Topic 10: Valuing a Derivative Using a One-Period Binomial Model
Learning Objectives Coverage
LO1: Explain how to value a derivative using a one-period binomial model
Core Concept
The binomial model values options by assuming the underlying asset can move to only two possible prices over one period. By creating a risk-free hedged portfolio combining the option and underlying asset, we can determine the option’s fair value through no-arbitrage pricing. This model operationalizes the replication concept: if two portfolios produce identical payoffs in every state, they must have the same price today. exam-focus
Key Components
Binomial Tree Structure:
S₁ᵘ = S₀ × Ru (up move)
/
S₀
\
S₁ᵈ = S₀ × Rd (down move)
Where:
Ru = Up gross return (> 1)
Rd = Down gross return (< 1)
Hedge Ratio Formula formula pricing
h* = (c₁ᵘ - c₁ᵈ)/(S₁ᵘ - S₁ᵈ)
Where:
h* = Hedge ratio (delta)
c₁ᵘ = Call value if price goes up
c₁ᵈ = Call value if price goes down
S₁ᵘ = Stock price if up
S₁ᵈ = Stock price if down
Practical Example
Tesla Call Option Valuation:
- Current price: $200
- Strike: $220
- Up move: +30% → $260
- Down move: -20% → $160
- Risk-free rate: 5%
Step 1: Calculate payoffs
c₁ᵘ = Max(0, $260 - $220) = $40
c₁ᵈ = Max(0, $160 - $220) = $0
Step 2: Calculate hedge ratio
h* = ($40 - $0)/($260 - $160) = 0.40
Step 3: Build hedged portfolio
Buy 0.40 shares, sell 1 call
Up scenario: 0.40 × $260 - $40 = $64
Down scenario: 0.40 × $160 - $0 = $64
Portfolio is risk-free with value $64
Step 4: Calculate option value
0.40 × $200 - c₀ = $64/1.05
$80 - c₀ = $60.95
c₀ = $19.05
DeFi Application defi-application
Uniswap v3 concentrated liquidity is analogous to binomial pricing ranges. Liquidity providers set price bounds that mirror the two-state structure of the binomial model. For example, in an ETH/USDC position with a current price of 2200 resembles S₁ᵘ and the lower tick at $1800 resembles S₁ᵈ. LP fees earned function as the option premium — the compensation for taking on the binary risk within the specified range.
LO2: Describe the concept of risk neutrality in derivatives pricing
Core Concept
Risk-neutral pricing values derivatives as if all investors are indifferent to risk, using adjusted probabilities that make the expected return of all assets equal to the risk-free rate. This powerful abstraction simplifies valuation by eliminating the need to know actual probabilities or investor risk preferences. Combined with the hedge ratio approach, it provides two equivalent paths to the same option value — a frequent exam verification technique. exam-focus
Risk-Neutral Probability Formula formula
π = (1 + r - Rd)/(Ru - Rd)
Where:
π = Risk-neutral probability of up move
1-π = Risk-neutral probability of down move
r = Risk-free rate
Ru = Up gross return
Rd = Down gross return
Option Value Using Risk-Neutral Pricing
c₀ = [π × c₁ᵘ + (1-π) × c₁ᵈ]/(1+r)
Interpretation:
Option value = PV of expected payoff using risk-neutral probabilities
Practical Example
Apple Put Option:
- Stock: $150
- Strike: $140
- Up: +20% → $180
- Down: -15% → $127.50
- Risk-free: 3%
Calculate risk-neutral probability:
Ru = 1.20, Rd = 0.85
π = (1.03 - 0.85)/(1.20 - 0.85)
π = 0.18/0.35 = 0.514
Calculate put payoffs:
p₁ᵘ = Max(0, $140 - $180) = $0
p₁ᵈ = Max(0, $140 - $127.50) = $12.50
Value put option:
p₀ = [0.514 × $0 + 0.486 × $12.50]/1.03
p₀ = $6.075/1.03 = $5.90
DeFi Application defi-application
GMX’s pricing model uses Chainlink oracles for price feeds and applies risk-neutral pricing principles to perpetuals. The BTC perp mark price adjusts to the funding rate, reaching equilibrium when longs equal shorts. The funding rate itself reveals an implied risk-neutral measure — a DeFi-native expression of the same probability framework used in the binomial model.
