Probability & Options — Cheat Sheet

Probability — Core Definitions

Three interpretations

Classical: P = favourable ÷ total equally-likely outcomes
Frequentist: P = lim_n→∞ (k/n) — long-run frequency
Bayesian: P = degree of belief, updated by evidence via Bayes' theorem

Bayes' Theorem

P(A|B) = P(B|A) · P(A) / P(B)

Prior belief × likelihood → posterior belief

Expected Value

E[X] = Σ pᵢ · xᵢ (discrete)

E[X] = ∫ x · f(x) dx (continuous)

Variance & Standard Deviation

Var(X) = E[(X − μ)²] = E[X²] − (E[X])²

σ = √Var(X)

Variance of sum of independent rvs: Var(X+Y) = Var(X) + Var(Y)
Over T periods: Var ∝ T, so σ ∝ √T

Normal Distribution N(μ, σ²)

f(x) = (1/σ√2π) · exp(−(x−μ)²/2σ²)

68-95-99.7 Rule

Range	Probability	Trading meaning
μ ± 1σ	68.3%	~172 days/yr within 1σ daily move
μ ± 2σ	95.4%	~12 days outside (≈ 1/month)
μ ± 3σ	99.7%	~1 day outside per year (in theory)

Log-Normal (stock prices)

ln(S_T/S_0) ~ N((μ − σ²/2)T, σ²T)

Use log-returns, not price levels. Ensures S_T > 0 always.

Central Limit Theorem

Sum of many independent rvs → Normal, regardless of individual distributions. Foundation of daily-returns normality assumption.

Fat Tails (Kurtosis)

Real returns: excess kurtosis > 0 (leptokurtic). Negative skew in equities. Crashes more frequent than N predicts — options markets price this via the volatility smirk.

Random Walk / GBM

dS = μS dt + σS dW_t

dW_t = Z√dt where Z ~ N(0,1) — Brownian motion increment

Term	Meaning
μS dt	Drift — deterministic expected return
σS dW_t	Diffusion — random shock (volatility)

Key properties

Independent increments (no memory)
Position uncertainty ∝ √T (not T)
σ√T = 1-std-dev range of log-return over T years
Discrete form: σ_T-day = σ_annual × √(T/252)

Kelly Criterion

f* = (bp − q) / b

f* = optimal fraction; b = net odds; p = win prob; q = 1−p

f* = μ/σ² = Sharpe ratio / σ (continuous)

Maximises E[log(wealth)] — geometric growth
f > f*: ruin probability → 1 over time
f < f*: suboptimal but safer
Practice: use ½ Kelly for robustness

Sharpe Ratio

SR = (E[R] − r_f) / σ_R

Risk-adjusted return. SR=1 is good. SR=2 is excellent. SR>3 is suspicious (overfitting).

Black-Scholes Formula

C = S·N(d₁) − K·e^(−rT)·N(d₂)

P = K·e^(−rT)·N(−d₂) − S·N(−d₁)

d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T)

d₂ = d₁ − σ√T

Inputs

Symbol	Meaning	Source
S	Current stock price	Live feed
K	Strike price	Contract
T	Time to expiry (years)	Calendar
r	Risk-free rate	Central bank
σ	Volatility (annualised)	YOU estimate

Probability interpretation

N(d₂) = P(call expires ITM) under risk-neutral measure
N(d₁) = Call delta ≈ hedge ratio

ATM Approximation

C_ATM ≈ S · σ · √(T/2π) (quick mental check)

Put-Call Parity

C − P = S − K·e^(−rT) (arbitrage identity)

Holds always — no distributional assumption needed.

Key assumptions (and violations)

Constant σ — violated (vol smile exists)
Log-normal returns — violated (fat tails)
No jumps — violated in practice
Continuous hedging — impossible (transaction costs)

The Greeks — Reference

Greek	Formula	Meaning	Range
Δ Delta	N(d₁) call N(d₁)−1 put	Option price change per ₹1 stock move. ≈ P(ITM).	Call [0,1] Put [−1,0]
Γ Gamma	N'(d₁)/(Sσ√T)	Rate of delta change. Risk of re-hedging. Peaks ATM near expiry.	≥ 0 always
Θ Theta	∂C/∂t	Daily time decay. Negative for long options. Accelerates near expiry.	≤ 0 (long)
V Vega	S√T · N'(d₁)	Price change per 1% change in IV. Largest for long-dated ATM.	≥ 0 (long)
ρ Rho	KTe^(−rT)N(d₂)	Sensitivity to interest rates. Smaller effect in short-dated options.	call >0

P&L Attribution (delta-hedged position, 1 day)

ΔP&L ≈ Θ·Δt + ½·Γ·(ΔS)² + V·ΔIV

Gamma-Theta Tradeoff

Θ + ½·Γ·S²·σ² = r·C (B-S PDE)

Long gamma costs theta. Short gamma earns theta. The options desk is always managing this tradeoff.

Volatility

Historical (Realized) Vol

σ_realized = stdev(daily log-returns) × √252

Implied Volatility

Solve B-S for σ given observed market price. No closed form — numerical root-finding (Newton-Raphson).

Market price = BS(S, K, T, r, σ_implied)

Vol Surface

Smile / Skew: IV varies across strikes — equities show left skew (OTM puts expensive)
Term structure: IV varies across expiries — spikes around events
Volatility risk premium: IV > RV on average — sellers collect premium

VIX / India VIX

Model-free implied vol index. Forward-looking 30-day vol. Spikes in crises. Used as fear gauge.

Options Intuition — Quick Reference

Scenario	What happens to option price
Stock rises (call)	↑ call value (delta effect)
Time passes	↓ value (theta decay)
Vol rises	↑ both calls & puts (vega)
Deeper ITM	Δ → 1, Γ → 0, Θ → 0
Deep OTM	Δ → 0, Γ → 0, Θ → 0
ATM, near expiry	Γ → ∞ (pin risk!)
Rates rise	↑ call, ↓ put (rho)

Long vs Short Greeks

	Long option	Short option
Gamma	Long Γ	Short Γ
Theta	Pay Θ daily	Earn Θ daily
Vega	Long V (vol ↑ = good)	Short V (vol ↑ = bad)
P&L driver	RV > IV (big moves)	RV < IV (quiet mkt)