Finance

Black-Scholes Options Pricer

European options on non-dividend-paying stock · continuous compounding · lognormal price dynamics

Call Price
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Put Price
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d1 / d2
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Put-Call Parity check
Greeks
GreekCallPut
Δ Delta——
Γ Gamma—
ν Vega (per 1% σ)—
Θ Theta (per day)——
ρ Rho (per 1% r)——

Binomial Option Tree — Cox-Ross-Rubinstein

Discrete-time lattice model · configurable steps · European vs American early-exercise comparison

Option Price
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Early Exercise Premium
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American − European
Euro price
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Amer price
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u / d
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p* / Δt
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Price tree (top = stock price, bottom = option value · max 5 steps shown)

Implied Volatility Solver

Inverse Black-Scholes via Newton-Raphson · solves σ such that BS(S,K,T,r,σ) = market price

Implied Volatility
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Convergence
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DCF Valuation

Two-stage discounted cash flow · terminal value via Gordon Growth Model · values in $ millions

Intrinsic Value
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Period FCF ($M) PV ($M)
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WACC — Weighted Average Cost of Capital

WACC = (E/V)·ke + (D/V)·kd·(1−τ)  ·  discount rate for unlevered free cash flows

WACC
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Equity weight
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Debt weight
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After-tax kd
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ke − WACC spread
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equity-holders earn this above blended rate

Bond Pricer & Duration

Annual coupon bond · Macaulay & modified duration · convexity · DV01 · price-yield curve

Bond Price
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DV01
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per 1bp rate move
Macaulay D
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Modified D
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Convexity
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Price vs YTM (±300bp around current yield)

Efficient Frontier — 2-Asset Portfolio

Mean-variance optimization · minimum-variance portfolio · tangency portfolio · Capital Market Line

Min-Variance Portfolio
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E(R) / σ  ·  Sharpe —
Tangency Portfolio
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E(R) / σ  ·  Sharpe —
Efficient frontier  ·  T = tangency  ·  MV = minimum variance  ·  dashed = CML

CAPM — Capital Asset Pricing Model

E(Ri) = Rf + βi·[E(Rm)−Rf]  ·  Security Market Line

Expected Return
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Market Risk Premium
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Asset Risk Premium
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Security Market Line
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Formula Reference

27 cards across derivatives, fixed income, portfolio & risk, corporate finance, and quantitative methods

