CFA Lv2 衍生品

CFA 2级 衍生品 框架图 习题，包含Key Concepts、M1 Pricing and Valuation of Forward Commitments、M1 Pricing and Valuation of Forward Commitments等。

CFA Lv2 衍生品

• Key Concepts
  • 画图
    • 先画图，后做题
    • 从图上判断用什么折现率
  • 价格
    • S-Benefit
  • 价值
    • S-F
      • 没签合同只能买S，签了合同就是F
    • Ft(T) - F0(T)
      • 没签合同只能Ft买，签了合同就是F0买
  • 利率
    • 做题用单利大部分都可以做出来
  • 做题
    • 衍生品一定要多做题！
• M1 Pricing and Valuation of Forward Commitments
  • General Formula
    • Carry Arbitrage Model
      • Cash-and-carry arbitrage
      • Reverse Cash-and-carry arbitrage
      • 例题
        • Torres explains to her deputy, Alejandro Gutiérrez, “The derivatives contracts we enter into are typically priced using a carry arbitrage model to have zero cost to us at origination. In addition to requiring current interest rates and coupon payments of the underlying bonds, the model makes three assumptions: (1) There are no transaction costs for buying and selling securities; (2) short-selling proceeds become available when the short is covered; and (3) we can borrow and lend at the same risk-free rate of interest. When discussing the assumptions of the carry arbitrage model, Torres is least likely correct regarding: A. borrowing costs. B. transaction costs. C. short-sale proceeds.
          • C is correct. Torres is incorrect regarding the availability of short-selling proceeds. The carry arbitrage approach assumes the proceeds from a short sale can be immediately used to purchase other securities. A is incorrect. Torres is correct that the carry arbitrage model assumes funds can be borrowed and lent at the same risk-free rate. B is incorrect. Torres is correct that the carry arbitrage model assumes no market frictions, such as transaction costs.
            • 这题的关键难点在于理解题目。题目说short is covered，意思是做空的时候，借股票，然后还，是covered，才可以动用相应资金。但是我们实际上，short要立刻获得资金进行套利，所以C是错的。
      • 例题
        • Doyle and Kemper discuss the carry arbitrage model and how they can take advantage of mispricing in bond markets. Specifically, they would like to execute an arbitrage transaction on a Eurodollar futures contract in which the underlying Eurodollar bond is expected to make an interest payment in two months. Doyle makes the following statements: Statement 1 If the Eurodollar futures price is less than the price suggested by the carry arbitrage model, the futures contract should be purchased. Statement 2 Based on the cost of carry model, the futures price would be higher if the underlying Eurodollar bond's upcoming interest payment was expected in five months instead of two. Which of Doyle's statements regarding the Eurodollar futures contract price is correct? A. Only Statement 1 B. Only Statement 2 C. Both Statement 1 and Statement 2
          • 答案：C 解析：C is correct. Doyle's first statement is correct. Unless the Eurodollar futures contract's quoted price is equal to the no-arbitrage futures price, there is an arbitrage opportunity. Moreover, if the quoted futures price is less than the no-arbitrage futures price, then to take advantage of the arbitrage opportunity, the Eurodollar futures contract should be purchased and the underlying Eurodollar bond should be sold short. Doyle would then lend the short sale proceeds at the risk-free rate. The strategy that comprises those transactions is known as reverse carry arbitrage. Doyle's second statement is also correct. Based on the cost of carry model, the futures price is calculated as the future value of the sum of the underlying plus the underlying carry costs minus the future value of any ownership benefits. If the Eurodollar bond's interest payment was expected in five months instead of two, the benefit of the cash flow would occur three months later, so the future value of the benefits term would be slightly lower. Therefore, the Eurodollar futures contract price would be slightly higher if the Eurodollar bond's interest payment was expected in five months instead of two months.
            • 这题关键是S2的理解。interest还有5个月到期，潜台词是benefit折现的价值更低。 但是我们要知道，远期价格是加成本减收益，所以我们减的收益少了，远期合同的价格应该更高。
    • 例题
      • Which of the following statements regarding futures and spot prices is most likely correct? Statement A: At expiration, futures prices and spot prices must converge. Statement B: If the spot price exceeds the futures price, then an investor could purchase the asset at the spot price and enter a short futures contract to sell it at the higher price, thus locking in a profit. Statement C: If the spot price is less than the futures price at expiration, then an investor could purchase the futures contract and execute the contract to purchase the underlying at the lower futures price, sell it at the higher spot price, and make an arbitrage profit. A. Statement A B. Statement B C. Statement C
        • 正确答案:A 解析: 期货和现货价格到期日的价格理论情况下是相同的,如果他们的价格不相同就存在套利机会如果现货价格大于期货价格，那么可以通过购买期货合约并执行，然后以较低的期货价格购买标的资产，以较高的现货价格出售，获得套利利润;如果在到期日期货价格超过了现货价格,那么投资者可以以现货价格购买资产,然后签订一个期货空头合约,以更高的价格出售从而锁定利润。
          • 送分题
  • Forwards Contracts
    • Fixed Income Forwards and Equity Forwards
      • Fixed Income Forwards
        • 例题
          • Suppose that one month ago, an analyst purchased a bond forward contract with a maturity of three months. The notional principal of the contract is $10,000 quoted as 135 percentage of par. Currently, the analyst finds that the forward price is priced at 141 percentage of par. Assumed the annualized risk-free rate is 2.5%. Calculate the current value of the forward. A. $0.0597 B. $5.98 C. $597.54
            • 正确答案:C 本题考查债券远期合约的估值计算。 由题目可知债券远期在合约期间的价值计算公式为 V=(141-135)/(1+2.5%)^(2/12) =5.9754， 因此对于一份合约的价值为5.9754/100x$10,000=$597.54。答案选择C选项。
              • 注意是1个月前买了1个合同期为3个月的合约，所以还有2个月到期
        • 例题
          • Three months ago (90 days), Kim purchased a bond with a 3% annual coupon and a maturity date of seven years from the date of purchase. The bond has a face value of US$1,000 and pays interest every 180 days from the date of issue. Kim is concerned about a potential increase in interest rates over the next year and has approached Riley for advice on how to use forward contracts to manage this risk. Riley advises Kim to enter into a short position in a fixed-income forward contract expiring in 360 days. The annualized risk-free rate now is 1.5% per year and the price of the bond with accrued interest is US$1,103.45. Based on a 360-day year, the price of the forward contract on the bond purchased by Kim is closest to: A. US$1,082. B. US$1,090. C. US$1,120.
