# CFA 2018 Level 2 Derivatives

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Derivative
Level 2 -- 2017

Instructor: Feng Brief Introduction Topic weight: Study Session 1-2 Ethics &amp; Professional Standards 10 -15% Study Session 3 Quantitative Methods 5 -10% Study Session 4 Economics 5 -10% Study Session 5-6 Financial Reporting and Analysis 15 -20% Study Session 7-8 Corporate Finance 5 -15% Study Session 9-11 Equity Investment 15 -25% Study Session 12-13 Fixed Income 10 -20% Study Session 14 Derivatives 5 -15% Study Session 15 Alternative Investments 5 -10%
Brief IntroductionContent:
Ø SS 14: Derivative Investments: Valuation and Strategies
ü Reading 40: Pricing and Valuation of Forward Commitments
ü Reading 41: Valuation of Contingent Claims
ü
考纲对比:
Ø
与2016年相比，2017年的考纲几乎全部改变。
ü
ü
Brief Introduction
推荐阅读:

Ø 期权、期货及其它衍生产品 ü
John C.Hull 著
ü
ISBN: 978-7-1114-8437-0
ü 机械工业出版社 Brief Introduction
学习建议:
Ø
本门课程难度比较大，计算公式很多，一定要着重理解 和总结；

Ø 知识点之间的类比关系比较强，建议把第一部分学透后，
在继续学后面的知识点；
Ø 可以适当多做一些题，熟悉解题步骤，提高做题速度； Ø 最重要的，认真、仔细的听课。
Brief Introduction
Brief Introduction
Review of Derivatives in Level 1Tasks:
Ø
Review the basics of derivative instrument;
Ø Review the fundamental of derivative pricing. Review of Derivatives in Level 1Forward commitment
Ø
Contracts entered into at one point in time that require both parties to engage in a transaction at a later point in time (the expiration) on terms agreed upon at the start.
ü
Forward, future, and swapContingent claim
Ø
Derivatives in which the outcome or payoff is dependent on Review of Derivatives in Level 1Forward
Ø
An over-the-counter derivative contract in which two parties agree that one party, the buyer, will purchase an underlying asset from the other party, the seller, at a later date at a fixed price (forward price) they agree on when the contract is signed.
ü
In addition to the (forward) price, the two parties also agree on several other matters, such as the identity and Review of Derivatives in Level 1Futures
Ø
Futures contracts are specialized forward contracts that have been standardized and trade on a future exchange .
ü
Future contracts have specific underlying assets, times to expiration, delivery and settlement conditions, and quantities.
ü
The exchange offers a facility in the form of a physical location and/or an electronic system as well as liquidity Review of Derivatives in Level 1Swap
Ø
An over-the-counter derivative contract in which two parties agree to exchange a series of cash flows whereby one party pays a variable series that will be determined by an underlying asset or rate and the other party pays either (1) a variable series determined by a different underlying asset or rate or (2) a fixed series.
ü A swap is a series of (off-market) forwards.
Ø
The fixed price or rate at which the underlying will be purchased at a later date.
ü
Generally may not change as the (expected) price of the underlying asset changes.
Ø
The difference of “with the position” from “without the position”.
Review of Derivatives in Level 1Price of forward commitmentValue of forward commitment
Review of Derivatives in Level 1Option
Ø
A derivative contract in which one party, the buyer, pays a

sum of money to the other party, the seller or writer, and
receives the right to either buy or sell an underlying asset at