Core Concepts Summary (80/20 Principle)
The 20% You Must Know:
- Binomial model assumes two possible price moves
- Hedge ratio creates risk-free portfolio
- Risk-neutral probability simplifies valuation
- No-arbitrage principle ensures unique price
- Actual probabilities don’t matter for pricing
The 80% That Matters Most:
- Option value independent of investor risk preferences
- Volatility (spread between up/down) drives option value
- Perfect hedge eliminates directional risk
- Risk-free rate affects present value calculation
- Model extends to multi-period trees
Comprehensive Formula Sheet
Price Movement
Up Move:
S₁ᵘ = S₀ × Ru = S₀ × (1 + u)
Down Move:
S₁ᵈ = S₀ × Rd = S₀ × (1 - d)
Gross Returns:
Ru = S₁ᵘ/S₀ > 1
Rd = S₁ᵈ/S₀ < 1
Option Payoffs
Call Payoffs:
c₁ᵘ = Max(0, S₁ᵘ - X)
c₁ᵈ = Max(0, S₁ᵈ - X)
Put Payoffs:
p₁ᵘ = Max(0, X - S₁ᵘ)
p₁ᵈ = Max(0, X - S₁ᵈ)
Hedge Ratio (Delta)
Call Hedge Ratio:
h*c = (c₁ᵘ - c₁ᵈ)/(S₁ᵘ - S₁ᵈ)
Put Hedge Ratio:
h*p = (p₁ᵘ - p₁ᵈ)/(S₁ᵘ - S₁ᵈ)
Note: Put hedge ratio is negative
Risk-Neutral Valuation
Risk-Neutral Probability:
π = (1 + r - Rd)/(Ru - Rd)
Call Value:
c₀ = [π × c₁ᵘ + (1-π) × c₁ᵈ]/(1+r)
Put Value:
p₀ = [π × p₁ᵘ + (1-π) × p₁ᵈ]/(1+r)
Expected Stock Price (Risk-Neutral):
E[S₁] = π × S₁ᵘ + (1-π) × S₁ᵈ = S₀(1+r)
Portfolio Replication
Hedged Portfolio Value:
V₀ = h*S₀ - c₀
Terminal Value (Risk-Free):
V₁ = h*S₁ᵘ - c₁ᵘ = h*S₁ᵈ - c₁ᵈ
No-Arbitrage Condition:
V₀(1+r) = V₁
HP 12C Calculator Sequences
Calculate Risk-Neutral Probability
Example: r = 5%, up = 25%, down = -10%
[f] [CLX]
1.05 [ENTER] // 1 + r
0.90 [-] // - Rd
0.15 [STO] 1 // Store numerator
1.25 [ENTER] // Ru
0.90 [-] // - Rd
[RCL] 1 [x<>y]
[÷] // π = 0.4286
Calculate Option Value
c₁ᵘ = $20, c₁ᵈ = $0, π = 0.4286, r = 5%
[f] [CLX]
0.4286 [ENTER]
20 [×] // π × c₁ᵘ = 8.572
0.5714 [ENTER]
0 [×] // (1-π) × c₁ᵈ = 0
[+] // Expected payoff = 8.572
1.05 [÷] // PV = $8.16
Calculate Hedge Ratio
c₁ᵘ = $15, c₁ᵈ = $0, S₁ᵘ = $60, S₁ᵈ = $40
[f] [CLX]
15 [ENTER]
0 [-] // c₁ᵘ - c₁ᵈ = 15
60 [ENTER]
40 [-] // S₁ᵘ - S₁ᵈ = 20
[÷] // h* = 0.75
Practice Problems
Basic Level
Problem 1: Calculate call option value:
- Stock: $100
- Strike: $105
- Up: +15% → $115
- Down: -10% → $90
- Risk-free: 4%
Solution:
Payoffs:
c₁ᵘ = Max(0, $115 - $105) = $10
c₁ᵈ = Max(0, $90 - $105) = $0
Risk-neutral probability:
π = (1.04 - 0.90)/(1.15 - 0.90) = 0.56
Option value:
c₀ = [0.56 × $10 + 0.44 × $0]/1.04
c₀ = $5.60/1.04 = $5.38
Problem 2: Find hedge ratio:
- Stock moves: 65 or $40
- Call payoffs: 0
Solution:
h* = ($10 - $0)/($65 - $40)
h* = $10/$25 = 0.40
Intermediate Level
Problem 3: Put option with hedging:
- Stock: €80
- Strike: €85
- Up: +25% → €100
- Down: -20% → €64
- Risk-free: 3%
Solution:
Put payoffs:
p₁ᵘ = Max(0, €85 - €100) = €0
p₁ᵈ = Max(0, €85 - €64) = €21
Hedge ratio:
h* = (€0 - €21)/(€100 - €64) = -0.583
Risk-neutral probability:
π = (1.03 - 0.80)/(1.25 - 0.80) = 0.511
Put value:
p₀ = [0.511 × €0 + 0.489 × €21]/1.03
p₀ = €10.27/1.03 = €9.97
Advanced Level
Problem 4: Arbitrage opportunity:
- Stock: $75
- Up: +30%, Down: -25%
- Risk-free: 6%
- Call (K=8
- Find arbitrage profit
Solution:
Calculate fair value:
S₁ᵘ = $97.50, S₁ᵈ = $56.25
c₁ᵘ = $17.50, c₁ᵈ = $0
π = (1.06 - 0.75)/(1.30 - 0.75) = 0.564
Fair value:
c₀ = [0.564 × $17.50]/1.06 = $9.31
Market price $8 < Fair value $9.31
Strategy: Buy call, sell replicating portfolio
Arbitrage profit = $9.31 - $8 = $1.31
DeFi Applications & Real-World Examples
1. Automated Market Makers
Uniswap v3 Range Orders:
ETH/USDC Pool Example:
Current: $2000
Range: $1800-$2200
Similar to binomial:
- Lower bound = S₁ᵈ
- Upper bound = S₁ᵘ
- LP position = Short straddle
- Fees earned = Option premium
Curve StableSwap:
- Concentrated around $1 peg
- Binomial-like bounded prices
- Example: DAI/USDC pool
- Range: 1.01
- High concentration = Low volatility
2. Options Protocols
Lyra Protocol:
Uses Black-Scholes with binomial approximation:
- Discretizes continuous model
- Updates every block (~12 seconds)
- Example: ETH option pricing
- Binomial steps for American exercise
