Selçuk J. Appl. Math. Selçuk Journal of Vol. 6. No.2. pp. 27-41, 2005 Applied Mathematics
Mathematical Models in Finance Halim Kazan1 and Ahmet Ergülen2
Department of Management, Faculty of Economy, Gebze Institute of High Technology ,41400 Gebze Kocaeli ,Turkey;
e-mail:halim kazan@ gyte.edu.tr
Department of Economics, Faculty of Economics and Administrative Sciences, Nigde University, Nigde, Turkey;
e-mail:aergulen @ nigde.edu.tr
Received : May 25, 2005
Summary. Finance is the corner stone of the free enterprise system. Good …nancial management is therefore vitally important to the economic health of business …rms, and thus the nation and the world. The …eld is relatively com-plex, and it is undergoing constant change in response to shifts in economic con-ditions say Brigham and Gapenski in the introduction of their Financial Man-agement book (Brigham, Eugene F., Gapenski, Louis C;1994). As they said the …eld is relatively complex since most of the …nancial decisions are involved with uncertainty and risk. This is where quantitative methods and …nance meets. In …nancial decision making process, like most of the decision making process, …nal decision made by managers, not by some mathematical tools. However, those mathematical tools, used in …nancial decision making process, contribute to managers’decision a lot. Finance is consist of three interrelated areas which are Money and Capital Markets, dealing with securities markets and …nancial institutions, Investments, focusing on the decisions of individuals, …nancial and other institutions while they choose securities for their investment portfolios; and Financial Management, involving the actual management of non …nancial …rms (Brigham, E. F., Gapenski, L. C;1994). In this study I tried to summarize mathematical methods that have been used in …nance historically.
Key words: Finance, Mathematical Modeling. 1. Mathematical Models in Finance
The mathematics of …nance has many applications of probability and optimiza-tion theory. Even though mathematical tools known to be di¢ cult, …nance the-ory over the last two decades has found its way into the mainstream of …nance
practice. Nowadays many …nancial institutions are using of mathematical mod-els on …nancial researches. It was not always thus. The scienti…c breakthroughs in …nancial modeling both shaped and were shaped by the extraordinary ‡ow of …nancial innovation which coincided with revolutionary changes in the structure of world …nancial markets and institutions during the past two decades (Merton, Robert C.;1994).
Here we need to de…ne which areas of mathematics interact with …nance. Seem-ingly the majority of the interactions is with theoretical and applied probability, and di¤erential equations. Much of the body of …nance theory developed to date is essentially linear, characterized by linear partial di¤erential equations or, in the case of American Options, which are of course nonlinear because they are free boundary problems, by a linear framework. The scope for mathematical modeling and analysis of …nancial time series to models of investor behavior in markets that may be ine¢ cient or illiquid (Howison, S. D.;1994). The …nance theory is all about behavior of economic institutions in allocating and deploying their resources, across time, in an uncertain environment. Time and uncer-tainty are the central elements that in‡uence …nancial behavior says Merton in his study. The complexity of their interaction brings intrinsic excitement to the study of …nance as it often requires sophisticated analytical tools to capture the e¤ect of this interaction. Indeed, the mathematical models of modern …nance contain some of the most beautiful applications of probability and optimization theory. But of course, all that is beautiful in science need not also be practical; and surely, not all that is practical in science is beautiful. Here we have both (Merton, Robert C., 1994).
Louis Bachelier’s dissertation on the theory of speculation is known to …rst study that brings mathematical models and …nance together which was completed at the Sorbonne in 1900. Bachelier’s work notes the continuoustime mathemat-ics of stochastic processes and the continuoustime econommathemat-ics of option pricing. In analyzing the problem of option pricing, Bachelier provides two di¤erent derivations of the Fourier partial di¤erential equation as the equation for the probability density of what is now known as a Wiener process/brownian motion. In one of the derivations, he writes what is now commonly called the Chapman-Kolmogorov convolution probability integral, which is surely among the earlier appearances of that integral in print. In the other derivation, he takes the limit of a discretetime binomial process to derive the continuoustime transition prob-abilities. This same approach using the binomial process is now applied as a numerical approximation method to solve complicated derivativesecurity pricing problems ( Merton, Robert C., 1992).