            • • 衍生品的难点不在背公式，也不在按计算器，其实在阅读理解。 这里原合同是90天前进的，但是题目又说新合同是360天后，所以我们要用新合同的价格1103.45来折到360天后。 因为题目特地提出with accrued interest，这里我们要理解折的时候是全价，但是中间的coupon最后都会降低合同的全价。 最后一个坑就是，虽然新合同是360天后进入，但是因为我们需要来做对冲，所以新合同应该和旧合同match，所以新合同给coupon的时间应该和旧合同一致，所以现在新合同已经又90天的AI。 这题还有另外一直做法，先从现在，到90天后： 1103.45*1.015^(90/360)=1107.564863 然后给coupon 1107.56-15=1092.56 然后继续向前跑180天 1092.56*1.015^0.5=1100.7237 再给coupon 1100.72-15=1085.72 再跑90天马拉松结束 1085.72*1.015^(90/360)=1089.768746 这种做法的好处是，你画个图，顺着时间轴和coupon一步步陪跑就行，比较简单直观。
        • 例题
          • After discussing Kemper's new investment ideas, Doyle and Kemper evaluate one of their existing forward contract positions. Three months ago, BestFutures took a long position in eight 10-year Japanese government bond (JGB) forward contracts, with each contract having a contract notional value of 100 million yen. The contracts had a price of JPY153 (quoted as a percentage of par) when the contracts were purchased. Now, the contracts have six months left to expiration and have a price of JPY155. The annualized six-month interest rate is 0.12%. Doyle asks Kemper to value the JGB forward position The value of the JGB long forward position is closest to: A. JPY15,980,823. B. JPY15,990,409. C. JPY16,000,000.
            • 答案：B 解析：B is correct. The value of the JGB forward position is calculated as (155-153)/(1+0.0012)^0.5=1.998801 Therefore, the value of the long forward position is 1.9988 per JPY100 par value. For the long position in eight contracts with each contract having a par value of 100 million yen, the value of the position is calculated as 0.019988 × (JPY100,000,000) × 8 = JPY15,990,409.
              • 这里要注意，合同是有份数的，而且是用英文写的8，容易看漏。一定要仔细读题。
      • Stock Forwards
        • 例题
          • Suppose an investor entered a one-year equity forward contract at $110, and the contract remains three months before its expiration. The spot price of the stock is $116, and the interest rate is 2% on an annual compounding basis. If there are no benefits of carry, the forward value of the existing contract is closest to: A. $12.2341 B. $6.5432 C. $10.3256
            • 正确答案: B 本题考查 equity forward 的估值。 116-100/(1+2%)^025=6.5432
              • 考虑有合约的话需要花多少钱，无合约的话需要花多少，差价就是value
        • 例题
          • Troubadour next considers an equity forward contract for Texas Steel, Inc. (TSI). Troubadour takes a short position in the TSI equity forward contract. His supervisor asks, "Under which scenario would our position experience a loss?" The most appropriate response to Troubadour's supervisor's question regarding the TSI forward contract is: A. a decrease in TSI’s share price, all else equal. B. an increase in the risk-free rate, all else equal C. a decrease in the market price of the forward contract, all else equal.
            • 答案：B 解析：B is correct. From the perspective of the long position, the forward value is equal to the present value of the difference in forward prices: Vt = PV[Ft – F0], where Ft = FV(St +CCt –CBt ). All else equal, an increase in the risk-free rate before contract expiration would cause the forward price, Ft , to increase. This increase in the forward price would cause the value of the TSI forward contract, from the perspective of the short, to decrease. Therefore, an increase in the risk-free rate would lead to a loss on the short position in the TSI forward contract.
              • 首先，这里是short position，不能弄反了 A说S降价，那么将来F也降价，short是赚钱。 C说这个合同降价，我们卖了个东西，然后降价了，也是赚钱的。 因为Equity的未来价格，相当于S乘以Rf，如果Rf增加，未来价格也会增加，因为我们是做空，所以亏钱，应该选B。 这题也可以换个角度，假设我们是Long，什么角度赚钱，也可以很容易把B选出来，而且不会绕。
      • Equity Index Futures
        • 除了连续复利，其他和FW一样
        • 例题
          • The current S&P 500 index level is 2285 with the continuous dividend yield of 2.5%. What is the price of the nine-month index futures contract if the continuously compounded annual interest rate is 0.2%? A. 2245.92 B. 2117.23 C. 2845.31
            • 正确答案: A 本题考查 equity futures 连续复利的定价 2285 * e^(0.2%-2.5%)x0.75 =2245.92 因此答案选择 A选项。
              • 送分题
        • 例题
          • Carter holds a long position in an equity forward contract for QuantumByte Technologies (QBT) that has a remaining maturity of two months. The continuously compounded dividend yield on the QBT is 0.8%, the current stock price is $82. The continuously compounded annual interest rate is 0.32%. The current no-arbitrage futures price of QuantumByte Technologies (QBT) is closest to: A. $81.35. B. $81.93. C. $82.03.
            • B is correct exp((0.0032-0.008)*2/12)*82=81.934426
        • 例题
          • MDCM holds a portfolio of equities denominated in Australian dollars (AUD). Bradford expects a temporary, sharp decline in stock prices. She hedges using 3-month futures contracts on the S&P/ASX 200™ Index (SPI). The index is currently at 6,165 and yields 4.06%. The annualized 3-month risk-free rate is 1.96%. The no-arbitrage 3-month SPI futures price is closest to: A. 6,037. B. 6,133. C. 6,258.
