Topic 8: Pricing and Valuation of Options
Learning Objectives Coverage
LO1: Explain the exercise value, moneyness, and time value of an option
Core Concept
Options have two components of value: exercise (intrinsic) value and time value. Moneyness describes the relationship between the underlying price and strike price, determining whether exercising the option would be profitable. Time value represents the additional premium above exercise value attributable to the possibility of favorable price movements before expiration. These concepts are foundational to both put-call parity and the binomial model. exam-focus
Key Formulas formula pricing
Call Exercise Value:
Max(0, St - X(1 + r)^(-(T-t)))
Put Exercise Value:
Max(0, X(1 + r)^(-(T-t)) - St)
Time Value:
Option Premium - Exercise Value
Total Option Value:
Option Value = Exercise Value + Time Value
Moneyness Classifications
| Status | Call Option | Put Option |
|---|---|---|
| In-the-Money (ITM) | St > X | St < X |
| At-the-Money (ATM) | St = X | St = X |
| Out-of-the-Money (OTM) | St < X | St > X |
Practical Example
Apple Call Option Analysis:
- Stock price: $150
- Strike price: $140
- Time to expiry: 3 months
- Risk-free rate: 3%
- Option premium: $15
Calculations:
Exercise Value = Max(0, $150 - $140/(1.03)^0.25)
= Max(0, $150 - $139.30)
= $10.70
Time Value = $15 - $10.70 = $4.30
Moneyness: ITM (stock > strike)
DeFi Application defi-application
The Opyn Options Protocol uses oTokens to represent options on-chain, with exercise automated via smart contracts. For example, an ETH 3500 has an exercise value of $500, while the premium includes time value reflecting crypto’s characteristically high volatility. Collateral is locked in the smart contract, replacing the traditional clearinghouse margin mechanism.
LO2: Contrast the use of arbitrage and replication concepts in pricing forward commitments and contingent claims
Core Concept
Forward commitments use static replication with fixed positions held throughout the contract life. Options require dynamic replication because their payoff is asymmetric and non-linear, necessitating continuous adjustment of the hedge ratio as the likelihood of exercise changes. This distinction between static and dynamic replication is a direct consequence of the no-arbitrage framework applied to different payoff structures. exam-focus
Key Differences
| Aspect | Forward Commitments | Options |
|---|---|---|
| Payoff | Linear, symmetric | Non-linear, asymmetric |
| Replication | Static (buy-and-hold) | Dynamic (adjust continuously) |
| Initial Cost | Zero (ignoring costs) | Premium paid upfront |
| Obligation | Must execute | Right but not obligation |
| Hedging | Perfect with static hedge | Requires dynamic hedging |
Replication Strategies
Call Option Replication:
1. Buy Δ shares of underlying
2. Borrow (1-Δ) × X/(1+r)^T
3. Adjust Δ as price changes
Where Δ = hedge ratio (0 to 1)
Forward Replication:
1. Buy 1 share of underlying
2. Borrow full PV of forward price
3. Hold until maturity (no adjustment)
DeFi Application defi-application
Ribbon Finance structured products use covered call strategies, dynamically hedging with perpetuals. For example, an ETH covered call vault deposits ETH, sells OTM calls weekly, and adjusts the hedge based on delta — implementing the dynamic replication concept on-chain.
LO3: Identify the factors that determine the value of an option and describe how each factor affects the value of an option
Core Concept
Six primary factors determine option values, with different directional impacts on calls versus puts. Understanding these relationships is crucial for option pricing and risk management.
Factor Impact Summary
Factor | Call Value | Put Value
------------------------|------------|----------
↑ Underlying Price | ↑ | ↓
↑ Strike Price | ↓ | ↑
↑ Time to Expiration | ↑ | ↑/↓
↑ Risk-free Rate | ↑ | ↓
↑ Volatility | ↑ | ↑
↑ Dividends/Income | ↓ | ↑
Detailed Factor Analysis
1. Underlying Price (S):
- Calls: Direct relationship - higher S increases value
- Puts: Inverse relationship - higher S decreases value
- Most significant factor for ITM options
2. Strike Price (X):
- Calls: Lower strikes are more valuable
- Puts: Higher strikes are more valuable
- Determines moneyness threshold
3. Time to Expiration (T):
- Calls: Always positive (more time = more value)
- Puts: Usually positive, can be negative for deep ITM with high rates
- Time decay accelerates near expiration
4. Risk-free Rate (r):
- Affects PV of strike price
- Higher rates favor calls over puts
- Impact through cost of carry
5. Volatility (σ):
- Increases both call and put values
- Higher volatility = wider outcome range
- Most important for ATM options
6. Dividends/Income:
- Reduces call values (stock price drops)
- Increases put values (benefits from drop)
- European options can’t capture dividends
DeFi Application defi-application
Crypto’s high volatility dramatically increases option premiums compared to traditional equities. For BTC 30-day ATM options, traditional equity volatility of 20% might produce a 3% premium, while BTC volatility of 80% yields a 12% premium. DeFi protocols like Lyra use dynamic volatility surfaces to price these elevated premiums, drawing on the same factor-sensitivity framework used in traditional option pricing models.
