Mathematics (Module 2)

Flashcards with vocabulary terms and definitions related to probability calculus and mathematical finance proofs, definitions, and theorems.

Set Function Definition

A rule that associates to each subset A of Ω one, and only one, scalar μ(A)

Set Function Properties Definition

1) Grounded if μ(∅) = 0; 2) Positive if μ(A) ≥ 0 ∀ A; 3) Monotone If μ(A) ≤ μ(B) when A ⊆ B; 4) Additive if μ(A∪B) = μ(A) + μ(B) when A∩B = ∅; 5) Normalized if μ(Ω) = 1

Measure Definition

A grounded, positive, and additive set function. μ: 2^Ω → [0, +∞)

Probability Measure Definition

A normalized measure. P : 2^Ω → [0, 1]

Simple Probability Definition

A probability measure P : 2^Ω → [0, 1] is simple if there exists a finite event E such that P(E) = 1

Countably Additive Probability Definition

A probability P : 2^Ω → [0, 1] is countably additive if: P(∪Am) = lim P(Am)

Random Variable Definition

A real-valued function f : Ω → ℝ defined on a state space Ω

P-a.e. Random Variables Definition

Two random variables f, g : Ω → ℝ are equal P-almost everywhere when P(ω ∈ Ω : f(ω) = g(ω)) = 1, or more compactly, when P(f = g) = 1

Expected Value Definition

The quantity Ep(f) = Σ f(ω)P(ω) where the sum is taken over all ω in supp(P)

Variance Definition

The quantity Vp(f) = Ep((f - Ep(f))^2) where the sum is taken over all ω in supp(P)

Standard Deviation Definition

The square root of the variance and is denoted by σ(f) = √Vp(f)

Covariance Definition

The quantity Covp(f,g) = Ep((f - Ep(f)) · (g - Ep(g)))

Cumulative Distribution Function Definition

The function F(x) = P(f ≤ x) ∀ x ∈ ℝ

Essentially Bounded Random Variable Definition

If there exist scalars m, M ∈ ℝ such that P(m ≤ f ≤ M) = 1

Integrable Density Function Definition

A positive function φ : ℝ → [0, +∞) is an integrable density function if: ∫φ(x)dx = 1

Stieltjes Expected Value Definition

The improper Stieltjes integral: Ep(f) = ∫ x dΦ. When it exists

Support of a Probability Definition

The set of all outcomes ω that have non-zero probabilities. Formally: supp(P) = {ω ∈ Ω : P(ω) > 0}

Carrier Definition

An interval [a, b] such that Φ(x) = 0 ∀ x < a and Φ(x) = 1 ∀ x ≥ b

Additivity Theorem

For every finite collection A1, A2, …, An of pairwise disjoint subsets of Ω, it holds: μ(∪Ai) = Σ μ(Ai)

Uniform Probability Theorem

Assigns the same probability to all spaces, so: P(ω) = 1/|Ω| ∀ ω ∈ Ω

Dirac Probability Theorem

Fix a state ω0 in a state space Ω (finite or infinite). The set function P : 2^Ω →ℝ defined by: P(A) = 1 if ω0 ∈ A, 0 if ω0 ∉ A. Is denoted by δω0, and called Dirac probability measure.

Poisson Probability Theorem

Take Ω = ℕ = {0, 1, 2, …m}. Fix a scalar λ > 0 and define the scalar sequence: Pr = λ^r / r! · e^(-λ) ∀ m ∈ ℕ. For each event A in Ω define the sequence an = 1 if m ∈ A or 0 otherwise. Now define the Poisson Probability P: 2^[0,1] by: P(A)= Σ Pr

Geometric Probability Theorem

Consider Ω = ℕ, let {qr} be a sequence of positive numbers such that the series Σ qr converges with sum R. Define the scalar sequence pm = qm / R. The geometric probability is defined by taking γm = q*u^m, q ∈ (0, 1). In this case, Pr = (1-q)·q^u. Define the set Function P: 2^[0,1] by P(A) = Σ pm

Probability of Any w Outside the Support Theorem

If ω not in supp(P) then P(ω) = 0

Probability of a Support of a Simple Probability

The support of a simple Probability P : 2^[0, 1] is a finite event with Probability 1, that is P(supp(P)) = 1. Moreover, P(A) = 1 implies supp(P) ⊆ A for all events in A

Probability as a Sum Over Support

For each event A: P(A) = Σ P(w)

