Introduction to Alternative Investments — Derivatives and Risk-Neutral Valuation

Introduction to Alternative Investments — Derivatives and Risk-Neutral Valuation

CAIA Level I Study Guide

What Is It?

1. What is this topic?
   Derivatives (forwards, futures, options, swaps) and risk-neutral valuation (RNV) are tools to price, hedge, and manage risk in alternative investments.
2. How is it tested, applied, or used?
   CAIA tests pricing models, arbitrage logic, and hedging strategies. Real-world use: portfolio risk management, structured products, and regulatory compliance.

Why Does the Exam Ask This?

CAIA assesses whether candidates can: - Price derivatives without arbitrage. - Apply risk-neutral probability to value contingent claims. - Distinguish real-world vs. risk-neutral measures for hedging and speculation. - Interpret how derivatives integrate into alternative asset strategies (e.g., commodities, private equity).

What Do I Need to Know First?

1. Time value of money (present/future value).
2. Basic probability (expected value, variance).
3. Payoff diagrams for forwards, calls, and puts.
4. No-arbitrage principle.
5. Binomial option pricing basics.

Topic Snapshot

Derivatives and RNV are core to CAIA’s quantitative toolkit. They bridge traditional finance (e.g., Black-Scholes) with alternative assets (e.g., commodity futures, volatility trading). Mastery is critical for Level I’s "Quantitative Foundations" and "Structured Products" sections.

Exam / Job / Audit Weighting

• Frequency: High (5–10% of Level I).
• Difficulty Rating: Intermediate.
• Question Type: Calculation (e.g., forward pricing), conceptual (e.g., risk-neutral vs. real-world), and scenario-based (e.g., hedging strategies).

Difficulty Level

Intermediate

Must-Know Rules, Formulas, Standards, or Principles

1. Forward Price Formula (Cost-of-Carry Model):
   [
   F_0 = S_0 \cdot e^{(r + c - y)T}
   ]
2. (F_0): Forward price.
3. (S_0): Spot price.
4. (r): Risk-free rate.
5. (c): Storage cost (as % of spot).
6. (y): Convenience yield or dividend yield.

7. (T): Time to maturity.

8. Risk-Neutral Valuation (RNV):

9. Price derivatives by discounting expected payoffs at the risk-free rate under the risk-neutral probability measure ((Q)).

10. Key insight: Investors are indifferent to risk in (Q), so expected returns = risk-free rate.

11. Put-Call Parity (European Options):
    [
    C_0 + PV(K) = P_0 + S_0
    ]

12. (C_0), (P_0): Call/put prices.
13. (PV(K)): Present value of strike price (K).
14. (S_0): Spot price.

Misconceptions

1. "Risk-neutral valuation ignores real-world probabilities."
2. Correction: RNV uses adjusted probabilities ((Q)) to simplify pricing, not real-world ((P)) probabilities.
3. "Derivatives are only for speculation."
4. Correction: Primary use is hedging (e.g., airlines locking in fuel prices via futures).
5. "Forwards and futures are identical."
6. Correction: Forwards are OTC (customizable, credit risk); futures are exchange-traded (standardized, marked-to-market).
7. "Options are always expensive."
8. Correction: Deep out-of-the-money options can be cheap but have low probability of payoff.

Common Mistakes

1. Ignoring cost-of-carry in forward pricing (e.g., forgetting storage costs for commodities).
2. Mixing up risk-neutral and real-world probabilities (e.g., using (P) instead of (Q) in option pricing).
3. Misapplying put-call parity (e.g., forgetting to discount the strike price).
4. Assuming all derivatives are zero-sum (e.g., ignoring transaction costs or counterparty risk).
5. Overlooking convenience yield in commodity forwards (e.g., treating gold and wheat identically).

The Common Trap

Confusing risk-neutral vs. real-world measures. - Trap: Using real-world probabilities ((P)) to price derivatives (e.g., "There’s a 30% chance the stock goes up, so the option is worth..."). - Fix: Always use risk-neutral probabilities ((Q)) for pricing. Real-world (P) is for investment decisions, not valuation.

