人民银行研究院-衍生金融工具讲义.ppt

编制期货标的指数的风险,何佳,可转换公司债券是一种较差的债券形式,何佳,代理人（Agency）和道德风险（Moral Hazard）问题对现代金融理论极其重要。融资工具的设计，除了为企业提供融资手段，为投资者提供投资品种，更重要的是用来解决或减低代理人和道德风险的问题，从而提高企业的社会效益和股东利益。以下通过可转换债券与银行贷款、普通公司债券三者的比较说明，可转换债券在解决代理人和道德风险方面的所起的作用是最差的，从而在提高企业的社会效益和股东利益方面，同银行贷款、普通公司债券相比是最差的结论。1 1关于银行贷款和普通债券的讨论可见Rajan（1992, Journal of Finance），以下关于可转债券的讨论是我个人的见解。,一、 代理人和道德风险问题,在不考虑股权融资的前提下，本文中的代理人问题是指股东与债权人之间的矛盾，一旦企业获得债权融资，企业总有意愿经营下去，那怕这种经营是亏损的，因为亏损的钱是债权人的。 本文所指道德风险的问题是一般意义上的，也是普遍存在的问题，就是企业在获得融资后，企业管理者/所有者的努力程度可以影响企业的业绩，而这种努力强度外部是看不到的，但是对企业管理者/所有者是一种机会成本。,二、 最优债券,理想中的最优债券应该起到两个作用：第一能够阻止企业亏损经营；第二能够刺激企业管理者/所有者尽最大努力工作。,三、 银行贷款,银行的功能就是通过和企业长期合作，从而了解企业的内部信息，这样银行就能发挥阻止企业亏损经营的作用。但是在另一方面由于这种长期合作关系，这家特定的银行对这家特定的企业形成了垄断关系，当企业经营状况较好时，银行会提出和企业分成。这样从某种意义上讲，银行贷款在企业经营状况好的时候，具有了股权性质，这样就扭曲了企业经营者/所有者的积极性。由于努力增加的收入要和别人分享，而努力的成本由企业经营者自己承担，显然这会使企业经营者减少努力。,四、 普通公司债券,普通公司债券持有者只掌握公开信息，再加上免费搭车问题（Free-ride problem），这种债券没有能力阻止企业亏损经营，也正是由于同样原因，当企业经营好时，它不会，也没有能力和企业所有者分享好处，所以也不存在扭曲企业所有者的意愿。,五、 可转换债券,可转换债券持有者，同样只掌握公开信息，再加上免费搭车问题（free-ride problem），也不可能阻止企业进行亏损经营。另外，当企业经营状况好时，由于债务是可换股的，这种转换就使得原来的债权人和原来的企业所有者分享好处，从而导致企业所有者减少应有的努力。,六、 结论,银行贷款和普通公司债券相比，各有利弊。前者有利于解决代理人问题，但会扭曲企业所有者的积极性；后者不会扭曲所有者的积极性，但不能解决代理人问题。而可转换债券则即不能解决代理人问题，同时扭曲企业所有者的积极性。,在我们国家由于存在严重的代理人和道德风险问题，目前尤其应注重发挥商业银行的真正作用，以及发展普通公司债券。 当然，可转换债券可能有一些特点为某一类投资者所喜爱，但是这绝不是问题的关键所在，在资本市场上，我们所讨论的以上三种债券，债权人并不承担任何由于代理人和道德风险造成的成本，债权人在资本市场上整体保持盈亏平衡（break even），他们得到他们所付出的。这是因为资本市场是完全竟争的。所有由于代理人和道德风险问题造成的成本，则是由企业所有者和社会所承担。,Basic concepts What is a derivative?, Contract A derivative is a contract between two parties that specifies conditions in particular, dates and the resulting value of underlying variables under which payments or payoffs are to be made between the parties. Example 1: Social security is a derivative which requires a series of payments from an individual to the government before age 65, and payoffs after age 65 from the government to the individual as long as the individual remains alive.,Example 2: Earthquake insurance is a derivative in which an individual makes regular annual payments in exchange for a potentially much larger payoff from the insurance company should an earthquake destroy his property. Derivative is also known as contingent claims since their payoffs are contingent upon the outcome of underlying variables., Payoff table,Earthquake insurance policy, Subjective probability, Statistics of payoff,Payoff from insurance (X1, X2, Xn) Subjective probabilities (Q1, Q2,Qn) Expected payoff, EX = Q1X1 + Q2X2+ QnXn Variance of payoff , Var(X) = Q1X1-E(X)2 + Q2X2-E(X)2 + + QnXn-E(X)2 Covariance of payoff, Cov(X,Y) = Q1X1-E(X) Y1-E(Y)+ Q2X2-E(X) Y2-E(Y)+ + QnXn-E(X) Yn-E(Y) In our insurance example, E(X)= 1,000, Var(X) 23,931,250, Std = 4,892., Insurance premium,Risk free rate is 5%. For the given expected value a year form now, how much are you willing to pay to by the insurance? 1,000? 1,000/1.05=952.38, adjust for time value of money? People are risk-averse, they would want to pay more, the premium!, Risk-aversion,Another dollar when you are already rich is simply not as valuable to you (in terms of your welfare or utility) as an extra dollar when you are poor. Say your entire wealth is HKD1,000,000. Taking the extreme case, the chance of making a profit of another 1 million dollars is not worth it if it comes with an equal chance of losing 1 million dollars., Risk neutral (risk adjusted) probabilities,A very simple way to adjust for risk-aversion is to weight dollars less than we have in “rich” states and more than we have in “poor” states. Risk-neutral probability = subjective probability * risk aversion adjustment,The expected value under Risk-neutral probability = 1,160 Homeowners present value: 1,160/1.05=1,104.76, Earthquake insurance (all risk diversifiable from insurance company side),Even through the homeowner is willing to pay a premium as high as 1,104.76, the insurance company is willing to sell the policy for a lower 952.38 premium. Competition among insurance companies serves to drive the premium down to 952.38. So the insurance is a good deal for the homeowner! The reason why the insurance company is content to charge 952.38 is that, by selling many policies in different parts of the country, it can diversify away almost all its risk unlike the homeowner who has