1. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

Define and construct a probability space (Ω, F, P) and understand the role of sigma-algebras in information filtration.
Compute conditional expectations and understand their properties (tower property, pulling out known factors).
Define martingales and their key properties (constant expectation, orthogonal increments).
Apply the Doob-Meyer decomposition to semimartingales.
Understand the Radon-Nikodym derivative and its role in changing probability measures.
Apply the Girsanov Theorem to transform Brownian motion from the real-world measure P to the risk-neutral measure Q.
Derive the risk-neutral pricing formula using the martingale representation theorem.
Understand the relationship between market completeness and the uniqueness of the risk-neutral measure.

2. THE PROBABILITY SPACE (Ω, F, P) AND FILTRATIONS

2.1 The Probability Space

A probability space is a triple (Ω, F, P) where:

Ω is the sample space (the set of all possible outcomes).
F is a sigma-algebra (the collection of events to which probabilities can be assigned).
P is a probability measure (a function P: F → [0, 1] satisfying P(Ω) = 1 and countable additivity).

2.2 Sigma-Algebra (F)

A sigma-algebra F is a collection of subsets of Ω such that:

Ω ∈ F.
If A ∈ F, then A^c ∈ F (closed under complements).
If A_1, A_2, … ∈ F, then ∪_{i=1}^∞ A_i ∈ F (closed under countable unions).

2.3 Filtration (F_t)

A filtration is an increasing sequence of sigma-algebras:

F_0 ⊆ F_1 ⊆ F_2 ⊆ … ⊆ F_T ⊆ F

F_t represents all information available up to time t. A stochastic process X_t is said to be adapted to F_t if X_t is F_t-measurable (i.e., the value of X_t is known at time t given the information in F_t).

2.4 Financial Interpretation

At time t = 0, we know only the initial conditions (F_0 = {Ω, ∅} for a trivial filtration).
As time progresses, we learn more information (stock prices, economic data, news).
At time T, we know everything (F_T = F).

3. CONDITIONAL EXPECTATION

3.1 Definition

The conditional expectation E[X | F_t] is the expected value of the random variable X given the information available at time t. It is the best prediction of X based on the information in F_t.

Key Properties:

Measurability: E[X | F_t] is F_t-measurable.
Tower Property (Law of Iterated Expectations): For s ≤ t:

E[ E[X | F_t] | F_s ] = E[X | F_s]
Pulling Out Known Factors: If Y is F_t-measurable, then:

E[ Y X | F_t ] = Y E[ X | F_t ]
Linearity: E[aX + bY | F_t] = aE[X | F_t] + bE[Y | F_t].
Jensen’s Inequality: If φ is convex, then:

φ( E[X | F_t] ) ≤ E[ φ(X) | F_t ]

3.2 Conditional Expectation as Projection

In the L² space (the space of square-integrable random variables), the conditional expectation E[X | F_t] is the orthogonal projection of X onto the subspace of F_t-measurable functions. This means it minimises the mean squared error:

E[ (X – E[X | F_t])² ] ≤ E[ (X – Y)² ] for all F_t-measurable Y

3.3 Financial Application – Pricing a Derivative

The price of a European derivative with payoff H at time T is:

V_t = E^Q[ e^{-r(T-t)} H | F_t ]

Where Q is the risk-neutral measure. The conditional expectation is taken with respect to the information available at time t.

4. MARTINGALES

4.1 Definition

A stochastic process {X_t, t ≥ 0} is a martingale with respect to a filtration F_t if:

X_t is adapted to F_t.
E[|X_t|] < ∞ for all t (integrability).
For all s ≤ t:

E[X_t | F_s] = X_s

Interpretation: A martingale is a “fair game.” The expected future value, given all past information, is equal to the current value. There is no systematic drift.

4.2 Examples of Martingales

Brownian Motion: W_t is a martingale (E[W_t | F_s] = W_s).
Exponential Martingale: For a Brownian motion W_t, the process:

M_t = exp( θ W_t – (1/2) θ² t )
is a martingale for any constant θ.
Discounted Asset Price: Under the risk-neutral measure Q, the discounted asset price:

S_t^* = e^{-rt} S_t
is a martingale (E^Q[S_t^* | F_s] = S_s^*).