            • B is correct Because the no-arbitrage 3-month SPI futures price can be calculated as: F0(T) = S0e(rf - γ0)T where, S0 = current spot/index price rf = discount rate (continuously compounded) γ0 = carry benefit (in this case the continuously compounded dividend yield) T = time to maturity (expressed as a fraction of a year). F0(T) = 6,165e(0.0196 - 0.0406) x 0.25 = 6,132.72 ≈ 6,133
              • 这里要理解，虽然说的是yield，但是表达的其实是dividend yield，是benefit。
    • Forward Rate Agreement (FRA)
      • Long是看涨，从标的上升赚钱，付固定收浮动
        • 例题
          • Dylan Carter, chief risk officer at Nexus Capital Asset Management Group, talks to his assistant Savage Olaf about derivatives strategies. Their first job is to settle an FRA contract. In 60 days, Nexus expects to make a bank deposit of $10,000,000 for a period of 90 days at 90-day MRR set 60 days from today. Nexus is concerned about a possible decrease in interest rates. Nexus enters into a $10,000,000 notional amount FRA contract with the FRA rate is priced at 0.65%. After 60 days, 90-day MRR is 0.55%. The payment received of FRA to settle it will be closest to: A. $2,496.57. B. $2,494.84. C. -$2,478.75.
            • A is correct (0.0065-0.0055)/4*10000000/(1+0.0055/4)=2496.56722
              • FRA用单利还是复利，不影响考试做题，简单点直接上单利
              • 要注意，就算题目没问，也要把现金流折到FRA到期日，因为FRA是提前确定利率，提前支付
              • 衍生品要注意领会题目意图，比如说担心利率下跌，那么衍生品的头寸肯定是利率下跌的时候赚钱，所以是FRA的空方，收固定，给浮动。
        • 例题
          • Gutiérrez mentions a major new contract requiring FabC to increase its working capital by borrowing USD35 million in six months. Torres states she is concerned interest rates will rise substantially in the near future and is considering entering a long position (pay fixed) in a 6 × 24 forward rate agreement to lock in the price of the new loan. She asks Gutiérrez what the fixed rate would be using a 30/360 convention and the spot interest rates found in Exhibit 1. Gutiérrez's fixed-rate calculation for the forward rate agreement should be closest to: A. 3.73%. B. 3.88%. C. 5.59%
            • A is correct. The 18-month fixed rate for a 6 × 24 FRA is calculated as, 1/(1+FR*1.5)=0.9372/0.9896 The forward rate is 3.7274%.
              • 这题的关键点在于，FRP不管是多久，都是用单利。 用复利只能找到一个接近的数，但是不能完全对的上。
      • 例题
        • Three months later, the 6 × 9 FRA in Exhibit 6 reaches expiration, at which time the three- month US dollar Libor is 1.10% and the six- month US dollar Libor is 1.20%. Johnson determines that the appropriate discount rate for the FRA settlement cash flows is 1.10%. Based on Exhibit 6 and the three-month US dollar Libor at expiration, the payment amount that the bank will receive to settle the 6 × 9 FRA is closest to: A. $19,945. B. $24,925. C. $39,781.
          • A is correct. Given a three-month US dollar MRR of 1.10% at expiration, the settlement amount for the bank as the pay-fixed (receive-floating) party is calculated as Settlement amount pay-fixed (receive floating) = NA × {[Lm – FRA0]tm}/[1 + Dmtm]}. Settlement amount pay-fixed (receive floating) = $20,000,000 × {[0.011 – 0.0070] × (90/360)]/[1 + 0.011*(90/360)]}. Settlement amount pay-fixed (receive floating) = $20,000,000 × (0.001)/1.00275 = $19,945.15. Therefore, the bank will receive $19,945 (rounded) as the receive-floating party.
            • 这题的关键难点是给的数字太多，需要仔细理清楚中间的逻辑。 另外就是FRA不管多久，都是用单利计算。 题目给了到期时候3个月的LIBOR，其实这个1.1%才是本题的关键，只要找准这个数，这题分就基本到手了。 另外，因为是advanced set, advanced settle，所以1.1%和0.7%的差，是期末的金额，在FRA到期的时候，还要折3个月，用1.1%的利率折到期初。
      • 例题
        • One month ago (30 days), Cline entered a pay floating 3 × 6 forward rate agreement (FRA) at a rate of 2.31% with a notional amount of US$5,000,000. At the time, the three-month MRR was 1.28% and the six-month MRR was 1.8%. Now, 30 days after entering the FRA, two-month MRR is 1.5% and the five-month MRR is 2.5%. The current value of Cline’s FRA is closest to: A. −US$10,625. B. −US$10,515. C. US$10,612.
          • • 这题其实小数点非常影响结果，比如FRA(t)正常算出来是3.166667%，答案用的是3.16%，导致保留到3.16%，或者3.17%，或者3.1667%，算出来是3个不同的答案。而AB选项又非常接近。 如果考试遇到这种题，送一分给CFA不影响考试通过。
  • Fixed Income Futures
    • Conversion Factor
    • Cheapest-to-Deliver
    • Pricing of Fixed Income Futures
      • 全程用带着AI的全价折
      • 买单价FP=QFP*CF
        • 进两元(QFP)店买东西，买单付五元(FP)
        • 考试不考，纯帮助理解
          • CF存在的原因是有cheapest-to-deliver，所以报价是QFP，但是真实的成交价，基金经理可以挑个最便宜的。但是报价不会改变，所以乘上一个CF，得到最后成交的future price
          • 可以理解成两元店全场2元，这个是报价。然后你去买单，有些东西老板说，这个贴黑色标签的是5元，这个贴黄色标签的是10元。因为中间有个conversion factor
      • 考试注意问FP还是QFP，可以用CF去猜答案
      • 例题
        • Next, they turn to the pricing of Treasury bond futures contracts. Carter provides Olaf with the following information for a Treasury bond and calculates the price of a futures contract on this bond. The bond has a face value of $100,000, pays a 7% semiannual coupon, and matures in 15 years. The bond is priced at $156,000, has no accrued interest, and yields 2.5%. The futures contract expires in 8 months, and the annualized risk-free rate is 1.5%. There are multiple deliverable bonds, and the conversion factor for this bond is 1.098. Based on the information provided by Carter, the futures price of the Treasury bond is closet to: A. $141,234. B. $139,236. C. $146,689.
