The Mathematics of Equity Derivatives, Markets, Risk and Valuation ANALYTICAL FINANCE VOLUME I JAN R M RÖMAN Analytical Finance: Volume I 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 ISBN 978-3-319-34027-2 (eBook) 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 3, 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 vii viii Preface 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 Xtị ẳ er Ttị EẵXT ị: 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) ¼ max{S(T) – K, 0} This means that the present value is given by Xtị ẳ er Ttị EẵXT ị ẳ er Ttị EẵmaxfST ị K, 0g: 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 Xtị ẳ S0ị: Q1 ðSðT Þ > K Þ À eÀr ðTÀtÞ K : Q2 ðSðT Þ > K Þ; where Q1(S(T) > K) and Q2(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 BlackScholesMerton formula as Preface ix Xtị ẳ S0ị: N ðd Þ À eÀr ðTÀtÞ K : N ðd Þ: Here d1 and d2 are given (derived) variables N(x) is the standard normal distribution with mean and variance 1, so N(d2) represent the probability for the stock to reach the strike price K The variables d1 and d2 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 xi 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 1 10 12 12 13 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) 21 21 23 25 27 29 38 43 65 78 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 91 91 91 93 121 xiii 476 Appendix: Some Source Codes Function BaroneAdesiWhaleyPut(s As Double, x As Double, _ v As Double, r As Double, _ T As Double, b As Double) As Double Dim sk As Double, n As Double, K As Double, d1 As Double Dim q1 As Double, a1 As Double sk ¼ kp(x, T, r, b, v) n ¼ 2*b/(v*v) K ¼ 2*r/(v*v*(1 - Exp(-r*T))) d1 ¼ (Log(sk/x) + (b + v*v/2)*T)/(v*Sqr(T)) q1 ¼ (-(n - 1) - Sqr((n - 1)*(n - 1) + 4*K))/2 a1 ¼ -(sk/q1)*(1 - Exp((b - r)*T)*CND(-d1)) If (s > sk) Then BaroneAdesiWhaleyPut ¼ BlackScholes(mPUT, s, x, T, r, b, v) _ + a1*(s/sk) ^ q1 Exit Function End If BaroneAdesiWhaleyPut ¼ x - s End Function Function kc(x As Double, T As Double, r As Double, b As Double, _ v As Double) As Double ’ Calculation of seed value, Si Dim M As Double, q2u As Double, Su As Double, h2 As Double Dim Si As Double, d1 As Double, q2 As Double, LHS As Double Dim RHS As Double, bi As Double, E As Double, K As Double M ¼ 2*r/(v*v) q2u ¼ + 2*M Su ¼ x/(1 - 1/q2u) h2 ¼ -(b*T + 2*v*Sqr(T))*x/(Su - x) Si ¼ x + (Su - x)*(1 - Exp(h2)) If (T ¼ 0) Then T ¼ 0.000000001 K ¼ 2*r/(v*v*(1 - Exp(-r*T))) d1 ¼ (Log(Si/x) + (b + v*v/2)*T)/(v*Sqr(T)) q2 ¼ + 2*K LHS ¼ Si - x RHS ¼ BlackScholes(mCall, Si, x, T, r, b, v) + _ (1 - Exp((b - r)*T)*CND(d1))*Si/q2 bi ¼ Exp((b - r)*T)*CND(d1)*(1 - 1/q2) bi ¼ bi + (1 - Exp((b - r)*T)*CND(d1)/(v*Sqr(T)))/q2 E ¼ 0.000001 Appendix: Some Source Codes ’ Newton-Raphson algorithm for finding critical price Si Do While (Abs(LHS - RHS)/x > E) Si ¼ (x + RHS - bi*Si)/(1 - bi) d1 ¼ (Log(Si/x) + (b + v*v/2)*T)/(v*Sqr(T)) LHS ¼ Si - x RHS ¼ BlackScholes(mCall, Si, x, T, r, b, v) + _ (1 - Exp((b - r)*T)*CND(d1))*Si/q2 bi ¼ Exp((b - r)*T)*CND(d1)*(1 - 1/q2) bi ¼ bi + (1 - Exp((b - r)*T)*CND(d1)/(v*Sqr(T)))/q2 Loop kc ¼ Si