4.3 Properties of Martingales

Constant Expectation: E[X_t] = E[X_0] for all t.
Orthogonal Increments: For s ≤ t, E[ (X_t – X_s) * Y ] = 0 for any F_s-measurable random variable Y.
Optional Stopping Theorem: If τ is a bounded stopping time, then:

E[X_τ] = E[X_0]

4.4 Submartingales and Supermartingales

Submartingale: E[X_t | F_s] ≥ X_s for s ≤ t (the process tends to increase).
Supermartingale: E[X_t | F_s] ≤ X_s for s ≤ t (the process tends to decrease).

4.5 Financial Application – The Martingale Property in Option Pricing

The fundamental theorem of asset pricing states that a market is arbitrage-free if and only if there exists a risk-neutral measure Q under which all discounted asset prices are martingales.

For a stock price S_t following Geometric Brownian Motion:

dS_t = μ S_t dt + σ S_t dW_t^P

Under Q, the drift becomes r (the risk-free rate):

dS_t = r S_t dt + σ S_t dW_t^Q

The discounted price S_t^* = e^{-rt} S_t is a martingale under Q.

5. THE DOOB-MEYER DECOMPOSITION

5.1 The Theorem

Any submartingale X_t can be uniquely decomposed as:

X_t = M_t + A_t

Where:

M_t is a martingale.
A_t is a predictable, increasing process (the “drift” or “compensator”).

5.2 Financial Application – Decomposing Asset Returns

A stock price can be decomposed into:

S_t = e^{rt} + (martingale component) + (drift component)

This decomposition is fundamental to understanding the risk-return tradeoff.

5.3 The Continuous-Time Version

For an Ito process:

dX_t = μ_t dt + σ_t dW_t

The martingale part is ∫ σ_t dW_t, and the drift part is ∫ μ_t dt.

6. THE RADON-NIKODYM DERIVATIVE

6.1 Changing Probability Measures

Suppose we have two probability measures P and Q on the same measurable space (Ω, F). If P is absolutely continuous with respect to Q (i.e., P(A) = 0 whenever Q(A) = 0), then there exists a Radon-Nikodym derivative:

dP/dQ

Such that for any random variable X:

E^P[X] = E^Q[ X * (dP/dQ) ]

6.2 Properties of the Radon-Nikodym Derivative

Positivity: dP/dQ > 0 almost surely.
Expectation: E^Q[ dP/dQ ] = 1.
Bayes’ Rule: For any event A:

P(A) = E^Q[ 1_A * (dP/dQ) ]

6.3 The Radon-Nikodym Derivative as a Martingale

Define the density process:

Z_t = E^Q[ dP/dQ | F_t ]

Then Z_t is a Q-martingale with E^Q[Z_t] = 1.

6.4 Financial Application – Pricing Under Different Measures

To price an option under the risk-neutral measure Q, we need to change from the real-world measure P using the Radon-Nikodym derivative:

dQ/dP = exp( -∫_0^T θ_t dW_t^P – (1/2) ∫_0^T θ_t² dt )

Where θ_t is the market price of risk (the Sharpe ratio).

7. THE GIRSANOV THEOREM

7.1 The Theorem

Let W_t^P be a Brownian motion under measure P. Let θ_t be an adapted process satisfying the Novikov condition:

E^P[ exp( (1/2) ∫_0^T θ_t² dt ) ] < ∞

Define the Radon-Nikodym derivative:

dQ/dP = exp( -∫_0^T θ_t dW_t^P – (1/2) ∫_0^T θ_t² dt )

Then, under Q, the process:

W_t^Q = W_t^P + ∫_0^t θ_s ds

is a Brownian motion.

7.2 Interpretation

Girsanov’s theorem tells us how to change the drift of a Brownian motion by changing the probability measure.

Under P: W_t^P is a Brownian motion (drift 0).
Under Q: W_t^P has drift -θ_t (i.e., W_t^P = W_t^Q – ∫_0^t θ_s ds).

7.3 The Novikov Condition

The Novikov condition ensures that the Radon-Nikodym derivative is a martingale (i.e., it has expectation 1).

E^P[ exp( (1/2) ∫_0^T θ_t² dt ) ] < ∞

This condition is satisfied for most financial applications where θ_t is bounded.

7.4 Financial Application – Changing from P to Q

We want to change from the real-world measure P to the risk-neutral measure Q.