Options & Derivatives
Black-Scholes Call
C = \(S\cdot N(d_1) - Ke^{-rT}\cdot N(d_2)\) d_1 = \(\dfrac{\ln(S/K)+(r+\tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}\) d_2 = \(d_1 - \sigma\sqrt{T}\)
N(x) is the standard normal CDF. Risk-neutral probability the call expires in-the-money is N(d₂).
Black-Scholes Put & PCP
P = \(Ke^{-rT}N(-d_2) - SN(-d_1)\) C-P = \(S - Ke^{-rT}\)
Put-Call Parity holds by no-arbitrage for European options. A violation implies a riskless profit — buy cheap side, sell rich side, hold to expiry.
Option Greeks
ΔC = \(N(d_1)\)   ΔP = \(N(d_1)-1\) Γ = \(\dfrac{N'(d_1)}{S\sigma\sqrt{T}}\) ν = \(SN'(d_1)\sqrt{T}\) ΘC = \(-\dfrac{SN'(d_1)\sigma}{2\sqrt{T}} - rKe^{-rT}N(d_2)\)
Γ and ν are identical for calls and puts. Θ is typically negative (option loses time value daily). Divide raw Θ by 365 for per-day decay.
CRR Binomial Tree
u = \(e^{\sigma\sqrt{\Delta t}}\), \(\;d = 1/u\) p* = \(\dfrac{e^{r\Delta t}-d}{u-d}\) V_i^j = \(e^{-r\Delta t}[p^* V_{i+1}^{j} + (1-p^*)V_{i+1}^{j+1}]\)
For American options, replace continuation with max(continuation, intrinsic value) at each node. Early exercise premium = American − European price.
Futures & Cost of Carry
F = \(S_0 e^{rT}\)   (no dividends) F = \(S_0 e^{(r-q)T}\)   (continuous yield q) F = \((S_0 - I)e^{rT}\)   (discrete income I)
Basis = S − F. Convenience yield y reduces futures price: F = S·e^((r−y)T). Backwardation occurs when F < S (y > r).
Implied Volatility
Solve \(\sigma^*\) s.t. \(\text{BS}(S,K,T,r,\sigma^*) = P_{\text{mkt}}\) Newton-Raphson: \(\sigma_{n+1} = \sigma_n - \dfrac{\text{BS}(\sigma_n)-P}{\nu(\sigma_n)}\)
Converges in ~5 iterations from σ = 25% initial guess. IV smile: OTM options often have higher IV than ATM — BS assumes flat vol, reality doesn't.
Second-Order Greeks
Vanna = \(\dfrac{\partial\Delta}{\partial\sigma} = -N'(d_1)\dfrac{d_2}{\sigma}\) Volga = \(\dfrac{\partial^2 V}{\partial\sigma^2} = \nu\dfrac{d_1 d_2}{\sigma}\) Charm = \(\dfrac{\partial\Delta}{\partial t}\)
Vanna matters for vol-delta hedging. Volga (vomma) is the convexity of option price in vol — positive for long options. Charm is Θ of delta.
Fixed Income
Bond Price & YTM
P = \(\displaystyle\sum_{t=1}^{n}\frac{C}{(1+y)^t} + \frac{F}{(1+y)^n}\)
YTM is the single discount rate y that makes the bond's PV equal to its market price. Par bond: coupon rate = YTM. Premium: coupon > YTM. Discount: coupon < YTM.
Duration & DV01
D = \(\dfrac{\sum t\cdot PV(CF_t)}{P}\)   [Macaulay] D* = \(\dfrac{D}{1+y}\)   [Modified] DV01 = \(\dfrac{D^*\cdot P}{10{,}000}\)
DV01 is the dollar change in price per 1bp move in yield. A 10-year par bond at 5% has D ≈ 7.7 yrs. Longer maturity → higher duration → more rate sensitivity.
Convexity & Price Approximation
C = \(\dfrac{1}{P}\sum_{t=1}^{n}\dfrac{t(t+1)\cdot PV(CF_t)}{(1+y)^2}\) \(\dfrac{\Delta P}{P}\approx -D^*\Delta y+\tfrac{1}{2}C(\Delta y)^2\)
Convexity corrects duration's linear approximation. Long bonds have positive convexity — beneficial in both rate directions. Callable bonds can have negative convexity.
Forward Rates
f(t_1,t_2) = \(\dfrac{(1+s_2)^{t_2}}{(1+s_1)^{t_1}}-1\) Continuous: \(f = \dfrac{s_2 t_2 - s_1 t_1}{t_2-t_1}\)
s₁, s₂ are spot (zero-coupon) rates for maturities t₁ < t₂. Forward rate represents the break-even rate for lending between t₁ and t₂ implied by the current spot curve.
Nelson-Siegel Yield Curve
\(y(\tau)=\beta_0+\beta_1\dfrac{1-e^{-\tau/\lambda}}{\tau/\lambda}+\beta_2\!\left(\dfrac{1-e^{-\tau/\lambda}}{\tau/\lambda}-e^{-\tau/\lambda}\right)\)
β₀ = long-run level, β₁ = slope (short-end tilt), β₂ = curvature (hump). λ controls location of hump. Svensson adds a second hump term (β₃, λ₂).
Portfolio & Risk
CAPM & SML
E(R_i) = \(R_f + \beta_i[E(R_m)-R_f]\) βi = \(\dfrac{\text{Cov}(R_i,R_m)}{\text{Var}(R_m)}\)
Assets above the SML have α > 0 (underpriced). β > 1 amplifies market moves; β < 0 provides hedging. Systematic risk only is priced — idiosyncratic risk diversifies away.
2-Asset Portfolio Variance
E(R_p) = \(w_1 R_1 + w_2 R_2\) σ²p = \(w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1w_2\sigma_1\sigma_2\rho\) w_1^{MV} = \(\dfrac{\sigma_2^2-\sigma_1\sigma_2\rho}{\sigma_1^2+\sigma_2^2-2\sigma_1\sigma_2\rho}\)
When ρ = −1, perfect hedge is possible. Diversification benefit: σ_p < w₁σ₁ + w₂σ₂ for any ρ < 1. Min-variance weight w₁^MV can be negative (short position).
