4
1、PRELIMINARIES

⑶. Continuous Compounding Compound interest is paid on the original principal and on the accumulated past interest.

Notation:
T: PV: FV: R: m:
股指期货定价
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Our objective is to link the price of the futures or forward contract to the price of the underlying instrument and to identify factors that influence the relationship between these prices.
Three kinds of investment assets: ⑴、Investment assets providing no income ⑵、Investment assets providing a known cash income ⑶、Investment assets providing a known dividend yield
close out short deposit proceeds
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1、PRELIMINARIES

⑸.Arbitrage and the Law of One price



All current methods of pricing derivative assets utilize the notion of arbitrage. Arbitrage is a type of transaction in which an investor seeks to profit when the same good sells for two different prices. If two investment opportunities offer equivalent outcomes,they must have equivalent prices.
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Continued

When the interest is compounded once a year for T years:
m=1
FV PV (1 R)T

What if interest is paid more frequently? Here are a few examples of the formula:
can only give an upper bound to forward prices
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Assumptions

In order for the pricing method presented in this lecture to work, we need for there to be a least some participants in the market for which the following are true, or at least are close to being true. 1. There are no transactions costs. 2. All trading profits (net of losses) are subject to the same tax rate. 3. The market participants can borrow or lend at the same risk-free interest rate. 4. Market participants can take advantage of arbitrage opportunities when they occur.
m=4
R 4T FV PV (1 ) （quarterly compounding ) 4 R 12T FV PV (1 ) （monthly compounding ) 12 m=12
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continued

Consider an amount PV invested for T years at an interest rate of R per annum.
但值得注意的是，对于期货合约来说，一般较少谈及“期货合约价值”这 个概念。基于期货的交易机制，由于期货每日盯市结算、每日结清浮动盈 亏，因此期货合约价值在每日收盘后都归零。
The value of a forward contract f is the profit on this contract. For forward contracts, no cash is paid out up front,so the contracts have zero value when first written.
m=1 m>1 m→
Compounding frequency(m) Future value after n years (FV)

PV(1 R)
T
R mT PV(1 ) m
PV e
RT
The limit as m tends to infinity is known as continuous compounding


Keep some things in mind: K and F are not the same thing. At time zero they are, but K is set in the contract and F moves over time.
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continued



When you “short” a stock you borrower the stock (get control) and then sell it (lose control.)
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short selling
use funds investor
buy replace broker borrow sell another investor


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有效年利率与计息次数
0.106 0.105 0.104 0.103 0.102 0.101 0.1 0.099 0 100 200 300 400 系列1
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continued

The limit as m tends to infinity is known as continuous compounding

Investment assets:
assets held by significant numbers of people purely for investment purposes (Examples: bond, gold)

Consumption assets:
assets held primarily for consumption purposes (Examples: copper, oil)
3
1、PRELIMINARIES


⑵ notation
We will use a lot of notation, so let’s be very clear about what each symbol means:

T = maturity date of the forward contract t = current time St = price of underlying at time t ST= Price of underlying at time T K = delivery price in the forward contract Ft= forward price at time t f = value of a long forward contract at time t. r = risk free rate
The standard future and present value formulas are:



PV→FV：PV* eRT＝ FV FV→PV：FV* e-RT ＝PV

For example, if your discount rate is 8%, and you are going to receive $200 in 2 years, the present value of thatrward and futures prices Forward prices Spot price of the underlying asset
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1、PRELIMINARIES

⑴ for many of the contracts that are traded in the market, it can be argued that the forward price and futures price of an asset are very close to each other when the maturities of the two contracts are the same.

Investment period(years) Present Value of initial investment Future Value of initial investment Nominal interest rate per annum Compounding frequency
PV 200e.08(2) 170.43
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