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. n denotes the period and i the state of the world
4. time is denoted by t or n , T * , T . t or n denote the calendar time as time
passes. T * is the maturity date or the expiration date in calendar time. T is the
time-to-maturity
5. S is capitalization of the firm or the stock price
6. F (n, i, T * , T ) is a forward contract or futures contract at time n state i ,
expiring at calendar time T * , delivering a zero coupon bond with the time-tomaturity T .
7. P(n, i, T ) denotes the n period, n state, time-to-maturity T bond i.e., the
discount function at the node ( n, i ) . The par value is $1.
8. B(n, i, T * ) is a default free bond that may have coupon at node ( n, i ) with a
maturity date T * . B(n, i, T ) is the default free bond that has a maturity T . The
par value is $100.
9. D(n, i, T * ) is the defaultable bond with maturity date at T * . The convention is
similar to B .
10. V is the firm value or project value
11. C (n, i, T * ) is the call option with expiration date T * .
12.
13.
14.
15.
16.
P is the put option
A is an asset
L is a liability
FA is a fixed asset
CF is cash flow, FCF is free cash flow, NCF is net cash flow,
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17. r (n, i, T ) is the yield of a zero-coupon bond with maturity T at node ( n, i ) .
Hence it is the yield curve or the transfer pricing curve. r is continuously
compounding rate. r1 is semi-annual compounding risk-free rate. rf is the
2
one period risk-free rate.
18. r f and r denote the foreign risk-free interest rate and the domestic risk-free
interest rate, respectively.
19. espot and e forward denote the spot exchange and the forward exchange rate,
respectively.
20. R(n, i, T ) is the yield of a zero-coupon bond, which is not a benchmark
representing time value.
21. fc is the fixed cost
22. m 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. p is the risk neutral probability
26. q is the market probability
27.  denotes volatilities.
28.  is the variance-covariance matrix
29.  is the correlation matrix
30. N ( ) is the cumulative normal distribution function with mean zero and standard
deviation 1.
31. e is random error or noise
32. dZ is the normal distribution over an (infinitely) short time dt , with mean zero
and standard deviation dt
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
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Specific Notations
Valuation
VaRasset : VaR of asset
S : the market capitalization or stock value
Si : stock price of state i
ST : the underlying stock price on the expiration data, T
S u : the upstate movement of the stock
S d : the downstate movement of the stock
Pi n (T ) : the price of a T time to maturity zero coupon bond, at time n and state i
P (t , T ) : the T -year bond values at time t without default risk
Pd (t , T ) : the probability of default at time t for a bond with T time to maturity
P(T ) : the initial discount function, prices of zero coupon bonds with a face value of $1
and with maturity T
P(t , T , T ) : the value of the swaption, which expires at time T * , on a swap with tenor
*
T at time t
B (t , T ) : the T -year bond values at time t without default risk
B j (n, i) : the entries in each lattice for double-up sinking fund where j denotes the j -th
case, n the period, and i the state
Bc (n, i ) : the callable bond price in the i -th state and the n -th period
B(n, i, T ) : bond with time to maturity T at time n and state i
B(T ) : the price of a zero coupon bond with maturity T
D ( n, i ) : the deposit level
D : the market value of the debt
IO (n, i ) : interest rate option value at time n and state i
coupon : coupon payment
principal : face value
div : dividend
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F : the forward or futures price
F (T * , T ) : the futures price or forward price at the initial with the expiration date T *
delivering T -year maturity bond. Here T * denotes the calendar time and T
denotes the time-to-maturity.
F (t , T * , T ) : the futures price or forward price at time t with the expiration date T *
delivering T -year maturity bond.