Core Concepts Summary (80/20 Principle)
The 20% You Must Know:
- Option Value = Exercise Value + Time Value
- Moneyness: ITM, ATM, OTM determines exercise likelihood
- Options have asymmetric payoffs unlike forwards
- Volatility increases both call and put values
- Time decay erodes option value approaching expiration
The 80% That Matters Most:
- Exercise value is immediate profit from exercising
- Time value exists due to potential for favorable moves
- Dynamic hedging required for option replication
- Six factors affect option values differently
- Options provide leverage and risk management
Comprehensive Formula Sheet
Value Components
Call Option Value:
ct = Exercise Value + Time Value
ct = Max(0, St - X(1+r)^(-(T-t))) + Time Value
Put Option Value:
pt = Exercise Value + Time Value
pt = Max(0, X(1+r)^(-(T-t)) - St) + Time Value
Payoff at Maturity
Call Payoff:
cT = Max(0, ST - X)
Put Payoff:
pT = Max(0, X - ST)
Profit at Maturity
Call Profit:
Π = Max(0, ST - X) - c0
Put Profit:
Π = Max(0, X - ST) - p0
Option Bounds
Call Bounds:
Lower: Max(0, St - X(1+r)^(-(T-t)))
Upper: St
Put Bounds:
Lower: Max(0, X(1+r)^(-(T-t)) - St)
Upper: X
Break-even Points
Call Break-even:
ST = X + c0
Put Break-even:
ST = X - p0
HP 12C Calculator Sequences
Calculate Exercise Value (Call)
Example: S = $50, X = $45, r = 4%, T = 0.5 years
[f] [CLX]
45 [ENTER]
1.04 [ENTER]
0.5 [y^x]
[÷] // PV of strike = $44.12
50 [ENTER]
[x<>y]
[-] // $50 - $44.12 = $5.88
Calculate Time Value
Option Premium = $7, Exercise Value = $5.88
[f] [CLX]
7 [ENTER]
5.88 [-] // Time Value = $1.12
Calculate PV of Strike for Bounds
X = $100, r = 3%, T = 0.25 years
[f] [CLX]
100 [ENTER]
1.03 [ENTER]
0.25 [y^x]
[÷] // PV = $99.26
Practice Problems
Basic Level
Problem 1: Calculate exercise and time value:
- Stock price: $80
- Strike: $75
- Call premium: $8
- Risk-free rate: 2%
- Time: 6 months
Solution:
Exercise Value = Max(0, $80 - $75/(1.02)^0.5)
= Max(0, $80 - $74.26)
= $5.74
Time Value = $8 - $5.74 = $2.26
Problem 2: Determine moneyness:
- Stock: $100
- Call strike: $110
- Put strike: $95
Solution:
- Call: OTM (stock < strike)
- Put: ITM (stock > strike)
Intermediate Level
Problem 3: Option bounds calculation:
- Stock: €120
- Strike: €115
- Rate: 3%
- Time: 3 months
Solution:
Call Lower Bound:
= Max(0, €120 - €115/(1.03)^0.25)
= Max(0, €120 - €114.14)
= €5.86
Call Upper Bound = €120
Put Lower Bound:
= Max(0, €115/(1.03)^0.25 - €120)
= Max(0, €114.14 - €120)
= €0 (cannot be negative)
Put Upper Bound = €115
Advanced Level
Problem 4: Factor sensitivity analysis: Initial: S = 50, σ = 30%, r = 5%, T = 1 year Call value = $6.50
Analyze impact of: a) Stock rises to $55 b) Volatility increases to 40% c) Time decreases to 6 months
Solution:
a) Stock to $55:
- Call becomes ITM
- Exercise value = $55 - $50/1.05 = $7.38
- Call value increases significantly
b) Volatility to 40%:
- Both calls and puts increase
- ATM call might rise to ~$8.50
c) Time to 6 months:
- Time decay reduces value
- Call might fall to ~$4.50
DeFi Applications & Real-World Examples
1. On-Chain Options Protocols
Hegic:
- Peer-to-pool options model
- Liquidity providers act as option writers
- Example: ETH call option
- Strike: $2000
- Premium: 0.05 ETH
- Auto-exercise at expiry if ITM
Dopex:
- Single Staking Option Vaults (SSOVs)
- Weekly/monthly expiries
- Example: rETH covered calls
- Deposit rETH
- Earn premiums + staking rewards
2. Structured Products
Ribbon Finance:
Theta Vault Strategy:
1. Deposit ETH
2. Sell weekly OTM calls (Δ ≈ 0.1)
3. Premium: ~2% weekly
4. Risk: Capped upside if ETH rallies