one house. The insurance company can then act as a risk-neutral investor and change a premium equal to the time-discount expected value., National catastrophe insurance (no risk diversifiable),A national catastrophe is an event such as an economic depression, (or a nuclear warfare), that negatively affects every individual in the economy simultaneously. No diversification is possible in the face of such an event. The law of large numbers does not apply since all individual outcomes are perfectly correlated. In that case, even competitive insurance companies would charge the maximum premium of 1,104.76. More generally, depending on the amount of risk that can be diversified away by the insurance companies, the premium charges will fall between 952.38 and 1,104.76., Inverse problem,Problem: Knowing the markets risk neutral probabilities, determine the market price of derivatives. Inverse problems: Knowing the market price of derivatives, determine the markets risk-neutral probabilities. The art of modern derivatives valuation is to learn as much as possible about the markets risk neutral probabilities from as few derivatives as possible., State-contingent claims and compete markets,Consider: 1. Asset: payoff = 1 2 3, available payoffs = a 2a 3a 2. Cash: payoff = 1 1 1 Total available payoffs = a+c 2a+c 3a+c One can purchase 0 1 2 since 1 2 3 1 1 1= = 0 1 2, but cant purchase 1 0 0,Consider: 1. Asset: payoff = 1 2 3, available payoffs = a 2a 3a 2. Cash: payoff = 1 1 1, available payoffs = a+c 2a+c 3a+c 3. Derivative: Payoff = 1 1 0, available payoffs = a+c+d 2a+c+d 3a+c One can now create “state-contingent claims” 1 0 0 = -1 2 3 + 31 1 1 1 1 0 0 1 0 = 1 2 3 - 31 1 1 +21 1 0 0 0 1 = 01 2 3 + 1 1 1 1 1 0 arbitrary payoffs now possible (”complete markets”) x y z = x1 0 0 + y0 1 0 +z0 0 1, Inverse problem,Asset: S = (1*P1 + 2*P2 + 3P3)/(1+r) Cash: 1/(1+r) = (P1 + P2 + P3)/(1+r) Derivatives: C = (P1 + P2 )/(1+r) P1 = 3- (1+r)(S+C) P2 = (1+r)(S+2C) 3 P3 = 1- (1+r)C, Fundamental theorems of financial economics,1. Risk neutral probabilities exist if only if there are no riskless arbitrage opportunities 2. The risk neutral probabilities are unique if and only if the market is complete 3. Under certain conditions, the ability to revise the portfolio of available securities over time can make up for the missing securities and effectively complete the market., Dynamic replication,Available securities: 1 2 3 and 1 1 1 only. Can we create 1 1 0 to complete the market? Suppose r =0 and S first moves down to 1.5 or up to 2.5 3 2.5 S 2 1.5 1 Replicating strategy (units of asset, dollar of cash) payoff 0 (-1, 3) (-.5, 1.75) payoff 1 (0, 1) payoff 1,Lecture #2: Pricing Forwards & Futures,Example: Hang Sheng Index Futures Specification of HIS futures contracts,We first focus on the similarities between the two and then study the differences between these two. I. Outline: A. No-Arbitrage Principle B. forward/Futures prices Forward/futures prices on stocks (with or without dividends), stock indices, and foreign currencies Commodity futures prices with cost-of-carry and convenience yield Treasury bill futures C. The relationship between forward and futures prices D. The differences between forward and futures contract E. Summary,II. The No-Arbitrage Principle: A. An arbitrage is any trading strategy requiring no cash input that has some probability of making profits, with no risk of loss. For example, a portfolio with zero price and strictly positive payoff, or one with a negative price and a non-negative payoff. B. No-Arbitrage Principle: There is no arbitrage opportunity in financial markets. C. In an efficiently functioning financial market arbitrage opportunities cannot exist (for very long). D. This rule implies that: Two securities that have the same payoff must have the same price.,E. Unlike equilibrium rules such as the CAPM, arbitrage rules require only that there be one intelligent investor in the economy. This is why derivatives security-pricing models (based on no arbitrage principle) do a better job of predicting prices than equilibrium based models. F. If arbitrage rules are violated, then unlimited risk-free profits are possible. G. To prove these arbitrage-rules, we will show that if they are