Quantitative Finance Formulas and Risk Management Models

Portfolio Theory and Expected Returns

1. Expected return of the portfolio: μ_P = w₁μ₁ + w₂μ₂.

Standard deviation of the portfolio return: σ_p = √(w₁²σ₁² + w₂²σ₂² + 2ρw₁w₂σ₁σ₂).

• Suppose two investments R₁ and R₂ with expected returns μ₁ and μ₂.
• w₁ is the proportion of money in the first investment.
• σ₁ and σ₂ are the respective standard deviations.
• ρ is the coefficient of correlation between the two investments.

Market Portfolio and Systematic Risk

2. Relationship between any investment and the market portfolio: R = a + βR_M + ε.

• R = return from investment.
• R_M = return from market portfolio.
• a and β are constants.
• ε is a random variable representing the regression error.
• βR_M: Systematic risk.
• ε: Non-systematic risk.

Capital Asset Pricing Model (CAPM)

3. CAPM Formula: E(R) = R_F + β[E(R_M − R_F)].

Spot-Forward Parity and Pricing

3. Spot-forward parity: The relationship between the current forward price F₀ and the spot price S₀ is defined as: F₀ = S₀e^rT.

• r = risk-free interest rate.
• T = time to maturity.
• This follows the no-arbitrage principle.

Risk-Neutral Pricing Principles

4. Risk-Neutral Pricing: Let p be the current value (or price) of a financial instrument, whose payoff at a future time T is P_T. Then:

p = E(e^−rTP_T)

Example of a call option: p = E[e^−rT(S_T − K)⁺], where S_T is the price of the underlying asset at time T, and K is the strike price.

Black-Scholes Model and Option Pricing

5. Black-Scholes Model:

• S_t: Price of the underlying asset at time t.
• σ: Volatility of the underlying asset.
• r: Risk-free interest rate.

S_t follows a Geometric Brownian Motion. Then: S_T = S₀ exp((r − σ²/2)T + σ√TN), where N is the standard normal distribution.

Option Pricing Formulas

Let C denote the price of a call option: C = SΦ(d₁) − Ke^−rTΦ(d₂).

Where:

• d₁ = [log(S/K) + (r + σ²/2)T] / σ√T
• d₂ = d₁ − σ√T
• S = current price of the underlying asset.
• Φ = standard normal cumulative distribution function.
• K = exercise price (&行使價).
• r = risk-free interest rate.
• t = maturity.

Let P be the price of a put option: P = Ke^−rTΦ(−d₂) − SΦ(−d₁).

Value at Risk (VaR) Analysis

6. VaR (Value at Risk): We are X percent certain that we will not lose more than V dollars in time T.

Let V be the Value-at-Risk of a portfolio at a time horizon T and confidence level 100α%. Let L be the random variable representing the portfolio loss over the time horizon. Then: Pr{L < V} = α. *(Note: Z_0.95 = 1.645).

Risk Diversification

In risk diversification, one may expect that: ρ(X + Y) ≤ ρ(X) + ρ(Y). However, VaR(X + Y) ≤ VaR(X) + VaR(Y) does not always hold.

Conditional Value at Risk (CVaR)

7. Conditional VaR: CVaR at a confidence level α is CVaR = E[L | L ≥ VaR], where L is the random P&L and VaR is the Value at Risk at confidence level α.

Example: X has a probability of 90% to be uniformly distributed between 0 and 1, and a probability of 10% to be uniformly distributed between 2 and 3. What is CVaR(X) at a 90% confidence level?

• Expected loss (tail): (2 + 3) / 2 = 2.5.
• Expected loss (body): (0 + 0.9) / 2 = 0.45.
• CVaR = 0.9 * 0.45 + 0.1 * 2.5.