To Accomplish This Task, Our Choice of Notations Follows Some Guidelines Stated Below

Glossary of Notations

To determine a set of consistent notations for the book is important. This book is about Financial Models. Appropriate choice of notations would enable the specifications of the models unambiguous and provide clarity to the readers. However, to determine such a set of notations is particularly challenging for this book. The financial models cover the capital markets and corporate finance. In each sub-field of finance, there are standard notations and therefore finding consistency across these sub-fields can be complex.

To accomplish this task, our choice of notations follows some guidelines stated below.

1.  Maintain the accepted notations in the standard literature as much as possible.

2.  Use capital letter to denote securities or present values of cash flows, with subscript u and d denoting upstate and downstate respectively.

3.  denotes the period and the state of the world

4.  time is denoted by or , , . or n denote the calendar time as time passes. is the maturity date or the expiration date in calendar time. is the time-to-maturity

5.  is capitalization of the firm or the stock price

6.  is a forward contract or futures contract at time state , expiring at calendar time , delivering a zero coupon bond with the time-to-maturity .

7.  denotes the period, state, time-to-maturity bond i.e., the discount function at the node. The par value is $1.

8.  is a default free bond that may have coupon at node with a maturity date . is the default free bond that has a maturity . The par value is $100.

9.  is the defaultable bond with maturity date at . The convention is similar to .

10.  is the firm value or project value

11.  is the call option with expiration date .

12.  is the put option

13.  is an asset

14.  is a liability

15.  is a fixed asset

16.  is cash flow, is free cash flow, is net cash flow,

17.  is the yield of a zero-coupon bond with maturity at node . Hence it is the yield curve or the transfer pricing curve. is continuously compounding rate. is semi-annual compounding risk-free rate. is the one period risk-free rate.

18.  and denote the foreign risk-free interest rate and the domestic risk-free interest rate, respectively.

19.  and denote the spot exchange and the forward exchange rate, respectively.

20.  is the yield of a zero-coupon bond, which is not a benchmark representing time value.

21.  is the fixed cost

22.  is for margins, like profit margin, or the number of sub-periods

23.  is for premiums, like default premium, risk premium

24.  is for tax rate with subscripts denoting the type of taxes.

25.  is the risk neutral probability

26.  is the market probability

27.  denotes volatilities.

28.  is the variance-covariance matrix

29.  is the correlation matrix

30.  is the cumulative normal distribution function with mean zero and standard deviation 1.

31.  is random error or noise

32.  is the normal distribution over an (infinitely) short time , with mean zero and standard deviation

33.  bold letter represents vector

34.  : the tilde letters mean the stochastic movement

35.  : the bar letters mean the expected value

36.  Euclid font type is used for book values

Specific Notations

Valuation

: VaR of asset

: the market capitalization or stock value

: stock price of state

: the underlying stock price on the expiration data,

: the upstate movement of the stock

: the downstate movement of the stock

: the price of a time to maturity zero coupon bond, at time and state

: the -year bond values at time t without default risk

: the probability of default at time t for a bond with time to maturity

: the initial discount function, prices of zero coupon bonds with a face value of $1 and with maturity

: the value of the swaption, which expires at time , on a swap with tenor at time t

: the -year bond values at time t without default risk

: the entries in each lattice for double-up sinking fund where j denotes the -th case, the period, and the state

: the callable bond price in the -th state and the -th period

: bond with time to maturity at time n and state

: the price of a zero coupon bond with maturity

: the deposit level

: the market value of the debt

: interest rate option value at time n and state

: coupon payment

: face value

: dividend

: the forward or futures price

: the futures price or forward price at the initial with the expiration date delivering -year maturity bond. Here denotes the calendar time and denotes the time-to-maturity.

: the futures price or forward price at time with the expiration date delivering -year maturity bond.

