)h
,
c
)
1
j
j
q
.
pj  −
P (C ∆,Γ (RT ) ≤ cq ) ≈
2
π
k
+
1/2
j=0
k=0
For this numerical method three different types of errors will have to be handled.
The truncation errors introduced by J, Kj < ∞, and the discretisation error
introduced by hj > 0. Assign an error tolerance, α, for the total error. This
gives each of the J + 1 terms an assigned error tolerance of α/(J + 1) divided
23
2.7. ERROR ANALYSIS
CHAPTER 2. THE FRAMEWORK
by pj . This tolerance is then split in two, half for the error introduced by hj
and the other half for the error introduced by Kj .
The truncation error
∞ introduced by J < ∞ is the easiest to handle. Simply
choose J so that j=J+1 pj is less than the desired accuracy. Moving on to the
error introduced by Kj , Davies (1980) handles it in the following manner.
If |ϕ(u)| ≤ B(u), and B(u) is a monotonically decreasing function in u (for
u ≥ U, U = (Kj + 1/2)hj ), the truncation error is bounded by
1
π
∞
k=Kj +1
Ij ((k + 1/2)hj , cq )
≤
k + 1/2
∞
u=U
B(u)
du .
πu
Davies considers three different bounds on I, resulting in
(1)
Bj (U )
2
Φj (U ) Π4(U ηjk )2 >1
=
πJ
(2)
Bj (U ) =
Φj (U )
,
πU 2 βj2
where B (3) only is valid if
n
k=1
(3)
Bj (U ) =
1 + 4(U ηjk )2
1/4
4(U ηjk )2
,
2.5
Φj (U ),
π
ln(1 + 4(U ηjk )2 ) + 2β 2 U 2 ≥ 1. The truncation
(1)
(2)
(3)
error is bounded by the minimum of (Bj , Bj , Bj ).
For managing the discretisation error the following expression, derived by Davies
(1973), is examined.
P (C ∆,Γ ≤ cq ) +
∞
(−1)n {P (C ∆,Γ < cq − 2πn/hj ) − P (C ∆,Γ > cq + 2πn/hj )}
n=1
=
∞
1
1
−
Im[ϕj ((k + 1/2)hj )e−i(k+1/2)hj cq ]/(k + 1/2).
2 π
k=0
A minimisation of the summation over n is desirable. This can be done by
choosing hj in a way that makes
max{P (C ∆,Γ < cq − 2πn/hj ), P (C ∆,Γ > cq + 2πn/hj )}
(2.6)
less than the allocated error. To find bounds on Equation (2.6), consider the
∆,Γ
moment generating function, E[euC ], of C ∆,Γ . Let ξ(u) denote its logarithm.
Conditional of j jumps and for C ∆,Γ ≥ x the following holds,
E [IC ∆,Γ >x − eu(C
∆,Γ
−x)
| N (T ) = j]
∆,Γ
= P (C
(RT ) > x | N (T ) = j) − eξj (u) e−ux ≤ 0
⇔ P (C ∆,Γ (RT ) > x | N (T ) = j) ≤ eξj (u) e−ux ,
where IX>x is the indicator variable of X > x. Let x = ξj (u). This gives us the
desired bounds on Equation (2.6),
P (C ∆,Γ (RT ) > x | N (T ) = j) ≤ eξj (u)−uξj (u) .
24
CHAPTER 2. THE FRAMEWORK
2.8. EXAMPLES
The logarithm of the moment generating function is computed by (see e.g. Johnson et al. (1994))
!
"
∆,Γ
ξj (u) = ln E[ euC
| N (T ) = j ]
=
n
uηjk (δjk )2
k=1
2.8
1 − 2uηjk
1
ln(1 − 2uηjk ).