          • B is correct (156000*1.015^(8/12)-3500*1.015^(2/12)-3500*2/6)/1.098=139235.66785
            • 注意和2023.08 Mock Exam B - Afternoon Session的原题的对比。个人建议记这个答案思路。
            • 这里要注意，FP=QFP*CF，一开始给的priced是正常的price，所以不需要考虑CF，但是最后问的future price其实是QFP，所以需要把算出来的FP/CF
            • 这题还是强烈建议画图，跟着时间轴跑一次，答案就出来了，这样最不容易错： 我们的起点T0是156000，没有AI。Coupon是6个月给一次，终点是8个月后： 先跑6个月： 156000*1.015^(6/12)=157165.645101 派息： 157165.65-3500=153665.65 再跑2个月： 153665.65*1.015^(2/12)=154047.434877 把AI扣了： 154047.43-3500*2/6=152880.763333 除以CF： 152880.76/1.098=139235.664845
      • 例题
        • Currently Sheroda is long a US Treasury futures position. Parisi notes the following information for the cheapest to deliver US Treasury bond for the contract; the bond has a face value of $100,000, pays a 7% semiannual coupon, and matures in 15 years. The bond is priced at $156,000, has no accrued interest, and a yield of 2.5%. The futures contract expires in 8 months, and the annualized risk-free rate is 1.5%. There are multiple deliverable bonds, and the conversion factor for this bond is 1.098. Based on the information provided by Parisi, the price of the bond futures contract Sheroda holds in Quantum is closest to: A. 141,234 B. 140,298 C. 146,689
          • B is correct
            • 这题的题目和答案来源是2023.08 Mock Exam B - Afternoon Session，上面一题的题目和答案来源是机构。最主要的差异是算future price的时候有没有AI(T)，我个人倾向于是有的，因为公式也有这部分。建议2题都看下，考试时候，如果有AI(T)算不出来答案，可以再用没AI(T)的情况再算一次。
      • 例题
        • The current futures price is lower than the no-arbitrage price calculated using this data, so Bradford constructs a riskless strategy using the futures contract and its underlying to exploit this arbitrage opportunity. A strategy to exploit the arbitrage opportunity in the 10-year US Treasury note futures contract would most likely include: A. selling the futures contract. B. investing the arbitrage profit. C. lending the short sale proceeds.
          • C is correct A. Incorrect because Bradford would not sell the futures contract as part of a reverse- carry arbitrage strategy. One of the four steps of reverse-carry arbitrage is to buy the futures contract. B. Incorrect because Bradford would not lend / invest an amount equivalent to the arbitrage profit to engage in reverse-carry arbitrage. This is true regardless of whether the arbitrage opportunity is carry or reverse-carry type. One of the four steps of reverse-carry arbitrage is to borrow the arbitrage profit. C. Correct because the current quoted futures price is lower than the no-arbitrage futures price. To exploit this opportunity, Bradford would engage in reverse-carry arbitrage (i.e. sell the expensive asset - the bond - and buy the cheap asset – the futures). The four (simultaneous) steps of reverse-carry arbitrage are 1) Buy the futures contract, 2) Sell the underlying asset (bond) short, 3) Lend the short-sale proceeds, and 4) Borrow the arbitrage profit.
            • 套利永远是期初不掏钱，期末赚钱无风险的套利收益。如果想提前获得套利收益，可以期初借PV(arbitrage profit)，所以不是invest，而是receive。B说反了。
            • C是对的，逻辑是，future便宜了，买。所以要现在把bond卖了。现在卖了的钱可以借出去。
      • 例题
        • Donald Troubadour is a derivatives trader for Southern Shores Investments. The firm seeks arbitrage opportunities in the forward and futures markets using the carry arbitrage model. Troubadour identifies an arbitrage opportunity relating to a fixed-income futures contract and its underlying bond. Current data on the futures contract and underlying bond are presented in Exhibit 1. The current annual compounded risk-free rate is 0.30%. Based on Exhibit 1 and assuming annual compounding, the arbitrage profit on the bond futures contract is closest to: A. 0.4158. B. 0.5356. C. 0.6195.
          • 答案：B 解析：B is correct. The no-arbitrage futures price is equal to the following: F0 = FV[B0 + AI0 – PVCI] F0 = (1 + 0.003)0.25(112.00 + 0.08 – 0) = 112.1640. The adjusted price of the futures contract is equal to the conversion factor multiplied by the quoted futures price: F0 = CF × Q0 F0 = (0.90)(125) = 112.50. Adding the accrued interest of 0.20 in three months (futures contract expiration) to the adjusted price of the futures contract gives a total price of 112.70. This difference means that the futures contract is overpriced by 112.70 – 112.1640 = 0.5360. The available arbitrage profit is the present value of this difference: 0.5360/(1.003)0.25 = 0.5356.
