CPPI strategies in discrete time [Elektronische Ressource] / vorgelegt von Michael Brandl

Michael Brandl

Informations

• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2009
• Nombre de lectures: 36
• Langue: Deutsch
• Poids de l'ouvrage: 1 Mo

Extrait

CPPI Strategies in Discrete Time
Inaugural-Dissertation
zur Erlangung des Grades eines Doktors
der Wirtschafts- und Gesellschaftswissenschaften
durch die
Rechts- und Staatswissenschaftliche Fakultät
der Rheinischen Friedrich-Wilhelms-Universität
Bonn
vorgelegt von
Michael Brandl
aus Daun
Bonn 2009Dekan: Prof. Dr. Erik Theissen
Erstreferent: Prof. Dr. Klaus Sandmann
Zweitreferent: Prof. Dr. Frank Riedel
Tag der mündlichen Prüfung: 4. Juli 2008
Diese Dissertation ist auf dem Hochschulschriftenserver der ULB Bonn
http://hss.ulb.uni-bonn.de/diss_online elektronisch publiziertDanksagung
Sehr herzlich bedanke ich mich an dieser Stelle bei allen, die mich bei der Anfertigung
dieser Arbeit direkt oder indirekt unterstützt haben. Zuallererst gilt mein besonderer
Dank meinem Betreuer, Prof. Dr. Klaus Sandmann, sowohl für seine fachliche und orga-
nisatorische Unterstützung als auch für seine umgängliche Art, die stets eine angenehme
Atmosphäre geschaffen hat. Ebenso bedanken möchte ich mich bei Prof. Dr. Frank Riedel
für kritische Hinweise und seinen Spielwitz.
Desweiteren danke ich Dr. Antje Mahayni und Sven Balder, sowohl für fachliche Dis-
kussionen und eine stimulierende Zusammenarbeit als auch für gelungene Skatabende.
Außerdem erwähnen möchte ich meine Kollegen aus der Betriebswirtschaftlichen Abtei-
lung III der Universität Bonn, An Chen, Haishi Huang, Simon Jäger, Birgit Koos, Xia
Su, Jens Wannenwetsch, Manuel Wittke und Anne Ruston, die es stets verstanden haben,
eine ungezwungene und freundliche Stimmung zu bewahren, die die Arbeit sehr erleichtert
hat.
Ganz besonderer Dank geht an Martina Blessing für ihre Geduld, ihre Rücksicht und
Rückhalt in schwierigen Zeiten.
Zu allergrößtem Dank verpflichtet bin ich meinen Eltern, die mich mein ganzes Leben auf
verschiedenste Arten und Weisen unterstützt haben.CPPI Strategies in Discrete TimeAbstract
In general, the purpose of portfolio insurance strategies is to limit the downside risk of risky
portfolios. The constant proportion portfolio insurance (CPPI) is a prominent example
of a portfolio insurance strategy. Based on a dynamic trading rule, the CPPI provides
payoffs greater than some minimum wealth level at some specified time horizon. The
great advantage of the CPPI is its particularly simple trading rule, which basically only
requires the knowledge of the current portfolio value and thus makes the CPPI applicable
to any kind of risky portfolio. Under the assumption of a complete financial market where
trading takes place in continuous time, it is well known that the payoffs provided by the
CPPI are greater than a pre-specified minimum wealth level with certainty. In this thesis
we are concerned with various sources of market incompleteness. One source of market
incompleteness are trading restrictions. Restricting the possibility of making changes
to the portfolio to a fixed set of trading dates allows for payoffs below the minimum
wealth level. The associated risk is called gap risk. The assumption of a fixed set of
trading dates is well suited for the derivation of various risk-measures related to gap
risk. Analyzing the gap risk is important with respect to the effectiveness of the CPPI
if trading in continuous time is not possible. One natural reason for the assumption
of trading restrictions are transaction costs. However, in the presence of transaction
costs the frequency of monitoring the portfolio is generally larger than the willingness
to rebalance the portfolio. With respect to transaction costs it is reasonable only to
rebalance the portfolio upon relevant changes in the portfolio value or the underlying
assets. This rationale leads to the notion of triggered trading dates. It turns out that
triggered trading dates are also better suited with respect to analyzing modifications of the
CPPI. The basic CPPI exhibits at least three structural problems. First, it requires the
assumption of unlimited borrowing which can be explicitly modelled with the introduction
of a borrowing constraint. Second, in the case of a good performance of the portfolio,
it is well possible that the minimum wealth level becomes insignificant in comparison to
the portfolio value. This can be modelled by increasing the minimum wealth level upon
good performances of the portfolio. Third, the exposure to the underlying risky assets
can become arbitrarily small such that portfolio may basically only consist of riskless