2
n
+ uγj −
k=1
Examples
In the following example the 1-day Value-at-Risk for a portfolio of 17, at-themoney, European call-options is reviewed. They are written on 17 different
underlying assets, whose price processes follows Equation (2.1). Today’s value,
S0 , is set to 100 units for all contracts, the annual jump arrival intensity, λ, is
set to 4 and the risk free interest rate is assumed to be 5 percent. The expected
jump amplitude is set so that µj + Qj,j /2 = 0, for j = 1, . . . , 17, M = 0 and let
Σ = 2λQ = Cov, where the covariance matrix, Cov, can be found in Appendix
A. Further, the error tolerance is set to 0.00001.
In Figure 2.2 the “actual” and the delta-gamma approximated Value-at-Risk
are plotted. In the “actual” approach a simulation of the underlying securities’
time T prices is done and the portfolio value is evaluated with the Black-Scholes
formula. In the delta-gamma approach the time T value of the portfolio is
simulated, using the delta-gamma approximation of the portfolio value. In both
cases 100 000 simulations were made. In Figure 2.3 the analytical and the
delta-gamma Value-at-Risk are plotted. Using the method laid out in Section
2.5 conditional VaR is computed to be 32 units, with a β-value of 4.67.
1-day
Analytic solution
Delta-Gamma simulation
1%
33.3367
33.3777
5%
22.4067
22.4106
1%
96.4767
96.4418
5%
65.8667
65.7183
10-days
Analytic solution
Delta-Gamma simulation
Table 2.1: A comparison between the analytical and the simulated delta-gamma VaR
for a long position in the portfolio (the results are taken from 100 000 simulations).
An example where the price process is driven by pure diffusion is also presented.
That is, the annual jump arrival intensity, λ, is zero. The results can be seen
in Figures 2.4 and 2.5. Figure 2.6 shows the analytical 5-day5 VaR with annual
jump intensities λ = 1, 2, 4. Otherwise the same data as above is being used.
More examples can be found in Appendix A.
5 5-days,
as in 5 trading days. That is one “ordinary” week.
25
2.8. EXAMPLES
CHAPTER 2. THE FRAMEWORK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
160
140
120
100
80
60
40
20
Value−at−Risk
Figure 2.2: 1-day VaR with an annual jump arrival intensity of 4. The dotted line is
the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma
approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
120
110
100
90
80
70
60
50
40
30
20
Value−at−Risk
Figure 2.3: 1-day VaR with an annual jump arrival intensity of 4. The dashed line is
the analytical VaR and the solid line the simulation of the delta-gamma approximation.
26
CHAPTER 2. THE FRAMEWORK
2.8. EXAMPLES
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
40
38
36
34
32
30
28
Value−at−Risk
26
24
22
20
Figure 2.4: 1-day VaR with pure diffusion (no jump component). The dotted line is
the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma
approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
40
38
36
34
32
30
28
Value−at−Risk
26
24
22
20
Figure 2.5: 1-day VaR with pure diffusion (no jump component). The dashed line is
the analytical VaR and the solid line the simulation of the delta-gamma approximation.
27
2.8. EXAMPLES
CHAPTER 2. THE FRAMEWORK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
95
90
85
80
75
70
65
Value−at−Risk
60
55
50
45
Figure 2.6: 5-day VaR with annual jump intensities λ = 1, 2, 4 (respectively, dotted-,
dash-dotted-, and solid line).
28
CHAPTER
3
Credit Risk
Credit risk concerns the risk of a financial loss because your counterparty not
is able to meet his, or hers, obligations. Such an event will be referred to as a
default event. A default event may be triggered by a number of reasons, some
stated in Section 1.2. The risk taker requires compensation for this risk. Financial institutions devote considerable resources to the pricing and management
of credit risk.
3.1
Portfolio Value
For the incorporation of credit risk the portfolio value is divided into two parts,
the value component and the default component. The value component is the
market value, conditional on no default within the time horizon, T , of the VaR
computation. All default-dependent payoffs can be priced by discounting them
with the short rate process r̄ = r + Lλ, where r is the deterministic short rate,
L the loss-given-default (LGD) and λ the intensity of the default process (see
e.g. Schönbucher (1997)). In the case of a portfolio of European options the
market value of option i with maturity at Ti is
#
$
Ti
λi (s) ds
.