            • 这题要注意一个细节点，我们一般说最后的价格，是除以CF，得到的一个报价。但是最后算套利的价差，不应该算和报价间的价差，而是应该算报价*CF最后实际交易价格的价差。
  • Swap Contracts
    • F=(1-Dn)/Sum(D1+D2+……+Dn)
      • 考试注意问annualized rate还是periodical rate，可以用payment frequency去猜答案
    • Interest Rate Swaps
      • Long是收浮动
        • Value = 浮动 - 固定
          • 浮动：Par Value
          • 固定：估值
        • 例题
          • A company entered into a EUR6 million four-year interest rate swap one year ago, the company's position is paying fixed and receiving floating with annual reset. The fixed-rate was 2% when the company entered into the swap. The current term structure of the interest rate for EUR is presented below. Based on the information above, what is the value of a paying-fixed rate swap? A. €916,596 B. €869876 C. €218462
            • 正确答案: A 本题考查 swap 的估值。 F = (1-D3)/(D1+D2+D3) = (1-0.7938)/(0.9709+0.9070+0.7938)=0.077179 因此根据swap的估值公式可得V=Notional Principal x (F1-F0) * (D1+D2+D3) = €6million * (7.72%-2%) x (0.9709+0.9070+0.7938)=€916,596，答案选择A选项。
              • 基本题，必须熟练掌握！
    • Currency Swaps
      • 期初期末都要交换本金，注意折汇率
      • 期初收什么币种，期间给相应的利率
      • 例题
        • Consider a one-year quarterly-reset swap involving two currencies, the US dollar($) and the Swiss franc(SF). The exchange rate is now $0.8/SF, and the notional principal would be $1million. At initiation, the US dollar and the Swiss franc term structures are shown as below. Calculate the fixed rate for US dollar and Swiss franc respectively. A. 3.69%, 5.56% B. 0.92%, 1.39% C. 3.69%, 1.39%
          • • 这题计算量比较大，主要用来熟悉知识点，考试的时候可能会简化计算，比如直接给你DF的表格，但是练习知识点的时候还是要完整练习一次。
    • Equity Swaps
      • Equity端直接在期初折算
        • 例题
          • Carter has entered a one-year equity swap 270 days ago. Under the terms of the swap, he would receive the return on the SCI 300 Index and pay a fixed annual interest rate of 4.8% on a notional amount of CNY 50,000,000. The swap payments are quarterly. At the time the swap was initiated 270 days ago, the value of the SCI 300 Index was 3,250. Today, the value of the SCI 300 Index is 3,738. Exhibit 1 provides present value factors based on the current CHY terms structure of interest rates.
          • Based on the information in Exhibit 1, the market value of Carter’s equity swap is closest to: A. ￥8,445,900. B. ￥7,611,000. C. ￥7,029,100.
            • C is correct Because this is a pay fixed rate receive equity swap the MV of the swap = PV equity receipts – MV of fixed rate bond. The market value of the equity swap is calculated as follows: (3738/3250-(1+0.048/4)*0.9976)*50000000=7029132.307692
              • 先画图，然后独立计算两倍的现金流，最后在T0轧差
        • 例题
          • Hancock gained exposure to the Italian market through an equity swap, where she pays fixed and receives equity. The swap has a EUR 10,000,000 notional amount, matures in 11 months, and resets quarterly. Hancock receives the return based on the FTSE MIB Index and pays 0.25% per quarter on the pay-fixed leg of the swap. She collects data on the FTSE MIB Index values in Exhibit 1 and estimates the current value of the pay-fixed leg of the swap to be EUR 10,097,400. Exhibit 1 Index value at initiation of the swap: 20.850 Index value at last reset: 18.920 Current index value: 19.970 The current no-arbitrage value (in EUR) of the equity swap is closest to: A. –519,000 B. 458,000 C. 555,000
            • B is correct Because to calculate the value of the equity swap (VEQ,t ), the current pay-fixed leg value is subtracted from the receive-equity leg value: VEQ,t = (St/St–1)NAE – VFIX(C0) – PV(Par – NAE) = 10,554,968 – 10,097,400 – 0 = 457,568 ≈ 458,000 where: (St/St–1)NAE = value of the receive-equity leg value = St (current equity index price) divided by St-1 (equity index price from last reset) multiplied by NAE (notional value of the swap) = (19,970/18,920) x 10,000,000 = 10,554,968 VFIX(C0) = value of the pay-fixed leg (fixed rate bond initiated with coupon C0) = 10,097,400 PV(Par – NAE) = 0 = present value function from the swap maturity date to Time t = 0 (bond par value = initial equity notional as PAR = NAE)
              • equity swap有2条腿，可以把当成2个独立的产品对待。分别算出这2个的价值，最后轧差就是equity swap的价值。
• M2 Valuation of Contingent Claims
  • Binomial Option Valuation Model
    • One-period binomila model
    • Risk-Neutral Valuation
      • Hedge ratio
        • 例题
          • Sousa's first task is to illustrate how to value a call option on Alpha Company with a one-period binomial option pricing model. It is a non-dividend-paying stock, and the inputs are as follows. ■ The current stock price is 50, and the call option exercise price is 50. ■ In one period, the stock price will either rise to 56 or decline to 46. ■ The risk-free rate of return is 5% per period.
            • The optimal hedge ratio for the Alpha Company call option using the oneperiod binomial model is closest to: A. 0.60. B. 0.67. C. 1.67.
              • A is correct. The hedge ratio requires the underlying stock and call option values for the up move and down move. S+ = 56, and S– = 46. c+ = Max(0,S+ – X) = Max(0,56 – 50) = 6, and c– = Max(0,S– – X) = Max(0,46 – 50) = 0. The hedge ratio is h = c+ - c– / S+ - S- = 6 - 0 / 56 - 46 = 6 / 10 = 0.60
                • 这种虽然是送分题，但是考试还挺爱考的，基本掌握本质Delta(option)/Delta(stock)。
            • The risk-neutral probability of the up move for the Alpha Company stock is closest to: A. 0.06. B. 0.40. C. 0.65
              • C is correct. For this approach, the risk-free rate is r = 0.05, the up factor is u = S+/S = 56/50 = 1.12, and the down factor is d = S–/S = 46/50 = 0.92. The risk-neutral probability of an up move is π = [FV(1) – d]/(u – d) = (1 + r – d]/(u – d) π = (1 + 0.05 – 0.92)/(1.12 – 0.92) = 0.13/0.20 = 0.65
                • 基础核心知识，必须熟练掌握
            • The value of the Alpha Company call option is closest to: A. 3.71. B. 5.71. C. 6.19.
              • A is correct. The call option can be estimated using the no-arbitrage approach or the expectations approach. With the no-arbitrage approach, the value of the call option is c = hS + PV(–hS– + c–). h = (c+ – c–)/(S+ – S–) = (6 – 0)/(56 – 46) = 0.60. c = (0.60 × 50) + (1/1.05) × [(–0.60 × 46) + 0]. c = 30 – [(1/1.05) × 27.6] = 30 – 26.286 = 3.714. Using the expectations approach, the risk-free rate is r = 0.05, the up factor is u = S+/S = 56/50 = 1.12, and the down factor is d = S–/S = 46/50 = 0.92. The value of the call option is c = PV × [πc+ + (1 – π)c–]. π = [FV(1) – d]/(u – d) = (1.05 – 0.92)/(1.12 – 0.92) = 0.65. c = (1/1.05) × [0.65(6) + (1 – 0.65)(0)] = (1/1.05)(3.9) = 3.714. Both approaches are logically consistent and yield identical values.
