The Curvature and Slope in Credit Spread Term Structure Curve.ppt

1、The Curvature and Slope in Credit Spread Term Structure Curve,2,Motivation,risk-free Interest rate term structure dynamics of expected future. pricing. risky YY* Credit Spread expected future credit risk (dt, lt) Credit spread term structure dynamics of expected future pricing. Pricing credit risk,

2、credit-risky assets, credit derivatives Curve shape credit quality, credit cycle, business cycle Credit risk management (e.g. buy a put option on its own credit spread to hedge),Asia financial crisis US high yield market,3,Example: A Risky Bond Rated BB, How much will it be worth next year? How much

3、 will I lose, if next year is bad? AAA AA c+1 A c c c c BBB BB BB B CCC CC C D,4,Example: Credit Option,Spread Shigh Slow,Time,5,Motivation,risk-free Interest rate term structure dynamics of expected future. pricing. risky YY* Credit Spread expected future credit risk (dt, lt) Credit spread term str

4、ucture dynamics of expected future pricing. Pricing credit risk, credit-risky assets, credit derivatives Curve shape credit quality, credit cycle, business cycle Credit risk management (e.g. buy a put option on its own credit spread to hedge),Asia financial crisis US high yield market,6,Literatures

5、Review - Prior Results,7,Literatures Review - Prior Results,Helwage 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

6、 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. Arb

7、itrage 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

8、? 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 I

9、BM 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 co

10、st 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

11、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:

12、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,

13、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

14、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

15、-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

16、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 fut

17、ures 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 German

18、y 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 toda

19、y. 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 commod

20、ity 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

21、) 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. T

22、his 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)

23、= 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

24、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

25、) = 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 ma

26、turing 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 forwa

27、rd/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 an

28、d 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 f

29、oreign 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 sto

30、rage 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

31、.,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

32、a long forward contract on the same spot security, both initiated on day 0 with the same delivery price 340 and 5 days to delivery. Take the following scenario of futures price movement:,2. What is the net cash flow for the forward contract? Same as that of the futures? 3. Since the main difference

33、between a futures and a (otherwise identical) forward is in the timing of cash flow, the two contracts are the same when the interest rate is zero. 4. In particular, for otherwise identical contract, the futures price is equal to the forward price: f0 =F0 5. An abstract proof in case of r=0:, This s

34、hows that, if r = 0, holding one long futures contract to maturity generates the same net cash flow as that of an otherwise identical forward contract. That is, if r=0, one long futures contract can replicate the cash flow of a long position on an otherwise identical forward contract. Question: If i

35、nterest rate is not zero but a constant, can you still use futures contract to replicate the cash flow of an otherwise identical forward contract? That is, can you use a trading strategy involving only the futures contract and generate a cash flow of ST-F0 at T?,B. Replicate a forward payoff using t

36、he futures contract can be done by extending ideas from the previous table. The following futures trading strategy replicates the payoff of an otherwise identical forward:,Explanation: The # Cs column is the number of contracts that should be purchased at the end of the preceding day. At the end of

37、day 0, we enter into e-r(T-1) futures contracts and hold them for one day; At the end of day 1, we close the above position with a net profit of e-r(T-1)(f1-f0), save this profit (or borrow against it if it is negative) in a money account until day T (which will become f1-f0 at time T). We then open

38、 a new position of e-r(T-2) futures contracts; We repeat these steps according to the table until T. Note that, since the time is in units of days here, the interest rate r must be a daily, c.c. rate.,Questions: Can we still replicate a forward payoff with futures if the interest rate is known (non-

39、stochastic) but not constant as we have assumed in our example? What if there is a dividend or storage cost associated with the spot? How does this affect our replication? C. If interest rates are known, forward and futures prices must be equal.,VI . Similarity and difference between forwards and fu

40、tures A. First, a potential source of confusion: When we use the phrase stock price, it means the value of one share of stock. Does forward price (or futures price) mean the value of one long forward (or futures) contract? What is the value of a forward contract?,B. We have shown that, when interest

41、 rates are deterministic, the forward price and the futures price are the same (for the same underlying and maturity.) That is, a forward contract and a futures contract have zero value for the same delivery price K=F, and have identical value at initiation. C. The difference between the two shows u

42、p after initiation. The market values of both contracts fluctuate with the spot price of the underlying security. A forward contract is, however, less sensitive with respect to spot fluctuation than a futures contract with the same maturity. That is, a $1 increase in the spot price corresponding to

43、a smaller change in the value of the forward contract than in the value of the futures contract.,VII . Value of forward and futures contract Example: The IBM stock, which will not pay any dividend within the next 6 months, is trading at $100 per share. The risk-free interest rate (annualized c.c.) i