Vi (Rt , λt ) = Ci (Rt ) · E exp −Li
t
The default component is the total loss from defaults. In case of a default
affecting firm i at τi , 0 < τi < T < Ti , the value of the portfolio is reduced
by the LGD multiplied with the time T value of that contract. Thus, the total
value of a portfolio consisting of n contracts at time T is expressed by
WT =
n
i=1
Vi (RT , λT ) −
n
i=1
29
Li Di (T )Vi (RT , λT )+ ,
3.2. STOCHASTIC INTENSITY
CHAPTER 3. CREDIT RISK
where Di (T ) is the indicator of a default event and where it will be assumed
that Li = L, ∀i. Let Vi (RT , λT )+ denote the positive part of Vi (RT , λT ) and, for
analytical tractability, approximate Vi (RT , λT )+ by Vi (R0 , λ0 )+ . Further, the
dependency between Di (T ) and λT is ignored.1 These approximations eliminates the dependency between the value component and the default component,
which enables us to write the characteristic function as ϕ(u) ≈ ϕV (u)ϕD (u).
Once the characteristic function is known, the Fourier inversion method laid out
in Section 2.4 is applicable.
3.2
Stochastic Intensity
Default events are modelled as the first jumps of a series of Cox processes2 with
intensities λ1 (t), . . . , λn (t). Each firm’s default intensity process is defined as
λi = λIi + pi λC
where λIi is the intensity of a default specific to firm i and λC is the intensity
of a common default which will affect firm i with probability pi . The individual
and common default intensities are defined as Cox-Ingersoll-Ross processes,
%
dλIi (t) = κIi (λ̄Ii − λIi (t))dt + σiI λIi (t) dBiI (t)
%
dλC (t) = κC (λ̄C − λC (t))dt + σ C λC (t) dB C (t)
where B C , B1I , . . . , BnI are standard Brownian motions and λ̄C , λ̄I1 , . . . , λ̄In denotes the long-run means. Assuming that pi = p, κIi = κC = κ, and that
√
σiI = pσ C = σ, λi is also a CIR process. Let λ̄i = λ̄Ii + pλ̄C , and Bi be a
Brownian motion, then λi is defined by
&
dλi (t) = κ(λ̄i − λi (t))dt + σ λi (t) dBi (t).
The probability of survival for each contract i to time T > 0 may now be
computed by (see Cox et al. (1985) for details)
#
$
T
λi (s)ds Ft
P (τi > T ) = E exp −L
t
= H1i (T − t, L)e−H2 (T −t,L)Lλi
(3.1)
where
#
H1i (T, L) =
1
2γe 2 (γ+κ)T
(γ + κ)(eγT − 1) + 2γ
$2κλ̄i /σ2
2(eγT − 1)
(γ + κ)(eγT − 1) + 2γ
&
γ = κ + 2Lσ 2 .
H2 (T, L) =
Both the individual and the common default intensities are initiated at their
long-run means, i.e. λIi (0) = λ̄Ii and λC (0) = λ̄C .
1 Duffie
and Pan (2001) shows that the covariance between DT and λT is of small order,
over short time horizons.
2 Roughly, a Cox process is a Poisson process with stochastic intensity.
30
CHAPTER 3. CREDIT RISK
3.3
3.3. D-G APPROXIMATION
Delta-Gamma Approximation
For the delta-gamma
approximation we Taylor expand the value component,
n
V (RT , λT ) = i=1 Vi (RT , λT ), around its initial values. Let YT = (RT , λT ) ,
then V (YT ) can be approximated in exactly the same way as in Section 2.2.
Delta and gamma are expressed as
∆=

∂2V
∂R12
...
..
.
...
 .
 .
 .
 ∂2V

 ∂Rn ∂R1
Γ= 2
 ∂ V ...
 ∂λ1 ∂R1
 .
..
 .
.
 .
∂2V
.
.
.
∂R1 ∂λn
∂V ∂V
,
∂Ri ∂λi
,
∂2V
∂R1 ∂Rn
∂2V
∂R1 ∂λ1
∂2V
2
∂Rn
∂2V
∂λ1 ∂Rn
..
.
..
.
..
.
..
.
∂2V
∂λ1 ∂Rn
∂2V
∂λ21
∂2V
∂λn ∂Rn
∂2V
∂λn ∂λ1
...