As I mentioned earlier, I’d like to give brief summary of mathematical models that has been used in …nance regardless which …eld of the …nance that mathe-matical tool used. For this purpose, I will give the chronological development of math models.
Finance was almost entirely a descriptive discipline with a focus on institutional and legal matters before 60s. Until this time …nance theory was pretty much a collection of accounting data, setting the rules. Actually, Bachelier’s work was not known to the …nance world until Samuelson discovered it in 60s (Samuelson, P. A;1965).
Samuelson suggested using instead geometric brownian motion de…ned as the pathwise solution
(1) St = So exp( W t + mt)
of the linear stochastic di¤erential equation
(2) dSt = StdW t + Stdt
where m = 1=2 2 and (Wt) denotes a standard brownian motion (Follmer, H., ;1994).
Although the mathematical modeling of bondprice sensitivity to interest rates (duration) had been developed by Macaulay (Macaulay, F.;1938) in 1938, there was little evidence of its use in practice more than 20 years later, even by issuer and trader specialists in the debt markets. Duration provides a measure of price sensitivity, and thus interest rate risk, that takes into account three important factors: coupon rate, time to maturity, and yield to maturity. A longer duration characterizes a more price sensitive bond. Finding the duration for a bond with a coupon rate not equal to zero is somewhat more complicated. For any bond, duration (D) is calculated as follows:
(3) D =Xt(CF t)(1 + r)P (b)1
where CF t is the cash ‡ow received in period t;T is the number of periods, and r is the appropriate yield to maturity.
Modern …nance is considered to begin in the 1960s. Mathematical models were developed in the …eld of capital markets and investments. The Markowitz (Markowitz, H., ;1952) mean variance theory of portfolio selection provided a tractable model for quantifying the riskreturn tradeo¤ for general assets with correlated returns. The basic technique employed by Markowitz …nds, for a given set of securities, what’s called the e¢ cient frontier. Portfolios that lie along the e¢ cient frontier o¤er investors the optimal risk/return combinations; these portfolios are called mean variance e¢ cient portfolios (Hearth, D., Zaima, J.;1995).
In order to illustrate Markowitz’s model, let us consider 3 assets shown in the table.
Assets: a b c Risk .12 .20 .20 Expected
Return .15 .30 .15
Here A and C have identical expected returns, while B and C have identical risk. It is obvious that A is preferable over C because its risk is less, and B is preferable over C because it has more expected return though they both have same risk. How to decide between A and B is the question to be answered. If we assume a line connects A and B in a X and Y graph (2 Dimensions), this line is called e¢ cient frontier. Any combination on this line is the optimum combination which minimize the risk and maximize the expected return. Mr. Markowitz explained the problem as: "We consider n securities whose returns r0 = (r1; r2; :::; rn) during the forthcoming period have expected values 0 = ( 1; 2; :::; n) and the covariance matrix C = ( ij). An investor is to select a portfolio X0 = (X1; X2; :::; Xn). The return R = r0X on the portfolio has expected value and variance, respectively,
(4) E = X; V = X0CX
The portfolio is to chosen subject to constraints
(5) AX = b ; X> 0
Wwhere A is mxn and b is mx1. Thus non-negative Xi are to be chosen subject to m>=1 linear inequalities.
A portfolio is feasible if it satis…es (5). An EV combination is feasible if it is the E and V of a feasible portfolio. A feasible EV combination (Eo, Vo) is ine¢ cient if there is another feasible EV combination (E1, V1) such that either
(i) E1> E0 and V16 V0 or (ii) V1< V 0 and E10> E o
12
Building on Markowitz’s fundamental work, Sharpe (Sharpe, W. F.;1964), who won the Nobel Prize with this study, and Lintner (Lintner, J.;1965)investigated the equilibrium structure of asset prices, and their Capital Asset Pricing Model (CAPM) became the foundational quantitative model for measuring the risk of a security. The CAPM would later form the basis for developing an entire industry to measure the investment performance of professional money managers (Merton, Robert C.;1994).
