Mathematical Finance Formula Sheet

Time Value of Money

$$PV = \frac{FV}{(1+r)^n}$$

Present Value of Future Cash Flow

$$FV = PV(1+r)^n$$

Future Value of Present Cash Flow

Compound Interest (Discrete)

$$A = P\left(1+\frac{r}{m}\right)^{mn}$$

Amount with m compounding periods per year

P = Principal
r = Annual rate
m = Compounding frequency
n = Years

Continuous Compounding

$$A = Pe^{rt}$$

Amount with continuous compounding

$$e^{r} = \lim_{n \to \infty}\left(1+\frac{r}{n}\right)^n$$

Euler's number relation

Ordinary Annuity

$$PV = PMT \cdot \frac{1-(1+r)^{-n}}{r}$$

Present Value of Annuity

$$FV = PMT \cdot \frac{(1+r)^{n}-1}{r}$$

Future Value of Annuity

Annuity Due

$$PV_{due} = PMT \cdot \frac{1-(1+r)^{-n}}{r} \cdot (1+r)$$

Payments at beginning of period

$$FV_{due} = PMT \cdot \frac{(1+r)^{n}-1}{r} \cdot (1+r)$$

Perpetuities

$$PV = \frac{PMT}{r}$$

Present Value of Perpetuity (constant)

$$PV = \frac{PMT}{r-g}$$

Growing perpetuity (g = growth rate)

Bond Pricing

$$P = \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{FV}{(1+y)^n}$$

Bond Price (C = coupon, y = yield)

Yield to Maturity

$$Price = \sum_{t=1}^{n} \frac{C}{(1+YTM)^t} + \frac{FV}{(1+YTM)^n}$$

Solve for YTM (iterative)

Bond Duration

$$D = \frac{\sum_{t=1}^{n} t \cdot PV(CF_t)}{Price}$$

Macaulay Duration

$$D_{mod} = \frac{D}{1+y}$$

Modified Duration

Dividend Discount Model

$$P_0 = \frac{D_1}{r-g}$$

Gordon Growth Model

$$P_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1+r)^t}$$

Multi-stage DDM

Portfolio Return

$$E(R_p) = \sum_{i=1}^{n} w_i E(R_i)$$

Expected Portfolio Return

Portfolio Variance

$$\sigma_p^2 = \sum_{i=1}^{n} w_i^2 \sigma_i^2 + 2\sum_{i

Variance with correlation

Covariance & Correlation

$$Cov(R_i, R_j) = E[(R_i - E(R_i))(R_j - E(R_j))]$$

Covariance between returns

$$\rho_{ij} = \frac{Cov(R_i, R_j)}{\sigma_i \sigma_j}$$

Capital Asset Pricing Model

$$E(R_i) = R_f + \beta_i[E(R_m) - R_f]$$

Expected return, Rf = risk-free rate

$$\beta_i = \frac{Cov(R_i, R_m)}{\sigma_m^2}$$

Efficient Frontier

$$\sigma_p^2 = w_a^2 \sigma_a^2 + w_b^2 \sigma_b^2 + 2w_a w_b \rho_{ab} \sigma_a \sigma_b$$

Two-asset portfolio frontier

Sharpe Ratio

$$S = \frac{E(R_p) - R_f}{\sigma_p}$$

Risk-adjusted return measure

Value at Risk (VaR)

$$VaR_{95\%} = \mu - 1.645 \sigma$$

Normal distribution approximation

5th-percentile return; loss magnitude $= 1.645\sigma - \mu$

Conditional VaR (CVaR)

$$CVaR = E[Loss | Loss \geq VaR]$$

Expected loss beyond VaR

Forward Price

$$F_0 = S_0 e^{rT}$$

Forward price on non-dividend stock

$$F_0 = (S_0 - PV(D)) e^{rT}$$

With dividends

Futures Pricing

$$f_t = S_t e^{r(T-t)} - PV_t(D)$$

Futures price at time t

Call Option Payoff

$$\text{Payoff} = \max(S_T - K, 0)$$

At expiration (K = strike)

Put Option Payoff

$$\text{Payoff} = \max(K - S_T, 0)$$

At expiration

Put-Call Parity

$$C - P = S_0 - Ke^{-rT}$$

European options relationship

Binomial Model

$$C = e^{-r\Delta t}[pC_u + (1-p)C_d]$$

Risk-neutral probability

$$p = \frac{e^{r\Delta t} - d}{u - d}$$

Black-Scholes Formula

$$C = S_0 N(d_1) - Ke^{-rT}N(d_2)$$

European Call Option

Black-Scholes Parameters

$$d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$$

$$d_2 = d_1 - \sigma\sqrt{T}$$

Delta (Δ)

$$\Delta_C = N(d_1)$$

Call option delta

$$\Delta_P = N(d_1) - 1$$

Put option delta

Gamma (Γ)

$$\Gamma = \frac{n(d_1)}{S_0 \sigma \sqrt{T}}$$

Rate of delta change

Theta (Θ)

$$\Theta = -\frac{S_0 n(d_1) \sigma}{2\sqrt{T}} - rKe^{-rT}N(d_2)$$

Time decay for call option

Vega (ν)

$$\nu = S_0 n(d_1) \sqrt{T}$$

Sensitivity to volatility changes

Rho (ρ)

$$\rho_C = KTe^{-rT}N(d_2)$$

Sensitivity to interest rate changes

Brownian Motion

$$W(t) - W(s) \sim N(0, t-s)$$

Wiener process properties

$$dW(t) \sim N(0, dt)$$

Geometric Brownian Motion

$$\frac{dS}{S} = \mu dt + \sigma dW$$

Stock price dynamics

Itô's Lemma

$$df = \left(\frac{\partial f}{\partial t} + \mu S \frac{\partial f}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 f}{\partial S^2}\right)dt + \sigma S \frac{\partial f}{\partial S} dW$$

Calculus for stochastic processes

Risk-Neutral Valuation

$$V(S,t) = e^{-r(T-t)} E^Q[V(S,T)]$$

Under risk-neutral measure Q

Vasicek Interest Rate Model

$$dr = a(b-r)dt + \sigma dW$$

Mean-reverting rate process

Hull-White Model

$$dr = (\theta(t) - ar)dt + \sigma dW$$

Time-dependent mean reversion

Credit Risk (Merton)

$$PD = N\left(\frac{-[\ln(V_0/D) + (r-\sigma_V^2/2)T]}{\sigma_V\sqrt{T}}\right)$$

Probability of default

Recovery Rate

$$LGD = 1 - \text{Recovery Rate}$$

Loss Given Default

$$ECL = PD \times LGD$$

Expected Credit Loss

Return on Assets (ROA)

$$ROA = \frac{\text{Net Income}}{\text{Total Assets}}$$

Profitability measure

Return on Equity (ROE)

$$ROE = \frac{\text{Net Income}}{\text{Shareholders' Equity}}$$

Return to equity holders

Debt-to-Equity Ratio

$$D/E = \frac{\text{Total Debt}}{\text{Total Equity}}$$

Leverage measure

Price-to-Earnings Ratio

$$P/E = \frac{\text{Market Price per Share}}{\text{Earnings per Share}}$$

Valuation multiple

Current Ratio

$$\text{Current Ratio} = \frac{\text{Current Assets}}{\text{Current Liabilities}}$$

Liquidity measure

Quick Ratio

$$\text{Quick Ratio} = \frac{\text{Current Assets} - \text{Inventory}}{\text{Current Liabilities}}$$

Acid-test liquidity