Derivatives Pricing
Understanding how derivatives are priced gives you a significant edge in trading. When you know the fair value of a derivative, you can identify when the market is overpricing or underpricing an instrument and make more informed decisions.
Futures Pricing
The Cost-of-Carry Model:
Futures prices are determined by the spot price plus the cost of carrying the underlying asset until the contract's expiration.
Formula: Futures Price = Spot Price x (1 + Carry Cost - Carry Yield)Components:
- Spot price: Current market price of the underlying asset
- Interest cost: The cost of financing (borrowing money to buy the asset)
- Storage cost: Physical storage for commodities (gold vaults, grain silos)
- Convenience yield: Benefit of holding the physical asset (for commodities)
- Dividends: Income from holding stocks (reduces futures price)
Example — Gold Futures:
- Gold spot: $2,000/oz
- Interest rate: 5% per year
- Storage cost: 0.5% per year
- Time to expiry: 6 months (0.5 years)
- Futures Price = $2,000 x (1 + (5% + 0.5%) x 0.5) = $2,000 x 1.0275 = $2,055
Contango vs Backwardation:
- Contango: Futures price > Spot price (normal for most commodities)
- Backwardation: Futures price < Spot price (indicates supply shortage or high demand for immediate delivery)
Options Pricing — The Black-Scholes Model
The Five Inputs:
The Black-Scholes model uses five variables to calculate the fair price of a European option:
- S: Current stock/asset price
- K: Strike price of the option
- T: Time to expiration (in years)
- r: Risk-free interest rate
- sigma: Volatility of the underlying (standard deviation of returns)
Key Insights from Black-Scholes:
Time Value: Options with more time until expiration are worth more (all else equal). Time value decays non-linearly — it accelerates as expiration approaches. Volatility: Higher volatility = higher option prices. A stock that moves a lot has a higher chance of reaching the strike price, making both calls and puts more valuable. Interest Rates: Higher rates increase call values and decrease put values slightly.Limitations of Black-Scholes:
- Assumes constant volatility (real markets have variable volatility)
- Assumes log-normal price distribution (real markets have fat tails — more extreme moves than predicted)
- Does not account for dividends (modified versions exist)
- Best for European options (cannot be exercised early)
Implied Volatility (IV)
What Is Implied Volatility?
IV is the market's expectation of future volatility, extracted from current option prices. It is the volatility input that makes the Black-Scholes price equal to the market price.
Why IV Matters:
- High IV: Options are expensive — the market expects big moves
- Low IV: Options are cheap — the market expects calm conditions
- IV Rank: Compares current IV to its historical range (0-100%)
- IV Percentile: What percentage of days had lower IV
IV and Trading Decisions:
| IV Level | Implication | Strategy |
|---|---|---|
| High IV (above 80th percentile) | Options expensive | Consider selling options |
| Low IV (below 20th percentile) | Options cheap | Consider buying options |
| Rising IV | Uncertainty increasing | Long volatility strategies |
| Falling IV | Uncertainty decreasing | Short volatility strategies |
The Volatility Smile/Skew:
In practice, IV is not the same for all strikes:
- OTM puts often have higher IV than ATM options (crash protection demand)
- This creates a "skew" — higher IV for lower strikes
- The shape of the skew provides information about market sentiment
Put-Call Parity
The Relationship:
Put-Call Parity is a fundamental equation linking the prices of calls, puts, the underlying, and a bond:
Call Price - Put Price = Stock Price - Strike Price x e^(-rT)Why It Matters:
- If put-call parity is violated, an arbitrage opportunity exists
- Market makers use it to ensure options are fairly priced relative to each other
- It explains why a call and put at the same strike and expiry move together
Practical Use:
If you know the call price, you can calculate the fair put price (and vice versa). This is especially useful when one option is more liquid than the other.
Pricing Exotic Options
Beyond Vanilla Options:
Exotic options have non-standard features:
- Barrier options: Activated or deactivated when price hits a level
- Asian options: Payout based on average price over time
- Lookback options: Payout based on the highest or lowest price during the contract
- Digital/Binary options: Fixed payout if condition is met
Pricing Methods:
- Analytical formulas: Extensions of Black-Scholes (for some exotics)
- Binomial trees: Step-by-step price modeling
- Monte Carlo simulation: Random path generation (for complex exotics)
- Finite difference methods: Numerical PDE solving
How Pricing Knowledge Helps You
For Retail Traders:
- Identify expensive options: If IV is at the 90th percentile, selling options has an edge
- Understand time decay: Know exactly how much your option loses per day
- Compare strategies: Calculate whether a spread is better than an outright option
- Avoid overpaying: If an option costs more than its Black-Scholes value, investigate why
For Futures Traders:
- Understand contango/backwardation: Know if the futures curve favors long or short
- Calculate fair value: Compare futures price to its theoretical value
- Arbitrage awareness: Recognize when futures are mispriced relative to spot
Key Takeaways
- Futures prices are based on spot price plus carry costs minus carry yields
- Options prices depend on five key variables: price, strike, time, rate, and volatility
- Implied volatility is the market's expectation of future movement
- High IV means expensive options; low IV means cheap options
- Put-Call Parity ensures calls and puts are priced consistently
- Understanding pricing helps you identify mispriced derivatives and choose better strategies