Using R for teaching financial mathematics and statistics

Transcription

1 MSOR Connections Vol No Spring Term 20 Julian Stander and John Eales Using R for teaching financial mathematics and statistics Julian Stander School of Computing and Mathematics University of Plymouth j.stander@plymouth.ac.uk We are grateful to our colleagues Paul Hewson and Rana Moyeed and to Neville Davies John Marriott and Kate Richards of the Royal Statistical Society Centre for Statistics Education for helpful conversations and encouragement. Abstract We discuss the use of R a free software environment for statistical computing and graphics in teaching financial mathematics and statistics. R can be used to demonstrate the ideas behind Brownian motion geometric Brownian motion and stochastic integration and to explore the dependence of the price of a financial product called a European option on other quantities such as the interest rate. The option price can also be estimated using a Monte Carlo approach and we discuss how R can be employed to illustrate features of this Monte Carlo methodology. Throughout we show how R can be used to provide insights into various financial and statistical concepts introduce a variety of relevant R packages and provide suggestions for further R related work. John Eales School of Computing and Mathematics University of Plymouth j.eales@plymouth.ac.uk. Introduction In a previous article [4] we discussed the use of Minitab [] for teaching statistics in higher education to students who may find mathematical subjects difficult and who may be studying statistics for the first time. Here we turn our attention to the use of another package R [9] in more advanced specialized teaching. R is a free software environment for statistical computing and graphics that is now widely adopted for teaching statistics and related modules in the UK and throughout the world. In recent years R together with the excellent and easy to use editor Tinn-R [5] or R s own script editor was used to enhance the student learning experience on a twenty credit module Financial Mathematics and Statistics offered on the MScs in Applied and Computational Mathematics and in Applied Statistics at the University of Plymouth. Ten credits of the material is taught in a somewhat simplified form to Stage 4 (final year) students on a variety of business related programmes. R has proved invaluable for illustrating a variety of complicated ideas and for engaging students in discussions about more standard statististical techniques. The use of R in this module has also provided students with additional opportunities to develop their programming abilities a very valuable transferable skill. In section 2 we describe some of the topics covered on the Financial Mathematics and Statistics module. The use of R to illustrate and explore one of these elementary stochastic calculus is discussed in section 3 whilst R s role in teaching another how to find the price of a financial product called a European option is discussed in section 4. The option price can be expressed as an expectation which can be estimated using the Monte Carlo method and in section 5 we discuss how we can use R to illustrate features of this Monte Carlo methodology. Finally in section 6

2 MSOR Connections Vol No Spring Term 20 we briefly present our conclusions. R code to produce all the figures in this paper is available from the authors. 2. Financial Mathematics and Statistics Module Much of the material taught on the Financial Mathematics and Statistics module is motivated by the problem of pricing option contracts and is loosely based on the excellent text by Mikosch [8]. A European call (or put) option is a contract purchased at time t = 0 which entitles the purchaser to buy (or sell) at expiry time t = T an underlying asset at a fixed price K known as the strike price. Examples of an underlying asset could be a share or some quantity of oil. The problem of pricing option contracts has received considerable attention in the literature and we will not revisit it here; see the seminal work by Black and Scholes [3] Merton [7] and Mikosch [8] amongst many others for a concise summary. The option contract price known as the rational or fair price can be found using arguments based on stochastic calculus via the Itô Lemma and associated results; alternatively a change of measure approach via Girsanov s Theorem can be employed. Full details are given in Mikosch [8] for example. Considerable time in the module is therefore given to developing and exploring some of the ideas behind elementary stochastic calculus. Once the fundamental concepts and definitions have been established they are used to derive the option rational price. The rational price itself depends on five quantities: the initial value of the asset; K the strike price; T the option expiry time; σ the volatility of the asset value; and r the interest rate. The dependence of the rational price on these five quantities is explored in detail. Next the alternative derivation of the rational price using a change of measures approach is presented. This leads to an expression for the rational price as an expectation which can be estimated using the Monte Carlo method; see Rizzo [0] for example. Other topics in the module include stochastic differential equations interest rate models foreign exchange and value at risk but we will not refer to them again. 