CAMP speci…es the relationship between risk and required rates of return on assets when they are held in well-diversi…ed portfolios. Camp theory has some assumptions for the sake of simplifying the model. These assumptions are:
Investors have homogeneous expectations. Frictionless capital markets.
Investors are rational and seek to maximize their expected utility func-tions.
All investors can borrow or lend at the risk free rate.
After stating the assumptions brie‡y, I want to give some information about Capital Market Line (CML) which is used in CAPM. In order to do this let us introduce a risk free asset, RF. Now combine the RF with a risky asset such as P1. Its expected return:
(6) ERp = (x)ERp1+ (1 X)RF
where X is the proportion of wealth invested in the risky portfolio P1 and (1-X) is the proportion invested in the risk free asset, RF. The standard deviation risk of the RF-P1 combination can be calculated as:
(7)
SDp = X2SD2pt + (1 X)2SD2RF + 2X(1 X)SDp1SDRF CORR(P1; RF ) The last two terms equal zero because SDRF equals zero. Thus:
(8) SDp = (X)2sd2p1= (X)SDpt
then CML equals:
(9) ERp = RF + (ERM RF=SDM ) SDp
where the y intercept (where the CML crosses the y axis) is RF and the slope equals (ERM-RF)/SDM.
Relative risk contribution of any security i = i: Then risk return relationship
(10) ERj = RF + (ERM RF )
Another important in‡uence of 1960s research on investment practice was the Samuelson (Samuelsom, P.A.;1965), Fama (Fama, E.;1938) e¢ cient markets hy-pothesis(random walk theory).This theory states that securities’prices ‡uctuate randomly around their respective intrinsic values. This theory implies that no
one or system can accurately and consistently predict short-term movements in securities prices.
The concept of market e¢ ciency evolved from research showing that stock price changes appeared to be independent over time, describing a random walk.
(11) P t = P t 1 + ap where a N (0; 2);
meaning that today’s price is equal to yestarday’s price, plus a random variable, a.
Earlier in 50s a statistician named Maurice Kendall (Kendall, M.;1953) was analyzing several economic time series using a new tool, the computer, when he discovered, to his surprise that changes in stack prices appeared to be almost random in nature. Furthermore, Kendall concluded that past price patterns couldn’t reliably predict future price patterns (Hearth, D., Zaima, J.;1995) It was Fisher & Lorie’s (Fisher, L., Lorie„ J.;1964) study of historical stock returns that …rst got attention to the e¢ cient market hypothesis. Using the newly created data base of the Chicago Center for Research in Security Prices, Fisher & Lorie showed that a randomly selected stock held from the mid1920s to the mid-1960s would have earned, on average, a 9.4% annual compound re-turn. Returns of this magnitude were believed to be considerably larger than those most professional managers earned for their clients during that period. Rigorous scienti…c con…rmation of this belief was provided by a host of empir-ical performance studies along lines set by Jensen who used the CAPM as a benchmark to test for superior performance among United States mutual funds in the postwar period (Merton, Robert C.;1994).
In early 1970s, mathematical models of …nance had more sophisticated, involv-ing both the inter-temporal and uncertainty aspects of valuation and optimal …nancial decision making. Markowitz’s mean variance model was enriched and extended in this time period. Inter temporal and international capital asset pricing models expanded the single risk measure, beta, of the SharpeLintner CAPM to multidimensional measures of a security’s risk. Solnik was the …rst to adopt these models into an international framework. Although static in its formal development, the arbitrage pricing theory of Ross also provides for mul-tiple dimensions in the measure of a security’s risk. Ross’ alternative security pricing model called Arbitrage Pricing Theory (APT). Ross uses the principle of arbitrage to develop a model with several factors and corresponding betas to price securities.