..
.
...
∂2V
∂R1 ∂λn





2
∂ V 
∂Rn ∂λn 
.
2
V 
. . . ∂λ∂1 ∂λ
n 
.. 
..

.
. 
∂2V
. . . ∂λ2
..
.
n
For analytical tractability we make the, quite considerable, approximation that
λT is Gaussian.3 This gives us V (RT , λT ) on a normally distributed form with
mean M and covariance matrix Θ. Further, assuming independency between λ
and R yields
M=
MR
Mλ
, Θ=
ΘR 0n×n
.
0n×n Θλ
The characteristic function may now be computed, with parameters
Miλ = E0 (λi (T )) = λi (0) + (1 − exp(−κT ))(λ̄Ii + pλ̄C − λi (0))
Θλi,i = Cov0 (λi (T ), λi (T ))
(1 − exp(−κT ))2 2 I
1 − exp(−κT ) 2
σ λi (0) +
σ (λ̄i + pλ̄C )
κ
2κ
= Cov0 (λi (T ), λj (T ))
(1 − exp(−κT ))2 C
1 − exp(−κT ) C
λ (0) +
= pσ 2 exp(−κT )
,
λ̄
κ
2κ
= exp(−κT )
Θλi,j
in the same way as in Section 2.3. For an incorporation of vega risk, let Vt depend
on Rt , λt and σt . Proceeding as in Section 2.6 will give the delta-gamma-vega
approximation.
3 By
modelling λt with an Ornstein-Uhlenbeck process, which is Gaussian, we could have
got around this approximation. However the usage of the Ornstein-Uhlenbeck process would
have given a positive probability of λt < 0.
31
3.4. DEFAULT COMPONENT
3.4
CHAPTER 3. CREDIT RISK
Default Component
A firm can default either from a common or an individual default event. In case
of a common default the firm has some given probability of survival. Let the
indicator variable ξi denote if firm i has survived a common default or not (where
zero denotes survival). This probability is already defined as P (ξi = 1) = p.
The default indicator can be expressed as
Di (T ) = DiI (T ) + ξi DC (T ).
The risk of double counting defaults is obvious, but the probability is of small
order and is shown by Duffie and Pan (2001) to have negligible effect. The
individual and common default indicators are triggered with the probability
defined by Equation (3.1). That is
#
$
T
pIi (T ) = P (DiI (T ) = 1) = 1 − E exp −
#
C
C
p (T ) = P (D (T ) = 1) = 1 − E exp −
0
0
T
λIi (s)ds
$
λ (s)ds
C
.
Letting K(T ) denote the default component, we get the characteristic function
as
ϕD (u) = E e−iuK(T )
≈ pC (T )E e−iuK(T ) DC (T ) = 1
+(1 − pC (T ))E e−iuK(T ) DC (T ) = 0
n !
n
" '
'
1 − p + pe−iuLVi (R0 ,λ0 )
ki ,
≈ 1 − pC (T ) + pC (T )
i=1
i=1
where
ki = 1 − pIi (T ) + pIi (T )e−iuLVi (R0 ,λ0 ) .
When the characteristic function, ϕ(u), is known Value-at-Risk can be calculated in the same way as in Section 2.4.
3.5
Examples
The 1-day VaR for the same portfolio as in Section 2.8 on page 25 is reviewed,
but we introduce credit risk this time. Each of the 17 default intensities are
assumed to have the same long-run means, λ̄i = 0.03, ∀i, and λ̄I /λ̄ = 0.2.
Further, let κ = 0.25, the loss-given-default, L, is assumed to be 50%, the
volatility is 0.22 and the probability for a common default to affect firm i is
80%. That is pi = p = 0.8, ∀i.
In Figure 3.1 the “actual” and the delta-gamma approximated Value-at-Risk
are plotted. In the “actual” approach a simulation4 of the underlying securities’
4 100
000 simulations of the price at time T were made.
