Chapter 3 The Determination of Forward and Futures Prices

3.1 Investment assets vs. consumption assets An investment asset is an asset that is held for investment purposes by significant numbers of investors. A consumption asset is an asset that is held primarily for consumption. Forward and Futures an agreement to buy or sell an asset at a certain time in the future for a certain price difference: forward contracts are settled at maturity but futures contracts are settled daily Forward price and futures prices are generally very close when the maturities of the two contracts are the same

3.2 Short selling(공매도) Def: Selling securities(유가증권) that are not owned. Your broker borrows the securities from another clients and sells them in the market in the usual way. At some stage you must buy the securities back so they can be replaced in the account of the client. You must pay dividends(배당금) and other benefits the owner of the securities receives.

Short selling(계속) An investor shorts 500 IBM shares in April when the share price = $120 and closes out his or her position by buying them back in July when the share price=$100. Suppose that a dividend of $1 per share is paid in May. The net gain is 500X$120-500X$1-500X$100 =$60,000-$500-$50,000 =$9,500

주가지수 (KOSPI 200) 선물 가격: (2002년 3월 8일 가격) 현물가격: 103.04 point 선물과 선도 가격의 결정요인 주가지수 (KOSPI 200) 선물 가격: (2002년 3월 8일 가격) 현물가격: 103.04 point 2002년 6월 만기 선물의 적정가격은 얼마일까? Price Quotes (2002년 3월 8일 한국경제 선물 가격표) KOSPI 200 주가지수 선물 시세 2002년 3월 물: 종가 – 103.25 2002년 6월 물: 종가 – 103.00 2002년 9월 물: 종가 - 108.80 위의 선물가격은 어떻게 결정되는가?

2002년 3월8일 현재 KOSPI200 3월선물가격 변동상황

3.3 Measuring interest rates 현재의 1,000원과 6개월 후의 1,000원은 같은 가치를 가지는가? 현재의 돈 1,000원을 투자 (은행에 예금)하면 6개월 후 1,000원 이상이 된다. 예) 금리가 연 10% 일 때, 1000원을 은행에 예금하면, 6개월 후 약 1050원이 된다. 즉 현재 1,000 원인 물품의 만기 6개월인 선물 가격은 1050원이어야 한다

Preliminaries on Interest Rate (I) 질문: 다음 세 은행 중 어느 은행에 예금을 할 것인가? 은행 A: 이자율 10%, 이자 한번 지급/ 년 (compounding annually) 은행 B: 이자율 10%, 이자 두 번 지급/ 년 (compounding twice per annum) 은행 C: 이자율 10%, 이자 네 번 지급/ 년 (compounding four times per annum) Suppose that an amount A is invested for a year at an interest rate R per annum If the rate is compounded once per year, the terminal value of the investment is If the rate is compounded m times per year, the terminal value of the investment is

Preliminaries on Interest Rate (II) Table 3.1 : Effect of compounding frequency. R = 0.1 Compounding Value of $100 at frequency end of year m = 1 $ 110.00 m = 2 $ 110.25 m = 4 $ 110.38 m =12 $ 110.47 m =52 $ 110.51 m =365 $ 110.52

Preliminaries on Interest Rate (III) 1년에 이자를 한 번 지급하는 이자율을 R 이라 하고, m 번 지급하는 년 이자율을 이라 하자 두 이자 지급 방법에 의한 1년 후 총 이자가 같기 위해서는 즉,

Preliminaries on Interest Rate (IV)

Continuous Compounding As we compound more and more frequently we obtain continuously compounding interest rate: The investment A grows to at a continuously compounding rate for time t .

Conversion Formulas 1 년에 m 번 지급하는 연 이자율을 이라 하고, continuous compounding interest rate 를 라 하자. 두 이자 지급 방법에 의한 이자 지급액 이 같기 위해서는 즉,

Conversion Formulas 예) R = 0.1 일 때, 일 때,

3.4 Assumptions and notation Repo Rate The relevant risk-free interest rate in the futures market is known as repo rate. A repo (repurchase agreement) rate is an agreement where one financial institution sells securities to another financial institution and agrees to buy them back later at a slightly higher price. The difference between the selling price and the buying price is the interest earned. The most common repo is an overnight repo, in which the agreement is renegotiated each day.

Assumptions on Forward prices The market participants are subject to no transaction costs when they trade. The market participants are subject to the same tax rate on all net trading profits. The market participants can borrow money at the same risk-free rate of interest as they can lend money. The market participants take advantage of arbitrary opportunities as they occur.

