Jan R M Röman Analytical Finance: Volume I The Mathematics of Equity Derivatives, Markets, Risk and Valuation Jan R M Röman Västerås, Sweden ISBN 978-3-319-34026-5 e-ISBN 978-3-319-34027-2 DOI 10.1007/978-3-319-34027-2 Library of Congress Control Number: 2016956452 © The Editor(s) (if applicable) and The Author(s) 2017 This work is subject to copyright All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Cover image © David Tipling Photo Library / Alamy Printed on acid-free paper This Palgrave Macmillan imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To my soulmate, supporter and love – Jing Fang Preface This book is based upon lecture notes, used and developed for the course Analytical Finance I at Mälardalen University in Sweden The aim is to cover the most essential elements of valuing derivatives on equity markets This will also include the maths needed to understand the theory behind the pricing of the market instruments, that is, probability theory and stochastic processes We will include pricing with time-discrete models and models in continuous time First, in Chap and we give a short introduction to trading, risk and arbitrage-free pricing, which is the platform for the rest of the book Then a number of different binomial models are discussed Binomial models are important, not only to understand arbitrage and martingales, but also they are widely used to calculate the price and the Greeks for many types of derivative Binomial models are used in trading software to handle and value several kinds of derivative, especially Bermudan and American type options We also discuss how to increase accuracy when using binomial models We continue with an introduction to numerical methods to solve partial differential equations (PDEs) and Monte Carlo simulations In Chap , an introduction to probability theory and stochastic integration is given Thereafter we are ready to study continuous finance and partial differential equations, which is used to model many financial derivatives We focus on the Black–Scholes equation in particular In the continuous time model, there are no solutions to American options, since they can be exercised during the entire lifetime of the contracts Therefore we have no well-defined boundary condition Since most exchange-traded options with stocks as underlying are of American type, we still need to use descrete models, such as the binomial model We will also discuss a number of generalizations relating to Black–Scholes, such as stochastic volatility and time-dependent parameters We also discuss a number of analytical approximations for American options A short introduction to Poisson processes is also given Then we study diffusion processes in general, martingale representation and the Girsanov theorem Before finishing off with a general guide to pricing via Black–Scholes we also give an introduction to exotic options such as weather derivatives and volatility models As we will see, many kinds of financial instrument can be valued via a discounted expected payoff of a contingent claim in the future We will denote this expectation E [ X ( T )] where X ( T ) is the so-called contingent claim at time T This future value must then be discounted with a risk-free interest rate, r , to give the present value of the claim If we use continuous compounding we can write the present value of the contingent claim as In the equation above, T is the maturity time and t the present time Example : If you buy a call option on an underlying (stock) with maturity T and strike price K , you will have the right, but not the obligation, to buy the stock at time T, to the price K If S ( t ) represents the stock price at time t , the contingent claim can be expressed as X ( T ) = max{ S ( T ) – K , 0} This means that the present value is given by The max function indicates a price of zero if K ≥ S ( T ) With this condition you can buy the underlying stock at a lower (same) price on the market, so the option is worthless By solving this expectation value we will see that this can be given (in continuous time) as the Black–Scholes–Merton formula But generally we have a solution as where Q ( S ( T ) > K ) and Q ( S ( T ) > K ) make up the risk neutral probability for the underlying price to reach the strike price K in different “reference systems” This can be simplified to the Black–Scholes–Merton formula as Here d and d are given (derived) variables N ( x ) is the standard normal distribution with mean and variance 1, so N ( d ) represent the probability for the stock to reach the strike price K The variables d and d will depend on the initial stock price, the strike price, interest rate, maturity time and volatility The volatility is a measure of how much the stock price may vary in a specific period in time Normally we use 252 days, since this is an approximation of the number of trading days in a year Also remark that by buying a call option (i.e., going long in the option contract), as in the example above, we not take any risk The reason is that we cannot lose more money than what we invested This is because we have the right, but not the obligation, to fulfil the contract The seller, on the other hand, takes the risk, since he/she has to sell the underlying stock at price K So if he/she doesn’t own the underlying stock he/she might have to buy the stock at a very high price and then sell it at a much lower price, the option strike price K Therefore, a seller of a call option, who have the obligation to sell the underlying stock to the holder, takes a risky position if the stock price becomes higher than the option strike price Acknowledgements I would like to thank all my students for their comments and questions during my lectures Special thanks go to Mai Xin who asked me to translate my notes into English I would also like to thank Professor Dmitrii Silvestrov, who encouraged me to teach Analytical Finance, and Professor Anatoly Malyarenko for his assistance and advice Notations B ( t ) The value of the money market account at time t r The risk-free interest rate R A short notation of + r Ω A sample space ω i Outcome i from a sample space Ω S ( t ) Price of a security (financial instrument, equity, stock) at time t F ( t ) The forward price of a security (financial instrument, equity, stock) at time t q The risk neutral (risk-free) probability of an increase in price p The objective (real) probability or the risk-free probability of an decreasing price Q The risk neutral probability measure P The objective (real) probability measure E Q [ ] The expectation value with respect to Q Var Q [ ] The variance with respect to Q ρ The risk premium X ( t ) A stochastic value/process I t The information set at time t u The binomial “up” factor with risk-neutral probability p u or q d The binomial “down” factor with risk-neutral probability p d or p Z A stochastic variable V ( t ) A value (process) μ,α The drift in a stochastic process σ The volatility in a stochastic process t Time T Time to maturity K The option strike price λ The market price of (volatility) risk (the sharp ratio) C A (call) option value Δ The change in the option value w.r.t the underlying price, S Γ The change in the option Δ w.r.t the underlying price, S ν The change in the option value w.r.t the volatility, σ Θ The change in the option value w.r.t time, t ρ The change in the option value w.r.t the interest rate, r d ,d Coefficients (variables) in the Black–Scholes model VaR Value-at-Risk F A set or subsets to the sample space Ω μ A finite measure on a measurable space W ( t ) A Wiener process N [ μ, σ ] A Normal distribution with mean μ and variance σ τ A stopping time (usually for American options) L t A likelihood function of time t Contents Trading Financial Instruments 1.1 Clearing and Settlement 1.2 About Risk 1.3 Credit and Counterparty Risk 1.4 Settlement Risk 1.5 Market Risk 1.6 Model Risk Time-Discrete Models 2.1 Pricing via Arbitrage 2.2 Martingales 2.3 The Central Limit Theorem 2.4 A Simple Random Walk 2.5 The Binomial Model 2.6 Modern Pricing Theory Based on Risk-Neutral Valuation 2.7 More on Binomial Models 2.8 Finite Difference Methods 2.9 Value-at-Risk (VaR) Introduction to Probability Theory 3.1 Introduction 3.2 A Binomial Model 3.3 Finite Probability Spaces 3.4 Properties of Normal and Log-Normal Distributions v As Double) As Double ' Calculation of seed