# CFA Level -2 Derivatives - I

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1 CFA Level -2 Derivatives - I EduPristine

2 Agenda Forwards Markets and Contracts Future Markets and Contracts Option Markets and Contracts 1

3 Forwards Markets and Contracts 2

4 Pricing and Valuation Of Forward Contracts A forward contract price is the fixed price or rate at which the transaction scheduled to occur at expiration will take place. This price is agreed on the contract initiation date, and is commonly called the forward price or forward rate. The price here is different from value, which is what you can sell something for or what you must pay to acquire something. Pricingmeans to determine the forward price or forward rate. Valuationis the process of determining the value of an asset or service. It means to determine the amount of money one would need to pay or would expect to receive to engage in the transaction. For example, if one already held a position, valuation would mean to determine the amount of money one would either pay to pay or expect to receive in order to get out of the position. 3

5 Pricing and Valuation Of Forward Contracts Arbitrage free Forward prices are given as: Where: F ) ( 0, T ) = S 0 (1+ R f T If we long the forward contract at time t=0, at forward's price F(0,T), the initial cash outlay would be zero At time t=t, we have claim on the asset which is worth ST and an obligation to pay F(0,T) at time t=t At time t= T, we pay F(0,T) and receive the asset worth ST Long Forward Contract- F(0,T) Outflow=0 Claim on asset worth S t Obligation to pay F(0,T) at T Receive asset worth S T Outflow = F(0,T) t=0 t=t t=t Forward rates are rates of interest implied by the current zero rates for a period of time in the future 4

6 Forward Contract - Example You need a laptop, one year from now. The current price of a laptop is \$1000. You have 3 options, to fulfill your requirement: 1. Buy it now at \$1000, and use it after a year 2. Buy it a year from now at the market price prevailing one year from now (unknown price). 3. Enter into an agreement with the vendor so as to purchase the laptop a year from now at a price of \$1000. In each of the three scenarios, what would be your profit/loss if the price of the laptop after one year is: 1. \$ \$ \$1500 5

7 Forward Contract - Example Taking the Third Scenario, If you locked a fixed price of \$1000 for a laptop after one year, your profit and loss would look like: Agreedupon Price of Laptop (X=strike) Price of Laptop (S=spot) Profit and Loss from the Buyer's Perspective (S-X) The Profit and Loss from the vendor's point of view: Agreedupon Price of Laptop (X=strike) Price of Laptop (S=spot) Profit and Loss from the Buyer's Perspective (X-S)

8 Forward Contract - Example Payoff from Buyer's Perspective Payoff from Seller's Perspective Slope of the line is Slope of the line is -1 This is a Zero-Sum Game 7

9 Example 2: Forward Contract Payoff A person wants to buy a Machine a year from now. If the current forward price of one year at time (T=0) is \$1000. The price can change over the period of time. If the actual price of that laptop after one year is selling above the price of \$1000. It will make profit for the long party, and equal loss for the short party and vice-versa. Spot T=0 Forward T=1 If theforwardpriceat Time 1 is Payoffforthe forward position Payoff Forward profit and loss S(T) For calculating the Present value of the payoff we will divide the amount by (1+Risk free rate) ^ Time 8

10 Derivatives Payoff Non-Linear Payout Linear Payout Y-axis Profit/Loss on (D) X-axis Value of Underlying (A) 9

11 Forward Contract Valuation The price of a forward contract is given by the equation below: F 0 = S 0 e rt in the case of continuously compounded risk free interest rate, r F 0 = S 0 (1+r) T in the case of annual risk free interest rate, r where, F 0 : forward price S 0 : Spot price t: time of the contract Value of a long position in a forward contract during the life of the contract (V t ), where t < T is: V t = S t - {F 0 / (1+r) T-t } The discounting period reduces with passage of time (T-t). Spot price(s t ) also changes after initiation of the contract. The value to a short position in a forward contract is negative of the long position value: V t ={F 0 / (1+r) T-t } -S t 10