Notations : time (in years) : maturity of a forward contract (선물/선도 만기일) : current price (at time t) of the underlying asset (현물가격) : delivery price (선물/선도 인도가격) : value (at time t) of a long position in the forward contract : forward price (at time t) : risk-free interest rate of continuous compounding

3.5 Forward price for an investment asset 현재시간 t 에서 이루어지는 선도(선물) 의 가격 (F) 은 얼마이어야 하는가? Example (Page 76) underlying asset : a non-dividend-paying stock Spot price : $40 Maturity : after 3 months risk-free interest rate : 5% per annum 위의 선도계약의 Delivery Price 는 얼마이어야 하는가?

Example Suppose that the delivery price of the forward contract is $43. What would you do? An arbitrageur can Borrow $40 to buy the stock at spot market. Take a short position in the forward contract.

What happen in three months? The arbitrageur has to pay off the loan. The required money is The arbitrageur delivers the stock and receives $43 So he makes a risk-free profit $2.50 at the end of the three months period. The forward price should be less than $40.50 Otherwise there is an arbitrage opportunity for investors who take short positions. And the forward price will go down.

Example Suppose that the delivery price of the forward contract is $39. What would you do? An arbitrageur can Short (sell)공매 the stock at spot market and invest the sale at 5% per annum for three months (What if he does not own it? Short selling) Take a long position in the forward contract.

What happen in three months? The investment grows to The arbitrageur pays $39 and receive the stock. The stock is used to close out the short (short selling) position. So he makes a risk-free profit $ 1.50 at the end of the three months period. The forward price should be higher than $40.50. Otherwise there is an arbitrage opportunity for investors who take long positions. And the forward price will go up.

Forward Prices (I): A generalization Current time : t, Spot price of the underlying asset: S Forward Contract Delivery price : K, Maturity : T Risk-free interest rate : r Then the forward price must be When a forward contract is initiated, the delivery price is set to the forward price so that K=F.

Forward Prices (II) If , an arbitrageur can borrow S for a period of time T-t at interest rate r and buy the asset, take a short position in the forward contract. Then he gains a risk-free profit

If , an arbitrageur can Short (sell) the stock at spot market and invest S at interest rate r for a period of time T-t take a long position in the forward contract. Then he gains a risk-free profit

Example Consider a four-month forward contract to buy a bond. The current price of the bond is $930. We assume that the four-month risk-free rate of interest is 6% per annum. The forward price is given by This would be the delivery price in a contract negotiated today.

3.6 Forward Price for an Asset That Provides a Known Cash Income Current time : t, Maturity : T Spot price of the underlying asset : S Risk-free interest rate : r The asset provides a cash income before maturity of a forward contract. 예) Stocks paying known dividends, Coupon-bearing bonds. 현재시간 t 에서 이루어지는 선도(선물) 의 가격 (F) 은 얼마이어야 하는가?

Example Example underlying asset : a bond Spot price : $900 Maturity : after 1 year six-month risk-free interest rate : 9% per annum one-year risk-free interest rate : 10% per annum The bond has coupon payments of $40 in six-months and one year (right before the delivery) 위의 선도계약의 Delivery Price 는 얼마이어야 하는가?

Suppose that the forward price is $930. What would you do? Example (continued) Suppose that the forward price is $930. What would you do? An arbitrageur can Borrow $900 to buy the bond at spot market. Take a short position in the forward contract. What is the best strategy to borrow $900? 1. Borrow $900 for one year. He needs to pay off

2. Note that the holder of the bond receives $40 in six months and another $40 in twelve months. Present value of $40 in six months: Borrow $38.24 at 9% for six months and remaining $861.76 at 10% for one year. He needs to pay off in six months and

What happen in one year? (cash flow) Example (continued) What happen in one year? (cash flow) The arbitrageur has to pay off the loan. The required money is $40 in six months and $952.39 in one year. The arbitrageur receives dividends $40 in six months (which is used to pay off the loan $38.24) and $ 40 in one year. He also receives $930 after delivering the bond.

Cash flow in six months : $40 (dividends) - $40 (payment for the loan) = 0 Cash flow in one year : $40 (coupon payment) + $930 (forward price) - $952.39 (payment for the loan) = $17.61 So he makes a risk-free profit $17.61 in one year. The forward price should be less than $930. Otherwise there is an arbitrage opportunity for investors who take short positions. And the forward price will go down.

Suppose that the forward price is $905. What would you do? Example (continued) Suppose that the forward price is $905. What would you do? An arbitrageur can Sell the bond at spot market and receives $900 Take a long position in the forward contract. Invest $38.24 at 9% interest rate for six months which grows $40 in six months. $40 is the amount of coupon that would have been paid on the bond. Invest remaining $861.76 at 10% interest rate for one year which grows $952.39 in one year.

