chapter4-1无文字金融工程专业.ppt

1,Chapter 4 Futures and Forwards Prices ,2,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.,Objective of this chapter,3,1、PRELIMINARIES, 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.,4,1、PRELIMINARIES, notation We will use a lot of notation, so lets 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,5,1、PRELIMINARIES,. Continuous Compounding Compound interest is paid on the original principal and on the accumulated past interest. Notation: T: Investment period(years) PV: Present Value of initial investment FV: Future Value of initial investment R: Nominal interest rate per annum m: Compounding frequency,6,Continued,When the interest is compounded once a year for T years: What if interest is paid more frequently? Here are a few examples of the formula:,m=1,m=4,m=12,7,continued,Consider an amount PV invested for T years at an interest rate of R per annum.,The limit as m tends to infinity is known as continuous compounding,8,示例,例：假设存款金额为100，名义年利率为10%，存款期限为1年，在年度计息的条件下，明年的期末存款余额为： 1001.1 = 110 半年计息一次： 1001.051.05 = 110.25 每季度计息一次： 1001.0254110.38 连续复利： 100e0.10=110.52,9,实际利率与计息次数,10,continued,The limit as m tends to infinity is known as continuous compounding The standard future and present value formulas are: PVFV：PV* eRT FV FVPV：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 that cash flow is:,11,continued,You should be aware of how to convert between interest rates of different compounding frequency. For example, if ry is the annually compounded rate of return what is the equivalent quarterly compounded rate of return? You can set this up easily enough by realizing that if you invest in both for a given time period (say 1 year) , you wind up with the same amount of cash, that is:,12,continued,In general to convert between a rate r1 that has compounding frequency m1 and rate r2 that has compounding frequency m2, you use this formula: So, if you had a semi-annual rate of 6% and wanted to know the equivalent quarterly rate:,13,continued,To convert between a discrete rate, rd with compounding frequency md and a continuously compounded rate rc, use: (this assumes t=1),A deposit account pays 12% per annum with continuous compounding, but interest is actually paid quarterly.How much interest will be paid each quarter on a $10,000 deposit?,14,1、PRELIMINARIES,.short selling Recall that when you take a short position in an instrument, it means that you are giving up control over that instrument. Short selling/融券: selling an asset that is not owned with the intention of buying it back later. When you “short” a stock you borrower the stock (get control) and then sell it (lose control.),15,short selling,investor,16,1.寻找卖空交易的股票. 要制定一个卖空交易，交易商A必须首先确认他能够借到的他计划卖空的股票数量。经纪人保持一个称为一箱列表（A Box List)，列出那些可以做卖空交易的股票清单。经纪人通过以下方式更新这个列表：经纪人自己的股票清单，经纪人客户同意借出的股票，以及其他第三方经纪人的股票。,17,2.执行一个卖空交易. 在确认了可以做卖空交易的股票后，交易商A在交易日，或称作“T”，执行一个卖空交易。大部分股票交易市场有为期三天的结算期。就是在交易日之后三天内，或“T3”，完成股票换现金的交易。,18,3.借到卖空交易的股票和完成卖空交易. 在T3的早上，交易商A的证券借贷部这一天实际交付做卖空股票的义务。他们查看他们的列表，用列表中的股票来完成卖空交易。,19,4.卖空收益的现金用做借来股票的抵押. 当交易结算时，从出售股票收到的现金作为交易商A的借来的股票的抵押。交易商A的经纪人用抵押的现金来投资。,20,5.利息付给卖空的出售人 抵押现金投资的利息收入的一部分用来支付管理费和股票借入费。由于管理费用的高成本，结余的利息通常只会付给那些大宗卖空交易的卖者。,21,6.通过卖空卖者支付的股息 （如果要求的话）. 如果卖空的股票支付股息的话，借出股票的人有收取股息的资格，因为他并没有卖掉这些股票，而仅仅是借出而已。,22,在未来的某个时候，或者是当交易商A决定回购他的卖空股票，或者股票借贷人要收回股票.在第一种情况下，卖空交易就结束了。在第二者情况下，经纪人会设法寻找另一个借贷方,借贷将转移到新的交易人或经纪人。交易商A的卖空交易将不受影响。如果无法找到替代的借贷方的话，经纪人可能通知交易商A借贷已经被收回，交易商A必须立刻平仓他的卖空。,23,24,卖空的特点,特点 被卖空证券是经纪商从其他客户账户中借用出来的。 在平仓之前，证券所发生的现金流入，比如红利或者利息等，都由卖空者代为支付。 卖空所得资金由经纪商进行保管，同时卖空者还需向经纪商缴纳一定比例的保证金。 当卖空者平仓时，经纪商要在市场上买回被卖空证券。 当被借证券的原所有者需要时，经纪人会设法寻找另一个借贷方,否则卖空者必须立即平仓，从市场上买回被卖空证券。,25,1、PRELIMINARIES,.