+02 / THE QUANT LAB

A little theory.A lot to play with.

Seven instruments to explore the mathematics of markets. Change an assumption. Watch the consequences.

Live calculations. Transparent assumptions. Built for exploration.

01Black-Scholes engine

EST. 1973, STILL USEFUL

SPOT ($)STRIKE ($)TIME (YEARS)VOLATILITY (%)RATE (%)OPTION TYPE

Fair value$5.04

delta0.464

gamma0.0255

vega0.281

Theta / day-0.022

rho0.207

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

Breakeven: $110.04 · Premium included in expiry P&L.

European options without dividends. Vega and rho are per 1 percentage point; theta is per calendar day. At expiry or zero volatility, Greeks use limiting conventions.

02Monte Carlo · 1,000 paths

A THOUSAND POSSIBLE TOMORROWS

START VALUE ($)DRIFT (% / YEAR)VOLATILITY (% / YEAR)HORIZON (YEARS)

1 years · 1,000 paths · 252 time steps · Change inputs, then simulate.

Mean terminal value$10,705

P5 / P95$7,939 / $14,238

VaR (95%, loss)$2,061

Expected shortfall (loss)$2,687

Probability of profit60.9%

Best / worst terminal$18,352 / $4,916

Geometric Brownian motion with constant drift and volatility. Each simulation draws a new sample; the white curve is the cross-sectional mean. Results describe this model, not a forecast.

03The efficient frontier

RISK, WITH A RETURN ADDRESS

View model assumptions

Equities: 8.5% expected return, 18% volatility · Bonds: 3.5000000000000004% expected return, 6% volatility · Gold: 5.5% expected return, 15% volatility · Crypto: 16% expected return, 60% volatility. These inputs are educational assumptions, not estimated future returns.

RISK-FREE RATE2%

MAXIMUM SHARPE

24.9%

52.6%

20.3%

2.2%

Return 5.4% · Risk 7.4% · Sharpe 0.46

MINIMUM VARIANCE

4.5%

83.7%

11.7%

0.0%

Return 4.0% · Risk 5.6% · Sharpe 0.35

Illustrative annual returns, volatilities, and correlations. E = equities, B = bonds, G = gold, C = crypto. Long-only portfolios; weights sum to one. Optima are solved across feasible asset subsets.

04Value at risk

LOOKING INTO THE LEFT TAIL

PORTFOLIO VALUE ($)ANNUAL VOLATILITY15%

HORIZON

CONFIDENCE

VaR (95%, 1 trading days)$15,542

Expected shortfall$19,491

VaR / portfolio1.55%

Horizon volatility0.94%

Normal quantile (z)1.645

Parametric Gaussian risk with zero mean, 252 trading days per year, and square-root-of-time scaling. Losses beyond VaR can be larger; expected shortfall measures the average in that tail.

05Discounted cash flow

THE LONG VIEW, DISCOUNTED

YEAR 1 FREE CASH FLOW ($M)GROWTH, YEARS 2–510%WACC10%TERMINAL GROWTH3%

Enterprise value$1.79B

PV of explicit cash flows$454.55M

PV of terminal value$1.34B

Terminal value / enterprise value74.6%

WACC ROWS / TERMINAL GROWTH COLUMNS

Enterprise value sensitivity to WACC and terminal growth
WACC ↓	2.0%	2.5%	3.0%	3.5%	4.0%
9.0%	$1.85B	$1.97B	$2.1B	$2.26B	$2.45B
9.5%	$1.73B	$1.82B	$1.93B	$2.07B	$2.22B
10.0%	$1.61B	$1.7B	$1.79B	$1.9B	$2.03B
10.5%	$1.51B	$1.59B	$1.67B	$1.76B	$1.87B
11.0%	$1.43B	$1.49B	$1.56B	$1.64B	$1.73B

Five annual cash flows followed by a Gordon-growth perpetuity. Year 1 FCF is entered directly; years 2–5 grow at the explicit rate. Terminal growth must be below WACC.

06Option payoff builder

A STRATEGY, LEG BY LEG

01LEG 1 TYPELEG 1 POSITIONLEG 1 STRIKE ($)$6.63

Net premium paid$6.63

Breakeven(s)$106.63

Max profitUnlimited

Max loss$6.63

European premiums from Black-Scholes: spot $100, 6 months, volatility 20%, rate 4%. One unit per leg. P&L includes premiums, excludes fees and financing. Max profit/loss considers all nonnegative expiry prices.

07Binomial tree pricer

ONE STEP CLOSER

STEPS10STRIKE100 $VOLATILITY20%OPTION TYPEEXERCISE

Risk-neutral up probability: 51.6% · 10 steps

Binomial price$9.7289

Black-Scholes$9.925

Difference-0.1962

Cox–Ross–Rubinstein lattice. Spot $100, maturity 1 year, rate 4%, no dividends. The lattice shows the first 5 levels of the selected tree; node labels are option values. Black-Scholes is the European benchmark, so an American put may remain above it.

END OF EXPERIMENTS / CHANGE AN ASSUMPTION, TRY AGAIN.