(12)
where EFk; t Eo; t is the risk premium for the kthfactor; Eo,t is the expected return value when all N factors equal zero; j; N equals to systematic risk or beta of security j with kth factor. This formula says that each factor Fk, captures an independently di¤erent e¤ect and its sensitivity is measured by its corresponding beta. The only requirement is that the number of factors must be less than the number of securities being evaluated (Roll, R., Ross, S.;1980). In early 70s one of the most important practical developments were done in option pricing theory by Black & Scholos (Black, F., & Scholes, M.;1973). This model incorporates the heading strategy described by the binomial pricing model, but it also allows for in…nite possible future stock prices. This model makes several important assumptions such as:
The stock underlying the call option provides no dividends or other dis-tributions during the life of the option.
There are no transaction costs in buying or selling either the stock or the option.
The short term, risk free interest rate is known and is constant during the life of the option.
Any purchaser of a security may borrow any fraction of the purchase price at the short term, risk free interest rate.
Short selling is permitted without penalty, and the short seller will receive immediately the full cash proceeds of today’s price for a security sold short.
The call option can be exercised only on its expiration date.
Trading in all securities takes place in continuous time, and the stock price moves randomly in continuous time.
(13)
V = P [N (d1)] Xe(kRF t)[N (d2)] d1= ln(P=X) + kRF + ( 2=2 t= t d2= d1 t
where;
V = current value of a call option with time t until expiration. P = current price of the underlying stock.
N (d1) = probability that a deviation less than di will occur in a standard normal distribution. Thus, N (d1) and N (d2) represent areas under a standard normal distribution function.
e = 2:7183
kRF = risk free interest rate.
t = time until the option expires (the option period). ln(P=X) = natural logarithm of P/X
2= variance of the rate of return on the stock.
In essence, the …rst part of the equation (), P [N (d1)] can be thought of as the expected present value of the terminal stock price, while the second term, Xe(kRF t)[N (d2)], can be thought of as the present value of the exercise price.
Virtually from the day it was published, this work brought the …eld to closure on the subject. The Chicago Board Options Exchange (OBOC) began trading the …rst listed options in the United States in April 1973, a month before the o¢ cial publication of the BlackSeholes model. By 1975, traders on the OBOE were using the model to both price and hedge their options positions. Indeed, Texas Instruments created a handhelci calculator which was specially programmed to produce BlackSeholes option prices and hedge ratios. Such a complete and rapid adoption of …nance theory into …nance practice was unprecedented, especially for a mathematical model developed entirely in theory. That rapid adoption was all the more surprising, as the mathematics used in the model were not part of the standard mathematical training of either academic or practitioner economists (Merton, Robert C.;1994).
The derivation of the Black & Scholes Option Pricing Model rests on the concept of a riskless hedge. By buying shares of a stock and simultaneously selling call options on that stock, an investor can create a risk-free investment position, where gains on the stock will exactly o¤set losses on the option, and vice versa. This riskless hedged position must earn a rate of return equal to the risk-free rate; otherwise, an arbitrage opportunity would exist, and people trying to take advantage of this opportunity would drive the price of the option to the equilibrium level as speci…ed by the Black & Scholes model.