32
CHAPTER 3. CREDIT RISK
3.5. EXAMPLES
time T prices is done and the portfolio value is evaluated with the Black-Scholes
formula. In the delta-gamma approach the time T value of the portfolio is
simulated 100 000 times, using the delta-gamma approximation of the portfolio
value. In Figure 3.2 the analytical and the delta-gamma Value-at-Risk are
plotted. Using the method laid out in Section 2.5 conditional VaR is computed
to be 31 units, with a β-value of 3.21.
1-day
Analytic solution
Delta-Gamma simulation
1%
32.9319
32.9790
5%
22.0819
22.0407
1%
94.9819
94.9311
5%
64.8219
64.5547
10-days
Analytic solution
Delta-Gamma simulation
Table 3.1: A comparison between the analytic and the simulated delta-gamma VaR
for a long position in the portfolio (the results are taken from 100 000 simulations).
An example where the price processes are driven by pure diffusion is also presented. That is, the annual jump arrival intensity, λ, is zero. The results can be
seen in Figures 3.3 and 3.4. Figure 3.5 shows the analytical 5-day5 VaR with
annual jump intensity λ = 4, and default intensities λ̄ = 0.003, 0.03, 0.06. Otherwise the same data as above is used. The options portfolio shows to be fairly
insensitive towards credit risk.
More examples can be found in Appendix B.
5 5-days,
as in 5 trading days. That is one “ordinary” week.
33
3.5. EXAMPLES
CHAPTER 3. CREDIT RISK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
160
140
120
100
80
Value−at−Risk
60
40
20
Figure 3.1: 1-day VaR. The dotted line is the simulation of the “actual” VaR and the
solid line the simulation of the delta-gamma approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
110
100
90
80
70
60
Value−at−Risk
50
40
30
20
Figure 3.2: 1-day VaR. The dashed line is the analytical VaR and the solid line the
simulation of the delta-gamma approximation.
34
CHAPTER 3. CREDIT RISK
3.5. EXAMPLES
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
40
38
36
34
32
30
28
Value−at−Risk
26
24
22
20
Figure 3.3: 1-day VaR with pure diffusion (no jump component). The dotted line is
the simulation of the “actual” VaR and the solid line the simulation of the delta-gamma
approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
45
40
35
30
Value−at−Risk
25
20
Figure 3.4: 1-day VaR with pure diffusion (no jump component). The dashed line is
the analytical VaR and the solid line the simulation of the delta-gamma approximation.
35
3.5. EXAMPLES
CHAPTER 3. CREDIT RISK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
120
110
100
90
80
70
Value−at−Risk
60
50
40
Figure 3.5: 5-day VaR with annual jump intensity λ = 4, and default intensities
λ̄ = 0.003, 0.03, 0.06 (respectively, dotted-, dash-dotted-, and solid line).
36
CHAPTER
4
Conclusions
We have presented an analytic framework for computing Value-at-Risk for a
portfolio exposed to market risk as well as credit risk. The price processes
for the underlying securities are modelled by jump-diffusions. The number of
jumps are Poisson distributed and the diffusion processes are driven by Brownian
motions. Credit risk is introduced by a default event indicator variable. In case
of a default a certain fraction of the defaulted option’s value is subtracted from
the portfolio value. The cumulative distribution function for the market value
of the portfolio is reached via Fourier inversion of the characteristic function. It
is shown that the characteristic function is easily computed once the expression
for the portfolio value is on a quadratic form of independent normal random
variables.
As an example, we look at a portfolio of European call options. In the preliminaries it is shown that the delta-gamma approximation under-estimates the loss
of a long position. This characteristic is clearly seen in the examples presented.
When the underlying is driven by a pure diffusion process the under-estimation
is not as rough as in the jump-diffusion case. Through the examples we can
also observe how insensitive a portfolio consisting of European call options is
towards credit risk. Value-at-Risk stays at, approximately, the same level although the long-run default intensity is increased 20 times. A considerably
higher sensitivity is observed towards market risk.
The analytic delta-gamma approach gives results which deviates approximately
±0.4% from the simulated delta-gamma VaR. This accuracy is satisfactory, but
the massive under-estimation made by the delta-gamma approximation when
the underlying securities are modelled by jump-diffusions is not. The focus in
this thesis has been on the accuracy of the analytic approximation. More effort
can be made to improve the accuracy of the delta-gamma approximation.