assets. Explicitly defining a minimum on the exposure to the risky assets provides another
modification. All modifications can be analyzed in a setup with triggered trading dates.
While the use of triggered trading dates allows for the modelling of transaction costs also
for the modifications of the CPPI, choosing small triggers allows for approximations of the
continuous-time case for which analytic expressions for the modifications are not known
in the literature so far either.Contents
Introduction 1
1 The Discrete CPPI with Fixed Trading Dates 7
1.1 ModelSetupandthesimpleCPPIincontinuoustime............ 9
1.2 Tradingrestrictions..................... 14
1.3 RiskMeasuresofDiscrete–TimeCPPI.......... 16
1.4 Effectivenesofthediscrete-timeCPPImethod............... 25
1.5 Convergence......................... 30
1.6 Conclusion..... 33
2 The Discrete CPPI with Triggered Trading Dates 35
2.1 BasicModelandDefinitions.......................... 38
2.2 TheSimpleDiscreteCPPIWithTriggeredTradingDates 44
2.3 LimitedBorowing-TheCappedCPPI.......... 59
2.4 Transactioncosts................................ 73
2.5 LongMaturities.. 78
2.6 Conclusion..... 83
3 Floor Adjustments on CPPI 85
3.1 TheCPPIwithFloorAdjustment....................... 87
3.2 Increasedinitialfloorlevels...... 97
3.3 Thecash-lockproblem........ 99
3.4 TheCPPIwithMinimumExposureRatio..................103
3.5 HedgingtheCPPIwithminimumexposureratio.....16
3.6 Conclusion..........................123
Appendix 125
iA Mathematical Prerequisites 125
A.1SomeaspectsaboutRandomWalks......................125
A.2BasicsaboutLaplaceTransforms.............13
A.3Someintegrals.............134
iiList of Figures
1.1 Expectedterminalvalueofasimplecontinuous-timeCPPI......... 13
1.2 Standard deviation of the terminal value of a simple continuous-time CPPI 13
1.3 Shortfall probability of the simple discrete-time CPPI with fixed trading
dates dependent on the number of rehedges for σ = 10% .......... 19
1.4 Shortfall probability of the simple discrete-time CPPI with fixed trading
dates dependent on the number of rehedges for σ = 30% 19
2.1 Sample path of Brownian motion with drift hitting twice the upper barrier
beforematuritytime.............................. 41
2.2 Distribution of the number of trading dates for different discretizations . . 45
2.3 Distribution of the number of trading dates fort volatilities .... 45
2.4 Density of the terminal value of the simple CPPI with triggered trading
datesfordifferentdiscretizations........................ 53
2.5 Distribution of the terminal value of the simple continuous-time CPPI con-
ditioned on the terminal value of the simple CPPI with triggered trading
dates ...................................... 57
2.6 Conditional distribution of the deviation of the terminal values of the simple
continuous-time CPPI and the simple CPPI with triggered trading dates . 57
2.7 Borrowing requirement of the simple CPPI with trading dates . . 59
2.8 Binomialtrewithmaximumlevelzero.................... 61
2.9 Binomialtrewithmaximumlevelgreaterzero ..... 61
2.10 Densities of the terminal values of the risky asset, the simple and capped
CPPI for G = 800................................ 68
2.11 Densities of the terminal values of the risky asset, the simple and capped
CPPI for G = 600. 68
2.12 Expected terminal value of the simple CPPI dependent on the number of
tradingdatesfordifferentvaluesofthetransactioncosts.......... 7
2.13 Expected terminal value of the capped CPPI dependent on the number of
tradingdatesfordifferentvaluesofthetransactioncosts 7
iii2.14 Probability of the capped CPPI outperforming the riskless asset dependent
on the maturity time for different borrowing limits and σ = 20% ...... 78
2.15 Probability of the capped CPPI outperforming the riskless asset dependent
on the maturity time for different borrowing limits and σ = 30% ...... 78
2.16 Probability of the simple CPPI outperforming the riskless asset dependent
onthematuritytimefordifferentproportionaltransactioncosts...... 80
2.17 Probability of the capped CPPI outperforming the riskless asset dependent
onthematuritytimefordifferentproportionaltransactioncosts...... 80
m 22.18 Combinations of m and σ such as to yield μ− r− σ =0for different μ.82
2
2.19 Expected yield of the capped CPPI for different values of the transaction
costs....................................... 82
3.1 Comparison of the expected yield of the capped CPPI and the CPPI with
floor adjustment as a function of the maturity time T ............ 91
3.2 Comparison of the standard deviation of the capped CPPI and the CPPI
with floor adjustment as a function of the maturity time T ......... 91
3.3 Comparison of the densities of the capped CPPI and the CPPI with floor
adjustmentinthequasi-continuous