3.1 Determination of Forward and Futures Prices Hull, chapter 3

3.2 Agenda We will study the relation between forward-, futures-, and spot prices. Discuss special characteristics of investment assets, consumption assets, currencies, dividend paying etc. We concentrate on forwards and ignore the effect of daily settlement. Home assignment: Go through/prove Appendix 3A.

3.3 Notation S0:S0:Current spot price F0:F0:Current futures- or forward- price T: Time to maturity/delivery r: Riskless interest rate for time T No transactions costs, no (or symmetric) taxes, unlimited borrowing/ lending at same interest rate, etc. Must hold at least for large investors.

3.4 Forward price for investment asset (Rep. chap 1) For a non-dividend paying investment asset (no storage costs) we have F 0 = S 0 (1 + r ) T Using continuous compounding: F 0 = S 0 e rT Continous compounding/discounting will be dominant throughout the course. I assume you a familiar with it – if not read about it in chap. 3 and/or ask questions now.

3.5 Why must these relations hold? Suppose: F 0 > S 0 e rT Forward price is relatively (too) high! –Short 1 forward, costs nothing now –Borrow S 0 –Buy underlying asset now and hold until T. Arbitrage profit at time T: F 0 - S 0 e rT >0.

3.6 Or conversely Suppose: F 0 < S 0 e rT Forward price is too low! –Buy forward (go long), costs nothing now –Short underlying and receive S 0 –Deposit S 0 at riskless interest rate until T. Arbitrage profit: S 0 e rT - F 0 >0.

3.7 Principle of no arbitrage Fundamental principle! Foundation of relative pricing. We have seen simple application here. Things will get more advanced but fundamental principle is the same. Most important premise of this course!

3.8 Extension to known (!) yields/dividends during contract period Claim: We have F 0 = (S 0 – I )e rT where I is present value of yields during contract period. Is it true? If so, why?

3.9 Discussion A lower forward price is intuitive. The forward contract does not give right to dividends/coupons from underlying during contract period. Therefore you will contract on a lower price, ceteris paribus. Arbitrage-argument: Consider the following strategy: Buy underlying and short 1 forward. That will cost you S 0 now. At time T you will receive F 0 for delivering the underlying. Present value of this: e -rT F 0. In addition you will receive certain dividend yields with a present value of I. No arbitrage  S 0 = I + e -rT F 0.

3.10 Example p. 47

3.11 When an Investment Asset Provides a Known Continuous Yield (Page 49, equation 3.7) Then we have F 0 = S 0 e (r–q )T where q is the average yield during the life of the contract (expressed with continuous compounding) Are there assets which pay out continuously??

3.12 Arbitrage-argument Buy e -qT units of the underlying and short one forward/future. That will cost e -qT S 0. Reinvest all dividends immediately in underlying. That yields e -qT e qT =1 unit of the underlying at maturity. Deliver underlying at maturity and receive F 0. Present value is = e -rT F 0 Conclusion : e -rT F 0 = e -qT S 0

3.13 Example p. 49

3.14 Valuing a Forward Contract Page 50 Suppose that K is delivery price in an in-progress forward contract F 0 is forward price that would apply to the contract today The value of a long forward contract, ƒ, is ƒ = (F 0 – K )e –rT Similarly, the value of a short forward contract is (K – F 0 )e –rT This is intuitive: A new contract (K = F 0 ) has value 0 today. If K is low, value is high etc. Note pricing as if assuming future spot price is F 0 !!!

3.15 Argument/proof Suppose you are long in forward with K. You can short a forward on F 0 at no cost. This results is a certain cash flow of F 0 - K at time T and no further obligations. No arbitrage requires f = e -rT (F 0 -K) Payoff must be riskless for the discounting to be correct. It is.

3.16

3.17 Forward vs Futures Prices Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different: A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price A strong negative correlation implies the reverse Work through App 3A.

3.18 Stock Indeces and futures Stock indeces are weighted average prices (S&P500, DJIA, NASDAQ100, KFX, KVX etc). Dividends are usually not taken into account. The indeces are not adjusted. They just mirror the prices. The index can therefore be seen/behaves as a dividend paying financial asset. Relation between spot and futures price is therefore F 0 = S 0 e (r–q )T where q is dividend rate on portfolio represented by the index.

3.19 Index Arbitrage When F 0 >S 0 e (r-q)T buy underlying index portfolio and short the future. When F 0 <S 0 e (r-q)T buy future and (short)sell underlying portfolio.

3.20 Index Arbitrage (continued) Index arbitrage involves simultaneous trades in futures and many different stocks Computers are typically used to generate the trades Occasionally (e.g., on Black Monday) simultaneous trades are not possible (execution of trades is delayed) and the theoretical no-arbitrage relationship between F 0 and S 0 does not hold

3.21 A foreign currency is analogous to a security providing a dividend yield The continuous dividend yield is the foreign risk-free interest rate It follows that if r f is the foreign risk-free interest rate Basically the international interest rate parity. We shall return to it later. Futures and Forwards on Currencies (Page 55-58)

3.22 Futures on Consumption Commodities (Page 59) No arbitrage ”only” implies F 0  S 0 e (r+u )T where u is the storage cost per unit time as a percent of the asset value. Alternatively, F 0  (S 0 +U )e rT where U is the present value of the storage costs. Corresponds to negative dividend yield. You “save” storage costs by holding future. You cannot exploit this inequality if you hold asset mainly for consumption purpose.

3.23 The Cost of Carry (Page 60) The cost of carry, c, is the storage cost plus the interest costs less the income earned For a non-d. paying investment asset F 0 = S 0 e cT = S 0 e rT Ie. –c=r for non-dividend paying asset –c=r-q dividend paying asset (ex. index) –c=r-r f for currencies –c=r+u for commodity w. storage costs. For a consumption asset F 0  S 0 e cT The convenience yield on the consumption asset, y, is defined so that F 0 = S 0 e (c–y )T

3.24 Futures Prices & Expected Future Spot Prices – Is there a relation?? (Page 61) Suppose k is the expected return required by investors on an asset (we have models like the CAPM to determine that). We can invest F 0 e –r T now to get S T back at maturity of the futures contract (Buy the future and deposit F 0 e –r T in bank account.) This shows that F 0 = E (S T )e (r–k )T

3.25 Futures Prices & Future Spot Prices (continued) If the asset has –no systematic risk, then k = r and F 0 is an unbiased estimate of S T –positive systematic risk, then k > r and F 0 < E (S T ) –negative systematic risk, then k E (S T ) What if the underlying is a well- diversified index?