Amir Sadr - Mathematical Techniques in Finance
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- Название:Mathematical Techniques in Finance
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Mathematical Techniques in Finance: краткое содержание, описание и аннотация
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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 future valueIRRinternal rate of returnPnLprofit and loss
present valueYTMyield to maturityp.a.per annum
discount factor, today's value unit payment at future date
dicount factor at
for unit payment at
interest rate
compounding interest rate with
compoundings per year
yieldAPRannual 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
price of an
‐year bond with coupon rate
, paid
times per year, with yield
accrual fraction between 2 dates according to some day count basis
clean price of a bond = Price
accrued interest
price of an
‐year zero‐coupon bond with yield
,
compoundings per year
price of an
‐year annuity with annuity rate of
, paid
times per year, with yield
price of
‐maturity Treasury Bill with discount yield
PV01present 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
balance of a level pay loan after
periods
principal and interest payments of a level pay loan in the
th period
price of
‐year level pay loan with loan rate of
, paid
times per year, with yield
ALaverage life
balance of a level pay loan after
periods with prepayments
principal and interest payments of a level pay loan within the
th period with prepaymentsSMMsingle monthly mortality rateCPRconstant prepayment ratio
periodic prepayment speed
utility of wealth
lottery
is preferred to
certainty‐equivalent of random payoff
,
absolute risk premium, relative risk premium
,
,
value, value of a portfolio, value of
th asset
quantity, price
weight of
th asset in a portfolio,
return of an asset over a period
:
. Can be divided by
to give rate of return
asset
's return, with mean
and standard deviation
, Cmean vector, standard deviation vector, and covariance matrix of asset returns
return of a risk‐free asset
market portfolio, return of the market portfolio
beta of an asset
,
empirical estimate of
arithmetic average of
samples of
,
forward date, future date
forward value of asset
at time
for forward date
‐value of a forward agreement on asset
for forward date
and price
simple (noncompounding) forward rate that can be locked at
for forward deposit period
. The first term may be omitted when
.
continuously compounding forward rate that can be locked at
for forward deposit period
. The first term may be omitted when
.FXforeign currency exchange rate
domestic and foreign interest rates for forward exchange rate calculations
the
‐forward exchange rate that can be locked at
today's value of an asset
today's value of a contingent claim
generic random sample path
value of a money‐market account at time
along sample path
normal or Gaussian random variable with mean
and variance
lognormal random variable whose log is
CDFcumulative ditribution functionpdfprobability density functionpmfprobability mass function
cumulative distribution function of a standard (
) normal random variable,
probability density function of a standard normal random variable,
BMBrownian motion
Brownian motion at time
along sample path
proportional, lognormal volatilityATM, ATMFat‐the‐money spot, at‐the‐money forward
Dirac's delta function,
absolute, normalized volatilityCMSconstant maturity swap rateAD,
Arrow‐Debreu price, today's price of unit payoff at state
on future date
i.i.d.independent and identically distributed
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