Black and Scholes, recognized that their replicatingportfolio approach could be applied not only to the option pricing, but to the pricing of general derivative securities with arbitrary nonlinear payo¤ contingent on one or more traded-security prices. Therefore, at the same time that their work was closing gates on fundamental research, on options, it was simultaneously opening new gates by setting the foundation for a new branch of …nance called contingentclaims analysis (CCA) says Merton. The applications of CCA range from the pricing of complex …nancial securities to the evaluation of corporate capital budgeting and strategic decisions and include, for instance, a uni…ed theory for pricing cor-porate liabilities and the evaluation of loan guarantees and deposit. Indeed, the theory and mathematical modeling of CCA for these applications have become even more important to …nance practice than the original options applications. Hans Follmer says the price ‡uctuation of a risky asset in a …nancial market is usually modeled as the trajectory of a stochastic process (St)t> 0 on some un-derlying probability space ( , F, P). Then he explains asset prices as temporary
equilibrium. In order to see his approach lets see the illustration given by him. Let A be a …nite set of agents who are active on the market for a single specula-tive asset. In reaction to proposed stock price p for period k and depending on circumstances summarized by a sample point ! in some probability space each agent a2A forms an excess demand ea,k(p,!). The actual stock price Sk(!) in period k is then determined by the equilibrium condition
(14) a 2 A Xea; k(Sk(!); !) = 0
then the sequence of temporary price equilibrium viewed as stochastic process
(15) Sk(!) (k = 0; 1; :::)
Let us assume that individual excess demand takes the simple log-linear form
(16) Ea; k(p; !) = a; klog(S0a; k(!)=p) + a; k(!);
where a,k may be viewed as a liquidity demand and S’a,k denotes an individual reference level of agent a for period k. Then the logarithmic equilibrium price in period k is determined via (14) as an aggregate,
LogSk(!) = X 0a; klogS0a; k(!) + a; k(!) (17)
a 2 Ak(!)
of individual price assessments and liquidity demands (Follmer, H.;1994). Davis and Clark talks about Discrete-time binomial models stating that option pricing based on a binomial stock price model was introduced earlier. Let us consider a 2-period model with time instants k=0,1,2 and a single stock whose price evolves as below:
S24 S12 S23 S0 S11 S22 S21 k=0 1 2
The initial price is S(0)=So and this moves to S(1)=S11 or S12 at time k=1, etc. It is not necessary to specify the probabilities of these transitions: the only requirement is that every path in the tree must have strictly positive probability. By the convention, S11<S12, S21<S22, S23<S24. There is also a riskless asset, the bank which yields a return per dollar over one period. To rule out arbitrage
we clearly must have S11<Rso<S12, S23<RS12<S24, S21<RS11<S22 as oth-erwise riskless pro…ts can be made by borrowing from bank to invest in stock or vice versa (Davis, M., Clark, J.;1994).
In the 1970s mathematical models had been primarily in equity markets and eq-uity derivative securities. On the other hand in 1980s, new applications were in the …xedincome arena. The models incorporated major multivariate extensions of the CCA methodology to price and hedge virtually every kind of derivative instrument, whether contingent or equities, …xedincome securities, currencies, or commodities. Dynamic models of interest rates were combined with CCA models to price both cashmarket and derivative securities simultaneously. The enormous U.S. national mortgage market could not have functioned e¤ectively without mathematical models for pricing and hedging mortgages and mortgage-backed securities whose valuations are especially complex because of prepayment options.
Even though mathematical …nancial models were used by U.S. institutional eq-uity investors, market makers and brokers trading U.S. eqeq-uity options, currency traders and a few …xed income traders in the 1970s, global and Investment banks and institutional investors of all types started using mathematical models during the 1980s. Meanwhile, practitioners in …nancial institutions actually took on a major role in applied research, including the creation of proprietary data bases, development of new numerical methods for solving partial di¤erential equations, and implementation of sophisticated estimation techniques for measuring model parameters.
During the 1970s, derivativesecurity exchanges were created to trade listed op-tions on stocks, futures on major currencies, and futures on U.S. Treasury bills and bonds. The success of these markets measured in terms of trading volume can be attributed in good part to the increased demand for managing risks in the volatile economic environment. This success in turn strongly a¤ected the speed of adoption of quantitative …nancial models. For example, experienced traders in the preceding overthecounter (OTC) dealer market had achieved a degree of success by using heuristic rules for valuing options and judging risk exposures. However, these rules of thumb were soon to be found to be in-adequate for trading in the fastpaced exchangelisted options market with its smaller price spreads, larger trading volume and requirements for rapid trad-ing decisions while monitortrad-ing prices in both the stock and option markets. In contrast, formal mathematical models along the lines of the BlackScholes model were ideally suited for application in this new trading environment.