value, Si Dim M As Double, q1u As Double, Su As Double, h1 As Double Dim Si As Double, d1 As Double, q1 As Double, LHS As Double Dim RHS As Double, bi As Double Dim E As Double, K As Double, n As Double n = 2*b/(v*v) M = 2*r/(v*v) q1u = (-(n - 1) - Sqr((n - 1)*(n - 1) + 4*M))/2 Su = x/(1 - 1/q1u) h1 = (b*T - 2*v*Sqr(T))*x/(x - Su) Si = Su + (x - Su)*Exp(h1) If (T = 0) Then T = 0.000000001 K = 2*r/(v*v*(1 - Exp(-r*T))) d1 = (Log(Si/x) + (b + v*v/2)*T)/(v*Sqr(T)) q1 = (-(n - 1) - Sqr((n - 1)*(n - 1) + 4*K))/2 LHS = x - Si RHS = BlackScholes(mPUT, Si, x, T, r, b, v) - _ (1 - Exp((b - r)*T)*CND(-d1))*Si/q1 bi = -Exp((b - r)*T)*CND(-d1)*(1 - 1/q1) bi = bi - (1 + Exp((b - r)*T)*CND(-d1)/(v*Sqr(T)))/q1 E = 0.000001 ' Newton Raphson algorithm for finding critical price Si Do While (Abs(LHS - RHS)/x > E) Si = (x - RHS + bi*Si)/(1 + bi) d1 = (Log(Si/x) + (b + v*v/2)*T)/(v*Sqr(T)) LHS = x - Si RHS = BlackScholes(mPUT, Si, x, T, r, b, v) - _ (1 - Exp((b - r)*T)*CND(-d1))*Si/q1 bi = -Exp((b - r)*T)*CND(-d1)*(1 - 1/q1) bi = bi - (1 + Exp((b - r)*T)*CND(-d1)/(v*Sqr(T)))/q1 Loop kp = Si End Function A VBA implementation of the Bjerksund-Stensland model is given below: Function BjerksundStenslandCall(s As Double, x As Double, _ v As Double, r As Double, _ T As Double, b As Double) _ As Double Dim Beta As Double, BInfinity As Double, B0 As Double Dim ht As Double, i As Double Dim alpha As Double, ss As Double If (b >= r) Then // Never optimal to exersice before maturity BjerksundStenslandCall = BlackScholes(mCall, s, x, T, r, b, v) Else Beta = (0.5 - b/(v*v)) + Sqr((b/(v*v) - 0.5)*(b/(v*v) - 0.5)_ + 2*r/(v*v)) BInfinity = Beta/(Beta - 1)*x B0 = Max(x, r/(r - b)*x) ht = -(b*T + 2*v*Sqr(T))*B0/(BInfinity - B0) i = B0 + (BInfinity - B0)*(1 - Exp(ht)) alpha = (i - x)*(i ^ (-Beta)) If (s >= i) Then BjerksundStenslandCall = s - x Exit Function End If ss = alpha*(s^Beta) - alpha*phi(s, T, Beta, i, i, r, b, v) ss = ss + (phi(s, T, 1, i, i, r, b, v) _ - phi(s, T, 1, x, i, r, b, v)) ss = ss - (x*phi(s, T, 0, i, i, r, b, v) _ - x*phi(s, T, 0, x, i, r, b, v)) BjerksundStenslandCall = ss End If End Function Function BjerksundStenslandPut(s As Double, x As Double, v As Double, r As Double, _ T As Double, b As Double) As Double BjerksundStenslandPut = _ BjerksundStenslandCall(x, s, v, r - b, T, -b) End Function Function phi(s As Double, T As Double, gamma As Double, _ h As Double,_i As Double, r As Double, b As Double,_ v As Double) As Double Dim lambda As Double, D As Double, kappa As Double, f As Double lambda = (-r + gamma*b + 0.5*gamma*(gamma - 1)*(v*v))*T D = -(Log(s/h) + (b + (gamma - 0.5)*(v*v))*T)/(v*Sqr(T)) kappa = 2*b/((v*v)) + (2*gamma - 1) f = CND(D) - ((i/s)^kappa)*CND(D - 2*Log(i/s)/(v*Sqr(T))) phi = Exp(lambda)*(s^gamma)*f End Function References Albanese, C., & Campolieti, G (2006) Advanced derivatives pricing and risk management San Diego: Elsevier Academic Press Barone-Adesi, G., et.al (1999) VaR without correlations for nonlinear portfolios Journal of Futures Markets , 19 , 583–602 Barucci, E (2003) Financial markets theory New York: Springer Berkowitz, J., & O’Brien, J (2002) How accurate are value-at-risk models at commercial banks? Journal of Finance, 57 , 1093– 1111 Björk, T (1998) Arbitrage theory in continuous time Oxford: Oxford University Press Black, F (1976) The pricing of commodity contracts Journal of financial economics, (1), 167–179 Black, F., & Scholes, M (1973) The pricing of options and corporate liabilities The Journal of Political Economy, , 637–654 [CrossRef] Campbell, S D (2005) A review of backtesting and backtesting procedures Finance and economics, discussion series Divisions of Research and Statistics and Monetary Affairs Federal Reserve Board, Washington, DC Clewlow, L., & Strickland, C (2000) Implementing derivative models London: Wiley Conroy, R M (2003) Binomial option pricing Darden Graduate School of Business Administration, University of Virginia UVA-F0943 Cox, J., Ross, S A., & Rubenstein, M (1979) Option pricing: A simplified approach Journal of Financial Economics, (1979), 145– 166 Curran, M (1994) Valuing Asian and Portfolio options by conditioning on the geometric mean price Management Science, 40 (12), 1705–1711 [CrossRef] Dana, R.