violated, arbitrage profits are possible.,III. Forward/Futures Prices A. Forward/Futures Prices (with no benefits or cost of holding the underlying security) 1. Assumptions There are no market frictions No counterparty risk Markets are competitive No arbitrage opportunities,2. Equilibrium Argument: Suppose you are interested in buying 1 hare of IBM stocks, which is trading at $100/share. You can either (1) buy it today; or (2) commit to buy in 6-month. Suppose the annualized, continuously compounded risk-free interest rate is 4%. What must the forward/futures price of the stock be? 1 IBM 1 IBM |_| Today 6-month, If you sign a futures contract today to purchase the share in 6-month, the cost will be F(t,T) in 6 months(t is now, and T is 6 month from now.) If you buy the share today on the spot market, you pay S(t) = 100 now. In equilibrium, you should be indifferent between the two methods: F(t,T) =S(t) er(T-t) =$100*e0.04*0.5 = $102.02,3. Arbitrage Argument: Why must the forward price be equal to $102.02? What if the forward price were $103.00/share? (Use Cash-and-carry strategy), What if the forward price were $101.00/share?,B. Pricing when underlying instrument pays a fixed dividend: 1. What if the stock pays a $1 dividend in 3 months? Now if we buy the stock early we capture the dividend. What must be the forward/futures price of the stock be? 1 IBM $1 1 IBM |_|_| Today 3-month 6-month Two ways of acquiring the IBM shares in 6-month: Use a (long) forward contract and spend F(t,T) in 6 months to buy 1 IBM in 6-onth; Spend PV(1 IBM in 6mos.) now to ensure that we have 1 IBM in 6 months $100 = S(t) = PV($1 in 3 mos.) +PV(1 IBM in 6 mos.). In equilibrium, we must have F(t,T) = S(t)-PV($1 in 3 mos.)er(T-t) =$100-$1*e-0.04*0.25 e0.04*0.5 = 101.01,2. Intuitively, the futures (forward) price is equal to the spot price plus the “cost of carry,“ which is interest cost in carrying the spot minus the dividend (benefit) of holding the spot 3. How to construct an arbitrage to prove that the futures price must be equal to $101.01? F(t,T) = S(t)-PV(D(t1)er(T-t),If LHS RHS, then we cab arbitrage: If the futures price is not the value given on the previous page, we see that arbitrage profits are possible. Assumptions made: No transaction cost, no short sale cost, and no borrowing constraints. These assumptions only have to hold for one individual.,C. Pricing when the underlying instrument pays a continuous/proportional dividend: 1. In some cases, the dividend/cost is (1) proportional to the price of the asset, and (2) is paid continuously. For example, we generally assume this when pricing stock index futures. Example: Consider a 6-month futures on the S if the dividend yield (q) is 3%/year, the index is currently at 1000, and the risk-free rate is 4%, what is the futures price?, The “underlying asset“ here is not 1 index: - If we buy the shares in the index today for $1000 and then sold shares and reinvested the dividends proportionally in the shares, at the end of the six months we will have eq(T-t)=1.01511 shares in the index at the end of six months. The underlying asset is e-q(T-t) indices: - If we purchase e-q(T-t) shares now, it will give us 1 index at futures maturity. The cost of the (today) is e-q(T-t)S(t) = e-.03*.51000 = 985.10. This means that the correct index futures price is F(t,T)=e-q(T-t)S(t)er(T-t) = S(t)e(r-q)(T-t) = $1005,2. If the price is different from this index arbitrage is possible. 3. What should you do to take advantage of the arbitrage opportunity if the index futures price were $1010?,A few points Index arbitrageurs assures that spot and futures prices are properly aligned The way we derive futures prices off the spot-market prices may cause you to believe that, in terms of price discovery, futures market prices are driven by spot market prices. This belief is, however, incorrect - price discovery usually occurs in the futures market first.,D. Pricing Foreign currency futures 1. Pricing foreign currency futures is very similar to pricing index futures - a unit of foreign currency can be thought of as a stock with a continuous dividend yield that is equal to the foreign interest rate. Example: Assume that the current USD-DM exchange rate is $0.67/DM.The US and Germany interest