Terms to Remember

1. Risk-Neutral Probability ((Q)): Adjusted probability measure where all assets earn the risk-free rate.
2. Convenience Yield: Non-monetary benefit of holding a physical asset (e.g., inventory for production).
3. Mark-to-Market: Daily settlement of futures contracts to reflect price changes.
4. Contango/Backwardation: Futures curve slopes upward (contango) or downward (backwardation) vs. spot price.
5. Delta ((\Delta)): Sensitivity of an option’s price to changes in the underlying asset’s price.

Step-by-Step Process

1. Pricing a Forward Contract

1. Identify the underlying asset (e.g., stock, commodity).
2. Note the spot price ((S_0)), risk-free rate ((r)), storage costs ((c)), and yields ((y)).
3. Apply the cost-of-carry formula:
   [
   F_0 = S_0 \cdot e^{(r + c - y)T}
   ]
4. Adjust for dividends (stocks) or convenience yield (commodities).

2. Risk-Neutral Valuation (Binomial Model)

1. Build a binomial tree for the underlying asset’s price.
2. Calculate risk-neutral probabilities ((q) and (1-q)):
   [
   q = \frac{e^{r\Delta t} - d}{u - d}
   ]
3. (u): Up-factor.
4. (d): Down-factor.
5. (\Delta t): Time step.
6. Compute option payoffs at expiration.
7. Discount expected payoffs at the risk-free rate to find the option price.

3. Hedging with Futures

1. Determine the exposure (e.g., long 1,000 barrels of oil).
2. Calculate the hedge ratio (e.g., 1:1 for perfect hedge).
3. Select the futures contract (e.g., NYMEX WTI crude).
4. Adjust for basis risk (mismatch between spot and futures).

Exam Answer Builder

1-Mark Question (MCQ)

What it tests: Recall of forward pricing formula. Example: An investor enters a 6-month forward contract on a non-dividend-paying stock currently trading at $50. The risk-free rate is 4%. What is the forward price? A) $50.00 B) $51.01 C) $52.02 D) $49.02

Key Tip: Memorize (F_0 = S_0 \cdot e^{rT}). Here, (F_0 = 50 \cdot e^{0.04 \cdot 0.5} = 51.01).

3-Mark Question (Calculation)

What it tests: Application of put-call parity. Example: A European call option on a stock (spot price = $100) has a strike of $95 and expires in 1 year. The risk-free rate is 3%, and the call price is $12. What is the price of a European put with the same strike and expiration?

Key Tip: 1. Use put-call parity: (C_0 + PV(K) = P_0 + S_0). 2. (PV(K) = 95 \cdot e^{-0.03 \cdot 1} = 92.19). 3. Rearrange: (P_0 = C_0 + PV(K) - S_0 = 12 + 92.19 - 100 = 4.19).

5-Mark Question (Scenario)

What it tests: Risk-neutral valuation in a binomial model. Example: A stock is priced at $40 today. In 1 year, it can rise to $50 (up) or fall to $30 (down). The risk-free rate is 5%. Calculate the price of a 1-year European call option with a strike of $42 using the binomial model.

Key Tip: 1. Calculate risk-neutral probability:
[
q = \frac{e^{0.05 \cdot 1} - 0.75}{1.25 - 0.75} = 0.5769
]
(where (u = 50/40 = 1.25), (d = 30/40 = 0.75)). 2. Compute payoffs: (C_u = \max(50 - 42, 0) = 8), (C_d = \max(30 - 42, 0) = 0). 3. Discount expected payoff:
[
C_0 = e^{-0.05 \cdot 1} \cdot (0.5769 \cdot 8 + 0.4231 \cdot 0) = 4.38
]

Case Study (Application)

What it tests: Hedging with futures in commodities. Example: A farmer expects to harvest 10,000 bushels of corn in 3 months. The current spot price is $4/bushel, and the 3-month futures price is $4.20. The farmer wants to lock in a price. Describe the hedge and calculate the effective price received if the spot price at harvest is $3.80.

Key Tip: 1. Short hedge: Sell 10 futures contracts (1 contract = 1,000 bushels). 2. At harvest:
- Sell corn in spot market: (10,000 \cdot 3.80 = \$38,000).
- Close futures position: Buy back at $3.80, profit = ((4.20 - 3.80) \cdot 10,000 = \$4,000). 3. Effective price: (38,000 + 4,000 = \$42,000) → \$4.20/bushel.

