Amir Sadr - Mathematical Techniques in Finance

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Explore the foundations of modern finance with this intuitive mathematical guide
Mathematical Techniques in Finance: An Introduction
Mathematical Techniques in Finance: An Introduction

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Option Pricing

In Chapter 6, we use the risk‐neutral framework to derive the Black‐Scholes‐Merton (BSM) option pricing formula by modeling asset returns as the continuous‐time limit of a random walk, that is a Brownian motion with risk‐adjusted drift. We recover and investigate the underlying replicating portfolio by considering the option Greeks: delta, gamma, theta. The interplay between these is shown by applying the Ito's lemma to the diffusion process driving an underlying asset and its derivative, leading to the BSM partial differential equation and its solution via methods from the classical boundary value heat equations.

We discuss the Cox‐Ross‐Rubinstein (CRR) model as a popular and practical computational method for pricing options that can also be used to compute the price of options with early exercise features via the backward induction algorithm from dynamic programming. For path‐dependent options such as barrier or averaging options, we present numerical models such as the Monte Carlo simulation models and variance reduction techniques.

Interest Rate Derivatives

Chapter 7introduces interest rate swaps and their derivatives used in structured finance. A plain vanilla swap can be priced via a static replication argument from a bootstrapped discount factor curve. In practice, simple European options on swaps and interest rate products are priced and risk‐managed via the normal version of Black's formula for futures. We introduce this model under the risk‐neutral pricing framework and show the pricing of the mainstream cap/floors, European swaptions, and CMS products. For complex derivatives, one needs a model for the evolution of multiple maturity zero‐coupon bonds in a risk‐neutral framework. We present the popular Hull‐White mean‐reverting model for the short rate and show the typical implementation methods and techniques, such as the forward induction method for yield curve inversion. We show the pricing of Bermudan swaptions via these lattice models. We conclude our discussion by presenting methods for calculating interest rate curve risk and VaR.

Exercises and Python Projects

The end‐of‐chapter exercises are based on real‐world markets and products and delve deeper into some financial products and highlight the details of applying the techniques to them. All exercises can be solved by using a spreadsheet package like Excel. The Python projects are longer problems and can be done by small groups of students as a term project.

It is my hope that by the end of this book, readers have obtained a good toolkit of mathematical techniques, methods, and models used in financial markets and products, and their interest is piqued for a deeper journey into quantitative finance.

—Amir Sadr

New York, New York

December 2021

Acknowledgments

One learns by teaching and I have learned much from my students at NYU. Many thanks to all of my students over the years who have asked good questions and kept me on my toes.

Thanks to my editors at John Wiley & Sons: Bill Falloon, Purvi Patel, Samantha Enders, Julie Kerr, and Selvakumaran Rajendiran for patiently walking me through this project and correcting my many typos. All remaining errors are mine, and I welcome any corrections, suggestions, and comments sent to asadr@panalytix.com.

A.S.

About the Author

Amir Sadrreceived his PhD from Cornell University with his thesis work on the Foundations of Probability Theory. After working at AT&T Bell Laboratories, he started his Wall Street career at Morgan Stanley, initially as a Vice President in quantitative modeling and development of exotic interest rate models, and later as an exotics trader. He founded Panalytix, Inc., to develop financial software for pricing and risk management of interest rate derivatives. He was a Managing Director for proprietary trading at Greenwich Capital, Senior Trader in charge of CAD exotics and USD inflation trading at HSBC, the COO of Brevan Howard U.S. Asset Management in the United States, and co‐founder of Yield Curve Trading, a fixed income proprietary trading firm. He is currently a partner at CorePoint Partners and is focused on crypto and DeFi.