Compare to 70s the development of sophisticated mathematical models and their adoption into …nance during the 1980s has grown considerably (Finnerty, J.D.;1992). A wave of deregulation in the …nancial sector as well as huge govern-ment budget de…cits, especially in the U.S., which increased several times the
amount of sovereign debt worldwide that required inter mediation, and place-ment were important factors driving innovation (Merton, Robert C.;1992). Though the research allocations for developing math models and …nancial data-bases were better in 1980s, we can’t say that those math models were as essential as those of them in 1960s and 1970s. In addition, the opportunities and feasi-bility of implementing these models in practice were also much greater. As it a¤ected to most of the scienti…c …elds, major developments in computing and telecommunications technologies made possible the formation of many new …-nancial markets and substantial expansions in the size of existing ones. Those same technologies made feasible the numerical solution of new complex CCA models with multivariate partial di¤erential equations.
The reduction in transaction costs for …nancial institutions was substantial dur-ing the 1980s. The costs of implementdur-ing …nancial strategies for institutions using derivative securities such as futures or swaps are often one tenth to one twentieth the cost of using the underlying eachmarket securities. This is espe-cially the ease for investments by foreign institutions which are often subject to withholding taxes on either interest or dividends.
Since we mention the transaction cost, I would like to talk about more detailed relationship between option valuation and the transaction cost. Valuing options when there are costs associated with trading the underlying leads to some in-teresting mathematical problems with important consequences. The value of an option depends on its pay o¤ at exercise or expiry. When you take the cost of transaction into consideration, you may face with 2 distinct problems: deter-mining the hedging strategy and valuing the option (Dewynne, J., Whalley, A., Wilmott, P;1994). Two approacvhes have been taken in the academic litera-ture: local in time and global in time. The former, for example Leland (Leland, H.;1985), Boyle & Vorst (Boyle, P., Vorst, T.;1992), Hoggard, and Whalley & Wilmott consider risk and return over a short interval of time. The latter, for example, Hodges & Neuberger, and Davis adopt optimal strategies in which risk and return are considered over the lifetime of the option.
The decline in costs does not derive only from reductions in bidask spreads and commissions. There are also cost savings from movement down the learning curve. With the cumulative experience of having built several new markets, innovators become increasingly more e¢ cient and the marginal cost of creating additional markets falls.
The same learningcurve e¤ect applies to the application of mathematical models. Beginning in the late 1980s and continuing to the current time, the volume of derivative securities business has shifted substantially from exchangetraded derivatives to morecustomized OTC contracts issued by …nancial institutions directly to their customers. I believe that this shift re‡ects a growing con…dence by institutions in their valuation models that comes not only from technical
improvements in the models but also from greater experience in their use. When a …rm operates in exchangetraded derivatives, it has current and historical prices to calibrate its valuation model. Competition among transactors in a centralized market provides at least some protection against losses from errors in any one transactor’s valuations. 1n a bilateral OTC transaction, the …rm must rely on its valuation models without the bene…t of market veri…cation. Hence, without su¢ ciently reliable mathematical models for valuation, much of the …nancial innovation that originates in the OTC market could not take place. At the same time such innovation increases the demand for more sophisticated models. Thus, we see that mathematical modeling both shapes and is shaped by the ‡ow of …nancial innovation (Merton, Robert C.;1994).