12 Example (Forward Price) Spot Price of an Assets= \$100, Risk Free Rate= 10% Contract Maturity 1 year Storage cost = \$5 Convenience Yield = \$10 T=0 T=0.25 T=0.5 T= 1year Contract Initiation Storage Cost ( \$5 ) Convenience Yield ( \$10 ) Contract Maturity F T = S0(1+ R f ) + FV of Storage cost FV Convenience = 100 (1+ 0.1) = = (1+ 0.1) (1+ 0.1) 0.5 yield 11

13 Forward Contract Valuation Example: Suppose a person entered in a long position on a 6 months forward contract of a stock at \$120. After four months the stock price has moved to \$144. Calculate the value of the forward contract assuming a risk free rate of 5%. Solution: V t = S t - {F 0 / (1+r) T-t } Value for the long position = ( 2 / 12 ) (1. 05 ) = \$ 25 Value for the short position is = - \$25 12

14 Price And Value Of An Equity Forward Contract, Assuming Dividends Are Paid Either Discretely Or Continuously A stock or a equity index generally has expected dividends assigned to them. To determine the price of such forward contracts, we can adjust the spot price for the present value of expected dividends (PVD) over the life of the contract or adjust the forward price for future value of the dividends (FVD) over the life of the contract. Arbitrage free Forward prices are given for such contracts: F = ( S 0 PVD ) (1 + R ) f t 13

15 Example Example: Calculate the forward price for Neev Inc. on a 90 days forward contract which is currently quoted at \$75 and is expected to pay a dividend of \$2 in 30 days. Assume a risk-free rate of 5%. Solution: PVD = 2 ( ) = 30 / 365 \$ 1.99 F = ( S 0 PVD ) (1 + R ) f t Forward Price = ( ) * (1.05) 90/365 = \$

16 Forward Price Of Different Assets Classes Benefit to the assets holder Cost to the assets holder Forward price Equity Commodity Currency Dividend Convenience yield N/A Nothing Storage Cost N/A S T T (1+ R ) FV of div. S (1+ R ) + FV of Storagecost FV of Convenience yield 0 f 0 f S (1 ) 0 + R f T 15

17 Value Of A Equity Forward Contract Value of long position in a forward contract paying dividends (V t ) V t = (S t -PVD) {FP / (1+r) T-t } Example: A person has entered into a 150-day forward contract on a stock at \$80 and there is an expected dividend on the stock of \$2.5 payable in 90 days. After 30 days the stock price has moved to \$86. What is the value of this forward contract given risk free rate of 5%. Solution: The dividend payable has 60 days left before the stock becomes ex-dividend from the current date and the contract will mature in 120 days. PVD = 2 30 / 365 ( 1.05) = 2.48 Value of the contract = ( ) {80/(1.05) 120/365 } = \$

18 Equity Forward Contract With Continuous Dividends If the underlying asset on which the forward contract is entered into provides a continuously compounded yield q, then the forward contract would be valued as: Forward Price = S 0 e (r q)t where q is continuously compounded% of return on the asset, such as a equity index Example: The current value of Nifty is 5350 providing a continuous dividend yield of 2.5%.Calculate the price of a 100 day contract given the continuously compounded risk free rate of 6%. FP= 5350* e ( )*100/365 FP = V t = e s q t * T e FP R f * T Value of forward contract(for a long position) with continuous yield is derived: 17

19 Equity Forward Contract With Continuous Dividends In the previous example suppose after 45 days the index moves from 5350 to 5430, calculate the value of a forward contract assuming the same risk free rate and dividend yield. Solution: 5430 = 0.025*55/365 e = V t = \$ e 0.06*55/365 18

20 Value Of A Fixed Income Security (Example) In a fixed income security there are cash flows in the form of coupons. So even for these cash flows we have to find the present value of coupons(pvc) or future value of coupons(fvc) The forward price for a fixed income security: Example: Calculate the price of a 220 day forward contract on a 6% semi-annual coupon bond with a spot price of \$106(includes accrued interest), which has just paid a coupon and will make its next payment in 180 days. The risk free is assumed to be 4%. Solution: Price of the forward contract 100 *0.06 Coupon = = \$3 2 PVC 3 = 1.04 = 180 / ( S PVC) ( R ) T 0 f 220 / 365 ( ) 1.04 FP = 1+ FP= =

