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Unformatted text preview: Lecture 6 Cash Flows, Forward Contracts and Futures We like to broaden our studies in arbitrage pricing from call and put options to other kinds of derivative securities in order to suit the need for quantitative analysis of the evergrowing financial markets. Cash flows, forward contracts, futures and their valuation will be considered in this lecture. Some previous concepts and results require modifications although the basic principle still holds. 6.1 Dividends and returns Assume the basic setup in Lecture 2. A cash flow can be thought of as a contract with a value V ( t ) at time t , in which one party of the contract will receive cash payments from the other party on certain future dates. A good example of cash flows is the dividends associated with a stock that we mentioned in Section 5.3. For n = 1 ,...,N and t = 0 , 1 ,...,T , let D n ( t ) be the dividend per unit of security n issued at time t , in particular D n (0) = 0. Let S n ( t ) represent the exdividend price of security n , i.e. the price right after any time t dividend payment. Assume the dividend process is adapted. To check whether arbitrage opportunities exist, it is better to look at returns rather than security prices. Note that a holder of one unit of security n at time t 1 will earn a profit Δ S n ( t ) + D n ( t ) over the period ( t 1 ,t ]. The return process R n = { R n ( t ) } is defined as follows: R n (0) = 0, and for t = 1 ,...,T , R n ( t ) = Δ S n ( t )+ D n ( t ) S n ( t 1) , if S n ( t 1) > , if S n ( t 1) = 0 . (6.1) In particular, R ( t ) = r ( t ) for the bank account. Moreover, the return process for S * n is defined by R * n (0) = 0, and for t = 1 ,...,T with Δ S * n ( t ) = S * n ( t ) S * n ( t 1), R * n ( t ) = Δ S * n ( t )+ D n ( t ) /B ( t ) S * n ( t 1) , if S * n ( t 1) > , if S * n ( t 1) = 0 . (6.2) Also, R * ( t ) ≡ 0 for the bank account. If X = { X ( t ) , t = 0 , 1 ,...,T } is a martingale under a probability measure Q and with respect to FF , then the increment process Δ X = { Δ X ( t ) } is called the corresponding martingale difference sequence , i.e. E Q (Δ X ( t )  F t 1 ) = 0 ∀ t = 1 ,...,T. Here is a modification of Theorem 3.1 in the case of dividendpaying securities. 1 Theorem 6.1 No arbitrage ⇐⇒ there is a probability measure Q with Q ( ω ) > ∀ ω ∈ Ω , such that R * n = { R * n ( t ) , t = 1 ,...,T } is a Qmartingale difference sequence, n = 1 ,...,N . We still call Q an EMM. The proof of Theorem 3.1 still applies if we replace Δ S n ( t ) there by Δ S n ( t ) + D n ( t ). 6.2 Forward contracts and prices A forward contract is an agreement made between two parties at time t in which the buyer agrees to buy an underlying asset on a certain specified future date τ (with t < τ ≤ T ) for a delivery price ; while the seller agrees to sell the asset on the same date τ for the same price. At the maturity date τ , the seller delivers the asset to the buyer in return for a cash payment equal to the delivery...
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 Spring '08
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 Forward contract

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