Ck : caplet
PWV (k ) : the present path-wise value where k denotes the index for the path
PWV p (k ) : the present path-wise value for the primary decomposition of the security
C : the value of a European futures option, a European call option or the value of a bond
call option
C (t , T * ) : the call price or the payoff to call option on the expiration date T
P : the put option price at the expiration date
CP : conversion price
y : income to the bank on a total return basis
v : the sale volume over the reporting period
I : the investment value of the bond
I *payout : the payout as expenses
V the project value
Vu : the project value at the upstate
Vd : the project value at the downstate
V : the value of the firm at each node
Vu : the (firm) value at the upstate
Vd : the (firm) value in the downstate
V fc : the fixed-cost firm
Vup : the primitive firm at the upstate
Vdp : the primitive firm at the downstate
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c (n) : the annual maintenance costs
CIF (n, i ) : the cash inflow
NCF : the net cash flow
FCF : free cash flow
FCF (n, i ) : the free cash flow at time n, state i
Casht : cash position
W : the wealth level or the warrant
 : a proportion in the portfolio
 : the market value of the fixed costs
 : the value of the growth option
FA : fixed assets
fc : fixed cost
V : firm value
VL : levered firm (value)
VU : un-levered firm (value)
V : the market observed value of the financial contract
V p the primitive firm value
V fc the fixed cost firm value
 : objective function
div : the target dividend for the next period
f : net funding
b : new borrowing
l paydown : paydown of liabilities
v : new volume sales
e : the operating expenses for the period
 : the present value of bankruptcy costs
PV : the present value
U () : utility function
divu : a cash dividend at the upstate
divd : a cash dividend at the downstate
I : investment size
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Rates
Ftk ,tk 1 : the forward rate for the period between time t k and tk 1
f (t , T * ) : one-period forward rate or the instantaneous forward rate
r (t , T ) : the yield curve movement
 s : continuously compounded stock return
m : the gross profit margin
Rreq : the required rate of return of the stock
r (T ) : the yield of the bond P(T )
YTM : the yield to maturity
r : continuously compounding risk free rate
r (n, i ) : the one-period interest rate or the discount factor
R D (n, i ) : the deposit rate determined by the bank
R(n, i ) : the market interest rate
rf : one period risk free rate
 : the discount rate
Ri : mean of return of asset i
r (i ) : interest rate time i
RA : total return of the assets
RL : total return of the liabilities
 : the corporate marginal tax rate
 : tax rate
k e : the cost of equity
RM : market return rate
rf (t ) : the transfer pricing curve
RtD : the deposit rate set at time t
RX : a cap rate
Rt : the market short term rate at time t
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K D : the cost of debt
 : the cost of capital
 c : the corporate tax
 i : the personal marginal tax rate on interests
 e : The personal marginal capital gain tax rate on equity
GRI : the gross return on investment
F
Rreal
: the constant real risk free rate
c c : the convenient yield
r (n, i, t ) : t-year interest rate at time n and state i
r (n, i ) : one-period interest rate for time n and state i
r (n, i, j ) : one-period interest rate for time n and state i , j
 : risk premium
g : growth rate
Rcoupon : coupon rate
r : the continuously compounding yield
r (T ) : a spot curve
rt ,T : the interest rate which will be applied during the period of t to T when t and T
denote calendar time
r (t , T ) : the yield to maturity of the zero coupon bond with maturity T at time t
RM : the return of market portfolio
Time
T : time horizon
T : the number of years to the payment date at time 0
T * : swap delivery date
T : the tenor of a swap
T * : the expiration date of the option
T : the maturity of the underlying bond
t k : a reset date
 k : the reset period tk  tk 1
T : time to maturity
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dt : an infinitesimal time interval
KRD(i ) : i -th key rate duration
ti : knot point
Statistics Operator
E ( Ri ) : the expected rate of return of an asset i
 
E CF : the expected value of risky cash flows
X : a normal distribution with mean zero and standard deviation of 1
 : covariance matrix of all risk sources
p : the risk neutral probability
q : the market probability
 i : the standard deviation of an asset i
 (t , T ) : the volatility of the T year rate at time t
 (i ) : volatility time i
 : the volatility of margin ratio
 asset : volatility of asset
 (i, j ) : the volatility of the j-year spot rate, i years from now, or the volatility surface
for i and j vary over a range of years
 (n) : the term structure of forward volatilities
 s (n) : 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 T
 k : annualized volatility for the k-th caplet
f
 : the instantaneous standard deviation (or volatility or risk) of the underlying asset or
the risk
 P : portfolio risk
 M : the market risk
 ( s ) : the term structure of volatilities of proportional change in bond price
 () : a standard deviation over dt
 : a variance-covariance matrix
 ): correlation matrix
p : the risk neutral probability
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t : increment time t
N () : the cumulative normal distribution
e the error term
v : an eigenvector
 : the instantaneous returns or the drift of the underlying asset or the drift
dZ : 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
N () : cumulative normal distribution
q : upstate market probability parameters
dt standard deviation
 : the critical value
b : the adjustment speed of the mean reversion process
 : the recovery ratio
 : the default premium
m : profit margin of the time deposits
 asset : beta of asset
f * : the reserve ratio
 : the hedge ratio
u : upward movement proportion
d : downward movement proportion
 : the eigenvalue or the market price of risk,
 r

Indices
N : the number of asset
n : the number of additional shares after conversion
m : the number of shares
i and j for the state of the world
m : the number of steps in one period
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