Under P:

dS_t = μ S_t dt + σ S_t dW_t^P

We set θ_t = (μ – r) / σ (the market price of risk).

Under Q, by Girsanov:

dW_t^Q = dW_t^P + θ_t dt

Substituting:

dS_t = μ S_t dt + σ S_t (dW_t^Q – θ_t dt)
dS_t = μ S_t dt + σ S_t dW_t^Q – σ S_t θ_t dt
dS_t = μ S_t dt – (μ – r) S_t dt + σ S_t dW_t^Q
dS_t = r S_t dt + σ S_t dW_t^Q

Therefore, under Q, the stock price has drift r (the risk-free rate).

8. THE MARTINGALE REPRESENTATION THEOREM

8.1 The Theorem

Let W_t be a Brownian motion and let F_t be its filtration. If M_t is a martingale with respect to F_t, then there exists a unique adapted process φ_t such that:

M_t = M_0 + ∫_0^t φ_s dW_s

8.2 Interpretation

Every martingale can be represented as a stochastic integral with respect to Brownian motion. This means that any contingent claim can be replicated by a dynamic trading strategy in the underlying asset.

8.3 Financial Application – Replication of a Derivative

Let H be the payoff of a European option at time T. Define the martingale:

M_t = E^Q[ e^{-rT} H | F_t ]

By the Martingale Representation Theorem, there exists φ_t such that:

dM_t = φ_t dW_t^Q

The replicating portfolio consists of φ_t units of the risky asset and (M_t – φ_t S_t^*) units of the risk-free asset.

8.4 Market Completeness

A market is complete if every contingent claim can be replicated by a self-financing trading strategy in the underlying assets.

Complete Market: There is a unique risk-neutral measure Q.
Incomplete Market: There are multiple risk-neutral measures.

In a complete market, the Martingale Representation Theorem guarantees that every payoff can be replicated.

9. THE FUNDAMENTAL THEOREM OF ASSET PRICING

9.1 Part 1 (No Arbitrage)

A market has no arbitrage opportunities if and only if there exists at least one risk-neutral measure Q under which all discounted asset prices are martingales.

9.2 Part 2 (Market Completeness)

A market is complete if and only if the risk-neutral measure Q is unique.

9.3 The Pricing Formula

Under the risk-neutral measure Q, the price of a derivative with payoff H at time T is:

V_t = E^Q[ e^{-r(T-t)} H | F_t ]

9.4 The Arrow-Debreu Prices

The state price density (or pricing kernel) is:

π_t = e^{-rt} (dQ/dP)

The price of any asset is:

V_t = E^P[ π_T / π_t * H | F_t ]

10. PRACTICAL IMPLEMENTATION

A. Simulating Brownian Motion and Verifying Martingale Property:

import numpy as np
import matplotlib.pyplot as plt

# Simulate Brownian motion
T = 1.0
n_steps = 252
dt = T / n_steps
n_paths = 1000

dW = np.random.normal(0, np.sqrt(dt), (n_paths, n_steps))
W = np.cumsum(dW, axis=1)

# Check martingale property
t1 = 100
t2 = 200
E_Wt2_given_Wt1 = W[:, t1]  # E[W_t2 | F_t1] = W_t1
actual_mean = np.mean(W[:, t2])
print(f"E[W_t2] = {actual_mean:.4f}")
print(f"W_t1 = {np.mean(W[:, t1]):.4f}")

B. The Exponential Martingale:

theta = 0.5
M = np.exp(theta * W - 0.5 * theta**2 * np.arange(0, T, dt))
# Check that E[M_t] = 1
print(f"E[M_T] = {np.mean(M[:, -1]):.4f}")

C. The Radon-Nikodym Derivative and Change of Measure:

# Real-world measure P: drift μ
mu = 0.08
sigma = 0.20
r = 0.03

# Market price of risk
theta = (mu - r) / sigma

# Radon-Nikodym derivative dQ/dP
RN = np.exp(-theta * W[:, -1] - 0.5 * theta**2 * T)
print(f"E^P[dQ/dP] = {np.mean(RN):.4f}")

