Opening up opportunities in uncertain economies » THEAMITOS

The intersection of financial theory and Python programming provides a solid framework for analyzing finance under conditions of uncertainty. This article explores financial theory with Python, a model economy of increasing complexity: from the simplest static two-state economy to a dynamic economy modeled using stochastic processes. These scenarios provide insight into fundamental financial concepts such as net present value, risk-free pricing, and portfolio optimization. Python enhances this analysis, enabling practical applications of these theories in a repeatable and scalable manner.

The static two-state economy: a fundamental framework

A static two-state economy is the simplest model for analyzing finance in a context of uncertainty. This model assumes only two relevant dates – today and a future date – and two possible outcomes (states) for the future. Despite its simplicity, this framework introduces essential financial concepts:

1. Net Present Value (NPV): The valuation of future cash flows discounted to present value.
2. Expected return: The weighted average of potential returns, based on their probabilities.
3. Volatility: A measure of uncertainty or risk in the return of an asset.
4. Conditional Claims: Financial instruments whose return depends on the state of the economy.
5. Replication of options: Construct portfolios to replicate options performance.
6. Arbitration rates: Exploit price gaps to make profits without risk.
7. Martingale measurement: A probability measure that discounted asset prices are martingales.
8. completeness of the market: A market is complete if every contingent claim can be reproduced through asset trading.
9. Risk-free pricing: Price assets as if investors were indifferent to risk.
10. Medium Variance Portfolios: Balancing expected returns and risk.

Python in a two-state economy

Python simplifies the practical implementation of financial theories in a two-state economy. For example, consider the replication of a conditional claim. If the gain depends on whether the economy transitions to State A or State B, Python can be used to construct a portfolio that reflects this gain.

Python Example: Replicating Options in a Two-State Economy

import numpy as np
# Assumptions
state_prices = np.array([0.4, 0.6]) # Probabilities of State A and B
payoff = np.array([50, 100]) # Payoff in each state
# Replicating portfolio
replicating_portfolio = np.linalg.solve(np.eye(2), payoff * state_prices)
print("Replicating Portfolio:", replicating_portfolio)

Expansion towards a static three-state economy

The introduction of a third possible state broadens the analytical scope of the model. This static three-state economy allows for the examination of more complex financial concepts:

1. Market incompleteness: Not all conditional claims can be perfectly reproduced.
2. Indeterminacy of martingale measurements: Several probability measures can align with market prices.
3. Super-Replication: Build portfolios that guarantee reimbursements greater than or equal to the claim reimbursement in all states.
4. Approximate replication: Achieve a close approximation of possible claim reimbursements.

This framework also facilitates the introduction of Capital Asset Pricing Model (CAPM)which explains asset prices by equilibrium risk-reward tradeoffs.

Python Example: Market Incompleteness Analysis

# Simulating three-state probabilities and payoffs
state_prices = np.array([0.3, 0.4, 0.3])
payoff = np.array([50, 75, 100])
# Attempt to solve for replication (not always feasible in incomplete markets)
try:
replicating_portfolio = np.linalg.solve(np.eye(3), payoff * state_prices)
print("Replicating Portfolio:", replicating_portfolio)
except np.linalg.LinAlgError:
print("Market is incomplete; exact replication is not possible.")

Presentation of agents and decision making

Agents faced with individual decision problems bring a new dimension to financial modeling. Their decisions are generally guided by the expected utility maximization paradigmwhich balances risk and return preferences. Incomplete markets complicate decision-making because not all risks can be hedged.

THE representative agent This concept simplifies this complexity by combining individual preferences into a single utility function. This makes it possible to derive notions of equilibrium and to make the link between optimality, martingale measures and risk-free pricing.

Example: utility maximization in Python

import scipy.optimize as opt
# Define utility function (e.g., logarithmic)
def utility(consumption):
return np.log(consumption)
# Constraints and optimization
def expected_utility(weights, returns, probabilities):
consumption = np.dot(weights, returns)
return -np.dot(probabilities, utility(consumption)) # Negative for maximization
# Example inputs
returns = np.array([[50, 100], [75, 125]])
probabilities = np.array([0.5, 0.5])
weights_guess = [0.5, 0.5]
# Optimize weights
result = opt.minimize(expected_utility, weights_guess, args=(returns, probabilities), bounds=[(0, 1), (0, 1)])
print("Optimal Weights:", result.x)