Convergent Events and Countable Additivity 1°

Let P : 2^[0, 1] be a Probability, The Following Statements Are Equivalent: i) P is Countably additive; ii) If Am ↑ A, then P(Am) ↑ P(A); iii) If Am ↓ A, Then P(Am) ↓ P(A)

Countably Additive Poisson Probability Theorem

The Poisson Probability is Countably Additive

No Countably Additive Uniform Probability Theorem

There is no Countably Additive Uniform Probability p : 2^Ω → [0, 1]

Perfect Correlation

Let P: 2^Ω→ℝ be a simple probability, for Random Variables f,g : Ω→ℝ, Non-constant P-a.e. it HOLDS: |ρp(f,g)| = 1

Right Continuity of Φ Theorem

If P is Countably Additive, Then Φ Is Right Continuous, With limx->-∞ Φ(x) = 0 AND limx->+∞ = 1

Continuity of Φ Theorem

If P is Countably Additive, Then ∀ Xo∈ℝ : Φ(Xo) - limx->Xo- (Φ(x)) = P(f = Xo). Moreover, Φ Is Continuous at Xo∈ℝ => P(f = Xo) = 0

Uniform Distribution Function and Gaussian Theorem

Given Any Two Scalars a < b, Consider the uniform DISTRIBUTION FUNCTION IS: Φ(x) = 0 IF x

Essentially Bounded Random Variable and Ep THEROREM

All Essentially Bounded Random variables have finite EXPECTED VALUE

Linearity and Monotonicity of Ep(f) Theorem

: i) Ep(af + βg) = αEp(f) + βEp(g) ∀α, β∈ℝ; ii) Ep(f) <=Ep(g) if f <= g.

Elementary Financial Operation Definition

An exchange between two amounts of money available on different dates.

Accumulation Operation Definition

An exchange between an amount of money available today C (principal or invested capital) with another one M (final value or accumulated value) available in the future (at a date t>0)

Discount Operation Definition

An exchange between an amount of money S (nominal value) available at t>0 With another one a (present VALUE or Discounted value) available Today (at t = 0).

Conjugated Financial Factors Definition

The financial factors ((t) And e(t) ARE SAID TO BE CONSUGATEd Financial Factors When: ∀t > 0 f(t) · e(t) = 1

Annual Interest Rate Definition

The INTEREST GENERATED BU €1 INVESTED FOR THE FIRST YEAR IS INDICATED WITH THE SIMBOL I AND IS CALLEdThe Annual INTEREST RATE

Simple Interest Definition

The INTEREST INDICATED WITH I IS A FUNCTION OF C, t And i And is defined by The Formula i = C · i · t WITH C , t >= 0 AND i>0

Simple Discount Definition

THE SIMPLE DISCOUNT IS THE CONJUGATE OPERATION OF SIMPLE ACCUMULATION, AND IS DEFINED BI THE DISCOUNT FACTOR : 1 e(t) = 1 + it For t >= 0

Compound Interest Definition

COMPOUND INTEREST IS THE INTEREST CALCULATED ON BOTH THE INITIAL PRINCIPAL ANDThe INTEREST ACCUMULATED FROM PREVIOUS PERIODS. THE FINAL VALVE M OF A PRINCIPAL C INVESTED AT Time O, at Maturite t, IS Given By: M = C (1 + i)^t For t >= 0

Compound Discount Definition

THE COMPOUND DISCOUNT IS THE CONSUGATED OPERATION OF COMPOUND ACCUMULATION, AND IS DEFINED BY THE DISCOUNT FACTOR: 1 e(t) = (1 + ist Fort >= 0

Continuously Compounded Interest

CONTINUOUSLY COMPOUNDED INTEREST ASSUMES INTEREST IS CAPITALIZED AT EVERY INSTANT, AND IS defined By The Formula: M = c · e^(δt) FOR t > =0 WHERE δ is CALLED The Force Of Interest

Force of Interest Definition

FOR THE FINANCIAL LAW f(t) WE DEFINE THE FORCE OF INTEREST AS: δ(t) = f’(t)/f(t). For Simple Interest: δ(t)= i/(1+it); For compound interest : δ(t) = Ln(1+i).

Decomposability Definition

LET f(t) Be An Accumulation FACTOR . The ACCUMULATION FACTOR f(t) is said to be decomposable When: f(t) = f(t-s) · f(s) ∀s.t SUCH THAT 0<=s<=t

Annuity Definition

THE SEQUENCE OF CASH FLOWS R1 . R2 …. RM WITH THE SAME SIGN AND MATURITIES, RESPECTIVELY, t1 . t2 … tr IS CALLed An Annuity . IF Rs>0 ∃s , THEN THE CASH FLOWs ARE ALL inflows, meanwhile if Rs<0 ∃s , THEN THE CASH FLOWS ARE ALL OUTFLOWS.