- Risk-neutral measure from volatility surface
Dopex SSOV:
- Weekly expiry options
- Binomial-like strike selection
- Example: rETH calls
- Strikes at ±10%, ±20% from spot
- Discrete outcomes like binomial
3. Lending Protocols
Aave Liquidation:
Binary outcome like binomial:
- Healthy: Collateral > Debt × LTV
- Liquidation: Collateral < Debt × LTV
Example: ETH collateral
- Current: $2000
- Liquidation: $1600 (80% LTV)
- Binary payoff structure
4. Real-World Corporate Examples
Convertible Bond Pricing:
Tesla Convertible:
- Face value: $1000
- Conversion price: $300
- Stock: $250
Binomial approach:
- Up: Stock > $300, convert
- Down: Stock < $300, hold bond
- Value using risk-neutral probabilities
Employee Stock Options:
Tech startup ESO:
- Strike: $10
- Current: $8
- IPO scenarios: $20 or $5
Binomial valuation:
- Up: $20 → Exercise for $10 profit
- Down: $5 → Expire worthless
- Risk-neutral value determines grant size
Common Pitfalls & Exam Tips
Common Mistakes to Avoid
-
Using real probabilities
- Risk-neutral ≠ actual probabilities
- Don’t need investor expectations
- Focus on no-arbitrage pricing
-
Wrong hedge ratio sign
- Calls: Positive hedge ratio
- Puts: Negative hedge ratio
- Check portfolio construction
-
Forgetting present value
- Always discount at risk-free rate
- One period = one discounting
- Multi-period compounds
-
Volatility confusion
- Volatility = spread between up/down
- Not the expected return
- Wider spread = higher option value
Exam Strategy Tips
-
Quick checks:
- π must be between 0 and 1
- If not, check Ru > 1+r > Rd
- Hedge ratio ≤ 1 for calls
-
Calculation sequence:
- Always start with payoffs
- Then hedge ratio or risk-neutral prob
- Finally, option value
-
Memory aids:
- Risk-neutral: “1+r sandwich” between Ru and Rd
- Hedge ratio: “Payoff spread over price spread”
- Value: “Expected payoff discounted”
Key Takeaways
Must Remember:
- Two-state model simplifies option pricing
- Hedge ratio creates risk-free portfolio
- Risk-neutral probability from no-arbitrage
- Real probabilities irrelevant for pricing
- Volatility drives option value
Critical Insights:
- Foundation for advanced models (Black-Scholes)
- Perfect hedge possible in discrete time
- Risk preferences don’t affect fair value
- Extends to multi-period trees
- DeFi protocols implement similar logic
Cross-References & Additional Resources
Related Topics:
- Topic 8: Option value components
- Topic 9: Put-call parity relationships
- Quantitative Methods: Probability trees
- Fixed Income: Embedded option valuation
Key Readings:
- Cox, Ross, Rubinstein (1979): “Option Pricing: A Simplified Approach”
- Hull: Options, Futures, and Other Derivatives Ch. 13
- Shreve: Stochastic Calculus for Finance I (Binomial Model)
Practice Resources:
- Option pricing calculators with binomial trees
- Excel binomial option pricing templates
- Python libraries: QuantLib, py_vollib
DeFi Protocols to Study:
- Lyra: Options AMM using volatility surface
- Uniswap v3: Concentrated liquidity as options
- Squeeth (Opyn): Power perpetuals
- Panoptic: Options on Uniswap v3
- Polynomial: Structured products with binomial logic
Review Checklist
Conceptual Understanding
- Can you explain the binomial model structure?
- Do you understand risk-neutral pricing?
- Can you describe hedge portfolio construction?
- Do you know why real probabilities don’t matter?
Calculations
- Can you calculate hedge ratios?
- Can you determine risk-neutral probabilities?
- Can you value calls and puts?
- Can you identify arbitrage opportunities?
Applications
- Can you extend to multi-period models?
- Do you understand DeFi implementations?
- Can you apply to real options?
- Can you relate to market making?
Exam Readiness
- Memorized key formulas
- Practiced calculation sequence
- Reviewed no-arbitrage principle
- Completed practice problems
DeFi Integration
- Understand AMM range positions
- Know options protocol mechanics
- Can explain liquidation as options
- Familiar with on-chain implementations