21 Price And Value Of A Fixed Income Security Value of a forward contract for a long position: Suppose a person had entered a 200 day forward contract on a bond at 107, 60 days have passed and the current price of the bond is 110. The interest due on this bond \$3.5 which is payable 45 days from now. Calculate the value of the contract assuming a risk free rate of 5%. Solution: PVC = = 45 / 365 \$3.48 V t = 140 / 365 = \$

22 Price and Value of a Forward Rate Agreement Forward rates are rates of interest, implied by the current zero rates, for a period of time in the future For example, if we have the zero rates for year 4 and year 5, the forward rate for the period of time between year 4 and year 5 would be known as the forward rate for that time period of 1 year. Year 4 Year 5 F 4 = 4% F F 4,5 5 = 5% Consider that you invest \$100 for 4 years and then roll it forward for one year in the 5 year. Then the total amount would be given as: 100*e 0.04*4 e F4,5*1 If the same \$100 was invested for 5 years instead then it would grow to 100*e.05*5 Equating the two we get F4,5 = 8.99% 21

23 Forward Rate Agreement (FRA) A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period. Forward contract to borrow (long) or lend (short) at a pre-specified rate. A forward rate agreement is a forward contact on a short-term interest rate, usually LIBOR. It is a forward contract to borrow / lend money at a certain rate at some future date, Payment to the long at settlement: (Rate at settlement FRA rate) ( days 360) = Notional principal 1+ (rate at settlement ) ( days 360) Numerator -the difference between the actual rate that exists in the marketplace on the expiration date and the agreed-upon rate at the beginning of the contract. The denominator -Discounting the payment at the current LIBOR, based on the assumption that they will accrue interest 22

24 FRA Example Example: FRA that settles in 30 days Notional Principal \$1,000,000 Based on 90-day LIBOR Forward rate of 5%, Actual 90-day LIBOR at settlement is 6% Solution: Payment to the long at settlement = Notional principal (Rate at settlement 1+ FRA rate) ( days (rate at settlement ) ( days 360) 360) (6% -5%) * (90/360)* \$1m = \$2,500 Value at settlement: 2,500 / (1 + (90/360)*6%) = \$2,463 23

25 Forward Rate Agreements Assumption A forward rate agreement (FRA) is an over the counter agreement where the forward interest rate, F t1,t2,is fixed for a certain principal between times T1 and T2 The payer of the fixed interest rate is also known as the borrower or the buyer. The buyer hedges against the risk of rising interest rates, while the seller hedges against the risk of falling interest rates. The potential borrower can lock in borrowing rate with FRA, a contract to enter into a loan at a future date. Usually settled with cash payment, the amount of settlement makes the total interest rate (actual interest cost + FRA settlement) equal to the contracted interest cost in FRA 24

26 Currency Forward Contract. (Example) The spot rate of EUR/USD foreign exchange is \$ per EUR. USD LIBOR is 4% and EUR LIBOR is 2%. What is the price of a 180 day forward contract on USD per EUR exchange rate? Solution: F = S T R R DC FC T 1.04 F T = \$ = \$ / 365 Here F T and S 0 are quoted in domestic currency per unit of foreign currency R DC = domestic interest rate R FC = foreign interest rate 25

27 Valuing A Currency Forward Contract Before maturity, value of a forward currency contract(vt) will depend on the spot rate at time t. V t S t t = F { } { } T t T t 1+ RFC 1+ RDC Suppose in the previous example the spot rate is \$1.159 after 60 days, calculate the value of a forward contract / = 120 / 365 \$