37
CHAPTER 4. CONCLUSIONS
38
APPENDIX
A
Examples: Market Risk
Below Value-at-Risk for a portfolio of 17, at-the-money, European call options
is presented. The 5 and 10-day VaR, using annual jump intensities 2 and 4,
is computed. The expected jump amplitude is set so that µj + Qj,j /2 = 0,
for j = 1, . . . , 17, and Σ = 2λQ = Cov where the covariance matrix, Cov, can
be found in Appendix A on page 44. 100 000 simulations were made for each
example. For the cause of simplicity, the following input data were used for all
options.
Option’s time to maturity (T )
Underlying’s value today (S0 )
Risk free interest rate (r)
1 year
100
5%
39
Error tolerance (α)
Mean return (M̄ )
0.00001
0
APPENDIX A. EXAMPLES: MARKET RISK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
200
180
160
140
120
100
Value−at−Risk
80
60
40
Figure A.1: 5-day VaR with λ = 2. The dotted line is the simulation of the “actual”
VaR and the solid line the simulation of the delta-gamma approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
130
120
110
100
90
80
Value−at−Risk
70
60
50
40
Figure A.2: 5-day VaR with λ = 2. The dashed line is the analytical VaR and the
solid line the simulation of the delta-gamma approximation.
40
APPENDIX A. EXAMPLES: MARKET RISK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
200
180
160
140
120
100
Value−at−Risk
80
60
40
Figure A.3: 5-day VaR with λ = 4. The dotted line is the simulation of the “actual”
VaR and the solid line the simulation of the delta-gamma approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
130
120
110
100
90
80
Value−at−Risk
70
60
50
40
Figure A.4: 5-day VaR with λ = 4. The dashed line is the analytical VaR and the
solid line the simulation of the delta-gamma approximation.
41
APPENDIX A. EXAMPLES: MARKET RISK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
220
200
180
160
140
120
Value−at−Risk
100
80
60
Figure A.5: 10-day VaR with λ = 2. The dotted line is the simulation of the “actual”
VaR and the solid line the simulation of the delta-gamma approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
140
130
120
110
100
90
Value−at−Risk
80
70
60
Figure A.6: 10-day VaR with λ = 2. The dashed line is the analytical VaR and the
solid line the simulation of the delta-gamma approximation.
42
APPENDIX A. EXAMPLES: MARKET RISK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
220
200
180
160
140
120
100
80
60
Value−at−Risk
Figure A.7: 10-day VaR with λ = 4. The dotted line is the simulation of the “actual”
VaR and the solid line the simulation of the delta-gamma approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
140
130
120
110
100
90
80
70
60
Value−at−Risk
Figure A.8: 10-day VaR with λ = 4. The dashed line is the analytical VaR and the
solid line the simulation of the delta-gamma approximation.
43
Covariance matrix for contracts 1–17 (A–Q):
A
0.33640
0.07888
0.01856
0.06264
0.09396
0.08816
0.09280
0.05046
0.08584
0.12470
0.10904
0.10440
0.06728
0.05220
0.03248
0.05394
0.09976
B
0.11560
0.02176
0.05508
0.03672
0.07752
0.09520
0.05916
0.01258
0.10234
0.11186
0.08568
0.06902
0.06120
0.03808
0.06324
0.08772
C
0.02560
0.01728
0.00864
0.01824
0.02560
0.01856
0.01184
0.02064
0.03008
0.01728
0.01856
0.03600
0.02688
0.02480
0.02064
D
0.07290
0.03645
0.08208
0.08640
0.04698
0.02997
0.08127
0.08883
0.07776
0.06264
0.04860
0.03024
0.05859
0.08127
E
0.07290
0.06156
0.05400
0.03132
0.01998
0.06966
0.06345
0.05832
0.03915
0.03645
0.01512
0.04185
0.05805
Contract
M
N
O
P
Q
M
0.08410
0.06525
0.03248
0.06293
0.09976
N
0.20250
0.10080
0.05580
0.07740
O
0.31360
0.03472
0.04816
P
0.09610
0.09331
Q
0.18490
F
0.14440
0.12160
0.06612
0.02812