Ordinary Annuity Definition

LET S BE A NATURAL NUMBER. THE SEQUENCE OF CASH FLOWS R1 . R2 …. Rm With The same sign and Maturities, respectively, 1, 2 …. m is CALLED An Ordinary annuity. If Rs>0 ∃s , THEN THE CASH FLOWs ARE ALL inflows, MEANWhiLe If Rs<0 ∃s , THEN THE CASH FLOWS ARE ALL OUTFLOWS.

Due Annuity Definition

LET S BE A NATURAL NUMBER. THE SEQUENCE OF CASH FLOWS R1 . R2 …. Rm With The same sign and Maturities respectively, 0. 1 …. m -1 Is called a due annuity . If Rs>0 ∃s , THEN THE CASH FLOWs ARE ALL inflows, MEANWhiLe If Rs<0 ∃s , THEN THE CASH FLOWS ARE ALL OUTFLOWS.

Ordinary Perpetuity Definition

the cash flows are infinitely many, And for simplicity We Assume That They All have THE SAME VALUE.

Due Perpetuity definition

a financial operation described where the cash flows are infinitely many, And for simplicity We Assume That They All have THE SAME VALUE

DCF Definition

THE FUNCTION g : (-1, + ∞)→ℝ defined by g(x) = Σ Qs/(1 + x)^ts THE DISCOUNTED CASH FLOW ASSOCIATED TO THE FINANCIAL OPERATION . IT DESCRIBES THE PRESENT VALUE OF THE CASH FLOW, WHERE THE COMPOUND ANNUAL INTERESTRATE IS TAKEN AS A VARIABLE.

NPV definition

GLI) = Σ Qs/(1 + il)^ts

Investment Definition

A Financial operation, with maturity T, described by: Q0

Loan Definition

A Financial operation, with maturity T, described by: Q0>0 AND Qs<0 , S = 1, 2 …., M WITH AT LEAST AN Qs < 0

Internal Rates Definition

THE INTERNAL RATE OF A FINANCIAL OPERATION IS ANY SOLUTION OF THE EQUATION G(X) = 0.

IRR AND Effective Cost Definition

THE INTERNAL RATE OF AN INVESTMENT IS CALLED THE INTERNAL RATE OF RETURN (OR IRR) , whiLE THE INTERNAL RATE Of A LOAN IS CALLEdThe effECTIVE cost Of THE LOAN .

Fixed-Income Bond Definition

A FIXED INCOME BOND IS A SECURITY THAT CONTRACTUALLY GUARANTEES FUTURE PAYMENTS BOTH IN TERMS OF THE AMOUNTS AND OF THE MATURITA DATES.

Yield to Maturity Of a ZCB Definition

THE MELD TO MATURITY, OR GROSS COUPON PIELD, OF A ZCB IsTHE GROSS COMPOUND RATE OF RETURN i = i (0,T) THAT CHARAC TERIZES THE INVESTMENT FROM O UNTIL THE MATURITY TIO OF THE ZCB: i(o, T) = (√M/c )^(1/T - 1

Nield to Maturity of a Bond With Coupons Definition

THE HELD TO MATURITY (ITM) OF THE BOND DESCRIBED BI THE CASH FLOWS: #Q0… Qm# IS THE COMPOUND INTEREST RATE X * WHICH SOLVES THE EQUATION : G(x) = -Po+ ΣQs/(1 + x)^ts = 0

Duration Definition

GIVEN THE CASH FLOW: (A) : #t,…tm# then D(i) = Σ ts · as(1 + i)^(-ts) / Σ 0s(1 + i)^(-ts)

Financial Immunization Definition

AN Investment at rate i is financially (Globalli) immunized at YEAR 2* AGAINST THE INTEREST RATE RISK IF: V(zi+i) = V(zi) wi AND The LIFETIME 2* IS CALLED THE DATE OF IMMUNIZATION .

Volatility Definition

THE VOLATILITY OF THE PRICE OF A BOND IS THE SENSITIVITY OF THE PRICE TO A CHANGE IN NELD RATE: ΔP(i) = [P(i + Δi) - P(i)] / P(i)

Modified Duration definition

Di / (1+i)

Portfolio Definition

A Portfolio Is a Vector X = [X1. X2 … Xm EIRMTHAT REPRESENTS THE QUANTITIES OF M PRIMARY ASSETS TRADED ON THE MARKET.