                • 基础核心知识，必须熟练掌握
    • Two-period binomial model
    • Binomial interest rate tree
      • 利率二叉树
        • 上涨下跌都是50%
      • 股票二叉树
        • 风险中性利率派
          • 派u=(1+rf-rd)/(ru-rd)
            • 例题
              • Hancock considers purchasing a 3-month European-style put option on Seanemos, a non-dividend paying stock. Based on the data in Exhibit 2, Hancock uses the one-period binomial model and the expectations approach to calculate the no-arbitrage value of the put option. Exhibit 2 Current share price (in USD): 82.20 Exercise price (in USD): 80.00 Up factor: 1.25 Down factor: 0.70 Risk-free rate (annualized): 1% The no-arbitrage value (in USD) of the put option on Seanemos is closest to: A. 10.08 B. 12.32. C. 18.68.
                • A is correct Because the value of an option using the one-period binomial model and the expectations approach is determined as follows: 1. Calculate the risk-neutral probability of an up move, 2. Calculate the put values at each node and 3. Calculate the put option value. 1. Risk-neutral probability π = [FV(1) – d]/(u – d) = [1 + 0.01 × (90/360) – 0.70]/(1.25 – 0.70) = 0.55 2. Put values at each node in 3 months S+ = uS = 1.25(82.20) = 102.75 S– = dS = 0.70(82.20) = 57.54 p+ = Max(0;X – uS) = Max(0;80.00 – 102.75) = 0 p– = Max(0;X – dS) = Max(0;80.00 – 57.54) = 22.46 3. Put option value using the expectations approach: p = PV[πp+ + (1 – π)p–] = 1 / [1 + 0.01 × (90/360)] × [0.55 × 0 + (1 - 0.55) × 22.46 = 0.9975 × 0.45 × 22.46 = 10.08 where: π = risk-neutral probability of an up move d = down factor = 0.70 u = up factor = 1.25 S+ = outcome when underlying goes up S- = outcome when underlying goes down p+ = put value when underlying goes up p- = put value when underlying goes down X = exercise price = 80 The no-arbitrage approach would also give the same answer: hedge ratio, h = (p+ – p-)/(S+ – S-) = (0 – 22.46)/(102.75-57.54) = -0.4968 p = hS + PV(–hS+ + p+) = -0.4968 × 82.20 + 0.9975 × (0.4968 × 102.75 + 0) = 10.08
                  • 这题要注意，二叉树是三个月，所以1+rf是1+0.01/4 = 1.0025，而不是1.01
      • 例题
        • In the illustration, Sousa is also asked to describe related arbitrage positions to use if the call option is overpriced relative to the model. For the Alpha Company option, the positions to take advantage of the arbitrage opportunity are to write the call and: A. short shares of Alpha stock and lend. B. buy shares of Alpha stock and borrow. C. short shares of Alpha stock and borrow
          • B is correct. You should sell (write) the overpriced call option and then go long (buy) the replicating portfolio for a call option. The replicating portfolio for a call option is to buy h shares of the stock and borrow the present value of (hS– – c–). c = hS + PV(–hS– + c–). h = (c+ – c–)/(S+ – S–) = (6 – 0)/(56 – 46) = 0.60. For the example in this case, the value of the call option is 3.714. If the option is overpriced at, say, 4.50, you short the option and have a cash flow at Time 0 of +4.50. You buy the replicating portfolio of 0.60 shares at 50 per share (giving you a cash flow of –30) and borrow (1/1.05) × [(0.60 × 46) – 0] = (1/1.05) × 27.6 = 26.287. Your cash flow for buying the replicating portfolio is –30 + 26.287 = –3.713. Your net cash flow at Time 0 is + 4.50 – 3.713 = 0.787. Your net cash flow at Time 1 for either the up move or down move is zero. You have made an arbitrage profit of 0.787. In tabular form, the cash flows are as follows:
            • 答案可能有点啰嗦，但是更多的是帮助理解。 金融就是剪刀加浆糊，replication怎么做要熟练掌握
    • 例题
      • Statement 1 The calculation involves discounting at the risk-free rate. Statement 2 The calculation uses risk-neutral probabilities instead of true probabilities. Which of Sousa's statements about binomial models is correct? A. Statement 1 only B. Statement 2 only C. Both Statement 1 and Statement 2
        • C is correct. Both statements are correct. The expected future payoff is calculated using risk-neutral probabilities, and the expected payoff is discounted at the risk-free rate.
          • 特别要注意S2，这里默认说的是期权二叉树，用的是风险中性概率。
    • 例题
      • Rocha asks Sousa why the value of a similar in-the-money interest rate call option decreases if the exercise price is higher. Sousa provides two reasons. Reason 1 The exercise value of the call option is lower. Reason 2 The risk-neutral probabilities are changed. Which of Sousa's reasons for the decrease in the value of the interest rate option is correct? A. Reason 1 only B. Reason 2 only C. Both Reason 1 and Reason 2
        • A is correct. Reason 1 is correct: A higher exercise price does lower the exercise value (payoff) at Time 2. Reason 2 is not correct because the risk-neutral probabilities are based on the paths that interest rates take, which are determined by the market and not the details of a particular option contract.
          • 送分题
    • 例题
      • Franco comments: "I think six-month call options on the six-month forward rate would probably be the cheapest solution. The price of the European-style option can be evaluated as the present value of the expected terminal option's payoffs using the riskadjusted periodic rate. Because Newport has indicated that its goal is to pay a maximum interest rate of 1.25% on the loan, we could also use interest rate put and call options. I believe the binomial model can be used to value interest rate options." Franco's understanding of the valuation of the European style six-month call option is most likely: A. correct with respect to the payoffs and the discount rate. B. correct with respect to the payoffs but incorrect about the discount rate. C. incorrect with respect to the payoffs but correct about the discount rate.