44、s 4%. The forward/futures price was shown to be $102.02. Consider a forward contract and a futures contract, the IBM stock price jumps up by $1. What is the value of the forward contract? What is the value of the futures contract?, The value of a forward contract, after initiation, can be calculated

45、 by comparing the following two portfolios (no benefits or costs of holding the underlying security: - Long a forward contract; - Long one underlying security (St) and borrow K*e-r(T-t); Since both portfolios pay ST-K at time T, they must cost the same at time t, which means that the value of a forw

46、ard contract at time t is St-K*e-r(T-t) Sensitivity of the value of a forward contract (called Delta in Wall Street jargons) is one., What is the value of a futures contract after initiation? - Consider a long futures contract, and suppose that the current futures price is ft and that the closing pr

47、ice on the previous day was fd-1. What is the present value of a long futures contract? - Answer: ft-fd-1, since one can always liquidate the futures contract now by entering a short position in the same futures and receive, at the end of the day d, a net payoff of (fd-fd-1)-(fd-ft) = ft-fd-1 = Ster

48、(T-t)-fd-1. - Note that this present value does not include past profits (or losses) that have occurred up to the end of day d-1.,Back to our numerical example: Let F(St,T-t) denote the maturity-T futures price on a non-dividend-paying stock with price St. If the IBM price increases from $100 to $10

49、1, then the value of the futures contract changes by F(101,0.5)-F(100,0.5) = 101er(T-t) - 100er(T-t) = er(T-t) = 1.02 1. The sensitivity of value of a futures contract (called Delta) : er(T-t) 1. Summary of similarity and difference between forward and futures contracts (with known interest rate):,

50、With otherwise identical terms, the forward price is equal to the futures price - the values of a forward contract and a futures contract is zero for the same delivery price. After initiation, however, a futures contract is more volatile than an otherwise identical forward contract, in terms of pres

51、ent value. The present-value calculation, however, does not capture a more important difference between the two kinds of contracts. Futures contracts, when used to hedge against an existing position that has payoff at T only, can also create short-term cash flow problem, if the market fluctuates aga

52、inst the contract holder, while a forward contract will not. (Example: Metallgesellschaft),Lecture # 3: Minimum-Variance Futures Hedging,I. Basis Risk, Basis = Spot price - futures price Basis risk: if the basis is not equal to zero at the time of closing out the futures position. Short hedge a cash

53、 position: The initial cost (time 0): S0 The outcome (time t): St-(ft-f0) = f0+(St-ft),II . Basis risk (St-Ft) arises from:, The asset being hedged is not the same as the asset underlying the futures. The exact date of transaction for the underlying position is not known for certain. The maturity of

54、 the futures cannot match the desired horizon so that the futures position needs to be closed out before its maturity, Examples: HIS futures (May 1, 1998),III . Minimum-variance hedge ratio, Static hedge Nature position: buy Na unites of asset at t Futures position: short futures at time 0 for Nf un

55、its of the same asset Hedge ratio: h=Nf/Na Payoff of the static hedge position: Yt =StNa -(ft-f0)Nf =S0Na+(St-S0)Na-(ft-f0)Nf = S0Na+StNa-ftNf = S0Na+Na(St-hft),e. Volatility of the hedge position: Var(Yt) =Na2Var(St-hft) = Na2(S2+h2f2-2hSf) F.O.C. = h* = S/f f. Hedging effectiveness Var(NaSt)-Var(Y

56、t)/Var(NaSt) = 2, Considerations for implementation Hedge horizon and data frequency Choice of the type of contract and the contract month Rolling hedge Example:,h* = 0.928*0.00262/0.00313 = 0.786 If Na=50,000 units and the futures contract size is 1,000 units, Nf* = h*Na=0.786*50,000=39,300 This is

57、 approximately 39 contracts.,IV . Hedging a stock portfolio, Measuring the stock portfolio in terms of index units Na=S0/I0, Hedge portfolios payoff Yt =St-(ft-f0)Nf =S0+(St-S0)-(ft-f0)Nf =I0Na+I0NaSt/S0-Nfft = I0Na+I0Na(+S,IIt/I0+t)-hNaft = I0Na(1+t)+ Na(S,IIt-hft) Optimal hedge ratio h* = N*f/Na =

58、 S,II,fI/f Optimal # of futures contracts (if ft=Ite(r-q)(T-t), then I,f=1 and f=Ie(r-q)(T-t).) Nf* = NaS,I e-(r-q)(T-t) =S0/I0S,I e-(r-q)(T-t),Example: On May 1, 1998, we want to hedge a stock portfolio of HKD5,000,000 with =1.1 over a six- month period. The HK risk-free rate is 6%,We use the HIS futures because there is no futures contract on this portfolio. The ideal futures contract would be the November contract. We use the December contract because the November contract is not available. 10720=10563.68e7*(0.06-q)/12 q=3.5% Nf* = (5,