Payoff Definition

THE PAYOFF OF AN ASSET IS ITS MONETARY OUTCOME IN EACH POSSIBLE STATE OF THE WORLD. FOR A PRIMARY ASSET - C The patoff is represented by a vector =

Contingent Claim Definition

WE CALL CONTINGENT CLAIM ANY STATE Contingent patoff weℝ

Complete Financial Market definition

THE MARRET Wis complete If WEIR , That Is, if Al CONTINCENT CLAIMS ARE REPLICABLE

Incomplete Financial Market definition

THE MARKET W IS INCOMPLETE IF WCIRE , THAT IS, IF NOT ALL CONTINGENT CLAIMS ARE REPLICABLE

Arrow Contingent claim definition

AN ARROW Corpore) contingent claim - e cris a Vector THAT PAYS ONE UNIT IF STATE S: OCCURS AND O IN AL OTHER STATES

Payoff Operator Definition

THE PAYOFF OPERATOR IS THE FUNCTION ASSOCIATING TO EACH PORTFOLIO X THE CLAIM I THAT IT INDUCES: = Σ XiYis ∀s ==13

Payoff Matrix Definition

Y11 Ye2 …Yem; Y21 Y22 … Yam; ……..Yr1 Yaz … YRm

Market Value Definition

THE MARKET VALUE OF THE PORTFOLIO X IS THE FUNCTION ASSOCIATING TO Each PORTFOLIO ITS PRICE Today : v: /R IP X = Σ XiPi

Law of One Price Definition

The Financial Market (L , P) satisfies The Law of one price (LOP) IF . ∀ PORTFOLIO X . XeIRR: R(x) = R(x') => v(x) = w(x).

Price of a Replicable Contingent Claim Definition

THE PRICE Pr Of A replicable CONTINCENT CLAim WE W isTHE VALUE Of A REPLICATING PORTFOLIO XelEYW)-. That is, Pe = ~(1).

Pricing Kernel definition

Suppose The Financial Market [L,S Satisfies The Lop , Then 7 a Unique VECTOR #E W SUCH THAT: f(E) = t · w ∀weWEIR *. SUCH VECTOR I IS CALLED THE PRICING BERNEL Of THE CLAIMS .

No Arbitrage Conditions Definition

R(x)>/= 0 => v(x)>/= 0

Capitalization Axioms

Axiom 1 :dependence of M, that is M =M(C.t); Axiom 2:Additivity With RESPECT To The Principal that isM(Ce + ( , t) =M((t) + M((z , t); Axiom 3:Increasing monotonicity with respect to time, that is M (C, te)

Cauchy) THEOreM

∫x, ye f(x + 2) = f(x) + f(x) => f(x = ax , a t1

Exponential Cauchy Equation Theorem

=> ((x = e^(ml), me ~

Axiom Satisfieting Functions Theorem

Let (10 , t>=0 be , respectively , Capital and Liftime ALL FUNCTIONS M (C . t) WhiCh Satisfy An-Ay Axions are M (c , t) = Cf(t) WITH THE FOLLOWING PROPERTIES: 1) ((a = 1; 2) fis Increasing

Equivalent Rates in Compound Interest

TWO INTEREST RATES ARE CALLED EQUIVALENT If IN THE SAMELIFETIME THEY PRODUCE THE SAME FINAL VALUE STARTING FROM THE SAME PRINCIPAL . •For simple Interest : im. (im m) •For compound Interest : (1 + i) = (1 + im)m

Remark on Ordinary Perpetuit1

In compound Interest , The present value A of an ordinary Perpetuity is defined By: A = R/(1+i)+…+… =R + Lim (1+ i )/i = R/i.

Remarr on Due Perpetuity

In compound Interest , the present value a of a Due Perpetuity is defined By: A: + 1) +…. + p +…+ Σ = R + & = R (1+i)/i.

Ries2-Markov TheoreM

A FUNCTION f : /R→/R is Linear And increasing=> 7 a Unique positive Vector 2 I Such That: l · x ∀ x∈ℝ

Representation Result for the Pricing Rule Theorem

Suppose The Financial Market (L, p) Satisfies The Lop. THEN 7 A UNIQUE VECTOR TE W SUCH THAT: ((m) = I ·w∀w∈ℝ