$a_{\overline{n\hspace{.2em}|}}$j

discount factor for n payments and rate j

B(t) or Bt

value of a risk-free security (e.g. a bond or a bank account) at time t

b(x; n, p)

probability mass function for the binomial law Bin(n, p)

ℬ(x; n, p)

(cumulative) probability distribution function for the binomial law Bin(n, p)

Bin(n, p)

binomial probability distribution with number of trials n and success probability p

CA

American call

CE

European call

CDF

cumulative (probability) distribution function

Corr(X, Y)

correlation coefficient of X and Y

Cov(X, Y)

covariance of X and Y

CRR

Cox–Ross–Rubinstein

D

downward move in a binomial tree

div

dividend

D(t, T)

discount factor from time t to time T

D(t) ≡ D(0, t)

discount factor from time 0 to time t

$\frac{\text{d}\mathbb{P}}{\text{d}\mathbb{Q}}$

Radon–Nikodym derivative of ℙ w.r.t. ℚ

${\left(\frac{\text{d}\mathbb{P}}{\text{d}\mathbb{Q}}\right)}_t$

Radon–Nikodym derivative process at time t

EMM

equivalent martingale measure

Exp(λ)

exponential probability distribution with rate λ

E[X]

mathematical expectation of X

$\tilde{\text{E}}\left[X\right]$

risk-neutral mathematical expectation of X

E[X |ℱ]

mathematical expectation of X conditional on a σ-algebra ℱ

Et[X]

mathematical expectation of X conditional on ℱt

${\tilde{\text{E}}}^{\left(g\right)}\left[X\right]$

mathematical expectation of X w.r.t. the probability measure ℙ(g)

${\tilde{\text{E}}}_t^{\left(g\right)}\left[X\right]$

mathematical expectation of X conditional on ℱt w.r.t. the probability measure ℙ(g)

Et,x[X]

mathematical expectation of X conditional on an underlying process having value x at time t

Et,x[X]

mathematical expectation of X conditional on a underlying vector process having value x at time t

εt(γ · W)

exponential martingale process of an adapted process γ w.r.t. Brownian motion W

εt(γ · W)

exponential martingale process of an adapted vector process γ w.r.t. vector Brownian motion W

ℱt

σ-algebra generated by information available at time t

$\mathbb{F}$

filtration

f(t; T, T′)

forward rate at time t for interval [T, T′]

f(t, T)

instantaneous forward rate at time t for maturity T

F(t, T)

forward price at time t for maturity T

fX, fD

probability density function (of random variable X or probability distribution D)

FX, FD

(cumulative) distribution function (of random variable X or probability distribution D)

Gamma(κ, λ)

gamma probability distribution with shape parameter κ and rate parameter λ

GBM

geometric Brownian motion

i(m)

nominal interest rate compounded at frequency m

$\mathbb{I}$A

indicator of event (or set) A

iff

if and only if

i.i.d.

independent and identically distributed

K

strike price

Λ (·)

payoff function

mtX

minimum over [0, t] of the process X

MtX

maximum over [0, t] of the process X

n(x)

probability density function for a standard normal law

n2(x, y; ρ)

joint probability density function for two standard normal random variables with correlation coefficient ρ

nn(x1, . . . , xn; ρ)

joint probability density function for n standard normal random variables with correlation matrix ρ

$\mathcal{N}$(x)

(cumulative) probability distribution function for a standard normal law

$\mathcal{N}$2(x, y; ρ)

joint probability distribution function for two standard normal random variables with correlation coefficient ρ

$\mathcal{N}$n(x1, . . . , xn; ρ)

joint probability distribution function for n standard normal random variables with correlation matrix ρ

Norm(μ, σ2)

normal probability distribution with mean μ and variance σ2

Normn(m, Σ)

n-variate normal probability distribution with mean vector m and covariance matrix Σ

NPV

net present value

ω

scenario (element of a state space)

Ω

state space

P

present value, or principal, or purchase price

ℙ

probability measure

$\tilde{\mathbb{P}}\equiv {\tilde{\mathbb{P}}}^{\left(B\right)}$

risk-neutral probability measure with bank account as numéraire

${\tilde{\mathbb{P}}}^{\left(g\right)}$

risk-neutral probability measure (or EMM) with asset g as numéraire

ℙ(A)

probability of event A

ℙ(A | B)

probability of event A conditional on event B

${\mathcal{P}}_t$

partition of Ω generated by information available at time t

$\mathcal{P}\left(X\right)$

partition of Ω generated by random variable X

$\mathcal{P}\left(X_1,...,X_n\right)$

partition of Ω generated by random variables X1, . . . , Xn

PA

present (discounted) value of an annuity

P A

American put

PE

European put

Π(t) or Πt

portfolio value at time t

$\overline{\text{Π}}\left(t\right)\text{or}{\overline{\text{Π}}}_t$

discounted portfolio value at time t

p(s, t; x, y)

transition PDF for a one-dimensional diffusion

p(t; x, y)

time-homogeneous transition PDF for a one-dimensional diffusion

p(s, t; x, y)

transition PDF for a multidimensional diffusion

p(t; x, y)

time-homogeneous transition PDF for a multidimensional diffusion

PDF

probability density function

Pois(λ)

Poisson probability distribution with rate λ

ϱ

Radon–Nikodym derivative

ϱt

Radon–Nikodym derivative process at time t

ρ

correlation coefficient

r

interest rate

r(t)

instantaneous interest rate at time t

$r_{\left[t_1,t_2\right]}$

rate of return from time t1 to time t2

$R_{\left[t_1,t_2\right]}$

total return from time t1 to time t2

ℝ

set of real numbers

ℝ+

set of nonnegative real numbers

σ(X)

σ-algebra generated by random variable X

σ({Xλ})

σ-algebra generated by a collection {Xλ}

S(t) or St

price of a risky asset (e.g. a stock) at time t

Si(t) or Sti

price of the ith risky asset at time t

SDE

stochastic differential equation

SQB

squared Bessel process

T

maturity time; expiry time; exercise time

$\mathcal{T}$bX

first hitting time of X at level b

${\mathcal{T}}_{\left(a,b\right)}^X$

first exit time of X from the interval (a, b)

U

upward move in a binomial tree

Unif(a, b)

uniform probability distribution on an interval (a, b)

V (t)

(accumulated) value function at time t

υ(τ, S)

derivative pricing function of time to maturity τ and spot S

V (t, S) or Vt(S)

derivative pricing function of calendar time t and spot S

$\overline{V}\left(t,S\right)\text{or}{\overline{V}}_t\left(S\right)$

discounted derivative pricing function

VA

future (accumulated) value of an annuity

Var(X)

variance of X

VaR

Value at Risk

w.r.t.

with respect to

W

Brownian motion

W(μ,σ)

scaled Brownian motion with a linear drift

y(τ)

yield rate for time to maturity τ

y(t, T)

yield rate at time t for maturity T

Z(t, T)

zero-coupon bond price at time t for maturity T

ZCB

zero-coupon bond