Finance

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abcd
The Importance of Being NormalStephen Carlin
and Andrew Smith

• Finance Investment Conference 2003
• 22-24 June 2003
• The Caledonian Hilton Hotel, Edinburgh

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Presentation Overview

• Evidence for Non-Normality
• skewness, kurtosis
• Multi-dimensional models Three Approaches
• Triangle, copula and joint
• Implications for Option Pricing
• Our objective to show that multi-dimensional fat
  tailed distributions are important, practical and
  fun!

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Part I Evidence for Non-Normality

• Examine historic returns on 16 stock markets
• weekly, from 1 January 1973 through 1 January
  2003
• source Datastream
• Calculate sample moments
• ln( return factor )
• holding period 1 week through 13 weeks
• standard deviation, normalised skewness and
  kurtosis
• Compare to simulations from normal random walk

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Standard Deviation Comparison

• Historic data

• Random walk

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Normalised Skewness Comparison

• Historic data

• Random walk

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Normalised Kurtosis Comparison

• Historic data

• Random walk

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Five Example Distributions Quarterly Ln Returns
normal
Case 1
Case 2
Case 3
Case 4
mean
0.01
0.1
stdev
skew
0
0
0
-1
-1
kurt
0
2
5
2
5
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Example Distributions Density (log scale)
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Conic Moment Ln(MGF) Conic Section
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Conic Moment Construction (Bivariate Distribution)
large building at x 1 covered in fly-paper
z
y
swarm of flies released at (0,0,0)
x
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Fly Paper Chart
market return (Singapore)
market return (Belgium)
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Conic Moments Parameters

• Each fly performs a random walk
• Drift m per unit time
• Variance-covariance V per unit time
• 3 dimensional vector m
• with mx gt 0
• drift towards the flypaper
• 3 3 positive semi-definite matrix V

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Conic Moments Univariate Examples
conic section
distribution fn ProbZltz
mgf Eexp(pZ) exp?(p)
normal
inverse Gaussian
Cauchy
conic moment
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Part II Approaches to the Multivariate Case

• Weve looked at stock markets individually
• How to model many markets at once?
• Triangular approach
• Copula approach
• Joint Approach

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The Triangular (Cascade) Approach

X1 µ1 L11Z1 X2 µ2 L21Z1 L22Z2 X3 µ3
L31Z1 L32Z2 L33Z3 X4 µ4 L41Z1 L42Z2
L43Z3 L44Z4 X5 µ5 L51Z1 L52Z2 L53Z3
L54Z4 L55Z5
Xi returns on five different asset classes

Zi independent, zero mean, unit standard
deviation.
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Applying a Normal Copula to CM Marginals
Step 3 Simulate from CM distribution and
compute empirical distribution function (green)
Step 1 simulate bivariate normal
with correlation ? (blue dots)
Step 4 map each normal observation to
the corresponding percentile of the CM
distribution
Step 2 for each component, calculate
empirical distribution function (pink)
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Joint Approach

• Multivariate CM Distributions
• Natural extension Ln MGF is conic section
• n dimensional vectors X, p
• Eexp(p.X) exp?(p)
• (n1) vector m, (n1)(n1) positive definite V
• corresponding random walk construction

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Example Bivariate Density
where
reader exercise verify the MGF
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Approach Comparison
copula
joint
triangular
data evidence
inconclusive tests not well developed
marginal distributions
common V00/m0 kurt q(skew)
unrestricted
unrestricted
uncorrelated components
limited for skew distributions
unlimited
unlimited
generalised hyperbolic
conditional distributions
simple down the triangle otherwise messy
messy
order/basis independent
order yes basis no
yes
no
closed form MGF
yes
yes
no
no
no
yes
closed form PDF
yes
no
yes
Levy process
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Recommendation Capital Market Returns
sim 5
sim 4
sim 3
sim 2
year 03 04 05 06
country Afghanistan Albania Algeria Andorra Angol
a
joint model
triangular model
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Part III Option Pricing Implications

• Hedging / arbitrage arguments fail for fat tailed
  distributions
• Notation forward price F,
• risk free discount factor v,
• strike price K
• Absence of arbitrage implies deflator exists (but
  not unique)
• joint distribution of share price S
• and deflator D
• E(D) v E(DS) vF
• Assume ln(S/F), ln(D/v) bivariate conic moment
• defined by 3-vector m and 33 matrix V

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Option Pricing Parameters
m0
V00
V0S
V0D
VSS
VSD
mS
mD
VDD
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Option Price Formulas (cf Black-Scholes)

• Call Option
• Put Option

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Option Prices Implied Quarterly Volatility
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Choosing Preference Effect V0D

• Cannot observe an empirical distribution
• Equilibrium argument difficult to compute
• or Escher transform easy but arbitrary
• or Minimal martingale (Follmer Schweizer)
• minimise E(DlnD)
• or No-Good-Deal Bounds

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Looking for a Good Deal

• Suppose I have 1 to invest
• A particular strategy gives value R at time 1
• A good deal is a strategy where
• E(R) is large
• stdev(R) is small
• But as D is a deflator, we know E(DR) 1

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Bounding the Good Deal

• Economic rationale to impose no good deal
• Example best investment has
• quarterly stdev 10
• expected return risk free 2
• Now remember correlations -1
• 1 vE(R) Cov(D,R) - stdev(D) stdev(R)
• E(R) v 1 v 1 stdev(D) stdev(R)
• So pick v 1 stdev(D) 20

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Efficiency and ATM Implied Quarterly Volatility
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Conclusions

• Real markets show fat tails
• important eg for economic capital
• modelling focus on low number, not right number
• Multivariate models require care
• practical obstacles not insuperable
• may result in simpler models overall as outliers
  no longer justify additional parameters
• Finance stuck in a rut normal distributions
• business implications of alternative models are
  not yet widely understood

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References
Barndorff-Nielsen, O. E., 1998, Processes of
Normal Inverse Gaussian Type, Finance and
Stochastics, 2(1), pp. 4168. Cochrane, J. H.,
and J. Saá Requejo, 2000, Beyond Arbitrage Good
Deal Asset Price Bounds in Incomplete Markets,
Journal of Political Economy, 108, pp.
79119. Jarvis, S., Southall, F. E. and Varnell,
E. M. 2001. Modern Valuation Techniques, Staple
Inn Actuarial Society www.sias.org.uk Smith, A.
D. 2003. Option Pricing with Deflators. In
Scherer, B (ed) Asset Liability Management
Tools, Risk Publications.
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stcarlin_at_bw-deloitte.comandrewdsmith8_at_bw-deloitt
e.comwww.conicmoments.com
abcd
The Importance of Being NormalStephen Carlin
and Andrew Smith