# Under Q, W_t^Q = W_t^P + θ t
W_Q = W + theta * np.arange(0, T, dt)
# Check that W_Q is a Brownian motion under Q

MODULE 3, LESSON 3.3

Random Variables, Probability Distributions, and Characteristic Functions

1. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

Define random variables and distinguish between discrete and continuous types.
Compute and interpret the Probability Density Function (PDF), Cumulative Distribution Function (CDF), and Quantile Function.
Compute the expected value, variance, and higher moments of random variables.
Derive and apply the Moment Generating Function (MGF) and Characteristic Function (CF).
Construct joint distributions, marginal distributions, and conditional distributions.
Understand and apply the Law of Total Probability and Bayes’ Theorem.
Compute the distribution of sums of independent random variables using convolution.
Identify and apply key distributions used in finance (Normal, Log-Normal, Poisson, Exponential, Gamma, Chi-Square, and Stable distributions).
Apply the Central Limit Theorem to approximate sums of random variables.

2. RANDOM VARIABLES – DEFINITION AND TYPES

2.1 Definition

A random variable X is a function from the sample space Ω to the real numbers R:

X: Ω → R

2.2 Types of Random Variables

Discrete Random Variable: Takes on a countable number of values (e.g., number of defaults, stock price movements measured in ticks).
Continuous Random Variable: Takes on an uncountable number of values in an interval (e.g., stock returns, interest rates).

2.3 Distribution Function (CDF)

For any random variable X, the Cumulative Distribution Function (CDF) is:

F_X(x) = P(X ≤ x)

Properties:

F_X is non-decreasing.
F_X is right-continuous.
lim_{x→-∞} F_X(x) = 0 and lim_{x→∞} F_X(x) = 1.

2.4 Probability Density Function (PDF)

For a continuous random variable X, the PDF is:

f_X(x) = dF_X(x) / dx

Properties:

f_X(x) ≥ 0 for all x.
∫_{-∞}^{∞} f_X(x) dx = 1.
P(a ≤ X ≤ b) = ∫_a^b f_X(x) dx.

2.5 Quantile Function (Inverse CDF)

The quantile function (or percentile function) is the inverse of the CDF:

Q(p) = F_X^{-1}(p) = inf{ x ∈ R : F_X(x) ≥ p }

This is used for calculating Value at Risk (VaR).

Financial Application – Value at Risk (VaR):
The 95% VaR is the 5th percentile of the return distribution:

VaR_{95%} = -Q(0.05)

3. EXPECTED VALUE, VARIANCE, AND MOMENTS

3.1 Expected Value (Mean)

For a discrete random variable X with probability mass function p(x):

E[X] = Σ_{x} x * p(x)

For a continuous random variable X with PDF f_X(x):

E[X] = ∫_{-∞}^{∞} x * f_X(x) dx

3.2 Expectation of a Function of X

For any function g(X):

E[g(X)] = ∫_{-∞}^{∞} g(x) * f_X(x) dx

3.3 Variance

Var(X) = E[(X – E[X])²] = E[X²] – (E[X])²

3.4 Standard Deviation

σ_X = sqrt(Var(X))

3.5 Higher Moments

Third Central Moment: E[(X – E[X])³] → Measures skewness.
Fourth Central Moment: E[(X – E[X])⁴] → Measures kurtosis.

Skewness:

Skewness = E[(X – E[X])³] / σ³

Skewness = 0: Symmetric distribution (Normal).
Skewness > 0: Right-skewed (positive skew).
Skewness < 0: Left-skewed (negative skew).

Kurtosis:

Kurtosis = E[(X – E[X])⁴] / σ⁴

Normal Distribution: Kurtosis = 3.
Excess Kurtosis = Kurtosis – 3.
Excess Kurtosis > 0: Fat tails (heavy tails).

4. MOMENT GENERATING FUNCTION (MGF) AND CHARACTERISTIC FUNCTION (CF)

4.1 Moment Generating Function (MGF)

The MGF of a random variable X is:

M_X(t) = E[e^{tX}]

Properties:

M_X(0) = 1.
The nth moment is E[X^n] = M_X^{(n)}(0) (the nth derivative at t = 0).
If X and Y are independent, M_{X+Y}(t) = M_X(t) * M_Y(t).

4.2 Characteristic Function (CF)

The CF of a random variable X is the Fourier transform of the PDF:

φ_X(t) = E[e^{itX}]

Where i = √(-1).

Properties:

φ_X(0) = 1.
|φ_X(t)| ≤ 1.
The CF uniquely determines the distribution (inversion theorem).
If X and Y are independent, φ_{X+Y}(t) = φ_X(t) * φ_Y(t).

