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Probability Theory and Mathematical
Statistics
On the stochastic behaviour of Bank profitability
Isobel Burger*, Mark A. Petersen and Marisa van der Walt
School of Computer, Statistical and Mathematical Sciences
North-West University
Potchefstroom
SAMS Subject Classification: 10
This conference talk is based on the paper [1] by Burger, Petersen and van der Walt. Our
contribution provides a brief discussion of key aspects of the profitability of commercial
banks (see, for instance, [2], [3] and [4]). We make a technical contribution to this debate
by constructing stochastic models for concepts related to the capacity of such banks to
make a profit. In this regard, two measures of profitability are investigated. The first
of these is the return on assets, that is intended to measure the operational efficiency of
the bank. The other measure of profitability that we consider is the return on equity
that involves the consideration of the bank owner’s returns on their investment. Furthermore, we derive stochastic models for the aforementioned measures of profitability by
incorporating relevant components from the bank’s balance sheet and income statements
associated with off-balance sheet items.
References
[1] Burger, Isobel, Mark A. Petersen and Marisa van der Walt (2004). On the Profitability of
Banking Systems, Submitted.
[2] Freixas, X., Rochet, J.-C., 1997. Microeconomics of Banking, Cambridge MA, London.
[3] Mishkin, F.S. (2004). The Economics of Money, Banking and Financial Markets (Seventh
Edition), Addison-Wesley Series, Boston, USA, ISBN: 0-321-20463-8.
[4] Petersen, M. (2004). Stochastic Approach to the Solvency, Profitability and Operational
Control of Commercial Banks, Working Paper.
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On pricing South African renewable commodities by means of
the Schwartz model
Hennie Fouche* and Mark A. Petersen
School of Computer, Statistical and Mathematical Sciences
North-West University
Potchefstroom
SAMS Subject Classification: 10
This conference talk is based on the paper [1] by Fouche and Petersen. In our contribution, we
discuss and apply a stochastic model that may be used for the pricing of renewable commodities
like (white and yellow) maize, wheat and sunflowers. More specifically, we conclude that an
appropriate model for determining the spot price of the aforementioned commodities is the
Schwartz model (see, for instance, [2] and [3]). An important feature of this model is that it
reflects reality in the marketplace by making allowances for spot prices and convenience yields
to be mean-reverting. In order to illustrate these ideas we provide simulations and numerical
examples using data from the South African renewable commodities market. When using the
Schwartz model, we find that the behaviour of the spot prices for the simulations and real data
bear a close resemblance to each other.
References
[1] Fouche, C.H. and M.A. Petersen (2004). On Pricing in the South African Renewable Commodities Market, Submitted.
[2] Schwartz, E.S. (1997). The Stochastic Behavior of Commodity Prices: Implications for
Valuation and Hedging. Journal of Finance, 52, 923-973.
[3] Schwartz, E.S. (1998). Valuing Long-Term Commodity Assets. Journal of Energy Finance
and Development, 3, 85-99.
Stochastic optimal control for option pricing
T.A. McWalter
Programme for Advanced Mathematics of Finance
University of the Witwatersrand
SAMS Subject Classification: 10, 11
In this talk, the classic Merton portfolio selection problem is recast to allow the pricing of
options. A portfolio consisting of an underlying asset, an option on that asset and a riskless
bond is constructed and a stochastic optimal control approach (without consumption) is used to
find the optimal portfolio choice. A no-arbitrage condition is used to derive the option pricing
PDE which is shown, under maximal risk aversion, to be compatible with the Black-Scholes
pricing approach. We consider the case where there is a scrip borrowing cost and show how the
writer of the option is able to gain a comparative advantage in pricing the option if he is willing
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to assume some risk in the underlying. Finally we look at some possible ways of extending this
approach to enable the modelling of incomplete markets.
Pricing multiple warrants in the
discrete model (II)
Petrus H. Potgieter
School of Economic Sciences
University of South Africa
SAMS Subject Classification: 10, 17
Warrants are call options written by companies on their own shares, for example employee or
executive share options (ESOs). ESOs are increasingly being used as a method of compensation.
Executives are granted call options which become vested over time and can then be exercised
by the employee. The valuation of these ESOs and their expensing in the income statements
of firms have become pressing issues, particularly in the light of the International Financial
Reporting Standards to become mandatory in 2005.
The feature of interest to us is the effect that warrant or ESO issuance has on the capital
structure of the firm:
• The dilution effect means that the warrant cannot simply be priced as a call option on
the equity value of the firm per share.
• The share price process itself changes after warrant issuance - it is no longer lognormal
or Markov. The stock return volatility also changes.
We discuss the theory in Part I and in Part II look at a binomial tree model for a simple warrant
process.