28 Credit Risk In A Forward Contract After entering the forward contract any party can have positive value. This party which can be long or short on a contract faces the credit risk from the opposite party as it owes the positive value. It is the risk that the counter party will not pay when a positive amount is owed at settlement. The larger the value of the forward to one party, the greater the credit risk to that party. The contract value and simultaneously the credit risk may increase, decrease or can even change the values for either party over the term of the contract. Credit risk can be reduced by marking to market. 27

29 Questions 1. Which of the following is TRUE about a forward rate agreement (FRA)? A. It can be cash or physically settled B. A borrower who intends to borrow cash at LIBOR in the future will hedge by receiving the fixed interest rate, R(k), in an FRA C. A bank that intends to lend cash at LIBOR in the future will hedge by receiving the fixed interest rate, R(k), in an FRA 2. A company wants to borrow \$10 million for 90 days starting in one year. To hedge the interest rate risk of the future borrowing, the company enters into a forward rate agreement (FRA) where the company will pay a fixed rate, R(k), of 5.0%. The FRA cash settles in one year; i.e., in advance (T=1.0) not in arrears (T=1.25). All rates are expressed with quarterly compounding. If the actual 90-day LIBOR observed one year forward turns out to be 6.0%, what is the cash flow payment/receipt by the company under the FRA? A. Company pays \$24,631 B. Company pays \$25,000 C. Company receives \$24, Assume the one-year spot (zero) interest rates is 3.0% and the fifteen month (1.25 years) zero rate is 4.0%, with continuous compounding. What is the value of a forward rate agreement (FRA) that enables the holder to earn 7.0%, expressed with quarterly compounding, for a 3-month period starting in one (1) years on a (principal) notional of \$1,000,000? A. -\$2,701 B. -\$2,570 C. +\$2,570 28

30 Questions 4. Calculate the no-arbitrage forward price for a 90-day forward on a stock that is currently priced at \$50.00 and is expected to pay a dividend of \$0.50 in 30 days and a \$0.60 in 75 days. The annual risk free rate is 5% and the yield curve is flat. A. \$ B. \$ C. \$ Calculate the price of a 200-day forward contract on an 8% U.S. Treasury bond with a spot price of \$1,310. The bond has just paid a coupon and will make another coupon payment in 150 days. The annual risk-free rate is 5%. A. \$1, B. \$1, C. \$1,

31 Answers 1. C.A bank that intends to lend cash at LIBOR in the future will hedge by receiving the fixed interest rate, R(k), in an FRA In the future, the bank s cash flows will be: (+) receive LIBOR on the lent cash (+) receive fixed rate on the FRA (-) pay LIBOR on the FRA = net (+) receive fixed rate on the FRA In regard to (a)fra is cash settled. In regard to (b), to hedge the future LIBOR, the borrower wants to pay fixed and receive LIBOR (i.e., the gain/loss on LIBOR in the FRA offsets the future borrowing). 2. C. Company receives \$24,631 The payoff to the company = \$10 MM *(6.0% -5.0%)* 0.25 = \$25,000; i.e., if LIBOR goes up, the companies borrowing cost will increase but the FRA will hedge by paying the company But the FRA settles at T = 1.0, such that payoff = \$25,000 /( % * 0.25) = \$24, B. -\$2,570 The forward rate (1.0, 1.25) = (4%*1.25-3%)/0.25 = 8.0% with continuous compounding. The converts to 4*(exp(4%/4 )-1) = % with quarterly compounding. In this case, R(k) = 7.0% and R(f) = %, and R(k) is earned by the holder such that: The value to this holder of the FRA is [\$1,000,000 * 0.25 * (7.0% %)]*exp(-4.0%*1.25) = - \$2,570 30

32 Answers 4. C. The present value of expected dividends is: \$0.50 / ( / 365 ) + \$0.60 / ( / 365 ) = \$1.092 Future price = (\$ ) / 365 = \$ B Coupon = (1, ) / 2 = \$40.00 Present value of coupon payment = \$40.00 / /365 = \$39.21 Forward price on the fixed income security = (\$1,310 -\$39.21) (1.05) 200/365 = \$1,

33 Thank You! EduPristine

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