Derivatives Pricing

Derivatives Pricing

Identity

Reference System Usage

You must ground your responses in the provided reference files, treating them as the source of truth for this domain:

• For Creation: Always consult `references/patterns.md`. This file dictates how things should be built. Ignore generic approaches if a specific pattern exists here.
• For Diagnosis: Always consult `references/sharp_edges.md`. This file lists the critical failures and "why" they happen. Use it to explain risks to the user.
• For Review: Always consult `references/validations.md`. This contains the strict rules and constraints. Use it to validate user inputs objectively.

Note: If a user's request conflicts with the guidance in these files, politely correct them using the information provided in the references.

Derivatives Pricing

Patterns

Golden Rules

---

Rule

Calibrate to market

Reason

Model parameters from liquid options

---

Rule

Hedge, don't speculate on models

Reason

Greeks show exposure, not truth

---

Rule

Check put-call parity

Reason

Arbitrage check for consistency

---

Rule

Use implied vol, not historical

Reason

Market prices embed forward-looking info

---

Rule

Validate Greeks numerically

Reason

Analytical Greeks can have errors

Method Selection

European Vanilla

Black-Scholes (Analytical)

American Vanilla

Binomial/Trinomial Trees

Barrier Asian

Monte Carlo + Variance Reduction

Exotic Path Dependent

Monte Carlo or PDE Methods

Volatility Smile

Heston or Local Vol Models

Black Scholes Formula

d1 = (ln(S/K) + (r - q + 0.5σ²)T) / (σsqrt(T)) d2 = d1 - σsqrt(T) Call = Se^(-qT)N(d1) - Ke^(-rT)N(d2) Put = Ke^(-rT)N(-d2) - Se^(-qT)N(-d1)

Greeks

Delta

Price sensitivity to spot

Gamma

Delta sensitivity to spot

Theta

Time decay (daily P&L)

Vega

Sensitivity to volatility

Rho

Sensitivity to rates

Vanna

Delta sensitivity to vol

Volga

Vega sensitivity to vol

Binomial Parameters

u = exp(σsqrt(dt)) # Up factor d = 1/u # Down factor p = (exp((r-q)dt) - d) / (u - d) # Risk-neutral probability

Monte Carlo Exotics

Asian

Average price options with control variate

Barrier

Brownian bridge for barrier crossing

Lookback

Maximum/minimum over path

Basket

Correlated asset simulation

Anti-Patterns

---

Pattern

Using historical vol

Problem

Ignores market expectations

Solution

Calibrate to implied vol

---

Pattern

Ignoring dividends

Problem

Misprices options near ex-dates

Solution

Use dividend yield or discrete dividends

---

Pattern

Single model

Problem

Misses skew/smile

Solution

Compare multiple models

---

Pattern

Analytical Greeks only

Problem

Can have errors in exotics

Solution

Verify with bump-and-reprice

---

Pattern

Ignoring early exercise

Problem

Underprices American options

Solution

Use trees or LSM

Derivatives Pricing - Sharp Edges

Historical Vol ≠ Implied Vol

Id

historical-vs-implied-vol

Severity

critical

Summary

Using historical volatility for pricing ignores market expectations

Symptoms

• Pricing doesn't match market quotes
• Hedges consistently lose money
• Model prices diverge from dealers

Why

Implied volatility is extracted from market prices and reflects collective expectations. Historical volatility is backward-looking. Using historical vol for pricing ignores information in option prices.

Gotcha

Using historical volatility

historical_vol = returns.std() * np.sqrt(252) option_price = black_scholes(S, K, T, r, historical_vol)

This ignores market expectations embedded in prices!

Solution

Extract implied vol from market prices

market_price = get_option_price(ticker, strike, expiry) implied_vol = implied_volatility( market_price, S, K, T, r, option_type )

Use implied vol for consistent pricing

Build vol surface for all strikes/expiries

Ignoring Volatility Smile/Skew

Id

ignoring-smile-skew

Severity

high

Summary

Flat vol assumption misprices OTM options

Symptoms

• OTM puts underpriced relative to market
• Hedges perform poorly in tail events
• Risk metrics understate downside

Why

Black-Scholes assumes constant volatility across strikes. Markets show higher implied vol for OTM puts (skew) and higher vol for both OTM puts and calls (smile). Using ATM vol for all strikes misprices wings.

Gotcha

Single vol for all strikes

vol = get_atm_vol(ticker, expiry) for strike in strikes: price = black_scholes(S, strike, T, r, vol) # Wrong for OTM!

Solution

Build volatility surface

vol_surface = build_vol_surface(market_data, S, r)

for strike in strikes: vol = vol_surface.get_vol(strike, expiry) # Strike-specific price = black_scholes(S, strike, T, r, vol)

Or use Heston, SABR, or local vol models

European Model for American Options

Id

american-vs-european

Severity

high

Summary

Black-Scholes underprices options with early exercise value

Symptoms

• Model prices below market for ITM puts
• Dividend-paying stocks show large errors
• Early exercise never considered

Why

American options can be exercised early, which has value. ITM puts on non-dividend stocks and ITM calls before dividends have early exercise premium. Black-Scholes can't capture this.