        • 答案：B 解析：B is correct. According to the expectations approach of options valuation, option values are simply the present value of the expected terminal option payoffs (based on risk-neutral probabilities) discounted at the estimated risk-free interest rate, rather than the risk-adjusted periodic rate. A is incorrect. The stated discount rate is correct. C. is incorrect. The valuation method approach is correct.
          • 这种定性题其实比定量难多了，需要逐字逐句推敲是否正确。比如这里，其实只错了个risk-adjusted。如果不确定，可以先随便选个，插上旗子，做完后再回来慢慢检查。
  • Black-Scholes-Merton (BSM) Model
    • 假设
      • 欧式
      • 连续
      • price -> log normal，r -> normal
      • r和volatility都是已知且恒定
      • 理想市场
        • 没摩擦，没交易成本，没税
    • S * N(d1) - X * N(d2)
      • S-X出发，加上行权概率
        • Put是X-S，但是公式位置懒得动，所以在d1,d2前面加负号
      • N(d1)是求导，即Delta
      • N(d2)是call的行权概率
        • N(-d2)是put的行权概率
      • N(-X) = 1-N(X)
      • 例题
        • Burr wants to assess Madisox’s comprehension of the components of the BSM. Madisox states that a call option can be viewed as a leveraged position in the underlying stock. To replicate a call option, the appropriate strategy is to purchase N(d1) shares and simultaneously borrow an amount e–rTXN(–d2). Madisox’s statement about the BSM model is least likely correct with respect to: A.purchasing NN(d1) shares. B.the leveraged position in a stock. C.borrowing an amount e–rTXN(–d2).
          • C is correct A.Incorrect. Madisox’s statement that the purchase of N(d1) shares is correct in terms of replicating a call option. B.Incorrect. Madisox’s statement that a call option can be viewed as a leveraged position in a stock is correct. C.Correct. With respect to a call option, Madisox is incorrect with respect to his comment to simultaneously borrow an amount e–rTXN(–d2). To create a leveraged position in a stock, the correct components are to purchase N(d1) shares by borrowing an amount e–rTXN(d2). The term e–rTXN(–d2) represents the amount lent when purchasing a put option.
            • 注意括号里的正负号！
      • 例题
        • • Based on Exhibit 1 and the BSM valuation approach, the initial portfolio required to replicate the long call option payoff is: A. long 0.3100 shares of TCB stock and short 0.5596 shares of a zero-coupon bond. B. long 0.6217 shares of TCB stock and short 0.1500 shares of a zero-coupon bond. C. long 0.6217 shares of TCB stock and short 0.5596 shares of a zero-coupon bond.
            • C is correct. The no-arbitrage approach to creating a call option involves buying Delta = N(d1) = 0.6217 shares of the underlying stock and financing with –N(d2) = –0.5596 shares of a risk-free bond priced at exp(–rt)(X) = exp(–0.0022 × 0.25) (55) = $54.97 per bond. Note that the value of this replicating portfolio is nSS + nBB = 0.6217(57.03) – 0.5596(54.97) = $4.6943 (the value of the call option with slight rounding error).
              • 送分题 考试基本数字对，选项就能选出来
          • To determine the long put option value on TCB stock in Exhibit 1, the correct BSM valuation approach is to compute: A. 0.4404 times the present value of the exercise price minus 0.6217 times the price of TCB stock. B. 0.4404 times the present value of the exercise price minus 0.3783 times the price of TCB stock. C. 0.5596 times the present value of the exercise price minus 0.6217 times the price of TCB stock.
            • B is correct. The formula for the BSM price of a put option is p = e–rtXN(–d2) – SN(–d1). N(–d1) = 1 – N(d1) = 1 – 0.6217 = 0.3783, and N(–d2) = 1 – N(d2) = 1 – 0.5596 = 0.4404. Note that the BSM model can be represented as a portfolio of the stock (nSS) and zero-coupon bonds (nBB). For a put, the number of shares is nS = –N(–d1) < 0 and the number of bonds is nB = –N(d2) > 0. The value of the replicating portfolio is nSS + nBB = –0.3783(57.03) + 0.4404(54.97) = $2.6343 (the value of the put option with slight rounding error). B is a risk-free bond priced at exp(–rt)(X) = exp(–0.0022 × 0.25)(55) = $54.97.
              • Call 是 N(d1） - N(d2) Put是括号里面和外面都加个负号
    • Black Model
      • Option on Futures
      • Interest Rate Option
        • 例题
          • Franco replies: "The Black model can be used to value options on the Eurodollar future. In this model, futures options have two components: a futures component and a bond component. When hedging against rising interest rates, according to the Black model, the Eurodollar futures option used can be viewed as the futures component minus the bond component. Franco's description of the Black model's approach to valuation of Eurodollar futures options used for hedging is: A. correct. B. incorrect, because he is describing a call option. C. incorrect, because he is describing a put option.
            • 答案：B 解析：B is correct. Franco is incorrect because he describes a long call option, which according to the Black model can be viewed as the futures component minus the bond component. Long put options hedge against rising interest rates. The Black model evaluates put options as the bond component minus the futures component. A is incorrect. The statement is incorrect. C is incorrect. The Black model evaluates put options as the bond component minus the futures component.
              • 这题不难，但是要仔细推敲。 首先担心利率上涨，所以我们需要一个头寸，在利率上涨的时候获利。 利率上涨的时候价格下跌，所以我们需要long一个put。 BSM的基本公式是call，也就是stock/future - bond/money，所以题目描述的是call，不是我们想要的put。 最后才能得到结论，这题选B。
      • Swaption
        • 例题
          • Weber comments: "We can also consider options on swaps, which the Black model views as having a bond component and a swap component. The swaption, used to hedge against rising interest rates, can be evaluated as the swap component minus the bond component." Is Weber's description of the swaption used for the hedge most likely correct? A. No, because it would be correctly evaluated as the bond component minus the swap component B. No, because he is describing a receiver swaption C. Yes
            • 答案：C 解析：C is correct. A payer swaption would hedge against rising interest rates. According to the Black model, the value of a payer swaption can be described as the swap component minus the bond component. B is incorrect. A receiver swaption hedges against falling interest rates and Weber is describing a payer swaption. A is incorrect. The receiver swaption is evaluated as the bond component minus the swap component.