End Function Function kp(x As Double, T As Double, r As Double, b As Double, _ 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 ¼ 2*b/(v*v) M ¼ 2*r/(v*v) q1u ¼ (-(n - 1) - Sqr((n - 1)*(n - 1) + 4*M))/2 Su ¼ x/(1 - 1/q1u) h1 ¼ (b*T - 2*v*Sqr(T))*x/(x - Su) Si ¼ Su + (x - Su)*Exp(h1) If (T ¼ 0) Then T ¼ 0.000000001 K ¼ 2*r/(v*v*(1 - Exp(-r*T))) d1 ¼ (Log(Si/x) + (b + v*v/2)*T)/(v*Sqr(T)) q1 ¼ (-(n - 1) - Sqr((n - 1)*(n - 1) + 4*K))/2 LHS ¼ x - Si RHS ¼ BlackScholes(mPUT, Si, x, T, r, b, v) - _ (1 - Exp((b - r)*T)*CND(-d1))*Si/q1 bi ¼ -Exp((b - r)*T)*CND(-d1)*(1 - 1/q1) bi ¼ bi - (1 + Exp((b - r)*T)*CND(-d1)/(v*Sqr(T)))/q1 E ¼ 0.000001 ’ Newton Raphson algorithm for finding critical price Si Do While (Abs(LHS - RHS)/x > E) Si ¼ (x - RHS + bi*Si)/(1 + bi) d1 ¼ (Log(Si/x) + (b + v*v/2)*T)/(v*Sqr(T)) LHS ¼ x - Si 477 478 Appendix: Some Source Codes RHS ¼ BlackScholes(mPUT, Si, x, T, r, b, v) - _ (1 - Exp((b - r)*T)*CND(-d1))*Si/q1 bi ¼ -Exp((b - r)*T)*CND(-d1)*(1 - 1/q1) bi ¼ bi - (1 + Exp((b - r)*T)*CND(-d1)/(v*Sqr(T)))/q1 Loop kp ¼ 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 >¼ r) Then // Never optimal to exersice before maturity BjerksundStenslandCall ¼ BlackScholes(mCall, s, x, T, r, b, v) Else Beta ¼ (0.5 - b/(v*v)) + Sqr((b/(v*v) - 0.5)*(b/(v*v) - 0.5)_ + 2*r/(v*v)) BInfinity ¼ Beta/(Beta - 1)*x B0 ¼ Max(x, r/(r - b)*x) ht ¼ -(b*T + 2*v*Sqr(T))*B0/(BInfinity - B0) i ¼ B0 + (BInfinity - B0)*(1 - Exp(ht)) alpha ¼ (i - x)*(i ^ (-Beta)) If (s >¼ i) Then BjerksundStenslandCall ¼ s - x Exit Function End If ss ¼ alpha*(s^Beta) - alpha*phi(s, T, Beta, i, i, r, b, v) ss ¼ ss + (phi(s, T, 1, i, i, r, b, v) _ - phi(s, T, 1, x, i, r, b, v)) ss ¼ ss - (x*phi(s, T, 0, i, i, r, b, v) _ - x*phi(s, T, 0, x, i, r, b, v)) BjerksundStenslandCall ¼ ss End If End Function Appendix: Some Source Codes Function BjerksundStenslandPut(s As Double, x As Double, v As Double, r As Double, _ T As Double, b As Double) As Double BjerksundStenslandPut ¼ _ 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 ¼ (-r + gamma*b + 0.5*gamma*(gamma - 1)*(v*v))*T D ¼ -(Log(s/h) + (b + (gamma - 0.5)*(v*v))*T)/(v*Sqr(T)) kappa ¼ 2*b/((v*v)) + (2*gamma - 1) f ¼ CND(D) - ((i/s)^kappa)*CND(D - 2*Log(i/s)/(v*Sqr(T))) phi ¼ Exp(lambda)*(s^gamma)*f End Function 479 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? 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approach to option pricing Journal of Futures Markets, 13(5), 564–577 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 Wilmott, P (2000) On quantitative finance Chichester: Wiley Index A absolute continuous, 118 adapted, 96–100, 112, 205, 245, 249, 261, 268, 276–9 all-or-nothing options, 293 American call option, 37, 48, 51, 229–35 American contract, 113, 233–5 American double-no-touch option, 301 American knock-in option, 303 American knock-out option, 303 American no-touch option, 301, 302 American option, vii, viii, 29, 56, 62, 77, 113–17, 188, 212, 228–39, 296, 369 American put option, 48, 54–6, 61, 77, 218, 228–31 American style digital, 296 arbitrage, vii, 21–3, 31–43, 92, 158, 160–2, 196, 204, 