125.710 Semester 2 2025 Final Assessment
Final exam
List of Topics Covered:
1. Derivative Contracts and Markets
2. Mechanics of Futures Markets
3. Hedging Strategies using Futures
4. Interest Rate and Interest Rate Futures
5. Pricing of Futures
6. Swaps
7. Stock Option
8. Trading Strategies Involving Options
9. Pricing of Options
10. Value at Risk & Credit Risk
1. Derivative Contracts and Markets
1) Overview of forward, futures, and options contract
- Futures/forwards: commitments where parties are obliged to buy/sell assets in the future at a pre-determined price.
- Futures: standardised, exchange traded, and daily settled, with default risk assumed by clearinghouse.
- Option: give the holder the right (not the obligation) to buy or sell at a certain price.
2) Payoffs from:
- A long/short forward
- A long call/put; a short call/put
3) Types of traders
- A trader is hedging when he/she has an exposure to the price of an asset and takes a position in a derivative to offset the exposure.
- In a speculation the trader has no exposure to offset. He/she is betting on the future movements in the price of the asset.
- Arbitrage involves taking a position in two or more different markets to lock in a profit.
2. Mechanics of Futures Markets
- Margins (initial margin, maintenance margin and variation margin)
- Open interest= number of long positions = number of short positions
3. Hedging Strategies using Futures
- Long (or short) futures to lock in the price when you will purchase (or sell) an asset in the future.
- Futures hedging effectively transfers the price risk to basis risk.
- Basis: the difference between the spot price and the futures price of a commodity.
• Basis (b2) = S2 - F2
• Net amount paid/received = F1 + b2
• Choice of the delivery month and/or asset underlying the futures
- Hedging investment asset with futures
• Optimal hedge ratio,
• Optimal number of contracts,
- Hedging portfolio with index futures
• Optimal number of futures contracts:
• Changing Beta (from β to β*)
If the face value of the asset is Va and face value of the future index is Vf, optimal number of short (or long) position is H*=(β- β*) (Va/Vf)
4. Interest Rate and Interest Rate Futures
1) Interest Rates
- Discrete and continuous compounding rates
- Spot/zero rate (for time T)
- Forward rate: (Note: R, and R2 are continuously compounded)
- Forward rate agreement (FRA)
- Contract value:
- Duration definition (a measure of the average life of a bond):
- Duration
- Theories of term structure: expectation theory, market segmentation, liquidity preference
2) Interest Rate Futures
- Duration-Based hedging
5. Pricing of Futures
- Arbitrage portfolio: IfF0 is too high: short forward + borrow and buy asset and ifF0 is too low: long forward + short asset. Generally,
- Prices:
-
Value of the contract?
6. Swaps
1) Swaps are used to transform. a liability or an investment (fixed vs. float)
2) Apply comparative advantage argument to design a swap; know how to calculate effective borrowing/investment rate.
3) Valuation of interest rate swaps
- Valuation as bonds: Vswap = Bfix - Bfloat
- Valuation as FRAs: Vswap = sum of VFRAs
7. Stock Option
1) Option positions
- Long call/put (premium is required)
- Short call/put (margins are required)
- Understand how cash dividends, stock dividends and stock splits affect (or not) the number of options and strike price.
2) Properties of Stock Options
- Payoffs of four option positions
- Effects of variables on option pricing
3) Derive arbitrage argument between two portfolios: (1) European call + zero-coupon bond that pays K at T and (2) European put + stock. (Put-call parity: c + Ke-rT = p + S0)
- If call price is too high: short call + (borrow and) buy put and stock
- If call price is too low: long call + short put and stock (and invest)
- European options (w/ dividends) c + D + Ke-rT = p + S0
- American options (w/dividends) S0 – D – K ≤ C – P ≤ S0 – Ke-rT
8. Trading Strategies Involving Options
- Bond plus option to create principal protected note
- Stock plus option
- Two or more options of the same type (a spread)
- Two or more options of different types (a combination)
9. Pricing of Options
1) Binominal Trees
- No-arbitrage argument
• Set up a riskless portfolio with long D shares and short 1 call:
• S0 u Δ – ƒu = S0 d Δ – ƒd
• ƒ = S0 Δ – (S0 u Δ – ƒu ) e–rT
- Risk-neutral valuation
• Find p that gives a return on the stock equal to the risk-free rate:
Risk-neutral probability
•
- Two-step binominal trees
• Need u, d, and p (and r and ∆t)
• For European options:
• For American options: Need to check the payoff from early exercise with the above formula result.
• For continuous distribution of stock return, we can get u and d based on volatility (σ) and length of time step (∆t):
2) The Black-Scholes-Merton Model
- In a short period of time of length Dt, the return on the stock is normally distributed:
- Then, the logarithm of ST is normal, ST is lognormally distributed
- The continuous compound return (x) is normally distributed, its expected return is (μ – σ2/2) not μ .
- The BS model starts from the instant risk-free hedging condition and reaches a differential equation.
- Applying the Options contract condition, such as CT=Max(ST-K,0), we can get a solution of the differential equation. The solution is the Block-Sholes model:
What is the risk neutral interpretation of the BS model?
10. Value at Risk & Credit Risk
1) Value at Risk: Three key words (%, T, K). Time units (from daily to monthly or yearly) in volatility.
- Two approaches to estimate default probability: (1) historical data, (2) bond spread
- Historical data provided by rating agencies
2) Credit risk: Arises from the possibility of a default by the counterparty.
• Hazard rate: probability of default (PD) conditional on no earlier default
• Conditional PD in the given year = survival rate until the end of previous year / unconditional PD during the given year
• Cumulative PD by time
• The expected default loss = Risk-Free bond - Risky bond
• The loss if defaulted
• (Risk-Free bond - Risky bond)/ Risk-Free bond =s =λ*(1-R) where s is defined as credit spread, or default loss rate
(=Risky bond value - Risk free bond)/Risk free bond value)
and λ: default rate (or probability), R: recovery rate.