FNCE 30007 Derivative Securities Lecture – Forward & futures pricing Outline Introduction Arbitrage Investment v consumption assets Short selling Investment assets Simple relation (no income) Known income (bond futures) Known yield (stock index futures, currency futures) Consumption assets Valuation Environmental, social and governance movement Futures & expected spot prices Reading: Chapter 5 Introduction: No arbitrage pricing Lecture 1 considered speculators, lecture 2 considered hedgers. We now focus on the other major participant – arbitrageurs Arbitrage takes advantage of a price differential between two or more markets No up front payment and a positive cash flow later Arbitrage possible if Same asset does not trade at the same price on all markets (the law of one price); or 2 assets with identical cash flows do not trade at the same price; or An asset with a known price in the future does not today trade at its future price discounted at the risk- free interest rate Introduction: Forward v Futures prices Forward and futures prices are usually equal. We therefore price futures contracts as if they are forwards Greatly simplifies the problem Slight differences may arise from Taxes Transaction costs Margins Futures more liquid & have no counterparty risk Introduction: Investment v Consumption assets Investment assets: held by significant numbers of people purely for investment purposes (gold, silver, stocks, bonds). Can price forwards & futures off spot via arbitrage. Consumption assets: held primarily for consumption & not usually for investment purposes (copper, oil, pork bellies, soybeans). Not able to price forwards & futures off spot via arbitrage. Introduction: Short selling Short selling: sell securities you do not own Your broker borrows the securities from another client and sells them in the spot market At some stage you buy the securities back & return them If the price falls: sell high, buy low. You pay dividends & other benefits to the owner of the securities. Diagram Introduction: Assumptions & notation Assume: No transaction costs Same tax rate Borrowing/lending at risk-free rate No-arbitrage opportunities (arbitrage opportunities taken advantage of immediately). S0: Spot price today F0: Futures or forward price today T: Time until delivery date (years) r: Risk-free interest rate for maturity T (continuously compounded rate p.a). Investment assets – Simple pricing The relationship between F0 and S0 is If F0 > S0erT, arbitrageurs buy the asset and short forward contracts If F0 < S0erT, arbitrageurs short the asset and buy forward contracts. rTeSF 00 = Investment assets - Example 1 Suppose: The spot price of gold is US$600 The quoted 1-year futures price of gold is US$650 1-year US$ interest rate is 5% p.a No income or storage costs for gold Is there an arbitrage opportunity Investment assets - Example 2 Suppose: The spot price of gold is US$600 The quoted 1-year futures price of gold is US$590 1-year US$ interest rate is 5% p.a No income or storage costs for gold Is there an arbitrage opportunity Short sale constraint What if short sales are not possible for all investment assets Short selling not needed for the forward price equation. Only require that at least some people hold the asset only for investment purposes. If the forward price is low, they sell the asset & take a long forward position. Investment assets - known $ income Investment asset has income during the life of a forward contract: F0 = (S0 – I)erT where I = present value of the income during life of forward contract If F0 > (S0 – I )erT, arbitrageurs buy the asset and short the forward contract. If F0 < (S0 – I )erT, arbitrageurs short the asset and buy the forward contract. Example: known $ income Consider a long forward contract to buy a coupon bond with a current price of $900. -The forward contract has a 9 month maturity & a coupon payment of $40 in 4 months. -The continuously compounded 4 and 9 month risk- free rates are 3% and 4% p.a The forward contract price should be $886.60. To show why we will consider 2 situations i) where the futures is overvalued at $910. ii) where the futures is undervalued at $870. Example: known $ income (i) The forward price is $910. Arbitrageur will Today: Borrow $900. The coupon payment has a present value of 40e(-0.03)(4/12) = $39.60. So, $39.60 is borrowed for four months and the rest ($860.40) is borrowed for nine months (at 4%). Buy the bond Enter into a forward contract to sell the asset for $910. In four months: Receive $40 coupon payment Use $40 to repay first loan with interest In nine months: Sell bond & receive $910 under the terms of the forward contract. Use $886.60 to repay second loan with interest. Profit: $910 – $886.60 = $23.40 Example: known $ income (ii) Forward price is $870. Arbitrageur will Today: Short the bond ($900) Enter into a forward contract to buy the bond for $870 in nine months. Of the $900 realized from shorting the bond, $39.60 is invested for four months at 3% per annum (grows to $40). The remaining $860.40 is invested for nine months at 4% per annum (grows to $886.60) In four months: Receive $40 from four-month inve