              • 这里要注意，swaption不再看call和put，而是看payer和receiver。 但是本质还是换汤不换药。N(d1) - N(d2)还是看涨，也就是其他的里面的call。但是这边的看涨，我们应该收固定支付浮动，也就是swap component - bond component。 如果记不住的话，还有个办法，这里担心利率上升，所以我们应该在利率上升的时候赚钱，我们肯定是收一个付一个，所以我们尽量付固定。一般bond是固定的，所以我们付固定，收益就是一个浮动的东西减去固定的东西。所以也描述的swap component - bond component也是一致的。
    • 标的如果不是stock，模型从BS变成B，原理不变
      • 调任何东西都是调X
    • Option Greeks
      • Delta
        • Call ~ (0,1)
        • Put ~ (-1,0)
        • N(option) = - N(stock) / Delta
      • Gamma
        • Gamma risk
          • 股价大幅跳跃
        • Long > 0; Short < 0
          • 例题
            • Lee also indicates that a long position in puts could be used to hedge larger moves in the GPX. She notes that although hedging with either puts or calls can result in a delta-neutral position, they would need to consider the resulting gamma. Lee's put-based hedge strategy for Solomon's ETF position would most likely result in a portfolio gamma that is: A. negative. B. neutral. C. positive.
              • C is correct. Because the gamma of the stock position is 0 and the put gamma is always non-negative, adding a long position in put options would most likely result in a positive portfolio gamma. Gamma is the change in delta from a small change in the stock's value. A stock position always has a delta of +1. Because the delta does not change, gamma equals 0. The gamma of a call equals the gamma of a similar put, which can be proven using put–call parity.
                • long的gamma大于0
        • 例题
          • Finally, Goldsworthy wants to determine the potential return of delta-hedged positions on Moonlight stock. Each position would require buying one of the options in Exhibit 2, and delta-hedging using Moonlight shares. Goldsworthy observes that Moonlight’s stock price often jumps and considers what consequences this has on the effectiveness of delta-hedging.
          • A delta hedge would be most effective for: A. Option 1. B. Option 2. C. Option 3.
            • A Delta hedge would be most effective when Gamma and Vega approach to zero, option 2 has lowest Gamma and Vega, so the most effective delta hedging will be using option 2.
              • Delta是一阶导，最有效的，找二阶导，Gamma。找最小的Gamma。
              • 如果Gamma一样，也可以看Vega，越小说明越不波动。
        • 例题
          • Burr asks Madisox to outline an appropriate hedging strategy. Madisox replies that to be fully hedged, an option trader will need to consider how changes in the stock price relative to the option exercise price affect the value of the call options. To be fully hedged against a small change in the stock price, Madisox suggests that the proper strategy to construct the hedge is to use call option delta and add the call option gamma to arrive at the number of shares required. Is Madisox's suggested hedging strategy for Weehawkin options most likely correct? A. Yes B. No, he should only use delta C. No, he should subtract gamma
            • A is correct A.Correct. Madisox's statement is correct. To be fully hedged against a small change in the stock price, the proper strategy to construct the hedge is to use call option delta and add the call option gamma to arrive at the number of shares required. The number of shares required is 0.606, based on the option delta of 0.587 plus the option gamma of 0.019. B.Incorrect. To be fully hedged against a small change in the stock price, the proper strategy to construct the hedge is to use call option delta and add the call option gamma to arrive at the number of shares required. C.Incorrect. You need to add, not subtract, option gamma to the option delta.
              • 注意fully hedge是delta + gamma
              • delta hedge是为了对冲线性变化。 gamma hedge是为了对冲非线性变化。
      • Theta
        • time passed
        • <0
          • 逝去时间越长，剩余时间越短，价值越小
        • 例题
          • After reviewing Exhibit 2, Solomon asks Lee which option Greek letter best describes the changes in an option's value as time to expiration declines. Which of the following is the correct answer to Solomon's question regarding the option Greek letter? A. Vega B. Theta C. Gamma
            • B is correct. Lee is pointing out the option price’s sensitivity to small changes in time. In the BSM approach, option price sensitivity to changes in time is given by the option Greek theta
              • 送分题 Theta和Time都是T打头。
      • Vega
        • Volatility
          • Volatility smile
            • volatility和exercise price的关系
          • 例题
            • Madisox notes that the implied volatility for the Weehawkin call option outlined in Exhibit 2 is 30%. With respect to other call options on Weehawkin stock, Madisox states the volatility surface provides a visualization of how implied volatility varies across both exercise price and time to maturity. Burr adds that implied volatility is useful in assessing the market price of risk since it is calculated on the basis of the historical volatility in the stock price. Jeffinsin concurs and adds that the volatility smile and skew typically have identical shapes whenever the market price of hedging is rising. Whose comment regarding implied volatility is most likely correct? A. Burr's B. Madisox's C. Jeffinsin's
              • B is correct A.Incorrect. Burr's statement is incorrect. Implied volatility is a measure of estimated future volatility, not historical volatility. Implied volatility is not calculated on the basis of historical volatility. Rather, implied volatility is a component of an option pricing model. B.Correct. Madisox's statement is correct. Implied volatility is a measure of future estimated volatility, which varies across both exercise price and time to expiration for various options. Accordingly, implied volatility is a measure of the market price of risk. C.Incorrect. Jeffinsin's statement is incorrect. Volatility skew tends to steepen whenever the market price of hedging is rising, which causes its shape to be different from the volatility smile.
                • 潜在波动是衡量未来，而不是过去。
                • 潜在波动衡量的是执行价格和到期时间与风险之间的关系
                • volatility skew说的是股价低的时候波动率高，股价高的时候波动率低，图像上波动率曲线是一个下行曲线。 volatility smile说的是股价很高和很低的时候波动率高，股价中间的时候波动率低，图像上波动率曲线是一个笑脸。 这2个图像是不同的。
      • Rho
        • Risk free rate
          • Call (+); Put (-)
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