OM-W2

Lecture Overview

• Course: FIN 9797: Options Markets

• Instructor: Lei Yu

• Semester: Fall 2025

• Topic: Forwards and Futures; No-Arbitrage Pricing, and Hedging with Futures

Agenda for Today

• Basic idea of No-arbitrage pricing: replication.

• Pricing of forwards and futures.

• Portfolio hedging with futures.

Arbitrage in a Simple Model

• Current asset prices:

  • Asset 1: $95

  • Asset 2: $43

• Two possible future states:

  • Good State: Prices go up.

  • Bad State: Prices go down.

• Seeking arbitrage opportunities:

  • To identify if it is possible to make a risk-free profit by exploiting price differences in the assets.

DCF versus No-arbitrage Pricing in the Simple Model

• Discounted Cash Flow (DCF) Method:

  • Valuations of assets depend on estimates of probabilities and discount rates.

• No-arbitrage Model:

  • Replicating Asset 1 with TWO units of Asset 2 is viable:

  • Payoff of Asset 1 aligns with payoff of $2 imes ext{Asset 2}.

  • Therefore,

  • Asset 1 price must equal $2 imes$ price of Asset 2 for no arbitrage to exist.

  • Example:

  • Asset 1 price: 95,

  • $2 imes$ Asset 2 price: $2 imes 43 = 86.

  • Application of arbitrage principle:

  • Buy Asset 2 (at 43) and sell Asset 1 (at 95) guarantees profit:

    • Profit calculation: $95 - 86 = 9.

Continuous Compounding Review

• Continuous compounding formulae:

  • Effective Annual Rate (EAR): $EAR = e^{APR} - 1$

  • Present Value (PV): $PV = FV imes e^{-Rt}$

  • Future Value (FV): $FV = PV imes e^{Rt}$

Examples of Continuous Compounding

• Present Value calculation:

  • For $100 to be received in 3 months with 8% annual return,

  • $PV = 100 imes e^{-0.08 imes (3/12)} = 98.02$.

• Future Value calculation:

  • For $500 to be received in 9 months at 4% annual return,

  • $FV = 500 imes e^{0.04 imes (9/12)} = 515.23$.

Replicating a Long Forward Contract

• Concept:

  • Buy the security now and carry it to maturity.

• Forward price equation:

  • $F(t,T) = S_t + ext{cost of carry} - ext{benefits of carry}$.

• Example of Gold Forward Pricing:

  • Spot price of gold: $300,

  • Cost of carry at 5% per annum leads to:

  • $F(t,T) = 300 imes e^{0.05} = 315.38$.

  • If contracted forward price is $340, arbitrage opportunity exists:

  • Buy gold at $300, short forward at $340.

Carrying Costs and Benefits

• Interest Rate Cost:

  • If buying now instead of at expiry, incur opportunity cost of interest.

  • Calculation with 5% compounding:

  • $Future Value = 300 imes (1 + 0.05) = 315$.

• Storage Costs:

  • Assumption: Zero for gold, but can apply to other assets (e.g., live hogs).

• Interest Rate Benefits:

  • When exchanging currencies; differences in interest rates impact forwards.

  • Formula for forward price in currencies:

  • $F(t,T)[GBPUSD] = S<em>t e^{(r</em>{d} - r_{f})(T-t)}$.

• Dividends:

  • If dividends are present, affects the valuation of forwards:

  • Continuous yield model: $F(t,T) = S_t e^{(r-q)(T-t)}$.

Examples of Arbitrage Opportunities

1. Example 2.1: Spot Gold Price at $300; Forward Price also at $300; 1-year interest rate at 5%.

2. Example 2.2: Spot Price of Oil at $19 and Futures Price at $25; interest at 5%, storage cost at 2%.

Covered Interest Rate Parity

• Key relation for currencies:

  • $F(t,T) = S<em>t e^{(r</em>{d} - r_{f})(T-t)}$.

• Logarithmic transformation:

  • $ext{ln}(F) - ext{ln}(S) = (r<em>{d} - r</em>{f})(T - t)$.

Using Futures vs. Spot

• Advantages of futures:

  • Easier to short with futures; immediate exposure without waiting.

  • Lower transaction costs.

  • Fund managers prefer to hedge using futures rather than stocks directly due to efficiency.

Margin Account Example

• Initial Long Position:

  • Contracts: 2, contract size: 100 oz, futures price: $1,250, total margin: $12,000.

Basis Risk

• Definition:

  • Basis = Spot Price - Futures Price.

  • Basis risk occurs when futures do not correlate logically with the underlying asset's price.

• Example:

  • Current gold spot at $1190; futures at $1195 indicate basis risk at $-5.

Long Hedge for Purchasing Asset

• Defined as a hedge where the purchase of an asset occurs at time t2 after initially hedging at time t1.

• Hedges with definitions:

  • $F_1: ext{Futures Price at Hedge Setup}$

  • $F_2: ext{Futures Price at Purchase}$

  • $S_2: ext{Asset Price at Purchase}$

  • $b_2: ext{Basis at Purchase}$

Short Hedge for Selling Asset

• Defined as hedge where the selling of an asset occurs at time t2 after initially hedging at time t1.

Practical Example of Basis Risk

• Short position by a wheat farmer:

  • Initial Spot Price: $50; Futures Price: $55.

  • Basis $-5. With cash price at $47 during sell, futures can lead to positive revenue.

Choice of Futures Contract

• Selecting a futures contract close to the hedging period end.

• In the absence of an exact asset futures, select by correlation, known as cross-hedging.

Optimal Hedge Ratio

• Optimal hedge ratio defined by the slope of linear regression relationships between changes in spot and futures prices:

  • $h^* = rac{<br>ho imes rac{ ext{SD}( ext{Change in } S)}{ ext{SD}( ext{Change in } F)}}$.

Optimal Number of Contracts

• Calculation based on:

  • Total units (Q) of asset position being hedged,

  • Units per futures contract (A),

  • Optimal number defined by $N^* = rac{F imes A}{h imes Q}$.

Additional Adjustments for Daily Settlements

• With daily settlements, adjustments for optimal hedge ratio and contract numbers must be made:

  • $ext{Adjusted Optimal } N = rac{A imes F }{h imes V_A}$.

In-Class Exercises

• Exercise 2.3: Calculate optimal hedging for 2 million gallons of jet fuel.

• Exercise 2.4: Adjust for daily settlements in similar hedging tasks.

Key Takeaways from Today

• Grasp the concept of arbitrage through replication and notice opportunities.

• Understand the characteristics of forward and futures contract payoffs:

  • Long: $(ST - K)$; Short: $(K - ST)$.

• Pricing formula relationship: $F(t,T) = S_t + ext{cost of carry} - ext{benefits of carry}$.

• Proficiency in calculating carry costs/benefits under varying compounding methods.

• Execute arbitrage opportunities effectively using cash and forward markets.

• Mastery of optimal hedging ratios and contract calculations.