211–15, 230–3, 249, 255, 267–78, 282, 313, 329, 337, 340, 359, 372, 380 arbitrage condition, 38, 92, 211, 215 arbitrage portfolio, 31 arbitrage strategy, 39, 271, 277 ARCH See autoregressive conditional heteroskedasticity (ARCH) Arrow-Debreu, 360 Asian options, 29, 289, 312–19 asset-linked FX option, 334 asset-or-nothing options, 294, 296 atom, 95, 117 autocorrelation, 184 autoregression, 184 autoregressive conditional heteroskedasticity (ARCH), 184–7 average price options, 312, 313, 315 average strike options, 312 B backwardation, 199 backward differences, 67, 69, 72 BaroneÀAdesi, Whaley, 236–8, 475–8 barrier options, 289, 297–311 basket options, 327–32, 343 benign barrier, 310 Note: Page numbers followed by “n” denote footnotes © The Author(s) 2017 J.R.M Röman, Analytical Finance: Volume I, DOI 10.1007/978-3-319-34027-2 485 486 Index bet options, 293 better of options, 327 big figure, 204 binary options, 218, 292–8 binomial distribution, 28, 48 binomial model(s), vii, 24, 29–39, 43–64, 77, 91–3, 106–17, 265, 308, 358, 461, 467 bivariate normal distribution, 473 Bjerksund, Stensland, 238–9, 478 Black, 17, 18, 163, 218, 238 Black-Scholes, vii, viii, 15n1, 33, 43, 48, 51, 60–8, 73–7, 121, 131–3, 145, 154–77, 182, 189, 197–200, 205–19, 228–36, 245–88, 308, 326, 352–5, 451, 465–9 Black-Scholes deflator, 352–6 Black-Scholes formula, 34, 49, 173, 197, 212–18, 237, 274, 281, 284, 294, 320, 386, 451 Black-Scholes smoothing, 49–52, 54, 464 bond(s), 6, 10–18, 21, 30, 38, 83–4, 155, 161, 212–14, 230, 252, 307, 340–4, 356, 366, 452 bonus knock-out option, 304 Borel algebra, 103 Borel measurable, 103 Borel set(s), 103 box algebra, 138 Boyle, 239–40 Brownian motion, 33, 43, 97, 113, 128–33, 155–7, 182, 189–98, 249, 328, 337, 348 buck, 203 bull spread, 402–4 butterfly spread, 324 C cable, 203 calendar spread, 323, 399, 438–41 capped option, 308, 340–2 caps, 15–18, 212, 342–4 cash-or-nothing digital options, 297 cash-or-nothing option(s), 289–94 Cauchy problem, 146 CBOE SKEW Index, 454–8 CBOE Volatility Index® (VIX® Index), 182, 452–9 central limit theorem, 9, 25–6, 28, 59, 76, 97, 130, 267 CFD See contract for difference (CFD) change of numeraire, 203 charm, 225, 226 chooser option, 289, 316–18 clearing, 1–7, 11, 161, 292, 366, 368, 451 cliquets, 319 coherent risk measures, 88–9 collar(s), 298 colour, 226, 332 commodity options, 209 complete, 35, 72, 96–8, 124, 226, 255–6, 266, 272–3, 297 complete market(s), 250, 265–7 complex chooser option, 316 compliance risk, compound options, 324–6 conditional value at risk (CVaR), 85–8, 88n3 conditioned contract(s), 270, 271 conditioned expectation, 106–8 confidence level(s), 78, 85–8 consumption, 115 contango, 199 contingent claim(s), viii, 35–41, 70, 93, 164, 255, 272, 275–7, 337, 344, 351–61 continuous compounding, viii, 33, 73, 161, 353 contract for difference (CFD), 290–2, 373–4 control variance, 77 conversion ratio, 327 correlation function, 184, 473 Index correlation option(s), 327–8, 332 counterparty risk, 8, 10–11, 292 co-variance, 83, 84 covered call, 383, 429, 440–3 Cox-Ross-Rubenstein, 461 Crank-Nicholson, 