4.3 Why the Characteristic Function is Preferred

The CF always exists (even when the MGF does not). This is because e^{itX} is bounded (|e^{itX}| = 1), whereas e^{tX} can grow exponentially.

4.4 Inversion Theorem

The PDF can be recovered from the CF:

f_X(x) = (1/2π) ∫_{-∞}^{∞} e^{-itx} φ_X(t) dt

5. JOINT, MARGINAL, AND CONDITIONAL DISTRIBUTIONS

5.1 Joint Distribution

For two random variables X and Y, the joint CDF is:

F_{X,Y}(x, y) = P(X ≤ x, Y ≤ y)

The joint PDF (for continuous variables) is:

f_{X,Y}(x, y) = ∂²F_{X,Y}(x, y) / ∂x∂y

5.2 Marginal Distribution

The marginal distribution of X is obtained by integrating out Y:

f_X(x) = ∫{-∞}^{∞} f{X,Y}(x, y) dy

5.3 Conditional Distribution

The conditional PDF of X given Y = y is:

f_{X|Y}(x | y) = f_{X,Y}(x, y) / f_Y(y), provided f_Y(y) > 0

5.4 Independence

X and Y are independent if:

f_{X,Y}(x, y) = f_X(x) * f_Y(y)

Equivalently:

F_{X,Y}(x, y) = F_X(x) * F_Y(y)

6. THE LAW OF TOTAL PROBABILITY AND BAYES’ THEOREM

6.1 Law of Total Probability

For a partition {A_i} of the sample space:

P(B) = Σ_i P(B | A_i) * P(A_i)

In continuous form:

f_X(x) = ∫ f_{X|Y}(x | y) * f_Y(y) dy

6.2 Bayes’ Theorem

P(A | B) = P(B | A) * P(A) / P(B)

In continuous form:

f_{X|Y}(x | y) = f_{Y|X}(y | x) * f_X(x) / f_Y(y)

6.3 Financial Application – Parameter Estimation

Bayes’ theorem is the foundation of Bayesian inference in finance:

Posterior ∝ Likelihood * Prior

7. CONVOLUTION – THE DISTRIBUTION OF SUMS

7.1 Definition

If X and Y are independent continuous random variables with PDFs f_X and f_Y, the PDF of Z = X + Y is:

f_Z(z) = ∫_{-∞}^{∞} f_X(x) * f_Y(z – x) dx

7.2 The Convolution of Normals

If X ~ N(μ_X, σ_X²) and Y ~ N(μ_Y, σ_Y²) are independent, then:

Z = X + Y ~ N(μ_X + μ_Y, σ_X² + σ_Y²)

7.3 Financial Application – Portfolio Returns

The return of a portfolio is the weighted sum of individual asset returns:

R_p = Σ_{i=1}^n w_i R_i

If each R_i is normally distributed, then R_p is also normally distributed with:

μ_p = Σ w_i μ_i
σ_p² = Σ_{i} Σ_{j} w_i w_j σ_{ij}

This is the foundation of Markowitz portfolio theory.

8. KEY DISTRIBUTIONS IN FINANCE

8.1 Normal Distribution (Gaussian)

PDF:

f_X(x) = [1 / (σ √(2π))] * exp( -(x – μ)² / (2σ²) )

MGF:

M_X(t) = exp( μt + (1/2) σ² t² )

CF:

φ_X(t) = exp( iμt – (1/2) σ² t² )

Properties:

Symmetric (skewness = 0).
Kurtosis = 3.
Tails decay exponentially fast (thin tails).
Used for asset returns (approximately) and Brownian motion.

Financial Application: Black-Scholes model assumes log-normal stock prices (which means normal log-returns).

8.2 Log-Normal Distribution

If Y = ln(X) ~ N(μ, σ²), then X is log-normally distributed.

PDF:

f_X(x) = [1 / (x σ √(2π))] * exp( -(ln(x) – μ)² / (2σ²) ), for x > 0

Mean:

E[X] = exp( μ + σ²/2 )

Variance:

Var(X) = exp(2μ + σ²) * (exp(σ²) – 1)

Financial Application: Stock prices are assumed to be log-normally distributed (positive prices, multiplicative returns).

