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Unformatted text preview: Rc = m ln (1 + ) Where Rc = rate of interest continuously compounding And Rm = Equivalent rate with compounding m times per annum. Lecture 2 (Ch5): Pricing Forwards & Futures Investment Assets vs. Consumption Assets When considering forward and futures contracts, it is important to distinguish between investment assets and consumption assets. An investment asset is an asset that is held for investment purposes by a significant number of investors. o Gold & Silver. A consumption asset is an asset that is held primarily for consumption. o Copper, oil & pork bellies. Assumptions & Notation Assume that the following is true for some market participants: 1. Market participants are subject to no transaction costs when they trade. 2. Market participants are subject to the same tax rate on all net trading profits 3. The market participants can borrow money at the same risk-free rate of interest as they can lend money. 4. The market participants take advantage of arbitrage opportunities as soon as they occur. Note: we do not require ALL assumptions to be true for ALL market participants. We require they be true for at least a few key market participants such as large derivative dealers. T : Time until delivery date in the forward or future contract (in years). S : Price of the asset underlying the forward or futures contract today. F : Forward or futures price today. r : Zero-coupon risk-free rate of interest per annum, expressed with continuous compounding, for an investment maturing at the delivery date. What Determines the Forward Price: No-arbitrage Argument There are two ways to acquire an asset for some date in the future. For example: o Purchase it now and store it o Take a long position in a forward contract Assumptions: o No transaction costs o Same tax rate for all trading profits o Borrowing rate = lending rate = risk-free rate of interest o Underlying asset pays no income , has no storage costs , no seasonal patterns in prices. Let us form two portfolios (A & B): Portfolio A : Buy the asset now and hold it until time T. Portfolio B : Take a long forward position today for delivery at time T with delivery price of F t (determined today). Since portfolios A & B can both acquire the same asset at time T, if there is no arbitrage opportunity, the cash flow either at time t or T must be the same : F t = S t e r(T t) r is risk-free rate of interest per annum at time t , with continuous compounding, for an investment maturing at T When: F t S t e r(T t) We now have an arbitrage opportunity....
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