69–70, 72, 467 credit risk, 3, 7–12, 78 cross, 203 currency-linked options, 333–9 currency options, 197–235, 289 currency-protected option, 334 currency risk, 8, 197, 334, 339 CUSIP, CVaR See conditional value at risk (CVaR) D deflators, 351–64 delayed-start options, 318 delta, 77, 82–4, 169, 173–4, 199–208, 219–25, 284, 294, 308–10, 379, 400, 440, 461, 467, 470 delta decay, 225 delta-gamma-hedging, 285–8 delta hedge, 77, 158, 161, 200, 204, 205, 219, 285, 294 delta-symmetric strike, 201 density function, 9, 105, 121–4, 165–7, 172, 193, 283 diffusion, 29, 96, 141, 171, 240–3, 245–88 diffusion process, viii, 28, 146, 147, 242, 245, 337 digital option(s), 216–18, 292–8, 308, 344–5 Dirac delta function, 169 distribution function, 98, 99, 236, 466 distribution measure, 98 dividend yield, 174, 214, 273–9 double no touch options, 300–1 Dow Jones Industrial Average, 182, 454 down-and-out calls, 307 down-and-out put option, 308 487 drift, 29, 33, 96, 125, 128, 141, 148, 155, 212, 242, 247, 249–51, 262, 268 E early exercise premium, 228, 236 equity price risk, ES See expected shortfall (ES) Eurodollar, 199 European call option(s), 37, 54, 59, 73, 92, 111, 164, 171, 173–6, 188, 196, 214, 231, 283, 299, 342, 461 European digital FX option, 302 European knock-in option, 304 European knock-out option, 304 European option(s), 29, 48, 56, 77, 115, 139, 158, 168, 180, 204, 212, 228–31, 236, 313, 322, 368 European option style, 207, 314–20, 334, 365, 453 European put option, 187 exchange option(s), 197, 327, 332–3 exercise region, 230 exotic options, viii, 35, 289–350 expectation value(s), viii, 23, 43, 100, 105–9, 141, 142, 147–51, 153, 164, 172, 190–2, 195, 314, 358 expected payoff, viii, 22, 247, 345, 351–4 expected shortfall (ES), 85–9 expected tail loss (ETL), 85 explicit finite difference method, 67–8 exploding spread option, 308 exponentially weighted moving average model (EWMA), 187 exponential weighted VaR, 83 extensible option, 336–7 F F-adapted, 96, 109, 142, 247 fair game, 23, 24, 37, 109, 246, 247 488 Index fair value, 38, 358, 454 fear gauge, 454 fence, 401–6 Fenics, 205 Feynman–Kac, 131, 148–50, 195, 198, 256 filtration, 95–100, 109, 112, 116, 119, 233, 249–52, 261–7, 279 finer, 94, 99, 109, 119, 128 first in first out (FIFO), 369 fixed domestic strike options, 335 fixed exchange rate option, 334 F–measurable, 96 FokkerÀPlanck, 29, 154 foreign exchange risk, 13 forward(s), 12–15, 27, 43, 66–77, 87, 135, 142, 153, 158–63, 184–7, 199–202, 212, 218, 273, 289, 300, 318–24, 333, 338, 348, 360, 367–9, 371–4, 382, 393, 398–401, 406–23, 430–6, 442–54 forward differences, 68, 142 forward options, 289, 318–24, 333 forward price, 43, 199, 367, 454 forward-start options, 318, 320 forward volatility, 322–4 future(s), viii, 6, 8–10, 13–16, 21–3, 34, 38, 43, 68, 69, 82, 135, 154, 161–3, 177, 182–6, 209, 215, 218, 230, 241, 265, 273, 278–80, 290, 316–21, 324, 333, 338, 340, 346, 358–62, 367–74, 380, 383, 387, 401, 406, 410, 416–19, 430–3, 438, 451–4, 459 F-Wiener process, 100 FX options, 197, 201–3, 209, 301, 334 G gain process, 275, 278–81 gamma (γ), 84, 220–1, 