Cash flow In six months : $40 (investments) - $40 (used to replace the coupon) = 0 In one year : $952.39 (investment) - $40 (used to replace the second coupon) – $905 (forward price) = $ 7.39 So he makes a risk-free profit $7.39 in one year.

First case (short forward contract) $40 + forward price - $952.39 Example (continued) Cash flow (in one year) First case (short forward contract) $40 + forward price - $952.39 Second case (long forward contract) $952.39 - $40 - forward price If forward price is not equal to $912.39, cash flow is not equal to zero for both cases. So there will be an arbitrage opportunity. Forward price should be $912.39

Current time : t, Maturity : T Spot price of the underlying asset : S Forward Prices Formula for an Asset with a Known Cash Income: A generalization Current time : t, Maturity : T Spot price of the underlying asset : S Risk-free interest rate : r The asset provides income with present value of I (미래소득의 현재가치 ).

An investor’s strategy Forward Prices Formula for an Asset with a Known Cash Income: A generalization An investor’s strategy Buy the asset on the spot market. Take a short position in the forward contract. Cash flow (if the forward price is equal to F ) outflow: S - I (at time t ) inflow : F ( at time T) at time t : total cash flow (at time t):

To make the total cash flow zero, the forward price should be Forward Prices Formula for an Asset with a Known Cash Income (continued) To make the total cash flow zero, the forward price should be This would be the delivery price in a contract initiated today.

For the previous example S = $900, T- t = 1 year r = 9% per annum (for six months), 10% per annum ( for one year) What is I? (The bond holder receives $40 after six months and one year) Hence

Example 10 months forward contract on a stock Spot price (S): $50 risk-free interest rate : 8% for all maturities dividends of $0.75 are expected after 3, 6, and 9 months

Example Present value of the dividends Forward price

3.7 Forward Prices for an Asset That Provides a Known Dividend Yield A dividends yield means the income which is expressed as a percentage of the asset price. q : (continuously compounding) annual rate of dividend yield. 예) When an asset price is $10 and q = 0.05, dividends in a small interval of time are paid at the rate of $0.5 per annum.

3.7 Forward Prices for an Asset That Provides a Known Dividend Yield Current time : t, Maturity : T Spot price of the underlying asset : S Risk-free interest rate : r Dividend yield : q% per annum What is the forward price?

Consider an investor’s strategy : Forward Prices for an Asset That Provides a Known Dividend Yield (continued) Consider an investor’s strategy : Buy unit of the asset on the spot market with all income (dividends) being reinvested in the asset. Take a short position in the forward contract. Since the dividends are reinvested in the asset and the yield rate is q, the asset grows to X S

Cash flow (if forward price is equal to F) outflow : (at time t) Forward Prices for an Asset That Provides a Known Dividend Yield (continued) Cash flow (if forward price is equal to F) outflow : (at time t) inflow: F ( at time T), present value : total cash flow (at time t): To make the total cash flow zero, the forward price should be This would be the delivery price in a contract initiated today.

Example (Page 51) 6 months forward contract on a security with dividend yield of 4% per annum which is 3.96% with continuous compounding Spot price (S): $25 risk-free interest rate : 10% per annum Forward price S = 25, r = 0.1, q = 0.0396, T - t = 0.5

3.8 Valuing Forward Contracts The present value of the long forward contract with delivery price K is given by The short forward contract :

Valuing Forward Contracts (continued) For a forward contract on an asset that provides no income, For a forward contract on an asset that provides an income I, For a forward contract on an asset that provides a dividend yield q,

Example 6 months long forward contract on a non-dividend-paying stock with delivery price $24 risk-free interest rate r : 10% current stock price S : $25 The current forward price is The value of the forward contract is

3.9 Are Forward and Futures Prices equal? Forward and futures prices are usually assumed to be the same. When interest rate are uncertain, they are in theory slightly different. Strong positive correlation between interest rates and the asset price implies that the futures price is slightly higher than the forward price. Strong negative correlation implies the reverse.

3.10 Stock index futures 예) KOSPI 200 Index, S&P 500 Index Stock index can be regarded as an asset paying continuous dividend yield. If the dividend yield rate is q, the futures price is In practice, the dividend yield on the portfolio underlying an index varies throughout the year. The chosen value of q should represent the average annualized dividend yield during the life of the contract.