Acronyms

bpbasis points, 1% of 1%, 0.0001 картинка 10future valueIRRinternal rate of returnPnLprofit and loss Mathematical Techniques in Finance - изображение 11present valueYTMyield to maturityp.a.per annum Mathematical Techniques in Finance - изображение 12discount factor, today's value unit payment at future date картинка 13 картинка 14dicount factor at картинка 15for unit payment at картинка 16 картинка 17interest rate картинка 18compounding interest rate with картинка 19compoundings per year yieldAPRannual percentage rate stated interest rate without any - фото 20yieldAPRannual percentage rate – stated interest rate without any compoundingsAPYannual pecentage yield – yield of a deposit taking compoundings into consideration: for compoundings per yearCFcash flow coupon rate - фото 21for Mathematical Techniques in Finance - изображение 22compoundings per yearCFcash flow Mathematical Techniques in Finance - изображение 23coupon rate Mathematical Techniques in Finance - изображение 24price of an картинка 25‐year bond with coupon rate картинка 26, paid картинка 27times per year, with yield картинка 28 картинка 29accrual fraction between 2 dates according to some day count basis Mathematical Techniques in Finance - изображение 30clean price of a bond = Price Mathematical Techniques in Finance - изображение 31accrued interest Mathematical Techniques in Finance - изображение 32price of an картинка 33‐year zero‐coupon bond with yield Mathematical Techniques in Finance - изображение 34, Mathematical Techniques in Finance - изображение 35compoundings per year Mathematical Techniques in Finance - изображение 36price of an картинка 37‐year annuity with annuity rate of картинка 38, paid Mathematical Techniques in Finance - изображение 39times per year, with yield Mathematical Techniques in Finance - изображение 40 Mathematical Techniques in Finance - изображение 41price of картинка 42‐maturity Treasury Bill with discount yield картинка 43PV01present value change due to an ”01” bp change in yieldPVBPpresent value change due to a 1 bp change in coupon, present value of a 1 bp annuity картинка 44balance of a level pay loan after картинка 45periods Mathematical Techniques in Finance - изображение 46principal and interest payments of a level pay loan in the Mathematical Techniques in Finance - изображение 47th period Mathematical Techniques in Finance - изображение 48price of картинка 49‐year level pay loan with loan rate of картинка 50, paid картинка 51times per year, with yield картинка 52ALaverage life картинка 53balance of a level pay loan after картинка 54periods with prepayments картинка 55principal and interest payments of a level pay loan within the картинка 56th period with prepaymentsSMMsingle monthly mortality rateCPRconstant prepayment ratio картинка 57periodic prepayment speed картинка 58utility of wealth картинка 59 картинка 60lottery картинка 61is preferred to картинка 62 Mathematical Techniques in Finance - изображение 63certainty‐equivalent of random payoff Mathematical Techniques in Finance - изображение 64, Mathematical Techniques in Finance - изображение 65 картинка 66absolute risk premium, relative risk premium картинка 67, картинка 68, картинка 69value, value of a portfolio, value of картинка 70th asset картинка 71quantity, price Mathematical Techniques in Finance - изображение 72weight of Mathematical Techniques in Finance - изображение 73th asset in a portfolio, Mathematical Techniques in Finance - изображение 74 return of an asset over a period Can be divided by - фото 75return of an asset over a period Can be divided by to give rate of return - фото 76: Mathematical Techniques in Finance - изображение 77. Can be divided by Mathematical Techniques in Finance - изображение 78to give rate of return Mathematical Techniques in Finance - изображение 79asset картинка 80's return, with mean картинка 81and standard deviation картинка 82 картинка 83, Cmean vector, standard deviation vector, and covariance matrix of asset returns картинка 84return of a risk‐free asset картинка 85market portfolio, return of the market portfolio Mathematical Techniques in Finance - изображение 86beta of an asset Mathematical Techniques in Finance - изображение 87, Mathematical Techniques in Finance - изображение 88 картинка 89empirical estimate of картинка 90 картинка 91arithmetic average of Mathematical Techniques in Finance - изображение 92samples of Mathematical Techniques in Finance - изображение 93, Mathematical Techniques in Finance - изображение 94 картинка 95forward date, future date картинка 96forward value of asset картинка 97at time Mathematical Techniques in Finance - изображение 98for forward date Mathematical Techniques in Finance - изображение 99 Mathematical Techniques in Finance - изображение 100 картинка 101‐value of a forward agreement on asset картинка 102for forward date Mathematical Techniques in Finance - изображение 103and price Mathematical Techniques in Finance - изображение 104 Mathematical Techniques in Finance - изображение 105simple (noncompounding) forward rate that can be locked at картинка 106for forward deposit period Mathematical Techniques in Finance - изображение 107. The first term may be omitted when Mathematical Techniques in Finance - изображение 108. Mathematical Techniques in Finance - изображение 109continuously compounding forward rate that can be locked at картинка 110for forward deposit period картинка 111. The first term may be omitted when картинка 112.FXforeign currency exchange rate картинка 113domestic and foreign interest rates for forward exchange rate calculations картинка 114the картинка 115‐forward exchange rate that can be locked at картинка 116 картинка 117today's value of an asset картинка 118 картинка 119today's value of a contingent claim Mathematical Techniques in Finance - изображение 120 Mathematical Techniques in Finance - изображение 121generic random sample path Mathematical Techniques in Finance - изображение 122value of a money‐market account at time картинка 123along sample path картинка 124 картинка 125normal or Gaussian random variable with mean Mathematical Techniques in Finance - изображение 126and variance Mathematical Techniques in Finance - изображение 127 Mathematical Techniques in Finance - изображение 128lognormal random variable whose log is картинка 129CDFcumulative ditribution functionpdfprobability density functionpmfprobability mass function cumulative distribution function of a standard normal random variable - фото 130cumulative distribution function of a standard ( normal random variable probability density function of a standard normal r - фото 131) normal random variable, probability density function of a standard normal random variable BMBrow - фото 132 probability density function of a standard normal random variable BMBrownian - фото 133probability density function of a standard normal random variable, Mathematical Techniques in Finance - изображение 134BMBrownian motion Mathematical Techniques in Finance - изображение 135Brownian motion at time картинка 136along sample path картинка 137 картинка 138proportional, lognormal volatilityATM, ATMFat‐the‐money spot, at‐the‐money forward Diracs delta function absolute normalized volatilityCMSconstant maturity s - фото 139Dirac's delta function, absolute normalized volatilityCMSconstant maturity swap rateAD ArrowDebr - фото 140 картинка 141absolute, normalized volatilityCMSconstant maturity swap rateAD, картинка 142Arrow‐Debreu price, today's price of unit payoff at state картинка 143on future date картинка 144i.i.d.independent and identically distributed

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