8.3 Poisson Distribution

For a discrete random variable X with rate λ:

P(X = k) = (e^{-λ} λ^k) / k!, for k = 0, 1, 2, …

Mean: E[X] = λ
Variance: Var(X) = λ

Financial Application: Number of defaults in a portfolio, number of trades in a time interval, number of operational risk events.

8.4 Exponential Distribution

PDF:

f_X(x) = λ e^{-λx}, for x ≥ 0

CDF:

F_X(x) = 1 – e^{-λx}, for x ≥ 0

Mean: E[X] = 1/λ
Variance: Var(X) = 1/λ²

Financial Application: Time between defaults, inter-arrival times for trades.

8.5 Gamma Distribution

PDF:

f_X(x) = (1 / Γ(α)) * β^α * x^{α-1} * e^{-βx}, for x ≥ 0

Where Γ(α) is the Gamma function.

Mean: E[X] = α/β
Variance: Var(X) = α/β²

Special Cases:

α = 1: Exponential distribution.
α = n/2, β = 1/2: Chi-Square distribution with n degrees of freedom.

Financial Application: Sum of squared normal variables (Chi-Square) is used in hypothesis testing and for testing the significance of regression coefficients.

8.6 Chi-Square Distribution (χ²)

If Z_1, Z_2, …, Z_n are independent standard normal variables, then:

X = Σ_{i=1}^n Z_i² ~ χ²(n)

PDF:

f_X(x) = [1 / (2^{n/2} Γ(n/2))] * x^{(n/2) – 1} * e^{-x/2}

Mean: E[X] = n
Variance: Var(X) = 2n

Financial Application: Testing the significance of portfolio performance (Jensen’s alpha), testing for autocorrelation (Ljung-Box test).

8.7 Stable Distributions

A distribution is stable if the sum of two independent copies of the distribution has the same shape (up to scaling and shifting).

Characteristic Function of a Stable Distribution:

φ_X(t) = exp( iμt – γ^α |t|^α [1 + iβ sign(t) Φ(t, α)] )

Where:

α ∈ (0, 2] is the tail index (stability parameter).
β ∈ [-1, 1] is the skewness parameter.
γ > 0 is the scale parameter.
μ ∈ R is the location parameter.

Special Cases:

α = 2: Normal distribution (β is irrelevant).
α = 1, β = 0: Cauchy distribution.
α = 1, β = 1: Lévy distribution.

Financial Application: Asset returns exhibit fat tails that are better captured by stable distributions than the normal distribution.

9. THE CENTRAL LIMIT THEOREM (CLT) AND ITS IMPLICATIONS

9.1 The Theorem

Let X_1, X_2, …, X_n be independent and identically distributed (i.i.d.) random variables with mean μ and variance σ² < ∞. Then:

(1/√n) Σ_{i=1}^n (X_i – μ) → N(0, σ²)

Equivalently:

(1/n) Σ_{i=1}^n X_i → N(μ, σ²/n)

9.2 Implications for Finance

Portfolio Returns: A well-diversified portfolio of n assets has returns that are approximately normal, regardless of the distribution of individual asset returns.
Risk Aggregation: Aggregated risk measures (e.g., total loss from a portfolio of loans) are approximately normal.
Asset Pricing: The CLT justifies the assumption of normality in many asset pricing models.

9.3 Limitations of the CLT in Finance

Tail Dependence: The CLT assumes independence, but financial returns exhibit tail dependence (crashes and contagion).
Infinite Variance: Some financial returns have infinite variance (e.g., Pareto-distributed losses), violating the CLT assumption.
Convergence Rate: The convergence to normality can be slow for skewed or fat-tailed distributions.

9.4 The Lindeberg-Feller CLT (for Non-Identical Distributions)

If X_1, X_2, …, X_n are independent but not identically distributed, the CLT still holds under the Lindeberg condition:

lim_{n→∞} (1/s_n²) Σ_{i=1}^n E[ (X_i – μ_i)² * 1_{|X_i – μ_i| > ε s_n} ] = 0

Where s_n² = Σ_{i=1}^n σ_i². This condition ensures that no single variable dominates the sum.