-A., & Jeanblanc, M (2007) Financial markets in continuous time Berlin: Springer Davis, M., & Etheridge, A (2006) Louis Bachelier’s theory of speculation: The origins of modern finance Princeton: Princeton University Press Derman, E (1996) Model risk Goldman Sachs Quantitative Strategies Research Notes DeRosa, D F (2000) Options on foreign exchange New York: Wiley Drezner, Z (1978) Computation of the bivariate normal integral Mathematics of Computation, 32 , 277–279 [CrossRef] Glasserman, P (2000) Monte Carlo methods in financial engineering New York: Springer Glasserman, P (2003) Monte Carlo methods in financial engineering New York: Springer Glasserman, P (2004) Monte Carlo methods in financial engineering New York: Springer Haug, E (1997) The complete guide to option pricing formulas New York: McGraw-Hill Ho, T S Y., & Lee, S B (2005) Securities valuations Oxford: Oxford University Press Hull, J Option futures and other derivatives (9th Edition) Prentice-Hall ISBN-13: 978-0133456318 Hunt, P J., & Kennedy, J E (2004) Financial derivative in theory and practise Chichester: Wiley Ingersoll, J E (1987) Theory of financial decision making Totowa: Rowman & Littlefield Jarrow, R., & Rudd, A (1983) Option pricing (pp 183–188) Homewood: Dow Jones-Irwin Jorion, P (2001) Value at risk: The new benchmark for managing financial risk New York: McGraw-Hill Kerry B (2005) A cource in derivative securities New York: Springer Kijima, M (2003) Stochastic processes with application to finance London: Chapman & Hall Kwok, Y.-K (2008) Mathematical models of financial derivatives New York: Springer Leisen, D (1998) Pricing the American put option: A detailed convergence analysis for binomial models Journal of Economic Dynamics and Control, 22 , 1419–1444 [CrossRef] Leisen, D., & Reimer, M (1996) Binomial models for option valuation – Examine and improving convergence Applied Mathematical Finance, , 319–346 [CrossRef] London, J (2005) Modeling derivatives in C++ London: Wiley Longstaff, F A (1989) Pricing options with extendible maturities Journal of Finance, 45 (3), 935–957 [CrossRef] Martin, J S (2001) Applied math for derivatives London: Wiley Merton, R C (1973) Theory of rational option pricing The Bell Journal of Economics and Management Science, (1), 141–183 [CrossRef] Merton, C R (1976) Option pricing when stock returns are discontinuous Journal of Financial Economics, 3, 167–179 Munk, C (2003) Fixed income analysis: Securities, pricing and risk management http://citeseerx.ist.psu.edu/viewdoc/download? doi=10.1.1.129.9912&rep=rep1&type=pdf Neftci, S N (2000) An introduction to the Mathematics of Financial Derivatives New York: Academic Press Neftci, S N (2004) Principles of financial engineering San Diego: Academic Press Pratt, J W (1968) A normal approximation for binomial, F, Beta, and other common, related tail probabilities, II The Journal of the American Statistical Association, 63 , 1457–1483 Rebonato, R (2001) Managing model risk In C Alexander (Ed.), Handbook of risk management FT-Prentice Hall Reiner, E., & Rubinstein, M (1991) Breaking down the barriers Risk, , 28–35 Rendleman, R J., & Bartter, B J (1980) The pricing of options on debt securities The Journal of Financial and Quantitative Analysis, 15, 11–24 Ross, S M (1983) Stochastic processes New York: Wiley Rubinstein, M (2000) On the relation between binomial and trinomial option pricing models The Journal of Derivatives, (2), 47–50 [CrossRef] Schweizer, M (2012) On Bermudan options ETH Zurich Shreve, Steven (2004) Stochastic calculus for finance II New York: Springer Shreve, Steven (2005) Stochastic calculus for finance I New York: Springer Sidenius, J (2000) LIBOR market models in practice Journal of Computational Finance, (3), 75–99 [CrossRef] Stoklosa, J (2007) Studies of barrier options and their sensitivities The University of Melbourne Tian, Y (1993) A modified lattice approach to option pricing Journal of Futures Markets, 13 (5), 564–577 [CrossRef] Trigeorgis, L (1991) A log-transformed binomial numerical analysis for valuing complex multi-option investments Journal of Financial and Quantitative Analysis, 26 (3), 309–326 [CrossRef] Wilmott, P (2000) On quantitative finance Chichester: Wiley Index A absolute continuous adapted all-or-nothing options American call option American contract American double-no-touch option American knock-in option American knock-out option American no-touch option American option American put option American style digital arbitrage arbitrage condition arbitrage portfolio arbitrage strategy ARCH See autoregressive conditional heteroskedasticity (ARCH) Arrow-Debreu Asian options asset-linked FX option asset-or-nothing options atom autocorrelation autoregression autoregressive conditional heteroskedasticity (ARCH) average price options average strike options B backwardation backward differences Barone−Adesi, Whaley barrier options basket options benign barrier bet options better of options big figure binary options binomial distribution binomial model(s) bivariate normal distribution Bjerksund, Stensland Black Black-Scholes Black-Scholes deflator Black-Scholes formula Black-Scholes smoothing bond(s) bonus knock-out option Borel algebra Borel measurable Borel set(s) box algebra Boyle Brownian motion buck bull spread butterfly spread C cable calendar spread capped option caps cash-or-nothing digital options cash-or-nothing option(s) Cauchy problem CBOE SKEW Index CBOE Volatility Index® (VIX® Index) central limit theorem CFD See contract for difference (CFD) change of numeraire charm chooser option clearing cliquets coherent risk measures collar(s) colour commodity options complete complete market(s) complex chooser option compliance risk compound options conditional value at risk (CVaR) conditioned contract(s) conditioned expectation confidence level(s) consumption contango contingent claim(s) continuous compounding contract for difference (CFD) control variance conversion ratio correlation function correlation option(s) counterparty risk co-variance covered call Cox-Ross-Rubenstein Crank-Nicholson credit risk cross currency-linked options currency options currency-protected option currency risk CUSIP CVaR See conditional value at risk (CVaR) D deflators delayed-start options delta delta decay delta-gamma-hedging delta hedge delta-symmetric strike density function diffusion diffusion process digital option(s) Dirac delta function distribution function distribution measure dividend yield double no touch options Dow Jones Industrial Average down-and-out calls down-and-out put option drift E E early exercise premium equity price risk ES See expected shortfall (ES) Eurodollar European call option(s) European digital FX option European knock-in option European knock-out option European option(s) European option style European put option exchange option(s) exercise region exotic options expectation value(s) expected payoff expected shortfall (ES) expected tail loss (ETL) explicit finite difference method exploding spread option exponentially weighted moving average model (EWMA) exponential weighted VaR extensible option F F-adapted fair game fair value fear gauge fence Fenics Feynman–Kac filtration finer first in first out (FIFO) fixed domestic strike options fixed exchange rate option F–measurable Fokker−Planck foreign exchange risk forward(s) forward differences forward options forward price forward-start options forward volatility future(s) F-Wiener process FX options G gain process gamma (γ) gap options GARCH Garman−Kohlhagen model Gaussian distributions geometric random walk Geske−Johnson Girsanov Girsanov kernel Girsanov theorem Girsanov transformation Greeks Green function H hedge parameters hedging portfolio Herstatt risk heteroscedasticity Hopscotch I implicit finite difference method implied volatility independent independent σ-algebras index linked bond indicated indicator function inflation linked claims information generated by information set initial margin integration schemas integration theory interest rate(s) interest rate risk International Swaps and Derivatives Association (ISDA) intrinsic value ISIN code Itô formula Itô integral Itô lemma J Jarrow-Rudd Jensen’s unlikeness Jones, Dow jump diffusion K kappa Knock-out and knock-in options Kolmogorov kurtosis L lambda least square Lebesgue integrals legal risk Leisen–Reimer