a fixed price either on a specific expiration date or at any time prior to the expiration date .
ü An option is a right, but not an obligation. ü
Default in options is possible only from the short to the
Ø
Option premium (c , p ): payment to seller from buyer. Ø Call option: right to buy. Ø Put option: right to sell. Ø
Exercise price/strike price (X): the fixed price at which the underlying asset can be purchased.
Ø
American option: exercisable at or prior to expiration.
Review of Derivatives in Level 1Option (Cont.)
Ø
Arbitrage is a type of transaction undertaken when two assets or portfolios produce identical results but sell for different prices.
Ø
Law of one price:
ü
Assets that produce identical future cash flows regardless of future events should have the same price;
ü
Review of Derivatives in Level 1Arbitrage
Review of Derivatives in Level 1Replication
Ø
Creation of an asset or portfolio from another asset, portfolio, and/or derivative.
Ø
An asset and a hedging position of derivative on the asset can be combined to produce a position equivalent to a risk- free asset.
ü
Asset + Derivative = Risk-free asset
ü
Asset - Risk-free asset = -Derivative Review of Derivatives in Level 1No arbitrage pricing
Ø
Determine the price of a derivative by assuming that there are no arbitrage opportunities (no arbitrage pricing).
ü
The derivative price can then be inferred from the characteristics of the underlying and the derivative, and the risk-free rate.

Pricing and Valuation of Forward ContractTasks:
Ø
Describe how forward contracts is priced and valued;
Ø Calculate and interpret the no-arbitrage value of Pricing and Valuation of Forward ContractPricing of forward
Ø
If the underlying asset generates no periodic cash flow, the forward price can be calculated as follows:
F (T) = S ×(1+r) T
ü
S : spot price;
ü r: risk free rate. Pricing and Valuation of Forward ContractCarry arbitrage model
T Ø
When the forward contract is overpriced , F (T) &gt; S (1+r) ,
Cash-and-Carry Arbitrage is available: ü
At initiation, borrowing money S at risk-free rate, buying (long) the spot asset, and selling (short) the forward at F (T);
Initial investment at initiation: \$0 ; ü
At expiration, settling the short position on forward contract by delivering the asset. Pricing and Valuation of Forward ContractCarry arbitrage model
T Ø
When forward contract is underpriced , F (T) &lt; S (1+r) ,
Reverse Cash-and-Carry Arbitrage is available: ü
At initiation, borrowing and selling (short) the spot asset, investing the proceed S at risk-free rate, and buying (long) the forward at F (T).
Initial investment at initiation: \$0 ; ü
At expiration, paying F (T) to settle the long position on forward contract, and delivering the spot asset to close the
Ø
If the underlying asset generates periodic cash flow, the forward price can be calculated as:
F (T) = (S - γ+ θ)(1+r) T
ü
γ: benefit of carrying the spot asset, in
present value form; ü
θ: cost of carrying the spot asset, in
present value form; ü γ - θ: net cost of carry.
Pricing and Valuation of Forward ContractPricing of forward
Ø
In the financial world, we generally define value as the value to the long position .
Ø At initiation, the forward contract has zero value .
Neither party to a forward transaction pays to enter the contract at initiation.
V (T) = 0
Pricing and Valuation of Forward ContractValuation of forwardValuation of forward (Cont.)
) - F (T)(1+r)
Pricing and Valuation of Forward Contract
Ø
During its life (t &lt; T), the value of a forward contract is:
(T-t)
V
(T) = (S
γ
θ
t
t
T
V T (T) = S
At expiration, the value of a forward contract is:
Ø
: present value of the benefit of holding an asset (t to T);
γ
t
ü
: present value of the cost of holding an asset (t to T);
t
t
ü
t
ϴ
F (T) Pricing and Valuation of Forward ContractExample
Ø
Assume that at Time 0 we entered into a one-year forward contract with price F (T) = 105. Nine months later, at Time t = 0.75, the observed price of the stock is S = 110 and the
0.75
interest rate is 5%. Calculate the value of the existing forward contract expiring in three months.
Ø Solution:
(T-t) -0.25 Summary ☆☆☆
Ø Importance:
Ø Content:
ü
Pricing and valuation of forward contract on underlying with/without cash flows.
Ø Exam tips:
ü
是forward pricing and valuation的一般形式，对后面的学 习非常重要，但考试一般都是靠后面具体的forward contract。

