Forwards, Swaps, Futures and Options

1 IEOR E4706: Foundations of Financial Engineeringc 2016 by Martin HaughForwards, Swaps, Futures and OptionsThese notes1introduce forwards, swaps, Futures and Options as well as the basic mechanics of their associatedmarkets. We will also see how to price forwards and swaps, but we will defer the pricing of Futures contractsuntil after we have studied martingale pricing. We will see how to price Options within the binomial the exception of the binomial model in Section 4, the underlying probability structure of the financialmarket plays only a small role in these notes.

2 Nonetheless, you should not be under the impression that theresults we derive only hold for deterministic models and are therefore limited in scope. On the contrary, many ofthe results we derive are very general and hold irrespective of the underlying probability structure that we mightfind ourselves working , we mention that it is easy to compute the value of a deterministic cash flow given the currentterm-structure of interest rates and we will often make use of this observation when pricing forwards and securities withstochasticcash-flows is more complicated and requires more sophisticated no-arbitrage orequilibrium methods.

3 The binomial model of Section 4, however, provides a simple yet important model forintroducing some of these methods. We will study them in more generality and much greater detail when westudy martingale pricing later in the ForwardsDefinition1A forward contract on a security (or commodity) is a contract agreed upon at datet= 0topurchase or sell the security at dateTfor a price,F, that is specified att= the forward contract is established at datet= 0, the forward price,F, is set in such a way that the initialvalueof the forward contract,f0, satisfiesf0= 0.

4 At the maturity date,T, the value of the contract is given2byfT= (ST F)whereSTis the timeTvalue of the underlying security (or commodity). It is veryimportant to realize that there are two prices or values associated with a forward contract at timet:ftandF. When we use the term contract value or forward value we will always be referring toft, whereaswhen we use the term contract price or forward price we will always be referring toF. That said, thereshould never be any ambiguity sinceftis fixed (equal to zero) att= 0, andFis fixed for allt >0so theparticular quantity in question should be clear from the context.

5 Note thatftneed not be (and generally is not)equal to zero fort > of forward contracts include: A forward contract for delivery ( purchase) of a non-dividend paying stock with maturity6months. A forward contract for delivery of a9-month T-Bill with maturity3months. (This means that upondelivery, the T-Bill has9months to maturity.) A forward contract for the sale of gold with maturity1year. A forward contract for delivery of10m Euro (in exchange for dollars) with notes draw heavily from David Luenberger sInvestment Science(Oxford University Press, 1997).

6 2If the contact specifies a purchase of the security then the dateTpayoff isST Fwhereas if the contact specifies a sale ofthe security then the payoff isF , Swaps, Futures and Computing Forward PricesWe first consider forward contracts on securities that can bestoredat zero cost. The origin of the term stored is that of forward contracts on commodities such as gold or oil which typically are costly to store. However, wewill also use the term when referring to financial securities. For example, while non-dividend paying stocks andzero-coupon bonds may be stored at zero cost, it is also the case that dividend paying stocks and coupon payingbonds can be stored at Contracts on Securities with Zero Storage CostsSuppose a security can be stored at zero cost and that short3selling is allowed.

7 Then the forward price,F, att= 0for delivery of that security at dateTis given byF=S/d(0,T)(1)whereSis the current spot price of the security andd(0,T)is the discount factor applying to the interval[0,T].Proof:The proof works by constructing an arbitrage portfolio ifF6=S/d(0,T).Case (i):F < S/d(0,T):Consider the portfolio that at datet= 0is short one unit of the security, lendsSuntil dateT, and is long one forward contract. The initial cost of this portfolio is0and it has a positive payoff,S/d(0,T) F, at dateT. Hence it is an (ii):F > S/d(0,T):In this case, construct the reverse portfolio and again obtain an 1(A Forward on a Non-Dividend Paying Stock)Consider a forward contract on a non-dividend paying stock that matures in6months.

8 The current stock priceis$50and the6-month interest rate is4%per annum. Compute the forward price, :Assuming semi-annual compounding, the discount factor is given byd(0,.5) = 1 = (1) then implies thatF= 50 = Contracts on Securities with Non-Zero Storage CostsSuppose now that we wish to compute the forward price of a security that has non-zero storage costs. We willassume that we are working in a multi-period setting and that the security has a deterministic holding cost ofc(j)in periodj, payable at the beginning of the period. Note that for a commodity,c(j)will generally representa true holding cost, whereas for a stock or bond,c(j)will be a negative cost and represent a dividend or Price for a Security with Non-Zero Storage Costs:Suppose a security can be stored forperiodjat a cost ofc(j), payable at the beginning of the period.

9 Assuming that the security may also be soldshort, then the forward price,F, for delivery of that security at dateT(assumed to beMperiods away) is givenbyF=Sd(0,M)+M 1 j=0c(j)d(j,M)(2)whereSis the current spot price of the security andd(j,M)is the discount factor between :As before, we could prove (2) using an arbitrage argument. An alternative proof is to consider thestrategy of buying one unit of the security on the spot market att= 0, and simultaneously entering a forwardcontract to deliver it at timeT. The cash-flow associated with this strategy is( S c(0), c(1).)

10 , c(j), .. , c(M 1), F)3 The act of short-selling a security is achieved by first borrowing the security from somebody and then selling it in themarket. Eventually the security is repurchased and returned to the original lender. Note that a profit (loss) is made if thesecurity price fell (rose) in value between the times it was sold and purchased in the , Swaps, Futures and Options3and its present value must (why?) be equal to zero. Since the cash-flow is deterministic we know how tocompute its present value and we easily obtain (2).Example 2(A Bond Forward)Consider a forward contract on a4-year bond with maturity1year.