An integral representation of the American put option
Sean Randell
Department of Computational and Applied Mathematics
University of the Witwatersrand
SAMS Subject Classification: 10, 16
Analytic solutions to the American put pricing problem invariably contain an integral representation of all or part of the value. One of these integral representations decomposes the
American put option value into a linear combination of the equivalent European put option
price and an early exercise premium. The early exercise premium is an integral that depends
on the unknown early exercise boundary. This talk will look at methods to approximate the
early exercise boundary, using the integral representation and hence also the early exercise
premium.
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10 Probability Theory and Mathematical Statistics
Bermudan swaption pricing in the
LIBOR market model
Nevena Šelić
Department of Computational and Applied Mathematics
University of the Witwatersrand
SAMS Subject Classification: 10
Callable LIBOR exotics, an example of which is a Bermudan swaption, is an important class of
interest rate derivatives that are best modelled using the LIBOR market model [3]. The class of
LIBOR market models provides a flexible framework in which the discrete forward LIBOR rates
are driven by a multi-dimensional diffusion. In the general setting the process is not Markov
and the high dimensionality renders the pricing of Bermudan contingent claims difficult to say
the least. A commonly used approach is the regression methodology of [2]. In this talk we
describe the duality approach proposed by [4] and [1] for pricing American options and examine
the way that this can be applied to the pricing of Bermudan swaptions in the LIBOR market
model.
References
[1] Haugh, M. B. and Kogan, L. (2004) “Pricing American Options: A Duality Approach”,
Operations Research, 52(2), 258–270
[2] Longstaff, F. A. and Schwartz, E. S. (2001) “Valuing American Options by Simulation: A
Simple Least-Squares Approach”, The Review of Financial Studies, 14(1), 113–147
[3] Piterbarg, V. V. (2003) “A Practitioner’s Guide to Pricing and Hedging
Callable Libor Exotics in Forward Libor Models”, SSRN Working Paper Series,
http://ssrn.com/abstract=427084
[4] Rogers, L. C. G. (2002) “Monte Carlo Valuation of American Options”, Mathematical
Finance, 12(3), 271–286
Pricing warrants in the continuous-time model (I)
Barbara Swart
School of Economic Sciences
University of South Africa
SAMS Subject Classification: 10, 17
Warrants are call options written by companies on their own shares, for example employee or
executive share options (ESOs). ESOs are increasingly being used as a method of compensation.
Executives are granted call options which become vested over time and can then be exercised
by the employee. The valuation of these ESOs and their expensing in the income statements
of firms have become pressing issues, particularly in the light of the International Financial
Reporting Standards to become mandatory in 2005.
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The feature of interest to us is the effect that warrant or ESO issuance has on the capital
structure of the firm:
• The dilution effect means that the warrant cannot simply be priced as a call option on
the equity value of the firm per share.
• The share price process itself changes after warrant issuance - it is no longer lognormal
or Markov. The stock return volatility also changes.
We discuss the theory in Part I, and in Part II look at a binomial tree model for a simple
warrant process.
Jump processes for option pricing
Isaac Takaidza
Department of Applied Mathematics
University of the North
SAMS Subject Classification: 10
When analyzing financial data it turns out that the prices process can jump: in particular
jumps become more visible as one samples the path more frequently.
Jump diffusion processes admit asymmetric and fat-tailed distribution of asset returns and thus
have similar impact on option prices compared to the Black-Scholes model when nonlinear
volatility structures are chosen. While asymmetric jump can induce distortion of option price
errors, the skewness of jump offers better explanations to empirical findings on implied volatility
curves.
Empirical data shows that the distributions of stock prices have peaks, a generic property of
models with jumps. Models with jumps allow for more realistic representation of price dynamics
and a greater flexibility in modelling.
We discuss some general properties of Levy processes, in particular the compound Poisson
processes. Fundamental results obtained in compound Poisson case are extended to a more
general setting.
Asian options of American type
G. Peskir and N. Uys*
Department of Computational and Applied Mathematics
University of the Witwatersrand
Johannesburg
SAMS Subject Classification: 10
We derive the free boundary problem for the early exercise Asian call with floating strike and
show that the optimal stopping boundary can be characterized as the unique solution of a
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10 Probability Theory and Mathematical Statistics
nonlinear integral equation arising from the early exercise premium representation (an explicit
formula for the arbitrage-free price in terms of the optimal stopping boundary). The key
argument in the proof relies upon a local time-space formula.
On short-term pension fund models and decision theory
Marisa van der Walt* and Mark A. Petersen
School of Computer, Statistical and Mathematical Sciences
North-West University
Potchefstroom
SAMS Subject Classification: 10
This conference talk is based on the paper [3] by van der Walt and Petersen. In our contribution, we employ a decision theoretic approach to solve an optimization problem related to
continuous-time pension funds whose dynamics are described for a short time period only (see,
for instance, [2] for other short-term problems). In this regard, our main objective is to minimize contributions made by members of the said fund and optimize asset allocation strategies
in order to maintain reasonable fund solvency. We find solutions to the aforementioned problem
by making use of principles from dynamic programming and optimal decision theory (see, for
instance, [1]). As part of this process, we analyze both quadratic and exponential value (cost)
functions and demonstrate how to find suitable decision rules that minimize them.
