CFA 2018 Level 2 Derivatives

Topic weight:

Study Session 1-2 Ethics & 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% Study Session 16-17 Portfolio Management 5 -10%

Weights: 100%

Brief Introduction

Content:

Ø SS 14: Derivative Investments: Valuation and Strategies üReading 40: Pricing and Valuation of Forward

Commitments

考纲对比:

Ø 与2016年相比，2017年的考纲几乎全部改变。

ü Reading 40 和 Reading 41的主要内容在2016年考纲中基 本也都有，但考纲的结构和表述几乎全部改变；

ü Reading 42是新增的，从2016年三级内容中调整到二级。

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

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。

Review of Derivatives in Level 1

Tasks:

Ø

Review

the basics of derivative instrument;

Review of Derivatives in Level 1

Forward 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. üForward, future, and swap

Contingent claim

Ø Derivatives in which the outcome or payoff is dependent on the outcome or payoff of an underlying asset.

Forward

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.

üIn addition to the (forward) price, the two parties also agree on several other matters, such as the identity and the quantity of the underlying.

Futures

Ø Futures contracts are specialized forward contracts that have been standardized and trade on a future exchange. üFuture contracts have specific underlying assets, times to

expiration, delivery and settlement conditions, and quantities.

üThe exchange offers a facility in the form of a physical location and/or an electronic system as well as liquidity provided by authorized market makers.

Swap

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.

üA swap is a series of (off-market) forwards.

Price of forward commitment

Ø The fixed price or rate at which the underlying will be purchased at a later date.

üGenerally may not change as the (expected) price of the underlying asset changes.

Value of forward commitment

Ø The difference of “with the position” from “without the position”.

üMay increase or decrease as the (expected) price of the underlying asset changes.

Option

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 long.

Option (Cont.)

Ø Option premium (c₀, p₀): 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. Ø European option: exercisable only at expiration.

Arbitrage

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:

üAssets that produce identical future cash flows regardless of future events should have the same price;

üTrader will exploit the arbitrage opportunity quickly (buy low and sell high), then make the prices converge.

Replication

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 ü Derivative - Risk-free asset = - Asset

• A “-” sign indicates a short position, or borrowing at Rf.

No arbitrage pricing

Ø Determine the price of a derivative by assuming that there are no arbitrage opportunities (no arbitrage pricing).

üThe 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 Contract

Tasks:

Ø

Describe

how forward contracts is priced and

valued;

Ø

Calculate and interpret

the no-arbitrage value of

Pricing and Valuation of Forward Contract

Pricing of forward

Ø If the underlying asset generates no periodic cash flow, the forward price can be calculated as follows:

F

₀

(T) = S

₀

×

(1+r)

T üS₀: spot price;

Carry arbitrage model

Ø When the forward contract is overpriced, F₀(T) > S₀(1+r)T,

Cash-and-Carry Arbitrage is available:

üAt initiation, borrowing money S₀ at risk-free rate, buying (long) the spot asset, and selling (short) the forward at F0(T);

• Initial investment at initiation: $0;

üAt expiration, settling the short position on forward contract by delivering the asset.

• Profit at expiration: F0(T) - S0(1+r)T.

Carry arbitrage model

Ø When forward contract is underpriced, F₀(T) < S₀(1+r)T,

Reverse Cash-and-Carry Arbitrage is available:

üAt initiation, borrowing and selling (short) the spot asset, investing the proceed S0 at risk-free rate, and buying (long)

the forward at F0(T).

• Initial investment at initiation: $0;

üAt expiration, paying F₀(T) to settle the long position on forward contract, and delivering the spot asset to close the short position on spot asset.

• Profit at expiration: S0(1+r)T- F0(T).

Pricing of forward

Ø If the underlying asset generates periodic cash flow, the forward price can be calculated as:

F

₀

(T) = (S

₀

-

γ

+

θ

)(1+r)

T

üγ: benefit of carrying the spot asset, in present value form;

üθ: cost of carrying the spot asset, in present value form; üγ - θ: net cost of carry.

Valuation 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.

V0(T) = 0

Valuation of forward (Cont.)

Ø During its life (t < T), the value of a forward contract is: Vt(T) = (St - γt + θt)- F0(T)(1+r)-(T-t)

üϴ_t: present value of the cost of holding an asset (t to T); üγ_t: present value of the benefit of holding an asset (t to T);

Ø At expiration, the value of a forward contract is: VT(T) = ST - F0(T)

Example

Ø Assume that at Time 0 we entered into a one-year forward contract with price F0(T) = 105. Nine months later, at Time t

= 0.75, the observed price of the stock is S0.75 = 110 and the

interest rate is 5%. Calculate the value of the existing forward contract expiring in three months.