224, 285, 379, 461, 467 gap options, 297–8 GARCH, 183, 185–7 GarmanÀKohlhagen model, 197–235 Gaussian distributions, 348 geometric random walk, 159 GeskeÀJohnson, 239 Girsanov, viii, 246–73, 281, 282, 353 Girsanov kernel, 248, 253, 257, 264, 282, 353 Girsanov theorem, 249, 253–9, 263, 269, 281 Girsanov transformation, 246–73 Greeks, vii, 57, 70, 173, 219, 224–8, 294, 308, 324, 379, 461–9 Green function, 170–1 H hedge parameters, 57, 173, 219 hedging portfolio, 36 Herstatt risk, 12 heteroscedasticity, 184 Hopscotch, 70–2 I implicit finite difference method, 68–9 implied volatility, 17, 33, 177, 178, 182, 183, 205–7, 320–2, 380, 451 independent, 26–8, 65, 97, 100, 106, 107, 128, 130, 136–9, 157, 175, 212, 231, 241, 253, 265, 266, 273, 320, 337, 343, 362, 447 independent σ-algebras, 106 index linked bond, 356 indicated, 102 indicator function, 102–5, 124, 153, 196 inflation linked claims, 355–6 information generated by, 96 information set, 23, 24 initial margin, 292, 366 integration schemas, 70 integration theory, 103–4 Index interest rate(s), viii, xi, 7–18, 21–3, 29–43, 57–9, 73–5, 83, 92, 98, 115, 128, 157–64, 174, 189, 195–223, 233, 247–56, 278–86, 325, 337–44, 352–467 interest rate risk, 8, 13, 251 International Swaps and Derivatives Association (ISDA), 6, intrinsic value, 56, 117, 215, 216, 296, 334, 376, 377 ISIN code, 5, Itô formula, 140, 147–51, 195, 210, 247 Itô integral, 126, 135 Itô lemma, 125–33, 150, 156, 165 J Jarrow-Rudd, 46, 47 Jensen’s unlikeness, 108 Jones, Dow, 182, 291, 454 jump diffusion, 240–3, 266 K kappa, 222 Knock-out and knock-in options, 289, 298–311 Kolmogorov, 131, 153 kurtosis, 83, 122 L lambda, 224 least square, 177, 180–2 Lebesgue integrals, 103 legal risk, Leisen–Reimer model, 48–54, 461–5 likelihood process, 252–9, 262–4 Lipschitz, 100 liquidity risk, log-normal, 19, 44, 73, 121–5, 132, 167, 198, 212, 247, 266, 315, 323, 344 489 log-normal distribution, 19, 44, 45, 73, 121, 133, 167 long butterfly, 436–9, 449 long call, 294, 309, 373, 379, 397, 415–20, 425, 459 long put, 318, 374, 379–89, 429, 459 long sloping synthetic forward, 416–19 long straddle, 378, 379, 443–6, 459 long strangle, 300, 444–9, 459 long synthetic forward, 415 Long-term capital management (LTCM), 18 lookback options, 110, 289, 311 M marked to the market, market indexes, 451–9 market price of risk, 40, 41, 249–52, 267, 352, 452 market price of volatility risk, 267, 353 market risk, 7, 8, 12–19, 78 marking-to-model, 14 Markov, 27, 97, 110–13, 241, 242 Markov process, 110–13 martingale, vii, viii, 23–8, 31, 37, 62, 93, 97, 109, 115, 120, 135, 171, 188, 233–5, 245–8, 252–9, 266, 269–79, 281, 337, 345, 363 martingale approach, 171–3 martingale measure, 24, 31, 37, 93, 233–5, 248, 254, 256–9, 266, 269–78, 281, 338, 345, 363 martingale representation, viii, 245, 255, 279 maximum likelihood, 261–5 maximum likelihood estimator, 187, 264 measurable, 9, 34, 95–103, 107–9, 119–20, 143, 255, 261, 271 measurable space, 95, 107, 261 model risk, 8, 13–20 mollification, 49 Monte Carlo, 11, 77, 138 490 Index Monte-Carlo simulations, vii, 72–82, 138, 308, 