References
[1] D.P. Bertsekas, (1976). Dynamic Programming and Stochastic Control. Mathematics in
Science and Engineering, 12, Academic Press.
[2] R. Dana and M. Jeanblanc, (2003). Financial Markets in Continuous Time. Springer.
Berlin.
[3] Mark A. Petersen and Marisa van der Walt (2004). On a Decision Theoretic Approach to
the Pension Fund Problem, Submitted.
Suitability of renewal processes and level crossings as models for
inter transaction arrival times
H.F. van Rooy II
Department of Mathematics and Statistics
Rand Afrikaans University
SAMS Subject Classification: 10
The last few decades have seen a great upsurge of research in the area of finance. Research
such as that done by Black, Scholes and Merton has popularized continuous time finance. In
contrast to this school of thought the author is of the opinion that discrete time finance offer a
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39
more realistic look at the practice of finance, as this is closer to how processes manifest in the
”real world”.
The author is currently investigating the applicability of modelling high frequency financial time
series, with specific focus on four intra day stock prices on the Johannesburg Stock Exchange,
as Marked Point Processes. This paper examines the realism of modelling the arrival times of
the ”points”, or the inter transaction ”waiting times”, as two special classes of Point Processes,
namely Renewal Processes and the Level Crossings of a more complicated, possibly continuous,
process. The former assumes independence, while research points to a time of day effect. This
can be addressed by means of a suitable transformation. The examination of the ”marks”, or
in this instance the stock prices, is deferred for later investigation.
References
[1] Cox, D.R and Isham, V (1980). Point Processes. London: Chapman and Hall.
[2] Gourieroux, C and Jasiak, J (2001). Financial Econometrics. Princeton: Princeton University Press. 341 - 408.
[3] Liptser, R.S. and Shiryayev, A.N. (1978). Statistics of Random Processes II. New York:
Springer-Verlag. 236-281.
The need for accurate cumulative normal approximations
in option pricing
Graeme West
School of Applied Mathematics
University of the Witwatersrand
SAMS Subject Classification: 10
There are plenty of ‘war stories’ of option calculators returning negative prices for slightly exotic options - for example, compound options, rainbow options, or partial time barrier options.
Upon investigation it turns out that the culprit is typically an insufficiently accurate univariate
or bivariate cumulative normal approximation. High degree accuracy is needed, because typically in option pricing the ‘small’ option value is the difference of two ‘large’ numbers - as an
illustrative example the Black Scholes option price V = SN (d1 ) − XN (d2 ) has this property.
More generally option prices might be some linear combination of ‘large’ numbers. Thus it is
necessary that the ‘coefficients’ N (·) be accurate. Furthermore, it is exactly these quantities
that typically will give the hedge ratios.
In this talk we show some double precision algorithms for univariate, bivariate and trivariate
cumulative normals, and show how this resolves the above problem. These algorithms are vb
translations of the FORTRAN routines of Genz (Statistics and Computing, 2004). We also
apply the trivariate case to pricing 3 asset rainbow options (Johnson, Journal of Financial and
Quantitative Analysis, 1987).
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10 Probability Theory and Mathematical Statistics
Periodicity and scaling of eigenmodes in an emerging market
Diane Wilcox*1 and Tim Gebbie2
Department of Mathematics and Applied Mathematics
1 University of Cape Town
2 FutureGrowth Asset Management
SAMS Subject Classification: 10
We investigate periodic, aperiodic and scaling behaviour of eigenmodes, i.e. daily price fluctuation time series derived from eigenvectors, of correlation matrices of shares listed on the
Johannesburg Stock Exchange (JSE) from January 1993 to December 2002. Periodic, or calendar, components are investigated by spectral analysis. We demonstrate that calendar effects
are limited to eigenmodes which correspond to eigenvalues outside the Wishart range. Aperiodic and scaling behaviour of the eigenmodes are investigated by using rescaled range methods
and detrended fluctuation analysis (DFA). We find that the eigenmodes which correspond to
eigenvalues within the Wishart range are dominated by noise effects. In particular, we find
that interpolating missing data or illiquid trading days with a zero order hold introduces high
frequency noise and leads to the overestimation of uncorrected (for serial correlation) Hurst
exponents. DFA exponents of the eigenmodes suggest an absence of long term memory.
Valuation of Asian options
Hong-Kun Xu
School of Mathematical Sciences,
Westville Campus
University of KwaZulu-Natal
SAMS Subject Classification: 10
Assume that an underlying asset (stock) {S(t)}t≥0 follows a geometric Brownian motion {W (t)}t≥0 .
A European Asian option on the stock is a claim whose payoff depends on some sort of averages
of the stock prices between the initiation and expiration time. The averages can be discrete
(arithmetic or geometric) and continuous. We are going to talk about the risk-neutral valuation of and PDE approach to Asian options. Stochastic volatility in Asian options is possibly
discussed. Some numerical methods will also be included.