model likelihood process Lipschitz liquidity risk log-normal log-normal distribution long butterfly long call long put long sloping synthetic forward long straddle long strangle long synthetic forward Long-term capital management (LTCM) lookback options M marked to the market market indexes market price of risk market price of volatility risk market risk marking-to-model Markov Markov process martingale martingale approach martingale measure martingale representation maximum likelihood maximum likelihood estimator measurable measurable space model risk mollification Monte Carlo Monte-Carlo simulations moving strike options multi-asset options N Nasdaq natural filtration negative back-spread negative price spread negative three-leg position negative time spread neutral time spread normal distribution normal random variable no-touch currency option NYSE O ODE OEX OMXS index one-touch currency option operational risk optimal exercise boundary options on commodities options on options outcomes outperformance option P parity relations partial differential equation (PDE) partition(s) path-dependent options pay-later option PDE See partial differential equation (PDE) Peizer–Pratt perpetual American put pip(s) P-martingale P–measurable Poisson distribution Poisson process portfolio strategy(ies) Portfolio VaR positive back spread positive price spread(s) positive stair positive three leg positive time spread premium-included delta pre-settlement risk probability density function probability distribution(s) probability measure probability space protective put put-call parity put-call symmetry Put Hedge Q quadratic integrable quantity-adjusted option quanto option quanto, quantos quid Q -Wiener process R Radon–Nikodym rainbow option random walk ranking the trades ratchet options ratchets ratio spread with call options ratio-spread with put options ratio spread with underlyings reachable rebate clause relative portfolio replicating portfolio resetting options rho Richardson extrapolation RIC-name Riemann integral risk averse risk neutral drift of volatility risk-neutral probabilities risk neutral valuation risk-neutral valuation formula risk premium Roll, Geske S sample space Scandies SDE See stochastic differential equation SEDOL See Stock Exchange Daily Official List self financed self-financing settlement Settlement risk sharpe ratio short butterfly short call short straddle short strangle Siegel’s Exchange Rate Paradox σ-algebra σ-algebra generated simple skewness soft call provision Standard & Poor’s (S&P) splitting index S(E) spread betting spread option SPX standard deviation standard error state prices stochastic differential equation (SDE) stochastic integral stochastic integration stochastic process stochastic variable stochastic volatility Stock Exchange Daily Official List (SEDOL) stopping times straddle strangle structured products sub martingale super martingale SWIFT Synthetic Call synthetic contracts Synthetic sold put T tau Taylor series expansion theta Tian Tigori time value touch options Tower property transition probabilities trinomial tree 20/20 Option 2X U Uhlenback process up-and-out puts V Value-at-Risk (VaR) value process, value processes Vanna VaR See Value-at-Risk (VaR) variance variance reduction variation margin vega VIX futures VIX volatility index VKN number volatility volatility surface volga VVIX W weather derivative Whaley, R E Wiener process window barrier knock-in option window barrier knock-out option WKN WPKN Y yard Footnotes Note: Page numbers followed by “n” denote footnotes ... 6.? ?10 Basket Options 6.? ?11 Correlation Options 6.? ?12 Exchange Options 6.? ?13 Currency-linked Options 6.? ?14 Pay-later Options 6.? ?15 Extensible Options 6.? ?16 Quantos 6.? ?17 Structured Products 6.? ?18 ... volatility quotation © The Author(s) 2 017 Jan R M Rưman, Analytical Finance: Volume I, DOI 10 .10 07/978-3- 319 -34027-2 _1 Trading Financial Instruments Jan R M Röman1 (1) Västerås, Sweden Financial instruments... process © The Author(s) 2 017 Jan R M Röman, Analytical Finance: Volume I, DOI 10 .10 07/978-3- 319 -34027-2_2 Time-Discrete Models Jan R M Röman1 (1) Västerås, Sweden 2 .1 Pricing via Arbitrage To