Mean-Variance Optimization
\(\min_{\mathbf{w}}\;\mathbf{w}^\top\boldsymbol{\Sigma}\mathbf{w}\) s.t. \(\mathbf{w}^\top\boldsymbol{\mu}=\mu_p,\;\mathbf{1}^\top\mathbf{w}=1\) Tangency: \(\mathbf{w}^*\propto\boldsymbol{\Sigma}^{-1}(\boldsymbol{\mu}-r_f\mathbf{1})\)
Tangency portfolio maximises the Sharpe ratio. All rational mean-variance investors hold a mix of the tangency portfolio and the risk-free asset (two-fund separation).
Fama-French 3-Factor
\(R_i - R_f = \alpha + \beta_m(R_m-R_f) + \beta_s\cdot\text{SMB} + \beta_h\cdot\text{HML} + \varepsilon\)
SMB = Small Minus Big (size factor). HML = High Minus Low book-to-market (value factor). FF5 adds RMW (profitability) and CMA (investment). Carhart adds UMD (momentum).
Sharpe · Sortino · Treynor
Sharpe = \((R_p-R_f)/\sigma_p\) Sortino = \((R_p-R_f)/\sigma_\downarrow\) Treynor = \((R_p-R_f)/\beta_p\)
Sortino uses downside deviation σ↓ (semi-variance below target). Treynor rewards systematic risk only — useful when portfolio is one component of a larger diversified fund.
Information Ratio · Calmar · Jensen α
IR = \((R_p-R_b)/\text{TE}\) Calmar = \(\bar{R}/|\text{MaxDD}|\) α = \(R_p-[R_f+\beta(R_m-R_f)]\)
IR measures active return per unit of tracking error — the active manager's Sharpe. Calmar uses maximum drawdown instead of volatility. TE = std(R_p − R_b).
Value at Risk
Parametric: \(\text{VaR}_\alpha = -(\mu-z_\alpha\sigma)P\) Historical: \(\text{percentile}_{1-\alpha}(\{R_t\})\cdot P\) ES/CVaR: \(E[\text{Loss}\mid\text{Loss}>\text{VaR}_\alpha]\)
z₉₅ = 1.645, z₉₉ = 2.326. ES is the mean loss beyond VaR — it is coherent (subadditive) where VaR is not. Basel III uses ES at 97.5%.
Kelly Criterion
Discrete: \(f^*=\dfrac{p(b+1)-1}{b}\) Continuous: \(f^*=\dfrac{\mu-r}{\sigma^2}\) Multi-asset: \(\mathbf{f}^*=\boldsymbol{\Sigma}^{-1}(\boldsymbol{\mu}-r)\)
Kelly maximises long-run log-wealth growth. In practice, half-Kelly (f*/2) is used to reduce drawdown volatility. Requires accurate probability estimates — overconfidence causes ruin.
Corporate & Valuation
WACC
WACC = \(\dfrac{E}{E+D}k_e + \dfrac{D}{E+D}k_d(1-\tau)\)
Use market values (not book) for E and D. k_e is typically estimated via CAPM. WACC is the appropriate discount rate for FCFF; FCFE uses cost of equity k_e directly.
Modigliani-Miller
MM I (no tax): \(V_L = V_U\) MM II (no tax): \(k_e = k_0 + (k_0-k_d)\dfrac{D}{E}\) MM I (tax): \(V_L = V_U + \tau D\)
Without taxes, capital structure is irrelevant. With taxes, debt provides a tax shield worth τD — incentivising leverage until offset by distress costs (trade-off theory).
Gordon Growth & DDM
P₀ = \(\dfrac{D_1}{k-g} = \dfrac{\text{FCF}_1}{r-g}\) Implied g = \(k - D_1/P_0\)
Requires r > g permanently — breaks for high-growth firms. Used as DCF terminal value: TV = FCF_{n+1}/(WACC − g_T). Rearrange for implied growth rate given market price.
FCFF & FCFE
FCFF = \(\text{EBIT}(1-\tau)+\text{D\&A}-\Delta\text{NWC}-\text{CapEx}\) FCFE = \(\text{FCFF}-\text{Int}(1-\tau)+\Delta\text{Net Debt}\)
FCFF is pre-financing; discount at WACC. FCFE is post-financing (available to equity only); discount at k_e. Consistent with MM — both approaches yield same equity value when applied correctly.
Quantitative Methods
Geometric Brownian Motion
SDE: \(dS = \mu S\,dt + \sigma S\,dW_t\) Solution: \(S_T = S_0\exp\!\bigl((\mu-\tfrac{1}{2}\sigma^2)T+\sigma W_T\bigr)\)
The (μ − ½σ²) drift term ensures E[S_T] = S_0·e^(μT). Log returns are normally distributed. The −½σ² correction is Itô's lemma in action: Jensen's inequality applied to log.
Ornstein-Uhlenbeck Process
SDE: \(dX_t = \theta(\mu-X_t)\,dt + \sigma\,dW_t\) Half-life: \(t_{1/2} = \ln 2\,/\,\theta\)
θ is mean-reversion speed; μ is long-run mean. Basis for Vasicek interest-rate model. Used in pairs trading: spread between cointegrated assets follows approximately OU dynamics.
GARCH(1,1)
r_t = \(\mu + \varepsilon_t\), \(\;\varepsilon_t = \sigma_t z_t\) σ²t = \(\omega + \alpha\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2\) Long-run var: \(\bar\sigma^2 = \omega/(1-\alpha-\beta)\)
Stationarity requires α + β < 1. Persistence α + β ≈ 0.97–0.99 is typical for daily equity returns (volatility clustering). EGARCH/GJR-GARCH extend this to capture leverage effect.