University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives FNCE 4004 – Derivatives Chapter 5 Determination of Forward and Futures Prices
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Consumption vs. Investment Assets • Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: stocks, bonds, gold, silver) • Consumption assets are held primarily for consumption (Examples: copper, oil, corn)
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives SHORT SELLING
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Short Selling • Forward pricing depends on short selling • Short selling involves selling securities you do not own • Reasons for short selling? – You are hedging an existing exposure – You are speculating that the security will fall in price
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Securities Lending • You cannot short sell if no one will lend you a security. This business is called securities lending. • Generally seen as low risk business • Lender can accept cash as collateral in which case they pay sub-market interest rates on the collateral – They invest the cash, earn market rates and earn the spread • Lender can also accept low-risk securities – for example Treasuries.
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Securities Lending Pitfalls • AIG set up AIG Securities Lending to centrally manage securities lending across insurance company subsidiaries. • Beginning in late 2005 AIG started to use the cash to invest in RMBS. At its peak AIG had $76bn invested of which 60% was in RMBS. The securities were AAA rated when they were purchased but fell in quality and price. • Part of the AIG bailout was to buy the MBS so that the securities that had been lent could be returned to the insurance companies. • Otherwise the securities lending subsidiary of AIG could have caused “collateral” damage to all of the insurance companies which were sound.
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Mechanics of Short Sale (www.interactivebrokers.com)
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Example • You short 100 shares when the price is $100 and close out the short position three months later when the price is $90 • During the three months a dividend of $3 per share is paid and there is no cost to borrowing the stock – What is your profit? – Do you earn interest on the money you’ve received? – Do you have to pay a borrowing cost?
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives SIMPLE FORWARD PRICES
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Forward Price • Consider an investment asset that does not pay any income. For example: – Non-dividend paying stock – Zero coupon bond – Treasury bill, strip • Assume that you can short the asset at no cost and that lending the asset does not provide income • Assume that there are no transaction costs or bid/offer spreads
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives An Arbitrage Opportunity? • Suppose that: – Spot price of a non-dividend-paying stock = $40 – The 3-month forward price is $43 – The continuously compounded risk free interest rate is 5% – The borrowing cost is 0%. • Is there an arbitrage opportunity? • Assume that the forward contract is uncollateralized.
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Arbitrage Opportunity 3 months Today • Deliver stock into • Sell stock forward in 3 months for $43 forward contract and receive $43 • Borrow $40 for 3 • Repay borrowing months $40 × 𝑓 0.05×0.25 • Buy Stock = 40.50 • Make $2.50 risk-free In these transactions nothing happens between today and 3 months time. If something can happen then we need to understand whether that affects the arbitrage.
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Arbitrage Opportunity – mistake! Today In 3 months • Buy stock forward in 3 • Receive $40 × 𝑓 0.05×0.25 = 40.50 months for $43 • Borrow $2.50 • Borrow stock for 3 months • Buy stock for $43 • Sell stock and receive $40 • Deliver stock into short • Invest $40 at risk-free rate • Lose $2.50 (!!!) If you make a mistake like this you simply have to reverse all of the trades
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Another Arbitrage Opportunity? • Suppose that: – The spot price of a non-dividend-paying stock is $40 – The 3-month forward price is USD 40.25 per share – The cont. compounded USD interest rate is 5% – All contracts are uncollateralized – Is there an arbitrage opportunity?
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Arbitrage Opportunity Today In 3 months • Buy stock forward in 3 • Receive $40 × 𝑓 0.05×0.25 months for $40.25 = 40.50 • Borrow stock for 3 • Buy stock for $40.25 months • Sell stock and receive • Deliver stock into short $40 • Earn $0.25 • Invest $40 at risk-free rate
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Some Notation Notation for Valuing Futures and Forward Contracts 𝑇 : Spot price today 𝐺 : Futures or forward price today 𝑈 : Time until delivery date 𝑠 : Risk-free interest rate for maturity T (cont. compounding)
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives The Forward Price • Consider an investment asset that has no yield and does not cost to borrow • Assume that the market does not require collateral for any transactions • If the spot price of an investment asset is 𝑇 and 𝑠 is the T-year risk-free rate of interest then the forward price for a contract deliverable in 𝑈 years is 𝐺 = 𝑇𝑓 𝑠𝑈
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives No Arbitrage Argument – part 1 Time T Today • Deliver stock into • Agree to sell stock forward at time T for F forward contract and receive F • Borrow S at risk-free • Repay borrowing rate S × 𝑓 𝑠𝑈 • Buy Stock for S • For there to be no arbitrage 𝐺 ≤ S × 𝑓 𝑠𝑈
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives No Arbitrage Argument – part 2 Today Time T • Agree to buy stock • Redeem investment for 𝑇𝑓 𝑠𝑈 forward at time T for 𝐺 • Borrow Stock today • Pay F and receive • Sell Stock and receive S stock • Invest $S at risk-free • Deliver stock to lender rate • For there to be no arbitrage 𝐺 ≥ S × 𝑓 𝑠𝑈 The no-arbitrage argument says that 𝐺 ≥ S × 𝑓 𝑠𝑈 and 𝐺 ≤ S × 𝑓 𝑠𝑈 , which means 𝐺 = 𝑇 × 𝑓 𝑠𝑈
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives FORWARD CONTRACTS
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Valuing a Forward Contract • A forward contract is a contract to buy/sell an asset at some time in the future. • A forward contract is worth zero (excluding bid-offer spreads) when it is first negotiated • Later it may have a positive or negative value • Suppose: – K is the delivery price – F is the forward price for a contract that would be negotiated today for settlement at time 𝑈
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Valuing a Forward Contract By considering the difference between a contract with delivery price 𝐿 and a contract with delivery price 𝐺 we can deduce that: • the value of a long forward contract, ƒ, is (𝐺 − 𝐿)𝑓 −𝑠𝑈 • the value of a short forward contract is (𝐿 − 𝐺)𝑓 −𝑠𝑈
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives COLLATERAL
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Collateral • Collateral is posted to protect the counterparties in a contract from non-delivery by the other counterpart. • We will assume for simplicity that collateral is in the form of US treasuries. • If a contract is collateralized then the discounting should be done at the same rate earned by the collateral. • What happens if it isn’t?
University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives Collateralized Lending/Borrowing • Assume that you can lend/borrow $95 today and have to return $100 tomorrow. • Assume that this transaction is collateralized with US treasuries and that there is a $100 strip with a maturity of 1 day trading at $97. • Is there arbitrage? • What if the maturity of all transactions is 1 year?
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