315, 328, 344, 355–8 moving strike options, 319, 321 multi-asset options, 326–8, 335, 339 N Nasdaq, 182, 213, 291, 365–8, 454 natural filtration, 96, 252, 267 negative back-spread, 391–5 negative price spread, 382–7, 395–9 negative three-leg position, 394–6 negative time spread, 387–9 neutral time spread, 438–40, 459, 460 normal distribution, 10, 19, 25, 26, 43–7, 73, 79–83, 121, 133, 167, 187, 199, 236, 352–7, 455–8, 466, 473 normal random variable, 172, 352 no-touch currency option, 299 NYSE, 182, 454 O ODE, 151, 191–5, 231, 280 OEX, 452–3 OMXS index, 367–8 one-touch currency option, 299 operational risk, optimal exercise boundary, 230 options on commodities, 209 options on options, 289, 324–6 outcomes, 9, 21–7, 36–8, 82, 86–8, 92–4, 104, 107, 119, 124, 133, 139, 140, 246, 320, 325, 334, 348 outperformance option, 327 P parity relations, 187–8 partial differential equation (PDE), vii, 29, 65–70, 145–68, 175, 198, 211, 228, 256, 273, 283 partition(s), 95–100 path-dependent options, 202, 298 pay-later option, 335–6 PDE See partial differential equation (PDE) Peizer–Pratt, 48, 465 perpetual American put, 231–3 pip(s), 204, 292, 299 P-martingale, 120 P–measurable, 99 Poisson distribution, 241 Poisson process, viii, 240–3 portfolio strategy(ies), 32, 254, 270–7 Portfolio VaR, 84–5 positive back spread, 411–14 positive price spread(s), 384–6, 401–7, 419, 427, 459 positive stair, 417–21 positive three leg, 424–31 positive time spread, 406–10 premium-included delta, 199 pre-settlement risk, 11 probability density function, 9, 122–4, 165, 172, 193 probability distribution(s), 26–8, 60–2, 97, 119–21, 132, 165, 184, 193, 261 probability measure, 22–5, 31, 37, 93, 104, 105, 118–20, 196, 247–62, 269, 275, 353, 358, 473 probability space, 93–121, 252, 274, 358 protective put, 400–2, 459 put-call parity, 60 put-call symmetry, 202, 203 Put Hedge, 400 Q quadratic integrable, 110, 271 quantity-adjusted option, 334 quanto option, 339 quanto, quantos, 334–9, 344–6 quid, 203 Q-Wiener process, 249–55, 279–81 Index R Radon–Nikodym, 105, 118–21, 248, 268 rainbow option, 327–32, 336 random walk, 27–9, 96, 158, 159, 250, 251, 266 ranking the trades, 369 ratchet options, 321–22 ratchets, 319–22 ratio spread with call options, 412, 420–4 ratio-spread with put options, 389–92 ratio spread with underlyings, 410–11 reachable, 36, 255, 273, 277–8 rebate clause, 307 relative portfolio, 156 replicating portfolio, 36, 62, 188, 338 resetting options, 319 rho, 57, 223–8, 380, 451, 461, 467 Richardson extrapolation, 48–55, 63 RIC-name, 368 Riemann integral, 126 risk averse, 38, 39, 290 risk neutral drift of volatility, 212 risk-neutral probabilities, 24, 33–5, 40, 93, 119, 353 risk neutral valuation, 31, 38–43, 98, 351–62 risk-neutral valuation formula, 31 risk premium, 23, 41, 158, 249, 352–4 Roll, Geske, 236, 472 S sample space, 21, 22, 91–3, 104, 119 Scandies, 203 SDE See stochastic differential equation SEDOL See Stock Exchange Daily Official List self financed, 156, 254–6, 270–2, 274–7 self-financing, 32, 271, 280 settlement, 1–7, 11, 12, 322, 367 Settlement risk, 12 491 