10. PRACTICAL IMPLEMENTATION

A. Computing Moments and Distributions:

import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

# Generate normal data
data = np.random.normal(loc=0, scale=1, size=10000)

# Compute moments
mean = np.mean(data)
var = np.var(data)
skew = stats.skew(data)
kurt = stats.kurtosis(data)

print(f"Mean: {mean:.4f}")
print(f"Variance: {var:.4f}")
print(f"Skewness: {skew:.4f}")
print(f"Excess Kurtosis: {kurt:.4f}")

# Compute PDF and CDF
x = np.linspace(-4, 4, 100)
pdf = stats.norm.pdf(x, 0, 1)
cdf = stats.norm.cdf(x, 0, 1)

# Quantile function (VaR)
VaR_95 = stats.norm.ppf(0.05, 0, 1)
print(f"95% VaR: {VaR_95:.4f}")

B. Fitting a Distribution to Data:

# Fit a normal distribution to data
mu_fit, sigma_fit = stats.norm.fit(data)
print(f"Fitted μ: {mu_fit:.4f}, Fitted σ: {sigma_fit:.4f}")

# Fit a stable distribution
alpha_fit, beta_fit, gamma_fit, delta_fit = stats.levy_stable.fit(data)
print(f"Fitted α: {alpha_fit:.4f}, β: {beta_fit:.4f}")

C. Verifying the Central Limit Theorem:

# Generate exponential data (skewed)
data = np.random.exponential(scale=1, size=(1000, 100))

# Compute sample means
sample_means = np.mean(data, axis=1)

# Plot histogram of sample means
plt.hist(sample_means, bins=50, density=True)
x = np.linspace(0, 2, 100)
plt.plot(x, stats.norm.pdf(x, 1, 1/np.sqrt(100)), 'r-')
plt.show()

D. Convolution of Independent Variables:

# Sum of two normals
X = np.random.normal(0, 1, 10000)
Y = np.random.normal(2, 0.5, 10000)
Z = X + Y

# Check the mean and variance
print(f"Mean of Z: {np.mean(Z):.4f}")
print(f"Variance of Z: {np.var(Z):.4f}")
print(f"Expected: μ = 2, σ² = 1.25")

11. SUMMARY FOR THE FINANCE PRACTITIONER

Random Variables: The building blocks of financial models. Asset returns are random variables.
Distributions: Normal for log-returns, Log-Normal for prices, Poisson for counts, Exponential for waiting times, Stable for fat tails.
Moments: Mean = expected return, Variance = risk, Skewness = asymmetry, Kurtosis = tail risk.
Characteristic Function: The Fourier transform of the PDF. Always exists and uniquely determines the distribution.
Joint Distributions: Modelling the dependence between assets (covariance, correlation).
Conditional Expectation: The best prediction of a future value given current information.
Central Limit Theorem: Diversified portfolios have approximately normal returns. Justifies the use of normality in many risk models.

Diploma in Financial Technology LESSON 3.2 Advanced Probability Theory – Conditional Expectation, Martingales, Radon-Nikodym Derivative, and Girsanov Theorem

Diploma in Financial Technology

1. LEARNING OBJECTIVES

2. THE PROBABILITY SPACE (Ω, F, P) AND FILTRATIONS

3. CONDITIONAL EXPECTATION

4. MARTINGALES

5. THE DOOB-MEYER DECOMPOSITION

6. THE RADON-NIKODYM DERIVATIVE

7. THE GIRSANOV THEOREM

8. THE MARTINGALE REPRESENTATION THEOREM

9. THE FUNDAMENTAL THEOREM OF ASSET PRICING

10. PRACTICAL IMPLEMENTATION

MODULE 3, LESSON 3.3

Random Variables, Probability Distributions, and Characteristic Functions

1. LEARNING OBJECTIVES

2. RANDOM VARIABLES – DEFINITION AND TYPES

3. EXPECTED VALUE, VARIANCE, AND MOMENTS

4. MOMENT GENERATING FUNCTION (MGF) AND CHARACTERISTIC FUNCTION (CF)

5. JOINT, MARGINAL, AND CONDITIONAL DISTRIBUTIONS

6. THE LAW OF TOTAL PROBABILITY AND BAYES’ THEOREM

7. CONVOLUTION – THE DISTRIBUTION OF SUMS

8. KEY DISTRIBUTIONS IN FINANCE

9. THE CENTRAL LIMIT THEOREM (CLT) AND ITS IMPLICATIONS

10. PRACTICAL IMPLEMENTATION

11. SUMMARY FOR THE FINANCE PRACTITIONER