S&P500지수 세계최대의 금융통계 및 투자정보서비스회사인 Standard and Poor’s사가 개발한 지수 1988년 4월부터 공업주 400종목, 공공주 40종목, 금융주 40종목, 운수주 20종목 으로 구성 지수산출공식: V0= 기준시점인 1941~1943년의 기준시가 Vt = t 시점에서의 S&P500 지수 Nit =i회사의 주식수, Pit = i회사 주식의 가격

KOSPI200 지수 주식시장의 상황을 정확히 반영할 수 있도록 200종목에 의한 시가 총액식 지수 시장대표성, 유동성, 업종대표성 등을 고려하여 선정 기준시점 및 지수: 1990년 1월 3일, 100.00 종목선정기준: 200종목의 시가총액이 전체상장종목시가 총액의 70%정도의 비중이 되도록. 산식: 100 ⅹ 지수 구성종목의 비교시점의 시가총액 합계 지수 구성종목의 기준시점의 시가총액 합계

Index Arbitrage 주가지수선물 차익거래 전략 When , an arbitrageur can make profits by buying the stocks underlying the index and selling futures. When , an arbitrageur can make profits by buying futures and selling the stocks underlying the index. Index arbitrage involves simultaneous trades in futures and many different stocks.

3.11 Forward and Futures Contracts on Currencies A foreign currency is analogous to security(유가증권, stocks and bonds) providing a continuous dividends yield. The continuous dividend yield is the foreign risk-free interest rate. is the foreign risk-free interest rate and r the domestic risk-free interest rate. Consider the following strategy: Buy spot of the foreign currency Short a forward contract on one unit of the foreign currency

Forward and Futures Contracts on Currencies(계속) If the foreign interest rate is greater than the domestic interest rate , F is always is less than S and F decreases as the maturity increases. If the domestic interest rate is greater than the foreign interest rate , F is always is greater than S and F increases as the maturity increases. 일반적으로 한국금리가 미국금리에 비해 높으므로, 미국달러 선물가격은 현물가격에 비해서 가격이 높다.

Forward and Futures Contracts on Currencies(계속) 한국 이자율 7%, 미국 이자율 5%, 1달러당 900원, 선도환율은 936.73원 이어야한다. 만약 920원이면 ? 미국 달러를 5% 로 100달러를 빌려 90000원을 0.07%로 투자한다. 100*Exp[0.05*2] = 110.517달러 (outflow) 90000* Exp[0.07*2] = 103,525원 (inflow)

Forward and Futures Contracts on Currencies(계속) 920원X110.517 달러 = 101676원으로 110.517달러를 매입하는 선도계약체결 이 돈으로 빌린 달러를 갚고 나면 103,525 – 101,676 = 1849원 남는다.

3.12 Futures on Commodities 예를 들어, 금 선물인 경우 선물 만기일 까지 금을 보관하는데 비용이 들어간다. 일반적으로 금과 같은 commodities에 대해서는 물품 보관비용 (storage cost)이 들어간다. If U is the present value of all the storage costs, the futures price is given by If the storage costs incurred at any time are proportional to the price of the commodity (u: storage cost per annum as a proportion of the spot price), they can be regarded as providing a negative dividend yield. So the futures price is

Example Contract size: 100oz, Spot price : $450/oz, Maturity: one year Risk-free interest rate : 7% per annum Storage cost : $2/oz per year (payable at the end of the year) present value of the storage cost : 위의 선물가격 (F) 은 얼마이어야 하는가? An investor’s strategy Borrow $45000 at risk-free interest rate to buy 100oz of gold. Take a short position in the futures contract. Cash flow inflow : F (at maturity of the futures contract) outflow: (payment for the loan) + $200 (storage cost) = $48,463 To make the total cash flow zero (otherwise, an arbitrage opportunity), the futures price should be $484.63.

Convenience yields(보유 편익률) The ownership of some commodity may provide benefits due to temporal local shortages or the ability to keep a production process running. The benefit may reflect the market’s expectations concerning the future availability of the commodity The benefit is called the convenience yield provided by the commodity If the storage cost is U, then the convenience yield, y, is defined as in

3.13 The cost of carry(보유비용) The cost-of-carry measures the storage cost plus the interest that is paid to finance the asset less the income earned on the asset. The cost-of-carry, c, is defined as in If there is a convenience yield (y) accruing

3.15 Futures prices and the expected future spot price Systematic risk: risk that can not be diversified away 증권시장 또는 증권가격전반에 영향을 주는 요인에 의해 발생하는 위험. 경제/정치/사회적요인들로 인해 발생하는 위험 Nonsystematic risk : risk that can be diversified away

Consider a speculator who takes a long futures position. The speculator puts the present value of the futures price into a risk-free investment (at time t) and simultaneously takes a long futures position (at time T, T시점의 자산가격 ) The present value of the investment: where k is the discount rate appropriate for the investment

Futures prices and the expected future spot price(계속) Hence we have If is positively correlated with the stock market as a whole, the investment has positive systematic risk. In this case Otherwise,

Summary Current time: t, Spot price: S, Maturity: T, risk-free interest rate: r Asset Forward/Futures Value of Long Forward Contract Price with Delivery Price K Provides no income Provides income with present value I Provides dividend yield q