Pricing and Valuation of Equity and Currency ForwardTasks:
Ø Describe how equity and currency forward contracts is priced and valued;
Ø Calculate and interpret the no-arbitrage value of Pricing and Valuation of Equity ForwardPricing and valuation of equity forward
Ø
If the underlying is a stock and has discrete dividends, then forward price can be calculated as:
T T
F (T) = (S - PVD ) or: F (T) = S - FVD ×(1+r) ×(1+r)
T ü
PVD: present value of expected dividends;
ü FVD: future value of expected dividends. Ø
The value of equity forward can be calculated as:
(T-t)
V (T) = (S - PVD ) - F (T)(1+r)
t t t Pricing and Valuation of Equity ForwardExample
Ø
Suppose Nestlé stock is trading for CHF70 and pays a CHF2.20 dividend in one month. Further, assume the Swiss one-month risk-free rate is 1.0%, quoted on an annual compounding basis. Assume that the stock goes ex-dividend the same day the single stock forward contract expires. Thus, the single stock forward contract expires in one month. Calculate the one-month forward price for Nestlé stock. Pricing and Valuation of Equity ForwardExample
Ø
Suppose we bought a one-year forward contract at 102 and there are now three months to expiration. The underlying is currently trading for 110, and interest rates are 5% on an annual compounding basis. If there are no other carry cash flows, calculate the forward value of the existing contract.
Ø Solution: - T - t
 
V T = (S - PVD ) - F T 1 + r t t t      Pricing and Valuation of Equity ForwardPricing and valuation of equity index forward
Ø
For equity index, the forward price is usually calculated as if the dividends are paid continuously: c c 
(R - δ ) T f  F (T) = S e c
R ü f
: continuously compounded risk-free rate; c
δ ü : continuously compounded dividend yield.
Ø
The value of equity index forward can be calculated as: c c  
δ (T - t) - R (T - t) f
  V (T) = S e - F (T) e t t Pricing and Valuation of Equity ForwardExample
Ø
The continuously compounded dividend yield on the EURO STOXX 50 is 3%, and the current stock index level is 3,500.
The continuously compounded annual interest rate is 0.15%. Calculate the three month forward price.
Ø Solution: ( R ) T   f c c    (0.15% 3%) 0.25
   
e e
F (T) = S 3500 3475.15
Ø
The price of currency forward can be calculated by covered interest rate parity (IRP):
ü
F (T) and S are quoted by direct quotation: DC/FC;
ü
R
DC
: interest rate of domestic currency;
ü
R
FC : interest rate of foreign currency. T DC FC 1 + R F (T) = S
1 + R      
Pricing and Valuation of Currency ForwardPricing of currency forward

Pricing and Valuation of Equity and Currency ForwardValuation of currency forward
Ø
The value of currency forward can be calculated as: -(T-t) -(T-t)  
V (T) = S (1+R ) - F (T) (1+R ) t t FC DC
Ø
For continuously compounded risk-free rate: c c  

-R (T - t) -R (T - t) FC DC
  V (T) = S e - F (T) e t t Pricing and Valuation of Equity and Currency ForwardExample
Ø
A corporation sold Euro(€) against British pound (£) forward at a forward rate of £0.8 for €1 at Time 0. The current spot market at Time t is such that €1 is worth £0.75, and the annually compounded risk-free rates are 0.80% for the British pound and 0.40% for the Euro. Assume at Time t there are three months until the forward contract expiration.
Calculate the forward price F (£/€, T) at Time t and the value

t Pricing and Valuation of Equity and Currency ForwardAnswer:
Ø
The forward price F (£/€, T) at Time t:
t T-t 0.25  
1 + R  1 + 0.8%  DC    
F (T) = S t t   0.75 0.7507  
1 + R  1 + 0.4%  FC  
Ø
The value of foreign exchange forward contract at Time t: S F (T)
V (T) = -[ ]
t
t T-t T-t
(1+R ) (1+R )
0.8
0.75 
= = 0.0499 £
0.25
0.25
Ø Importance:
☆☆☆ Ø
Content: ü
Pricing and valuation of equity forward;
ü Pricing and valuation of currency forward. Ø
Exam tips: ü
常考点：计算题。
Summary