2. Further Scope
A successful framework for analyzing issues involving the global …nancial system in the future must address endogenously di¤erences in institutional structure across geopolitical boundaries and in the dynamics of institutional change. The neoclassicaleconomies perspective addresses the dynamics of prices and quan-tities. But it is largely an ’institutionfree perspective in which only functions matter. It thus has nothing to say directly about crosssectional or inter temporal di¤erences in the institutions that serve these functions. In contrast, there are the institutional perspective in which institutions not only matters but also the conceptual ’anchor’. This perspective takes as given the existing institutional structure and views the objective of public policy as helping the institutions currently in place to survive and ‡ourish. Framed in terms of the banks or the insurance companies, managerial objectives are similarly posed in terms of what can be done to make those institutions perform their particular …nancial services more e¢ ciently and pro…tably. The institutional perspective addresses crosssectional di¤erences across borders but is static in focus. Hence, institu-tional change is exogenous within this perspective. (Merton, Robert C.;1994). In summary, in the vast bulk of the past, mathematical models have had a lim-ited and ancillary impact on …nance practice. But during the last two decades, these models have become central to practitioners in …nancial institutions and markets around the world. In the future, mathematical models are likely to have an indispensable role in the functioning of the global …nancial system including regulatory and accounting activities.
References
1. Bensoussan A., Crouthy M., Galai D.,( 1994) : Stochastic Equity Volatilityand the Capital Structure of the Firm , The Royal Society Philosophical Transactions: Physical Sciences and Engineering, Volume 347, Number 1684.
2. Bernstein P. (1992): Capital Ideas: the improbable origins of Modern Wall Street, New York, Free Press.
3. B lack F., Scholes M. (1973): The Pricing of Options and Corporate Liabilities , J. of Political Economics, V.82, pp. 637-654.
4. Boyle P., Vorst T. (1992): Option Replication in Discrete Time with Transaction Cost , J. of Finance, V 47, pp. 271
5. Brigham Eugene F., Gapenski Louis C. (1994): Financial Management Theory and Practice, 7thEdition, The Dryden Press, pp5-7-8-9
6. Brown R., Schaefer S. (1994): Interest rate volatility and the Shape of the Term Structure, The Royal Society Philosophical Transactions: Physical Sciences and En-gineering, Volume 347, Number 1684.
7. Cushman J. (1993): Investing in Loans to Business, New York Times", March 19. 8. Davis M., Clark J. (1994): A Note On Super Replicating Startegies, The Royal Society Philosophical Transactions: Physical Sciences and Engineering, Volume 347, Number 1684, pp. 485-494.
9. Davis M., Panas V., Zariphopoulou T. (1993): European Option Pricing with Transaction Costs, SIAM Jl. Control, V 31.
10. Dewynne J., Whalley A., Wilmott P. (1994): Path dependent options and trans-action costs, The Royal Society Philosophical transtrans-actions: Physical Sciences and Engineering, Volume 347, Number 1684, pp. 517-525.
11. Dimand R. (1993): The Case of Brownian Motion: a Note on Bachelier’s Contri-bution , J. of History Science, V.26.
12. Du¢ e D., Kan R. (1994): Multifactor term Structure Model, The Royal Society Philosophical Transactions: Physical Sciences and Engineering, Volume 347, Number 1684.
13. Eckl S., Robinson J., Thomas D. (1990): Financial Engineering: A Handbook of Derivetive Products, Oxford.
14. Fama E. : The Behavior of Stock Prices, Journal of Business 38. pp. 134-135. 15. Finnerty J.D. (1964): An Overview of Corporate Securities Innovation, J. of Appl. Corporate Finance, V 4, pp. 23-39.
16. Fisher L., Lorie J. (1964): Rates of Return on Investments in Common Stocks, Journal of Business 37, pp. 1-24.
17. Freeman A. (1993): A Survey of International Banking: New Tricks to Learn, The Economist, April 10.
18. Follmer H. (1994): Stock Price Fluctuation as a Di¤usion in a Random Environ-ment, The Royal Society Philosophical Transactions: Physical Sciences and Engineer-ing, Volume 347, Number 1684, pp. 471-483.
19. Grundfest J. (1990): Internationalization of the World’s Securities Markets: Eco-nomic Causes and Regulatory Consequences, J. of Financial Services Res. V.4. 20. Harding D. (1994): Money Making from Mathematical Models, The Royal Society Philosophical Transactions: Physical Sciences and Engineering, Volume 347, Number 1684.