: caplet

: the present path-wise value where k denotes the index for the path

: the present path-wise value for the primary decomposition of the security

: the value of a European futures option, a European call option or the value of a bond call option

: the call price or the payoff to call option on the expiration date

: the put option price at the expiration date

: conversion price

: income to the bank on a total return basis

: the sale volume over the reporting period

: the investment value of the bond

: the payout as expenses

the project value

: the project value at the upstate

: the project value at the downstate

: the value of the firm at each node

: the (firm) value at the upstate

: the (firm) value in the downstate

: the fixed-cost firm

: the primitive firm at the upstate

: the primitive firm at the downstate

: the annual maintenance costs

: the cash inflow

: the net cash flow

: free cash flow

: the free cash flow at time n, state

: cash position

: the wealth level or the warrant

: a proportion in the portfolio

: the market value of the fixed costs

: the value of the growth option

: fixed assets

: fixed cost

: firm value

: levered firm (value)

: un-levered firm (value)

: the market observed value of the financial contract

the primitive firm value

the fixed cost firm value

: objective function

: the target dividend for the next period

: net funding

: new borrowing

: paydown of liabilities

: new volume sales

: the operating expenses for the period

: the present value of bankruptcy costs

: the present value

: utility function

: a cash dividend at the upstate

: a cash dividend at the downstate

: investment size

Rates

: the forward rate for the period between time and

: one-period forward rate or the instantaneous forward rate

: the yield curve movement

: continuously compounded stock return

: the gross profit margin

: the required rate of return of the stock

: the yield of the bond

: the yield to maturity

: continuously compounding risk free rate

: the one-period interest rate or the discount factor

: the deposit rate determined by the bank

: the market interest rate

: one period risk free rate

: the discount rate

: mean of return of asset

: interest rate time

: total return of the assets

: total return of the liabilities

: the corporate marginal tax rate

: tax rate

: the cost of equity

: market return rate

: the transfer pricing curve

: the deposit rate set at time

: a cap rate

: the market short term rate at time

: the cost of debt

: the cost of capital

: the corporate tax

: the personal marginal tax rate on interests

: The personal marginal capital gain tax rate on equity

: the gross return on investment

: the constant real risk free rate

c: the convenient yield

: t-year interest rate at time n and state

: one-period interest rate for time n and state

: one-period interest rate for time n and state ,

: risk premium

: growth rate

: coupon rate

: the continuously compounding yield

: a spot curve

: the interest rate which will be applied during the period of t to when t and denote calendar time

: the yield to maturity of the zero coupon bond with maturity at time

: the return of market portfolio

Time

: time horizon

: the number of years to the payment date at time 0

: swap delivery date

: the tenor of a swap

: the expiration date of the option

: the maturity of the underlying bond

: a reset date

: the reset period

: time to maturity

: an infinitesimal time interval

: -th key rate duration

: knot point

Statistics Operator

: the expected rate of return of an asset

: the expected value of risky cash flows

: a normal distribution with mean zero and standard deviation of 1

: covariance matrix of all risk sources

: the risk neutral probability

: the market probability

: the standard deviation of an asset

: the volatility of the year rate at time

: volatility time

: the volatility of margin ratio

: volatility of asset

: the volatility of the j-year spot rate, years from now, or the volatility surface for and vary over a range of years

: the term structure of forward volatilities

: the volatility of the n-th year rate of the spot curve for each n = 1, 2, …

: annualized volatility of the bond price at time

: annualized volatility for the k-th caplet

: the instantaneous standard deviation (or volatility or risk) of the underlying asset or the risk

: portfolio risk

: the market risk

: the term structure of volatilities of proportional change in bond price

: a standard deviation over

: a variance-covariance matrix

): correlation matrix

: the risk neutral probability

: increment time

: the cumulative normal distribution

the error term

: an eigenvector

: the instantaneous returns or the drift of the underlying asset or the drift

: a unit of risk, a normal distribution over an (infinitely) short time interval viewed as a unit of risk, which has a zero mean and the standard deviation

: cumulative normal distribution

: upstate market probability parameters

: the critical value

: the adjustment speed of the mean reversion process

: the recovery ratio

: the default premium

: profit margin of the time deposits

: beta of asset

: the reserve ratio

: the hedge ratio

: upward movement proportion

: downward movement proportion

: the eigenvalue or the market price of risk,

Indices

: the number of asset

: the number of additional shares after conversion

: the number of shares

andfor the state of the world

: the number of steps in one period

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