Ø Solution:

Vt(T) = St - F0(T)(1+r)-(T-t) = 110 – 105(1+5%)-0.25 = 6.273

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

Forward

Tasks:

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 forward

Ø If the underlying is a stock and has discrete dividends, then forward price can be calculated as:

F0(T) = (S0- PVD0)×(1+r)T or: F0(T) = S0×(1+r)T - FVDT

üPVD: present value of expected dividends;

üFVD: future value of expected dividends.

Ø The value of equity forward can be calculated as: Vt(T) = (St - PVDt)- F0(T)(1+r)-(T-t)

Example

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.

 

T 112

0 0

F (T) = S  1+r - FVD_T= 70 (1 + 0.01 ) - 2.2 = 6 . 67 8 Ø Solution:

Example

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.

 

 



 

Pricing and valuation of equity index forward

Ø For equity index, the forward price is usually calculated as if the dividends are paid continuously:

ü : continuously compounded risk-free rate;

ü : continuously compounded dividend yield.

Ø The value of equity index forward can be calculated as:

c c

Example

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.

( ) (0.15% 3%) 0.25

0 0

F (T) = S _e Rfcc T _3500_e   _3475.15 Ø Solution:

Pricing of currency forward

Ø The price of currency forward can be calculated by covered interest rate parity (IRP):

üF₀(T) and S₀ are quoted by direct quotation: DC/FC;

üR_DC: interest rate of domestic currency; üR_FC: interest rate of foreign currency.

Ø For continuously compounded risk-free rate:

c c

Valuation of currency forward

Ø The value of currency forward can be calculated as:

Ø For continuously compounded risk-free rate:

c c

Example

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 Ft(£/€, T) at Time t and the value

of foreign exchange forward contract at Time t.

Answer:

Ø The forward price F_t(£/€, T) at Time t:

Ø The value of foreign exchange forward contract at Time t:

T-t _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。

Pricing and Valuation of FRA

Tasks:

Ø

Describe

how interest rate forward contracts is

priced and valued;

Ø

Calculate and interpret

the no-arbitrage value of

Forward 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).

üLong 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; üShort position can be viewed as the obligation to make a

loan at the contract rate (i.e., lend at the fixed rate, fixed receiver); gains when reference rate decrease.

The notation of FRA

Ø The notation of FRA is typically “a×b FRA”:

üa: the number of months until the contract expires;

üb: the number of months until the underlying loan is settled. Ø Example: 3×9 FRA

Today 3 months

(a) 9 months(b)

3×9 FRA

The uses of FRA

Ø Lock the interest rate or hedge the risk of borrowing or lending at some future date.

üOne 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 < spot rate, the long receives payment; • If forward rate > spot rate, the short receives payment.

Pricing of FRA

Ø The “forward price” in FRA is actually a forward rate, it can be calculated from the spot rates.

üFRA 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.

Pricing of FRA (Cont.)

Ø Forward rate models show how forward rates can be extrapolated from spot rates.





Example

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)] = [1+(1.75%×270/360)]

So, FRA rate = 2.22%

Valuation 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. üFor “a×b FRA”, the “interest saving” due to the FRA

position comes at “Time b”, but is settled at “Time a”;

üSo the “interest saving” need to be discounted to “Time a” to calculate the value of FRA.

t

Days NP (Underlying rate - Forward rate)

360

Example: 1



4 FRA

Ø Specification of 1  4 FRA: üTerm = 30 days

üNotional amount = $1 million üUnderlying rate = 90-day LIBOR

üForward rate = 7%

At t = 30 days, 90-day LIBOR = 8%, clarify the payment (value) of this FRA.

Solution: 1



4 FRA

Ø Underlying floating rate > fixed rate, so long position receives payment.

T₀ T30 T120

Forward

rate: 7% _{90-day Libor: 8%}Expiry of FRA; _{(8%-7%) x 90/360 x $1m}Interest saving: = $2,500

Discount at LIBOR for 90 days $2,500/[1+(8% x 90/360)]

Payment = $2,450.98

Example

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 advanced set, advanced settled.

Example (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.

[10M×(0.006 – 0.0055)×0.25]/[1 + 0.0055×0.25] = £1248.28

Valuation of FRA prior to expiration (t < a)

Ø Step 1: calculate the new FRA rate (FR_t):

Ø Step 2: calculate the value of FRA as:

0

Initiation date Evaluation datet FRA expiresa Underlying maturesb





b t





a t





b a

Example

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.

Answer:

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:

Vt = 10M×(1.46% -0.86%)×0.25/(1+1.35%×90/360)

= 14900

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 Forward

Tasks:

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 forward

Ø Similar to equity forward, the forward price of fixed income forward can be calculated as:

F0(T) = (S0- PVC0)×(1+r)T or: F0(T) = S0×(1+r)T - FVCT

üPVC: present value of expected coupon payment;

üFVC: future value of expected coupon payment.

Ø The value of fixed income forward can be calculated as: Vt(T) = (St - PVCt)- F0(T)×(1+r)-(T-t)

Or: Vt(T) = [Ft(T) - F0(T)]×(1+r)-(T-t)

Example

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.