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. the Law of One Price,26,continued,Asset prices are obtained from conditions that preclude arbitrage opportunities. We say that the price of a security is a “fair” level,or that the security is correctly priced,if there are no arbitrage opportunities. Determining arbitrage-free prices is at the center of valuing derivative assets.,27,1、PRELIMINARIES,. Investment Assets vs Consumption Assets 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),arbitrage arguments can be used to obtain exact forward prices in case of investment assets,can only give an upper bound to forward prices,28,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. There are no transactions costs. All trading profits (net of losses) are subject to the same tax rate. The market participants can borrow or lend at the same risk-free interest rate. Market participants can take advantage of arbitrage opportunities when they occur.,Assumptions,29,2、The concept of price versus value,The price in a forward contract is known as the delivery price(K). The forward price (F) for a particular forward contract at a particular time is the delivery price (k) that would apply if the contract were entered into at that time (F=K at time zero). 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.,30,远期价格是指使远期合约签订时价值为零的交割价格。远期价格是理论上的交割价格。关于远期价格的讨论也要分远期合约签订时和签订后两种情形。 一份公平合理的远期合约在签订的当天应使交割价格等于远期价格。如果实际交割价格不等于这个理论上的远期价格，该远期合约价值对于多空双方来说就都不为零 ，实际上隐含了套利空间。 在远期合约签订之后，交割价格已经确定，远期合约价值不一定为零，远期价格也就不一定等于交割价格。,30,类似地，在期货合约中，我们定义期货价格（Futures Prices）为使得期货合约价值为零的理论交割价格。,31,32,continued,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.,At time T (long position),How do we determine the value of the forward at time t0?,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。,33,34,Buy the old futures contract,Sell the new one,Deliver the two contracts and make a profit of FtK,f new=0,f t=(Ft - K) e-r(T-t),35,The value of forward contract,The present value of this difference is the value of the forward/futures contract (long position) at time t, ft, ft = (Ft-K)e-r(T-t),36,3. forward price for an investment asset,Three kinds of investment assets: 、Investment assets providing no income 、Investment assets providing a known cash income 、Investment assets providing a known dividend yield,37,2.1 forward price for an investment asset providing no income,Intuition It is very easy to get bogged down in the exact details of the various pricing formulas that we will use, so lets first consider the economic intuition that underlies all of these formulas. Consider when two parties enter into a forward contract on a non-income producing asset. The long party (LP) agrees to pay the short party (SP) K dollars at time T in exchange for the underlying asset. From the viewpoint of SP, SP is holding the asset on their behalf until the maturity date. Assuming no credit risk this means that it is the same as if SP bought the underlying asset today, held on to it for T years, and then sold it at price K.,38,Intuition,From the shorts perspective there is no risk in this contract. They pay the current market (spot) price for the asset (S0), they hold it for T years, and then they sell it for K dollars. This cost S0 and is certain to lead to a cash inflow of K/F0 at time T.,So how would the short party determine the delivery price that they would be willing to enter into? Look at it this way, they have to pay S0 dollars today, and they want to earn the risk-free rate on those dollars, so the delivery price K or the forward price at time zero F0 must be F0 = K = S0erT!,39,A formal prove,Consider the following portfolios(at time 0 ): Portfolio A: a long forward contract and a cash amount of on hand. Portfolio B: one unit of underlying asset.,A unit of asset,buy,40,A formal prove,Notice, then, that both portfolios A and B each give you the same end result or valuethe value of the security at maturity. On condition that there are no arbitrage opportunities, at some earlier time, such as time 0, these two portfolios should be identical too. If not, if AB or BA, then you would short the expensive one and go long the inexpensive