Black-Scholes Model Inputs and Outputs

The Black-Scholes model transforms five inputs into an option price and a set of risk sensitivities (Greeks). Understanding input data hygiene, the model's assumptions, and how to interpret each Greek for desk risk is essential for accurate pricing and hedging.

Inputs and Data Hygiene

ATM Example:

• S = $100
• K = $100
• σ = 30% (0.30)
• r = 2% (0.02)
• T = 60 days (60/365 = 0.164)
• q = 0%

Call Price: $4.73 Put Price: $4.41

Assumptions and Limitations

Black-Scholes assumes:

1. Lognormal returns: Stock prices follow geometric Brownian motion with constant volatility
2. Continuous trading: No gaps, no transaction costs
3. Constant rate and volatility: Neither r nor σ changes during option life
4. European exercise: No early exercise permitted
5. No dividends or continuous yield: Discrete dividends require adjustment

Limitation impact:

These limitations don't invalidate Black-Scholes but require awareness when applying it to real markets.

Greek-by-Greek Risk Interpretation

Delta (Δ)

Definition: Change in option price per $1 change in underlying.

Formula result for ATM call: Delta ≈ 0.53

Desk interpretation:

• Long 10 calls with delta 0.53 = net long 530 shares equivalent
• Hedge by shorting 530 shares
• Delta changes as spot moves (gamma effect)

Rate sensitivity: A 10 bp rate increase raises delta slightly for calls.

Gamma (Γ)

Definition: Change in delta per $1 change in underlying.

Formula result for ATM call: Gamma ≈ 0.039

Desk interpretation:

• Position becomes 0.039 more delta for each $1 up-move
• High gamma = delta changes quickly = more frequent hedge adjustments
• Gamma peaks for ATM options near expiration

Example: If stock moves +$5, delta increases by 5 × 0.039 = 0.195. New delta ≈ 0.725.

Theta (Θ)

Definition: Change in option price per day of time decay.

Formula result for ATM call: Theta ≈ -$0.08/day

Desk interpretation:

• Position loses ~$0.08 per day from passage of time
• For 10 contracts (1,000 shares): -$80/day
• Theta accelerates near expiration for ATM options

Weekend effect: Options decay over weekends but markets are closed. Some desks adjust for this.

Vega (ν)

Definition: Change in option price per 1% change in implied volatility.

Formula result for ATM call: Vega ≈ $0.19 per 1% vol

Desk interpretation:

• If IV rises from 30% to 31%, option gains ~$0.19
• Long options = long vega = benefit from vol increase
• Vega peaks for ATM options with longer expiration

10 bp rate sensitivity example: A 10 bp rate increase has minimal vega impact.

Rho (ρ)

Definition: Change in option price per 1% change in risk-free rate.

Formula result for ATM call: Rho ≈ $0.07 per 1% rate

Desk interpretation:

• If rates rise 100 bps (1%), call gains ~$0.07
• Puts have negative rho (benefit from rate decreases)
• Rho matters more for longer-dated options

10 bp rate sensitivity: 0.10 × $0.07 = $0.007 price change per contract. For short-dated options, rho is typically the smallest Greek concern.

Rate Sensitivity and Reporting

10 bp Rate Shift Example:

For a 10-lot position (1,000 shares exposure):

• P/L from 10 bp rate rise: +$7
• P/L from 1% vol increase: +$190
• P/L from 1 day time decay: -$80

Rate risk is small relative to delta and vega for short-dated equity options.

Operational Notes

• Data source verification: Use exchange mid-prices, not broker marks. Stale quotes create phantom arbitrage signals.
• Rate curve matching: Match Treasury rate tenor to option expiration. Using 3-month rate for 6-month option introduces error.
• Volatility input: For pricing, use implied volatility from market prices. For theoretical analysis, use historical or forecasted volatility.

Calibration Hygiene and Reporting

Calibration converts market prices to implied volatility:

1. Collect market prices: Bid/ask for range of strikes and expirations
2. Calculate mid prices: (Bid + Ask) / 2
3. Invert Black-Scholes: Solve for σ that produces observed price
4. Build surface: Organize by strike and expiration
5. Validate: Check for arbitrage violations (call spread, butterfly)

Report calibration timestamp, data source, and any exclusions (illiquid strikes). Stale calibration leads to incorrect Greeks and poor hedges.

Greeks Summary Table

Next Steps

For understanding how implied volatility varies across strikes, see Implied Volatility Surface Basics.

To compute Greeks numerically when closed-form isn't available, review Estimating Greeks Numerically.