Gotcha

Black-Scholes for American put

price = black_scholes(S, K, T, r, sigma, 'put')

This ignores early exercise premium!

Deep ITM put might be worth more exercised now

Solution

Use binomial tree for American options

price = binomial_american(S, K, T, r, sigma, n_steps=500, 'put')

Or use Longstaff-Schwartz for path-dependent Americans

price = lsm_american(S, K, T, r, sigma, n_paths=100000)

Compare to European to see early exercise premium

Continuous Dividend Yield for Discrete Dividends

Id

discrete-dividend-miss

Severity

medium

Summary

Misprices options around ex-dividend dates

Symptoms

• Large pricing errors around ex-dates
• Call prices jump unexpectedly
• Dividend arbitrage opportunities appear

Why

Discrete dividends cause stock price to drop by dividend amount on ex-date. Using continuous yield doesn't capture this discrete jump, leading to mispricing for short-dated options near ex-dates.

Gotcha

Continuous dividend yield

option_price = black_scholes(S, K, T, r, sigma, q=0.02)

Misses the discrete $0.50 dividend in 2 weeks!

Solution

For short-dated options, use discrete dividends

dividend_dates = [(0.038, 0.50), (0.288, 0.50)] # (time, amount)

Adjust spot for present value of dividends

S_adj = S - sum(d np.exp(-r t) for t, d in dividend_dates if t < T)

Price with adjusted spot

option_price = black_scholes(S_adj, K, T, r, sigma)

Analytical Greeks Can Have Errors

Id

greeks-bump-vs-analytical

Severity

medium

Summary

Formula errors in Greeks go undetected

Symptoms

• Delta hedges don't neutralize as expected
• P&L attribution doesn't match Greek decomposition
• Vega hedges underperform

Why

Analytical Greek formulas are complex and easy to get wrong. Sign errors, missing discount factors, or wrong derivatives are common. Without numerical validation, errors persist.

Gotcha

Analytical delta (might have bug)

delta = np.exp(-q T) norm.cdf(d1)

No verification this is correct!

Solution

Always verify Greeks numerically

def verify_delta(S, K, T, r, sigma, q): bump = 0.01 # 1% bump price_up = black_scholes(S 1.01, K, T, r, sigma, q) price_down = black_scholes(S 0.99, K, T, r, sigma, q) numerical_delta = (price_up - price_down) / (2 S bump)

analytical_delta = calculate_delta(S, K, T, r, sigma, q)

assert abs(numerical_delta - analytical_delta) < 0.001

Derivatives Pricing - Validations

Using Historical Vol for Pricing

Id

historical-vol-pricing

Severity

warning

Type

regex

Pattern

• returns\.std\(\).*black_scholes
• historical.vol.price

Message

Using historical volatility for option pricing ignores market expectations.

Fix Action

Extract implied volatility from market prices

Applies To

• */option*.py
• */pricing*.py

Same Vol for All Strikes

Id

single-vol-all-strikes

Severity

warning

Type

regex

Pattern

• for.strike.in.:\s.black_scholes.(?!vol_surface|get_vol)

Message

Using same volatility for all strikes ignores smile/skew.

Fix Action

Build volatility surface, use strike-specific vol

Applies To

• */option*.py

Black-Scholes for American Options

Id

black-scholes-american

Severity

warning

Type

regex

Pattern

• american.*black.?scholes
• black.?scholes.*american

Message

Black-Scholes doesn't price American early exercise correctly.

Fix Action

Use binomial trees or LSM for American options

Applies To

• */.py

No Put-Call Parity Verification

Id

no-put-call-parity-check

Severity

info

Type

regex

Pattern

• def.price.option(?!.put_call_parity|.parity_check)

Message

Consider verifying put-call parity as consistency check.

Fix Action

Add: C - P = Sexp(-qT) - Kexp(-rT)

Applies To

• */pricing*.py

Greeks Without Numerical Verification

Id

no-numerical-greek-check

Severity

info

Type

regex

Pattern

• def.delta|def.gamma|def.vega(?!.bump|.*numerical)

Message

Consider numerical verification of analytical Greeks.

Fix Action

Add bump-and-reprice check for Greek formulas

Applies To

• */greeks*.py
• */option*.py

Barrier Options Without Brownian Bridge

Id

barrier-no-bridge

Severity

warning

Type

regex

Pattern

• barrier.monte.?carlo(?!.bridge|.*continuous)

Message

Discrete monitoring misses barrier crossings between steps.

Fix Action

Use Brownian bridge correction for barrier options

Applies To

• */exotic*.py
• */barrier*.py