3. Illustrating Brownian motion geometric Brownian motion and elementary stochastic calculus with R Brownian motion is a key concept in stochastic calculus and hence in finance. A stochastic process where time t Є [0 ) is called Brownian motion if it starts at zero i.e. B 0 = 0; if it has stationary independent increments i.e. if the increment B s has the same distribution as +h B s+h s < t h > 0 and if 2... B are independent for n tn t < <t n ; if for every t > 0 has a N(0 t) distribution and hence B N (0 t t t ) i = 2... n; () i ti i i and if has a continuous sample path i.e. there are no gaps. This definition can be quite daunting for students meeting it for the first time due to its rather abstract nature. It is however straightforward to get students to write a simple R function to plot (an approximation of) a Brownian motion sample path. This can be set as an exercise once the algorithm to generate sample paths has been explained. Students will become familiar with R functions such as rnorm and cumsum and with some of R s plotting capabilities when doing this exercise. Fig shows 500 Brownian sample paths t Є [0 ] together with a probability density estimate (unbroken curve) based on the 500 values of B with the theoretical N(0 ) density (broken curve) also shown. Fig clearly illustrates the random nature of Brownian motion the fact that B ~ N(0 ) and the fact that the variance of Brownian motion increases with t. It also provides the instructor with the opportunity to discuss density estimation and the use of R s density function; see Silverman [] and Venables and Ripley [4]. Fig 500 Brownian motion sample paths t Є [0 ] together with (right) a probability density estimate (unbroken curve) based on the 500 values of B with the theoretical N(0 ) density (broken curve) also shown. Brownian motion is the basic ingredient of geometric Brownian motion a commonly adopted model for the value X t of an asset at time t Є [0 T]. Geometric Brownian motion takes the form: X t = exp(µt + σ ) where is the initial value and µ is known as the drift and σ the volatility. There are various ways of estimating µ and σ one of which uses historical data. Students can easily modify their Brownian motion sample path plotting function to plot geometric Brownian motion sample paths. Their modified function can be used to explore the roles of µ and σ as illustrated in Fig 2 where the effect of different drifts and volatilities can easily be seen. Theoretic results about the mean and variance of geometric Brownian motion X t can be compared with their sample counterparts. A natural question is whether real asset values follow geometric Brownian motion. The function get.hist.quote of the R package tseries [3] can be used to download historical financial data such as share values X t t = 2... over the web from a given data provider such as Yahoo! Finance. The same function can

3 MSOR Connections Vol No Spring Term 20 using mean square error. This is illustrated in Fig 3 which also provides the instructor with an opportunity to discuss how the quality of the approximation depends on n. Students can also investigate the effect of changing the definition of the sum in (2) to n B (B B ) for example. A challenging i = ti ti ti theoretical and practical exercise would be to ask them to transform the scales in Fig 3 so that points are distributed uniformly across them rather than being bunched towards lower values. Fig 2 Four sample paths of geometric Brownian X t = exp(µt + σ ) t Є [0 ] with different drifts µ (upward in the top row downward in the bottom row) and volatilities σ (low in the left column high in the right column). also be used for downloading indices such as the FTSE and exchange rates. Downloaded share values can be transformed to Y t = log X t log X t t = where log is to the base e which should follow a normal distribution if X t follows geometric Brownian motion. At this point the instructor can engage students in a discussion about checking goodness-of-fit and introduce them to the qqplot function from the car package [2] as an example of a tool for doing this. Students can also be asked to provide estimates of µ and σ from the downloaded historical data providing the instructor with further opportunity to discuss statistical estimation concepts. The rules of stochastic calculus are different from the rules of ordinary calculus. Following the approach taken in Riemann integration we may define the random integral " 0 as the following limiting sum: & 0 Bt := lim n n" i = B (B ) (2) i ti ti in which 0= t 0 < t < < t n = is a partition of the interval [0 ]. The notion of a random integral is a difficult one to grasp when met for the first time. Further complications arise when it