0.13072
0.14288
0.12312
0.09918
0.06840
0.02128
0.08246
0.13072
G
0.16000
0.08120
0.01480
0.13760
0.15040
0.11520
0.09280
0.07200
0.04480
0.08680
0.13760
H
0.08410
0.01073
0.08729
0.09541
0.07308
0.05887
0.05220
0.01624
0.05394
0.08729
I
0.13690
0.04773
0.03478
0.02664
0.02146
0
0.04144
0.02294
0.03182
J
0.18490
0.14147
0.12384
0.08729
0.07740
0.07224
0.07998
0.14792
K
0.22090
0.13536
0.10904
0.10575
0.05264
0.11656
0.16168
L
0.12960
0.09396
0.06480
0.04032
0.07812
0.13932
APPENDIX A. EXAMPLES: MARKET RISK
44
Contract
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
APPENDIX
B
Examples: Credit Risk
Below Value-at-Risk for a portfolio of 17, at-the-money, European call options
exposed to market risk as well as credit risk is presented. Value-at-Risk is
computed with time horizons 5 and 10 days, and annual jump intensities 2 and
4. Let the mean return vector, M̄ , be the null vector, and Σ = 2λQ = Cov
where the covariance matrix, Cov, can be found in Appendix A on the facing
page. 80% of the defaults are individual, i.e. λ̄I /λ̄ = 0.8. 100 000 simulations
were made for each example. For the cause of simplicity, the following input
data were used for all options.
Underlying’s value today (S0 )
Risk free interest rate (r)
Option’s time to maturity (T )
Error tolerance (α)
Vol. def. intensities (σλ )
100
5%
1 year
0.00001
0.22
45
Common def. prob. (p)
Mean reversion (κ)
Loss-given-default (L)
Long-run mean (λ̄)
0.2
0.25
0.5
0.03
APPENDIX B. EXAMPLES: CREDIT RISK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
200
180
160
140
120
100
Value−at−Risk
80
60
40
Figure B.1: 5-day VaR with λ = 2. The dotted line is the simulation of the “actual”
VaR and the solid line the simulation of the delta-gamma approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
130
120
110
100
90
80
Value−at−Risk
70
60
50
40
Figure B.2: 5-day VaR with λ = 2. The dashed line is the analytical VaR and the
solid line the simulation of the delta-gamma approximation.
46
APPENDIX B. EXAMPLES: CREDIT RISK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
180
160
140
120
100
Value−at−Risk
80
60
40
Figure B.3: 5-day VaR with λ = 4. The dotted line is the simulation of the “actual”
VaR and the solid line the simulation of the delta-gamma approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
120
110
100
90
80
70
Value−at−Risk
60
50
40
Figure B.4: 5-day VaR with λ = 4. The dashed line is the analytical VaR and the
solid line the simulation of the delta-gamma approximation.
47
APPENDIX B. EXAMPLES: CREDIT RISK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
220
200
180
160
140
120
Value−at−Risk
100
80
60
Figure B.5: 10-day VaR with λ = 2. The dotted line is the simulation of the “actual”
VaR and the solid line the simulation of the delta-gamma approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
140
130
120
110
100
90
Value−at−Risk
80
70
60
Figure B.6: 10-day VaR with λ = 2. The dashed line is the analytical VaR and the
solid line the simulation of the delta-gamma approximation.
48
APPENDIX B. EXAMPLES: CREDIT RISK
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
200
180
160
140
120
Value−at−Risk
100
80
60
Figure B.7: 10-day VaR with λ = 4. The dotted line is the simulation of the “actual”
VaR and the solid line the simulation of the delta-gamma approximation.
5
4.5
4
Probability (%)
3.5
3
2.5
2
1.5
1
0.5
0
130
120
110
100
90
Value−at−Risk
80
70
60
Figure B.8: 10-day VaR with λ = 4. The dashed line is the analytical VaR and the
solid line the simulation of the delta-gamma approximation.
49
APPENDIX B. EXAMPLES: CREDIT RISK
50
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