rate are r(US)=4%, r(DM)=6%. What is the futures price of DM for a 6-month futures contract?,F(t,T)=S(t)*e-r(DM)(T-t)er(US)(T-t) =S(t)e(r(US)-r(DM)(T-t) =$0.67/DM*e-.02*.5=$.663/DM The “underlying asset“ of a 6-month USD-DM futures contract is not 1 DEM today, but rather e-r(DM)(T-t)DM today. Notice that the DM interest rate is treated like a continuous proportional dividend paid on the DM. Again, an arbitrage opportunity exists if this relationship is not satisfied.,3. How could you take advantage of the arbitrage opportunity if the futures price of a DM were $0.65?,E. Pricing commodity futures with storage cost and convenience yield: 1. Storage costs include the cost of spoilage, ect. A higher storage cost increases the futures price level to the spot. 2. If, at initiation, we know that the PV of total storage cost from now to the maturity date is U, we can treat this (roughly) like a negative, known dividend: F(t,T) = S(t)+Uer(T-t),3. In many cases we assume that the cost is (1) proportional to the price of the asset, and (2) is paid continuously at rate . This means that the cost of storage can be treated like a negative dividend yield: F(t,T) = S(t)*e(T-t)er(T-t) 4. This means that futures prices should increase with maturity at the rate of interest plus the cost of storage. Is this what we observe in reality? If we do not, is there an arbitrage opportunity?,5. Convenience yield is defined to be the “fudge factor“ that makes the above relation an equality: F(t,T) = S(t)*e(r+-y)(T-t) The convenience yield can only be positive What is the convenience yield for a financial futures?,F. Pricing Treasury Bill futures 1. We price these like futures on non-dividend paying stocks 2. Example: What is the four months (Sept) futures price for a 3-month T-Bill, assuming that: The annualized 4-month interest rate, from May to September (4 months), is 4% (c.c.) and The annualized 7-month interest rate, from May to December, is 5% (c.c.) $100 |_|_| t T T+3/12 (May) (September) (December),The standard futures/spot relation can be used to price the T-Bill futures: F(t,T) = S(t)* er(t,T)*(T-t)= =$100*e-0.5*7/12*e.04*4/12= = $97.13*1.0134 = 98.42 3. And again, an arbitrage opportunity exists if this relationship is not satisfied. To see this, note that the following two transactions each pay $100 in December: A. Buy a $100 T-Bill maturing in 7-month B. Buy T-Bill maturing in 4-month, which pay F(t,T) = $98.42, and then in September use the proceeds from the T-Bill to buy a new 3-month T-Bill at a (guaranteed) price of $98.42.,IV . Summary of Forward/Futures Price Formulas: A. Equilibrium intuition for financial forward/futures: Our formulas for financial forward/futures prices can be summarized as: F(t,T) = t er(t,T)(T-t) where F(t,T): forward/futures price r(t,T): effective rate between t and T t: amount of money needed at t for a strategy that generates one share of the underlying security at T.,Strategies used: Stock (no dividend): buy a stock at t and hold until T. Stock (known dividend): borrow PV(dividend) and buy a stock at t, pay debt with dividend and hold stock till T. Stock (known dividend yield): buy e-q(T-t) shares of stocks and reinvest dividends into the stock and end up with one share at T. Foreign currency: buy e-r*(T-t) units of foreign currency and earn foreign interest and end up with one unit of foreign currency at T. Treasury Bill: buy one T-bill at t at price e-r(t,T*)(T*-t) and hold it until T.,B. General formulas (including commodity futures): F(t,T) = (St+U-D)e(r+u-q-y)(T-t) Where St : Current spot price U: PV of storage cost between t and T D: PV all dividends between t and T r: risk-free rate of interest (c.c.) between t and T u: proportional storage cost rate between t and T q: proportional dividend yield between t and T y: convenience yield between t and T Note: You will use either U or u, and either D or q.,V . Cost of carry in proportional terms F(t,T) = St*e(c-y)(T-t) Where c=r+u-q is the (proportional) cost of carry. III. Relation between forward and futures prices A. If interest rate are zero, forward contracts and futures contracts are basically the same. 1. Consider a long futures contract and a long forward contract on the same spot security, both initiated on day 0