comes to evaluating the integral. A naive application of the rules of ordinary calculus may evaluate " 0 as (B 2 B2)/2 = 0 B2/2 since B = 0. It can however be 0 proved in several ways that " 0 = (B 2 )/2. The students can quite easily modify their Brownian motion R function to evaluate the sum in (2) for a finite value of n and to compare the random value of this approximating sum with the random limit (B 2 )/2. R s function diff will play a part in evaluating n B (B B ) as will i = ti ti ti vector subsetting using [ ] to omit. Students can get an n appreciation of the effect of choosing different values of n on the accuracy of this approximation and this can be quantified Fig 3 Values of the approximating sum n B (B B ) against the i = ti ti ti limiting value (B 2 )/2 as n " are shown for n = and 000 in the approximating sum. The mean square error (MSE) associated with this approximation is also given. 4. Exploring the rational price of an option with R We have already mentioned that arguments from stochastic calculus via the Itô Lemma and associated results can be used to find the price of an option. It turns out that the price of a European call option is: Φ(g(T )) K exp( rt)φ(h(t )) (3) in which Φ is the cumulative distribution function of a standard normal random variable supplied by the R function pnorm log b l + ar + g(t ) = K 2 σ 2 k T σ T and h(t ) = g(t ) σ T. This price is seen to depend on five quantities: the initial value of the asset; K the strike price; T the option expiry time; σ the volatility of the asset value; and r the interest rate. The price does not however depend on drift µ. Students can be set an exercise of writing an R function that takes the five dependent quantities K T σ and r as arguments and returns the rational price (3). Students can check the prices returned by their function with those given by the GBSOption function of

4 MSOR Connections Vol No Spring Term 20 the foptions [6] package. They can use their function or the GBSOption function to produce plots that explore in detail how the rational price depends on K T σ and r as shown in Fig 4. The instructor can use Fig 4 to engage students in discussions about the limiting values of (3) and about intuitive reasons for these rational price behaviours. A dynamic version of Fig 4 that makes use of the tcltk (part of R) and TeachingDemos [2] packages and that enhances student exploration of the effect of K T σ and r on option price is available from the authors. Fig 4 The dependence of the rational price of a European call option on the initial value of the asset (vertical line at K dotted line has slope ) on K the strike price (vertical line at ) on T the option expiry time on σ the volatility of the asset value and on r the interest rate (horizontal line at K asymptote at for the last three graphs). Meeting the GBSOption function of the foptions [6] package has the spin-off of introducing students to general option pricing software. They can be asked to explore functions for pricing more complicated or exotic options such as barrier options in similar packages such as fexoticoptions [5]; see Joshi [6] for the definition of barrier and other options. 5. Estimating the rational price of an option using a Monte Carlo approach with R We have already stated that a change of measure approach via Girsanov s Theorem can be employed to price options. For the European call option the rational price can be shown to equal: E :exp ( rt) (X T K) + exp c qb T 2 q 2 TmE (4) in which X T = exp(µt + σb T ) as before (Y) + = max(0y) and q = µ + 2 σ 2 r. (5) σ The expectation in (4) is taken over the random variable B T ~ N(0T); see Mikosch [8] for example for full discussion. As the value of the rational price does not depend on drift µ µ is a free parameter and can be set to any value. A commonly used choice for µ is r 2 σ 2 yielding q = 0 in (5) and a simplification of (4) which upon substitution of X T takes the form: E :exp ( rt) c exp 'cr 2 σ 2 m T + σ T Z Km + E (6) in which the expectation is over the random variable Z ~ N(0 ); the expression (6) makes use of the fact that B T and T Z have the same N(0T) distribution. To avoid working out this expression analytically we can estimate (6) using the Monte Carlo method as: n n i = :exp ( rt) c exp 'cr 2 σ 2 m T + σ T Z i Km + E (7) in which Z i ~ N(0 ) i =...n are a random sample of size n that can be generated using R s rnorm function; see Rizzo [0] for detailed discussion of Monte Carlo estimation. Such a Monte Carlo estimation procedure can be used to price more complicated options for which an analytic treatment may not be available. Students can be set an exercise of writing an R function that allows them to evaluate (7). They can then compare the resulting rational price estimate with the price yielded by their own rational price function or the GBSOption function of the foptions [6] package with which they are already familiar. By running their function many times using a for loop or via replicate they can get a feel for the variation associated with (7) and for the effect of