11 Appendix C: Some Properties of Bessel Functions
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236
CHAPTER 3
. Advanced topics in pricing theory
b = a in equations (3.360) and (3.361).
xJ ax I bx dx =
xI ax Y bx dx =
xY ax K bx dx =
xJ ax K bx dx =
x
aJ
a2 + b2
x
bI ax Y
b2 + a2
x
+1
bx + aI
+1
bx
(3.362)
ax Y bx
(3.363)
+1
aY
+1
ax K bx − bY ax K
+1
bx
(3.364)
x
aJ
a2 + b2
+1
ax K bx − bJ ax K
+1
bx
(3.365)
a2 + b2
x2 2
J ax − J
2
−1
ax J
xY 2 ax dx =
x2 2
Y ax − Y
2
−1
ax Y
x2
2J ax Y ax − J
4
−J
ax Y
−1
+1
+1
+1
+1
ax
(3.366)
ax
(3.367)
ax Y
−1
ax
ax
(3.368)
x2 1+
2 1+2
Y 2 x + Y 2+1 x
(3.369)
J x dx =
x2 1+
2 1+2
J 2 x + J 2+1 x
(3.370)
J x Y x dx =
x2 1+
2 1+2
J x Y x +J
x2
+1
x2
+1 2
+1
ax I bx + bJ ax I
xJ 2 ax dx =
xJ ax Y ax dx =
x2
+1
Y 2 x dx =
+1
xY
+1
x
(3.371)
The Wronskian W I x K x = −1/x leads to other useful indefinite integrals:
dx
x aI x + bK x
2
=
1/b I x
aI x + bK x
b=0
(3.372)
2
=
− 1/a K x
aI x + bK x
a=0
(3.373)
or equivalently:
dx
x aI x + bK x
Analogous integral identities involving the ordinary Bessel J Y pair also obtain from the
Wronskian W J x Y x = 2/ x.
Differential equations:
1
Z x + Z x − 1 + 2 /x2 Z x = 0
x
1
Z x + Z x + 1 − 2 /x2 Z x = 0
x
Z =I K
(3.374)
Z =J Y
(3.375)
3.11 Appendix C: Some Properties of Bessel Functions
237
Recurrence relations:
2I x = I
x +I
−1
−2K x = K
2 /x I x = I
−1
− 2 /x K x = K
x +K
x −I
−1
−1
x
+1
x
+1
+1
xI x = ± I x + xI
(3.377)
x
+1
x −K
(3.376)
(3.378)
x
(3.379)
x
(3.380)
±1
xK x = ± K x − xK
±1
x
(3.381)
xZ x = ± Z x ∓ xZ
±1
x
(3.382)
2 /x Z x = Z
+1
x +Z
x
−1
(3.383)
where Z = J Y . Combining the Wronskian with recurrence relations gives
J xY
+1
x −J
+1
−2
x
xY x =
Leading-order asymptotic expansions for z →
:
ez
I z ∼√
2 z
K z ∼
2z
(3.384)
(3.385)
e−z
(3.386)
Jump discontinuities across the complex branch cut z = ei x → e−i x, x > 0:
I ei x − I e−i x = 2i sin
K e x −K e
i
−i
x = −i
I ei x + I e−i x = 2 cos
I x
(3.387)
I x + I− x
K x
(3.388)
(3.389)
Leading order expansions for small argument z → 0
1
+1
I z ∼
1
1−
I z ∼
K z ∼
1
2
z
2
z
2
z
2
+O z
−
−
+2
for complex
+ O z2−
for
+ O z2−
for real
= −1 −2 −3
(3.390)
= −1 −2 −3
(3.391)
=0
(3.392)
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CHAPTER
.4
Numerical Methods for Value-at-Risk
Portfolios of financial assets are exposed to many types of risks, future events that if they
occurred would result in financial losses. The purpose of risk management is to quantify
and control these dangers. Value-at-risk (VaR) is a measure of the market risk, the chance
of a loss in a company’s portfolio caused by unfavorable changes in prices and rates.
Minimum risk management standards for financial institutions are set and enforced by national
regulators. The Basel Accord [Bas88], the market risk amendment [Bas96a, Bas96b], and
the recent update [Bas88] contain the international guidelines implemented by the national
agencies.1 Value-at-risk has become the industry standard for quantifying market risk, partly
because of its intuitive appeal and, more importantly, because it is endorsed in the Basel
Accord.
For a given portfolio, value-at-risk is defined as the maximum loss forecast over a
specified holding period and within a given confidence level (see Figure 4.1). In other words,
it is a percentile of the distribution for changes in portfolio value. If
is the change in
portfolio value during the holding period, then value-at-risk is the solution to a nonlinear
equation:
P
≤ −VaR = 1 −
(4.1)
where is the confidence level. Another interpretation is that in the long term we expect
losses exceeding value-at-risk with frequency 1− . For = 99%, we expect losses exceeding
value-at-risk 1 out of every 100 days. Regulators require value-at-risk to be computed daily
with a confidence level of 99% and for a holding period of 10 days. However, since the
rules allow for value-at-risk for 1 day to be scaled to approximate the risk for 10 days, we
choose to consider daily holding periods in our examples. There are many review papers
about value-at-risk simulation: Stambaugh [Sta96] gives a high-level introduction; for more
1
The Basel Accord and related documents are available from the Bank of International Settlements (www.bis.org).
239
240
CHAPTER 4
. Numerical methods for value-at-risk
value-at-risk
FIGURE 4.1 The probability that a loss is greater than value-at-risk, the density of the shaded region,
is equal to 1 − .
in-depth, general, algorithmic, and mathematical discussions, we have a personal preference
for [Mor96a, Hul00, DP97].2
In financial markets, risk is caused by uncertainty about the value of an investment in
the future. The value of a portfolio is a function of a set of risk factors. Risk factor is the
generic term for a financial variable related to market prices of selected reference securities,
for example, equity indices, interest rates, foreign exchange rates, and commodity futures
prices. Market risk is the risk that the value of a portfolio declines as a consequence of
changes in the risk-factor values. Therefore, to model market risk we need to understand how
risk factors evolve over time.
Consistently with the hypothesis of absence of arbitrage, we will assume that the changes
in risk factors are random. Although historical data is of limited use to predict changes in risk
factors, it can be used to estimate statistical models to model risk factors and their correlations.
In our examples, we use stocks as elementary risk factors, although the methodology applies
to a wide range of financial instruments.
A simple formula for value-at-risk can be obtained in the case where an n × 1 vector
of relative changes R in the market risk factors is a multivariate normal random variable
with mean vector and covariance matrix C, and if one assumes that the change in portfolio value can be approximated by an affine function of the relative changes in the risk
factors:
≈
+
T
R
(4.2)
Throughout this chapter we shall use superscript T to denote the transpose. Note: We are
using
to denote the change in portfolio, i.e.,
= t − 0 for a time lapse t, whereas
in the dot product is the vector of sensitivities w.r.t. the returns (i.e., the delta Greeks of
the portfolio), as defined later.
2
The Web site www.gloriamundi.org is an excellent source for information and links to papers on value-at-risk.