Ø Describe how interest rate forward contracts is priced and valued;
Ø Calculate and interpret the no-arbitrage value of Pricing and Valuation of FRAForward rate agreement (FRA)
Ø
A FRA is an over-the-counter (OTC) forward contract in which the underlying is an interest rate (e.g. Libor).
ü
Long position can be viewed as the obligation to take a
loan at the contract rate (i.e., borrow at the fixed rate,
floating receiver); gains when reference rate increase;
ü
Short position can be viewed as the obligation to make a
loan at the contract rate (i.e., lend at the fixed rate, fixed Pricing and Valuation of FRAThe notation of FRA
Ø
The notation of FRA is typically “a ×b FRA”:
ü
a: the number of months until the contract expires;
ü b: the number of months until the underlying loan is settled. Ø
Example: 3×9 FRA
3 ×9 FRA
Today 3 months 9 months Pricing and Valuation of FRAThe uses of FRA
Ø
Lock the interest rate or hedge the risk of borrowing or lending at some future date.
ü
One party will pay the other party the difference (based on notional value) between the interest rate specified in the FRA and the market interest rate at contract settlement.
If forward rate &lt; spot rate, the long receives payment;If forward rate &gt; spot rate, the short receives paym Pricing and Valuation of FRAPricing of FRA
Ø
The “forward price” in FRA is actually a forward rate, it can be calculated from the spot rates.
ü
FRA rate is just the unbiased estimate of the forward rate; Recall the forward rate model in “Fixed Income Level 2”;
But we use simple interest for money market instrument.
Note: Libor rates are add-on rate and quoted on a
30/360 day basis in annual terms.
Ø Forward rate models show how forward rates can be extrapolated from spot rates.
  b a 30 b a 30 b 30 a
1 S
1 S
1 FR 360 360 360      
             
   
a b b
30 b
1 S 360   
Pricing and Valuation of FRAPricing of FRA (Cont.)
Ø
Based on market quotes on Canadian dollar (C\$) Libor, the six-month C\$ Libor and the nine-month C\$ Libor are presently at 1.5% and 1.75%, respectively. Assume a 30/360- day count convention. Calculate the 6
×9 FRA fixed rate.
Ø Solution:
[1+(1.5% ×180/360)]×[1+(FRA rate×90/360)]
Pricing and Valuation of FRAExample
Pricing and Valuation of FRAValuation of FRA at expiration (t = a)
Ø
Although the interest on the underlying loan comes at the end of the loan, the FRA is settled at the expiration of FRA.
ü
For “a ×b FRA”, the “interest saving” due to the FRA position comes at “Time b”, but is settled at “Time a”;
ü
So the “interest saving” need to be discounted to “Time a” to calculate the value of FRA.
 Days   NP (Underlying rate - Forward rate)  

Ø
Specification of 1  4 FRA:
ü
Term = 30 days
ü
Notional amount = \$1 million
ü
Underlying rate = 90-day LIBOR
ü
Forward rate = 7% At t = 30 days, 90-day LIBOR = 8%, clarify the payment (value) of this FRA.
Pricing and Valuation of FRAExample: 1  4 FRASolution: 1  4 FRA
Ø
Underlying floating rate &gt; fixed rate, so long position receives payment.
T T
30 T 120
Forward rate: 7% Expiry of FRA;
90-day Libor: 8% Interest saving:
(8%-7%) x 90/360 x \$1m = \$2,500
Payment = \$2,450.98 Pricing and Valuation of FRA Pricing and Valuation of FRAExample
Ø
In 30 days, a UK company expects to make a bank deposit of £10M for a period of 90 days at 90-day Libor set 30 days from today. The company is concerned about a decrease in interest rates. Its financial adviser suggests that it negotiate today, at Time 0, a 1 ×4 FRA, an instrument that expires in 30 days and is based on 90-day Libor. The company enters into a £10M notional amount 1
×4 receive-fixed FRA that is Pricing and Valuation of FRAExample (Cont.)
Ø
After 30 days, 90-day Libor in British pounds is 0.55%. If the FRA was initially priced at 0.60%, the payment received by the UK company to settle it will be closest to?Solution:
Ø
Because the UK company receives fixed in the FRA, it benefits from a decline in rates.Valuation of FRA prior to expiration (t < a)
t t b t Days from a to b NP (FR - FR )
Pricing and Valuation of FRA