increasing sample size n on the bias and standard deviation of the estimates (standard errors). This can provide them with a much better understanding of the Monte Carlo estimation method itself. The European put option (recall that the put option is the right to sell an asset while the call option is the right to buy it) can be easily priced by changing (X T K) + in (4) into (K X T ) + and making the associated changes in (6) and (7). Fig 5 encapsulates all these ideas for n = 00 and n = 0000 and can inform discussion about the effect of sample size on Monte Carlo error. An extended exercise could investigate the effect of the choice of the free parameter µ on Monte Carlo error. A different exercise could modify (4) and the resulting Monte Carlo methodology to price more complicated options such as barrier options. Both these would enhance student understanding and extend knowledge. A more advanced task would involve investigating the effect of making changes to the underlying distributional assumption () and could lead to interesting discussions about the Central Limit Theorem especially if distributions with heavier tails than the normal are considered. 6. Conclusions In this paper we have illustrated some of the ways in which R [9] a free software environment for statistical computing and graphics can be used in teaching financial mathematics and statistics. In particular R can be employed to give students a better understanding of Brownian motion and geometric Brownian motion the commonly adopted model for the value of an asset. R can also be used to demonstrate 0

5 MSOR Connections Vol No Spring Term 20 Fig 5 Histograms of 5000 Monte Carlo estimates of two option prices. The left column is for a European call option while the right column is for a European put option. The top row is for a sample size of n = 00 in the Monte Carlo estimation procedure while n = 0000 in the second row. The bias and the standard deviation of the estimates (standard error SE) are also given. The vertical line shows the exact rational price calculated using (3) for the call option and the analogous formula for the put option. some of the ideas of elementary stochastic calculus such as stochastic integration. It can then be utilized to calculate the rational price of a financial product called an option and to explore how rational price depends on other quantities such as interest rate. The rational price can also be expressed as the expectation of a random quantity. This can be estimated by using a Monte Carlo approach and we have described how R can be employed to give students an enhanced understanding of this Monte Carlo estimation method. In addition we have introduced a selection of packages and mentioned some useful available functions and we have provided suggestions for exercises and further work to extend student knowledge and understanding. In conclusion we believe that the use of R on our Financial Mathematics and Statistics modules has substantially enhanced the student learning experience by providing a sophisticated vehicle for the illustration and discussion of a range of financial and statistical concepts. It has also provided students with additional opportunities to develop valuable programming skills. We owe a great debt of gratitude to all who have volunteered their time and expertise to create the wonderful resource that is R and its contributed packages Black F. and Scholes M. (973) The pricing of options and corporate liabilities. J. Political Economy 8: Eales J. and Stander J. (2009). Using Minitab for teaching statistics in higher education. MSOR Connections Aug 2009 Vol. 9(No. 3):39 4. Available via: j_and_stander_j_minitabteach.pdf [Accessed 6 August 200]. Faria J. C. (2009) Resources of Tinn-R GUI/Editor for R Environment. UESC Ilheus Bahia Brasil. Joshi M. S. (2008) The Concept and Practice of Mathematical Finance. Cambridge University Press Cambridge second edition. Merton R.C. (973) Theory of option pricing. J. Econ. Manag. Sci. 4:4 83. Mikosch T. (998) Elementary Stochastic Calculus with Finance in View. World Scientific New Jersey. 9. R Development Core Team (200) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing Vienna Austria. 0. Rizzo M. L. (2008) Statistical Computing with R. Chapman & Hall/CRC Boca Ratony.. Silverman B. W. (986) Density Estimation for Statistics and Data Analysis. Chapman and Hall London. 2. Snow G. TeachingDemos: Demonstrations for teaching and learning 200. R package version Trapletti A. and Hornik K. tseries: Time Series Analysis and Computational Finance 20. R package version Venables W. N. and Ripley B. D. (2002) Modern Applied Statistics with S. Springer New York fourth edition. 5. Wuertz D. many others and see the SOURCE file. fexoticoptions: Exotic Option Valuation R package version Wuertz D. many others and see the SOURCE file. foptions: Basics of Option Valuation 200. R package version References. 2. Minitab. Available via: [Accessed 6 August 200]. Fox J. and Weisberg S. (200) An R Companion to Applied Regression. Sage Thousand Oaks CA second edition.