Jones DAO:
- Leveraged option strategies
- Combines options with perpetuals
- Example: Bull strategy using call spreads
3. Option AMMs
Lyra Protocol:
- Automated market maker for options
- Dynamic volatility surface
- Uses Black-Scholes pricing
- Example spread:
- Buy price: IV + 5%
- Sell price: IV - 5%
4. Real-World Corporate Examples
Tesla Convertible Bonds:
Embedded Call Option:
- Bond: $1000 face value
- Conversion: 10 shares
- Effective strike: $100
- If TSLA > $100, convert to equity
Apple Stock Buyback:
Accelerated Share Repurchase:
- Sell puts to buy shares cheaper
- Strike: $140 when stock at $150
- Collect premium if not exercised
- Buy shares at discount if exercised
Common Pitfalls & Exam Tips
Common Mistakes to Avoid
-
Confusing American vs European
- European: Only at expiration
- American: Anytime before expiry
- American ≥ European value
-
Time value always positive
- Before expiration: Yes
- At expiration: Zero
- Never negative
-
Volatility affects both equally
- Calls and puts both increase
- ATM most sensitive
- Deep ITM/OTM less sensitive
-
Forgetting interest rate effects
- Higher rates help calls
- Higher rates hurt puts
- Through PV of strike
Exam Strategy Tips
-
Quick moneyness check:
- Call ITM: S > X
- Put ITM: S < X
- Both ATM: S = X
-
Factor impact memory:
- Volatility: Only factor that helps both
- Time: Usually helps both (except deep ITM puts)
- Underlying: Opposite effects
-
Calculation shortcuts:
- Deep ITM call ≈ S - PV(X)
- Deep ITM put ≈ PV(X) - S
- ATM option ≈ 40% × volatility × √time
Key Takeaways
Must Remember:
- Option Value = Exercise + Time Value
- Moneyness determines exercise likelihood
- Six factors affect option values
- Volatility increases both calls and puts
- Asymmetric payoffs distinguish from forwards
Critical Insights:
- Time decay accelerates near expiration
- Dynamic hedging required for replication
- Options provide leverage and flexibility
- DeFi options often European style
- High crypto volatility increases premiums
Cross-References & Additional Resources
Related Topics:
- Topic 9: Put-Call Parity relationships
- Topic 10: Binomial option pricing
- Fixed Income: Bond options and embedded options
- Portfolio Management: Protective puts and covered calls
Key Readings:
- Hull, J.: Options, Futures, and Other Derivatives Ch. 10-11
- Black & Scholes: Original 1973 paper
- CBOE: Options Institute educational materials
Practice Resources:
- CBOE Options Calculator
- OptionStrat: Visual strategy builder
- TastyTrade: Options education platform
DeFi Protocols to Study:
- Lyra: Full-featured options AMM
- Hegic: Peer-to-pool options
- Ribbon: Structured products
- Dopex: Options vaults
- Opyn: Squeeth and options infrastructure
Review Checklist
Conceptual Understanding
- Can you explain exercise vs time value?
- Do you understand moneyness classifications?
- Can you identify how each factor affects values?
- Do you know option bounds?
Calculations
- Can you calculate exercise value?
- Can you determine time value?
- Can you compute option bounds?
- Can you calculate break-even points?
Applications
- Can you identify appropriate strategies?
- Do you understand replication differences?
- Can you explain DeFi options protocols?
- Can you analyze factor sensitivities?
Exam Readiness
- Memorized factor impact table
- Practiced moneyness identification
- Reviewed calculation formulas
- Completed practice problems
DeFi Integration
- Understand on-chain options mechanics
- Know major options protocols
- Can explain AMM pricing models
- Familiar with structured products