  
 
   
 
1 S
360      
V Days from t to b
360
Ø
Step 1: calculate the new FRA rate (FR
1 S
1 S
      b t a t b a b t a t t
Step 2: calculate the value of FRA as:
Ø
):
t
   1 FR        Pricing and Valuation of FRAExample
Ø
We entered a long 6 ×9 FRA at a rate of 0.86%, with notional amount of C\$ 10M. The 6-month spot C\$ Libor was 0.628%, and 9-month C\$ Libor was 0.712%. After 90 days have passed, the 3-month C\$ Libor is 1.25% and the 6-month C\$ Libor is 1.35%. Calculate the value of the receive-floating
6 ×9 FRA. Pricing and Valuation of FRAAnswer:
Ø
Step 1: [1 + (1.25%
×90/360)]×[1 + (new FRA rate×90/360)] = [1 + (1.35%
×180/360)] So, new FRA rate = 1.46%
Ø
Step 2: V = 10M
×(1.46% -0.86%)×0.25/(1+1.35%×90/360)
t
= 14900 Summary ☆☆☆
Ø Importance:
Ø Content:
ü Pricing and valuation of FRA. Ø
Exam tips: ü
常考点：FRA value 的计算。 Pricing and Valuation of Fixed-Income ForwardTasks:
Ø
Describe how fixed income forward contracts is priced and valued;
Ø Calculate and interpret the no-arbitrage value of Pricing and Valuation of Fixed Income ForwardPricing and valuation of fixed income forward
Ø
Similar to equity forward, the forward price of fixed income forward can be calculated as:
T T
F (T) = (S - PVC ) or: F (T) = S - FVC ×(1+r) ×(1+r)
T ü
PVC: present value of expected coupon payment;
ü FVC: future value of expected coupon payment. Ø
The value of fixed income forward can be calculated as:
(T-t)
V (T) = (S - PVC ) - F (T) ×(1+r)
t t t Pricing and Valuation of Fixed Income ForwardExample
Ø
One month ago, we purchased five euro-bond forward contracts with two months to expiration and a contract notional of €100,000 each at a price of 145 (quoted as a percentage of par). The euro-bond forward contract now has one month to expiration and the current forward price is 148. Assume the risk-free rate is 0.1%, calculate the value of the euro-bond forward position.
(T-t)
Ø
V
t
(T) = [F
t
(T) - F (T)] ×(1+r)
1/12
= (148 - 145) ×(1+0.1%)
= 2.9997 So, the value of the forward position is: 0.029997
×€100,000×5 = €14998.5

Pricing and Valuation of Fixed Income ForwardAnswer:
Pricing and Valuation of Fixed Income ForwardPricing of fixed income futures
Ø
In terms of fixed income futures, there are several unique issues:
ü
The prices of bonds are often quoted without accrued interest (i.e. flat price, clean price).
ü
Bond futures contracts often have more than one bond that can be delivered by the short (delivery option), and

conversion factor (CF) is used in an effort to make all Pricing and Valuation of Fixed Income ForwardPricing of fixed income futures (Cont.)
ü
When multiple bonds can be delivered for a futures contract with particular maturity, a cheapest-to-deliver