sharpe ratio, 250, 356–7 short butterfly, 448–51 short call, 371–4 short straddle, 431–5 short strangle, 433–7 Siegel’s Exchange Rate Paradox, 259–61 σ-algebra, 94, 108 σ-algebra generated, 96 simple, 14, 21, 27–9, 34, 36, 39, 61, 79, 83, 102–5, 125, 133, 142, 146, 160–1, 171, 180, 185, 189–92, 214–15, 230, 235, 251, 262–6, 292, 313–19, 340, 352, 358 skewness, 83, 122, 206, 457 soft call provision, 307 Standard & Poor’s (S&P), 6, 182, 183, 291, 332, 452–5, 457, 458 splitting index S(E), 95 spread betting, 291 spread option, 308, 327 SPX, 452, 453 standard deviation, 9, 10, 74, 81–3, 139, 155, 177, 249, 349–52, 454–7, 473 standard error, 75–6 state prices, 358–64 stochastic differential equation (SDE), 100, 127, 146–51, 165, 210, 243, 349 stochastic integral, 135, 142, 147 stochastic integration, 133–43 stochastic process, 23, 27, 33, 43, 73, 96, 97, 107–9, 125, 131, 140, 146, 155, 210, 254, 276, 348, 353 stochastic variable, 30, 36, 43, 91, 96, 99, 105–7, 109, 116, 212, 255, 270 stochastic volatility, 16, 20, 72, 210–12, 250, 267 Stock Exchange Daily Official List (SEDOL), stopping times, 113–17, 234–5 straddle, 201, 300, 316–18, 324, 378, 379, 431–5, 443–7 492 Index strangle, 206, 300, 324, 433–7, 444–6 structured products, 340, 344 sub martingale, 24, 109, 235 super martingale, 24, 109, 115 SWIFT, 2, Synthetic Call, 400–2 synthetic contracts, 372–4 Synthetic sold put, 440–3 T tau, 223 Taylor series expansion, 243 theta, 221–6, 295, 309, 379, 461, 467 Tian, 47 Tigori, 47 time value, 50, 215, 220, 229–31, 296, 376–9, 383, 386, 399, 403, 442 touch options, 308 Tower property, 101 transition probabilities, 152–3 trinomial tree, 67, 239–40 20/20 Option, 332 2X, 99 U Uhlenback process, 348–50 up-and-out puts, 307–8 V Value-at-Risk (VaR), 13, 18–20, 27, 78–89 value process, value processes, 30, 32, 35, 62, 92, 113, 115, 155, 156, 255, 270, 272, 280 Vanna, 225 VaR See Value-at-Risk (VaR) variance, ix, 10, 26–9, 43, 47, 73, 76–8, 81–4, 87, 97, 101, 121–3, 128–30, 133, 137–9, 155, 172, 183, 184, 186, 198, 207, 329, 352, 356 variance reduction, 76–7 variation margin, 4, 292 vega, 57, 222–4, 227, 295, 309, 322–4, 380, 461, 467 VIX futures, 452, 454 VIX volatility index, 182 VKN number, volatility, viii, 8, 33, 125, 148, 247, 300, 353, 376 volatility heat map, 454, 456 volatility surface, 177–82, 206 volga, 224 VVIX, 454, 455 W weather derivative, viii, 346–50 Whaley, R E., 236, 472 Wiener process, 33, 96–100, 113, 129, 136, 140, 147, 155, 190, 210, 245, 249, 253–8, 262–81, 348 window barrier knock-in option, 305 window barrier knock-out option, 305 WKN, WPKN, Y yard, 203 ... considering model risk? In this case, the risk factors include the risk of model mis-specification (leaving out important sources of risk, mis-specifying the dynamic of the risk factors), and the. .. probabilities This can be further divided: i A priori risk, such as the outcome of the roll of a dice, tossing coins, etc ii Estimable risk, where the probabilities can be estimated through statistical.. .Analytical Finance: Volume I 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