This vs That

Time-Saver Hack

Eliminate wrong options in forward pricing: - If the spot price is higher than the forward price, the market is in backwardation (likely due to high convenience yield or dividends). - If the forward price is higher, it’s contango (typical for storable commodities with no yield).

Mini Scenarios

1. Basic

A gold miner wants to lock in a selling price for 1,000 oz of gold in 6 months. The spot price is $1,800/oz, and the 6-month futures price is $1,850. What should the miner do? - Notice: Short futures to hedge price risk. Effective price = $1,850/oz.

2. Applied

A hedge fund holds a portfolio of stocks and buys put options to protect against a market crash. The put delta is -0.4. How should the fund adjust its stock holdings to maintain a delta-neutral position? - Notice: Sell 40% of the stock portfolio to offset the put’s negative delta.

3. Tricky

A trader observes that the 1-year forward price of oil is $80, while the spot price is $75. The risk-free rate is 2%, and storage costs are 1%. What is the implied convenience yield? - Notice: Rearrange the cost-of-carry formula: [ 80 = 75 \cdot e^{(0.02 + 0.01 - y) \cdot 1} \implies y = 0.02 + 0.01 - \ln(80/75) = -0.033 \text{ (or -3.3%)} ] Negative convenience yield implies contango (e.g., oversupply).

Diagnostic MCQ Bank

Easy

Question: Which of the following is NOT a derivative? A) Futures contract B) Swap agreement C) Treasury bond D) Call option

Correct Answer: C) Treasury bond Explanation: - Why right: A Treasury bond is a direct debt instrument, not a derivative. - Trap: Swaps and options are derivatives, tempting learners to overlook bonds.

Medium

Question: A stock is priced at $50. A 1-year European call option with a strike of $55 costs $3. The risk-free rate is 4%. What is the price of a European put with the same strike and expiration? A) $5.00 B) $6.19 C) $7.00 D) $8.19

Correct Answer: B) $6.19 Explanation: - Why right: Use put-call parity: [ P_0 = C_0 + PV(K) - S_0 = 3 + 55 \cdot e^{-0.04 \cdot 1} - 50 = 6.19 ] - Trap: Forgetting to discount the strike price (option D ignores discounting).

Hard

Question: A commodity’s spot price is $100, and the 6-month futures price is $105. The risk-free rate is 3%, and storage costs are 2%. What is the implied convenience yield? A) -1% B) 0% C) 1% D) 2%

Correct Answer: B) 0% Explanation: - Why right: Rearrange cost-of-carry: [ 105 = 100 \cdot e^{(0.03 + 0.02 - y) \cdot 0.5} \implies y = 0 ] - Trap: Assuming convenience yield is always positive (option C).

Real-World Patterns

1. Hedging in Agriculture:
   Farmers use futures to lock in crop prices, avoiding volatility (e.g., corn futures for ethanol producers).
2. Structured Products:
   Banks embed options in notes (e.g., "autocallables") to offer yield enhancement, using RNV to price the embedded derivatives.
3. Regulatory Arbitrage:
   Hedge funds exploit differences between risk-neutral and real-world measures to structure trades (e.g., volatility arbitrage).

30-Second Cheat Sheet

1. Forward price = Spot × (e^{(r + c - y)T}) (adjust for costs/yields).
2. Risk-neutral valuation discounts expected payoffs at the risk-free rate.
3. Put-call parity: (C + PV(K) = P + S).
4. Contango = futures > spot; backwardation = futures < spot.
5. Delta hedging: Adjust underlying position to offset option delta.

Related Concepts

1. Binomial Option Pricing (Level I).
2. Black-Scholes Model (Level II).
3. Hedging Strategies (Level I).

Verified Source List

1. CAIA Association. CAIA Level I Curriculum (2025–2026).
2. Hull, J. Options, Futures, and Other Derivatives (11th ed.).
3. McDonald, R. Derivatives Markets (4th ed.).
4. CFA Institute. Derivatives (Level II Reading).
5. NYMEX/COMEX. Futures Contract Specifications.