stment Use $40 to pay coupon on the bond In nine months: Receive $886.60 from nine-month investment Buy the bond for $870 under the terms of the forward contract Close out short position in the bond Profit: $886.60 – $870 = $16.60 Investment assets – known yield If the asset underlying a forward contract has a known yield F0 = S0e(r–q)T where q = average yield over the life of the contract Two examples Stock index futures Currency futures Known yield – stock index futures Investment asset paying a dividend yield The price relationship is F0 = S0 e(r–q )T where q = dividend yield (over life of contract) on the portfolio represented by the index Index must represent an investment asset changes in the index must correspond to changes in the value of a tradable portfolio Known yield – stock index futures F0 > S0e(r-q)T arbitrageur buys the stocks underlying the index and sells futures F0 < S0e(r-q)T an arbitrageur buys futures and sells the stocks underlying the index Index arbitrage involves simultaneous trades in futures & many different stocks Very often computer used to generate trades Known yield – stock index futures Lets consider the E-mini S&P500 futures contract S&P500 close Feb 3/2020 3225.52 4 week US T-bill rate 1.53%p.a Dividend yield 1.79%p.a Contract specs reveal expires 3rd Friday of the month. Therefore 34 trading days prior to expiration T=34/252 [ ]3225.52exp (0.0153 0.0179) 34 252 3224.39oF = × = Known yield – currency Foreign currency provides a continuous yield - the foreign risk-free interest rate rf Underlying asset is 1 unit of foreign currency. F S e r r Tf0 0= ( ) Known yield – currency example 2year interest rates in Australia and the U.S. are 5% and 7%. Spot exchange rate is 0.62 USD per AUD. Find the two year forward rate. Given that fx rate is 1 unit of foreign currency, we consider from perspective of US investor. Can also redo from the perspective of an Australian investor. Consider when 2year forward rate is 0.63 & 0.66. Consumption assets F0 ≤ S0 e(r+u )T where u is the storage cost per unit time as a percent of the asset value. Alternatively, F0 ≤ (S0+U )erT where U = present value of the storage costs. Consumption assets - example Consider a 1 year copper futures contract. Assume no income and that it costs $2 per ounce per year to store copper, with payment being made at the end of the year. The spot price is $600 and the risk-free rate is 5% p.a for all maturities. Find the futures price. Consider when futures price is $700 & $610 Consumption assets – convenience yield Convenience yield is the benefit from holding the physical asset. Reflects market expectations of future availability High inventory expectations low conv yield Low inventory expectations high conv yield Convenience yield is F0eyT = (S0+U )erT If U is proportional to the spot price F0eyT = S0e(r + u)T Example Cost of carry Cost of carry, c, is the storage cost plus the interest costs less the income earned Investment asset F0 = S0ecT Consumption asset F0 ≤ S0ecT Convenience yield on consumption asset, y, is defined so that F0 = S0 e(c–y )T Forward valuation The value of a futures contract is zero - value reflected in the margin account. K is delivery price in a forward contract entered into previously. F0 is forward price that would apply to the contract today. Value of a long forward contract, , is = (F0 – K )e–rT Value of a short forward contract is = (K – F0 )e–rT Forwards have a value of zero at the time first entered into because F0=K. As time passes, the forward price and the value of the contract change. Forward valuation: example Consider a long forward contract on a non-dividend paying stock entered into some time ago that has 5 months left to maturity. The risk-free rate with continuous compounding is 9% p.a. The current stock price is $30 and the delivery price is $28. The value of the contract is: F0 = 30e(0.09)(5/12) = $31.15 = (F0 – K )e–rT = (31.15 – 28)e(-0.09)(5/12) = $3.03 or = S0 – Ke–rT = 30 - 28e(-0.09)(5/12) = $3.03 Will use this in the BSM lecture ESG Environmental, social and governance (ESG) movement is now emerging in the derivatives markets. ESG companies typically less exposed to environmental and regulatory tail risks (Value at Risk lecture) Eurex and Nasdaq have launched ESG futures Similar plans for many other markets China will soon open an exchange solely dedicated to the trading of carbon finance futures ESG Eurex Futures on the STOXX Europe 600 ESG-X index Index screens out companies with low ESG rankings Index enables investors to easily switch portfolio to an ESG compliant benchmark with low cost and tracking error Futures on the index now traded Futures can be used for hedging and speculative purposes This should add liquidity to the underlying index Nasdaq In October 2019 Nasdaq launched a futures based on the OMXS30 responsible index Also excludes companies with poor ESG standards Futures Prices & Expected Spot Prices So far we have focused on the contemporaneous relation between F0 and S0. What about the relation between F0 and E (ST ) If F0 = E (ST ) F0 unbiased estimate of ST F0 F0>E (ST ) contango