(CTD) bond typically emerges after adjusting for the conversion factor. Pricing and Valuation of Fixed Income ForwardPricing of fixed income futures (Cont.)
Ø
Calculation of accrued interest (AI):
t AI= PMT T
T PMT PMT+F t
… … … n settlement
1 date Pricing and Valuation of Fixed Income ForwardPricing of fixed income futures (Cont.)
Ø
The quoted price of fixed income futures can be calculated as:
T
Quoted futures price = [(S - PVC) - AI ]/CF ×(1+r)
T T
or: Quoted futures price = [S - AI - FVC]/CF ×(1+r)
T ü
AI : the accrued interest at maturity of the futures contract;
T ü
S : bond full price; S = Quoted price + AIAI : the accrued interest at initiation of the future contr Pricing and Valuation of Fixed Income ForwardExample
Ø
Suppose the underlying of Euro-bond futures is a German bond that is quoted at €108 and has accrued interest of €0.083. The euro-bond futures contract matures in one month. At expiration, the underlying bond will have accrued interest of €0.25 and have no coupon payments due until the futures contract expires. Assume the conversion factor of the underlying bond is 0.729535 and the current one-month Pricing and Valuation of Fixed Income ForwardAnswer:
Ø
According to the example:
ü
CF = 0.729535; T = 1/12; FVC = 0; r= 0.1%;
ü
S = €108 + €0.083 = €108.083;
ü
AI = €0.25;
T Ø
So the futures price is:
1/12
[108.083 - 0.25]/0.729535 = €147.82 ×(1 + 0.1%) Pricing and Valuation of Forward and FuturesA brief summary
Ø
The forward or futures price is simply the value of the underlying adjusted for any carry cash flows;
Ø
The forward value is simply the present value of the difference in forward prices at an intermediate time in the contract;
Ø
The futures value is zero after marking to market because profits and losses are settled daily. The time value of money
Ø Importance:
☆☆ Ø
Content: ü Pricing and valuation of fixed income forward. ü Pricing and valuation of fixed income futures. Ø
Exam tips: ü
常考点：fixed income forward price 和 value的计算。
Summary

Pricing and Valuation of Interest Rate SwapTasks:
Ø
Describe how interest rate swap is priced and valued;
Ø Calculate and interpret the no-arbitrage value of Pricing and Valuation of Interest Rate SwapSwap
Ø
There are three kinds of swaps:
ü Interest rate swaps
If A loans money to B for a fixed rate of interest and B loans the same amount to A for floating rate of interest.
ü Currency swaps
If the loans are in two different currencies.
ü Equity swaps Pricing and Valuation of Interest Rate SwapInterest rate swap
Ø
Plain Vanilla interest rate swap is an interest rate swap in which one party pays a fixed rate (fixed-rate payer) and the other pays a floating rate (floating-rate payer).
ü
Notional amount is not exchanged at the beginning or the end of the swap, because both loans are in same currency and amount;
ü
On settlement dates, interest payments are netted; Pricing and Valuation of Interest Rate SwapPricing of interest rate swap
Ø
Principle: the fixed rate in swap (FS, swap rate) should makes the contract value zero at initiation.
Ø Methodology:
ü
A receive-floating, pay-fixed swap is equivalent to being long a floating-rate bond and short a fixed-rate bond;
ü
If both bonds are priced at par, the initial cash flows are zero and the par payments at the end offset each other; Pricing and Valuation of Interest Rate SwapExample of receive-floating, pay-fixed interest rate swap
… S - FS
n-1
S - FS S - FS
t t t
1 2 n =
Long floating- …
S S S +Par
1 n-1
rate bond Short fixed-
…
FS - FS - FS-Par rate bond Pricing and Valuation of Interest Rate SwapPricing of Plain Vanilla interest rate swap
Ø
At initiation, the floating-rate bond has a value equal to its par value, what we should do is to find a fixed-rate bond with a value equal to the same par value at initiation.
PV = PV = Par value Fixed rate bond Floating rate bond
Ø
Assume F as the periodic coupon payment of the n-period fixed-rate bond with par value of \$1. wherein:
D
n
= discount factor or PV factor, the price of zero-coupon bond with par value of \$1 and maturity of n periods.
Ø
Then, we have:      1 2 3 n 1 = F D + F D + F D + ...+ F D + 1 D n
Pricing and Valuation of Interest Rate SwapPricing of Plain Vanilla interest rate swap (Cont.)
Ø
Suppose we are pricing a five-year Libor-based interest rate swap with annual resets (30/360 day count). The estimated present value factors are given in the following table. Calculate the fixed rate of the swap.
Maturity (years) Present value factors 1 0.990099 2 0.977876 3 0.965136
Pricing and Valuation of Interest Rate SwapExample