21. Hearth D., Zaima J. (1995): Contemporary Investments Security and Portfolio Analysis, Dryden, pp.188, 426.
22. Hodges S., Neuberger A. (1993): Optimal Replication of Contingent Claims under Transaction Costs, Rev. Future Markets, V.8.
23. Hodges S. (1994): Dynamic Asset Allocation: Insights from Theory, The Royal Society Philosophical Transactions: Physical Sciences and Engineering, Volume 347, Number 1684.
24. Hoggard T., Whalley A., Wilmott P. (1993): Hedging Option Portfolios in the Presense of Transaction Costs, Adv. Fut. Opt. Res.
25. Howison S. D. (1994): Applied Mathematics and Finance, The Royal Society Philosophical Transactions: Physical Sciences and Engineering, Volume 347, Number 1684, pp.465-469.
26. Kendall M. (1953): The Analysis of time Series, Part 1: Prices, Journal of Royal Statistical Society 96, pp. 11-25.
27. Leland H. (1985): Option pricing and Replication with Transaction Cost , J. of Finance, V 40, pp. 1283-1301.
28. Lintner J. (1965): The Valuation of Risk Assets and The Selection of Risky Investments in Stock Portfolios and Capital Budgets , Rev. Econ. Statistics, V 47, pp.13-37.
29. Macaulay F. (1938): Some Theoretical Problems Suggested by the Movement of Interest Rates, Bond Yields, and Stock Prices since 1856, National Bureau of Economic Research.
30. Markowitz H. (1952): Portfolio Selection , Journal of Finance 7, pp.77-91. 31. Markowitz H. (1994): The General Mean-Variance Portfolio Selection Problem, The Royal Society Philosophical Transactions: Physical Sciences and Engineering, Volume 347, Number 1684.
32. Merton Robert C. (1994): In‡uence of Mathemetical Models in Finance on Prac-tice: Past, Present, and Future, The Royal Society Philosophical Transactions: Phys-ical Sciences and Engineering, Volume 347, Number 1684, pp. 451-463.
33. Merton Robert C. (1992): Continuos -Time Finance, Revised Edition, Oxford. 34. Merton Robert C. (1992): Financial Innovation and Economic Performance , J. of Applied Corporate Finance V.4.
35. Miller, M.H. (1986): Financial Innovation: the last20 years and the next, J. of Financial Quantitative Analysis, V.21, pp. 459-471.
36. Miller M.H. (1992): Financial innovation: achievements and prospects, J. of appl. Corp. …nance, V 4, pp. 4-11.
37. Pliska S., Selby M. (1994): On a Free Boundary Problem that arises in Portfolio Management, The Royal Society Philosophical Transactions: Physical Sciences and Engineering, Volume 347, Number 1684.
38. Roll R., Ross S. (1980): An Emprical Investigation of the Arbitrage Pricing Theory, Journal of Finance,pp. 1073-1103.
39. Samuelson P. A. (1965): Rational Theory of Warrant Pricing, Industrial Manage-ment Review, pp.13-31.
40. Samuelsom P.A. (1965): Proof that Properly Anticipated Prices Fluctuate Ran-domly, Industrial Management Review 6, pp. 41-49.
41. Sharpe W. F. (1964): Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance, V.19, pp. 425-442.
42. Whalley A., Wilmott P. (1992): A Hedging Strategy and option Valuation Model with Transaction Costs, OCIAM, Oxford.
43. Williams J. (1989): The Economic Function of Future Markets, Cambridge Uni-versity Press.
44. Wilmott P., Howison S., Dewynne J. (1995): The Mathematics of Financial Deriv-atives, Cambridge University Press.
45. Ziemba W., Hensel C. (1994): World-wide security market anomalies, The Royal Society Philosophical Transactions: Physical Sciences and Engineering, Volume 347, Number 1684.