Answer:

Ø V_t(T) = [F_t(T) - F₀(T)]×(1+r)-(T-t)

= (148 - 145)×(1+0.1%)-1/12 _{= 2.9997}

So, the value of the forward position is: 0.029997×€100,000×5 = €14998.5

Pricing of fixed income futures

Ø In terms of fixed income futures, there are several unique issues:

üThe prices of bonds are often quoted without accrued interest (i.e. flat price, clean price).

üBond 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 deliverable bonds roughly equal in price.

• Price paid = Futures price×CF

Pricing of fixed income futures (Cont.)

üWhen 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 of fixed income futures (Cont.)

Ø Calculation of accrued interest (AI): t

Pricing of fixed income futures (Cont.)

Ø The quoted price of fixed income futures can be calculated as: Quoted futures price = [(S0 - PVC)×(1+r)T - AIT]/CF

or: Quoted futures price = [S0×(1+r)T - AIT - FVC]/CF

üAI_T: the accrued interest at maturity of the futures contract;

üS₀: bond full price;

• S0 = Quoted price + AI0

• AI0: the accrued interest at initiation of the future contract.

üCF: the conversion factor.

Example

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

risk-free rate is 0.1%, calculate the price of the Euro-bond futures.

Answer:

Ø According to the example:

üCF = 0.729535; T = 1/12; FVC = 0; r= 0.1%;

üS₀ = €108 + €0.083 = €108.083; üAI_T = €0.25;

Ø So the futures price is:

[108.083×(1 + 0.1%)1/12 - 0.25]/0.729535 = €147.82

A 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 makes it not equivalent to forward value, but the differences tend to be small.

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。

Pricing and Valuation of Interest Rate Swap

Tasks:

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 Swap

Swap

Ø There are three kinds of swaps: üInterest 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. üCurrency swaps

• If the loans are in two different currencies. üEquity swaps

Interest 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).

üNotional amount is not exchanged at the beginning or the end of the swap, because both loans are in same currency and amount;

üOn settlement dates, interest payments are netted;

üFloating rate payments are typically made in arrears.

Pricing of interest rate swap

Ø Principle: the fixed rate in swap (FS, swap rate) should makes the contract value zero at initiation.

Ø Methodology:

üA receive-floating, pay-fixed swap is equivalent to being long a floating-rate bond and short a fixed-rate bond; üIf both bonds are priced at par, the initial cash flows are

zero and the par payments at the end offset each other; üSo, the coupon rate of fixed-rate bond should equal the

swap rate.

Example of receive-floating, pay-fixed interest rate swap

Pricing 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.

Fixed rate bond Floating rate bond

PV = PV = Par value

Pricing of Plain Vanilla interest rate swap (Cont.)

Ø Assume F as the periodic coupon payment of the n-period fixed-rate bond with par value of $1.

wherein:

Dn = 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:

Example

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

4 0.951529

5 0.937467

Answer:

0.990099 + 0.977876 + 0.965136 + 0.951529 + 0.937467

D

= 1.3%

Valuation of Plain Vanilla interest rate swap

Ø The value of a swap is the difference of value between the floating-rate bond and the fixed-rate bond at any time during the life of the swap.

ü For fixed-rate payer (floating-rate receiver): Vt(T) = PVFloating-rate bond – PVFixed-rate bond

ü For fixed-rate receiver (floating-rate payer): Vt(T) = PVFixed-rate bond – PVFloating-rate bond

Valuation of Plain Vanilla interest rate swap (Cont.)

Ø Note: the value of a floating rate bond will be equal to the par value at each settlement date.

üAt each settlement date, the coupon rate of a floating rate will be reset to the market rate, so the bond will be price at par.

Example

Ø Two years ago, we entered a annual-reset €100M 7-year receive-fixed interest rate swap with fixed swap rate of 2%. The estimated PV factors are given in the following table. We know the current equilibrium fixed swap rate is 1.3%. Calculate the value for the party receiving the fixed rate.

Maturity (years) PV Factors

1 0.990

2 0.978

3 0.965

4 0.952

5 0.938

Answer:

Ø Because the value of the floating rate bond is equal to the new fixed rate bond, so the value of the swap is the

difference of value between the old fixed rate bond and the new fixed rate bond:

 

) 78+0.965+0.952+0.938) 100M

=3.376M

Ø Importance: ☆☆☆ Ø Content:

ü Pricing and valuation of interest rate swap. Ø Exam tips:

ü _{常考点：计算题。}

Pricing and Valuation of Currency and Equity Swap

Tasks:

Ø

Describe

how currency and equity swap is priced

and valued;

Ø

Calculate and interpret

the no-arbitrage value of

Currency swap

Ø Currency swap involves two different currencies.

üThe principle amount of currency swap is exchanged at the beginning according to the exchange rate, and returned at termination;

üOn settlement dates, interest payments are not netted; üFloating rate payments are typically made in arrears.

Currency swap (Cont.)

Ø There are four possible structures for currency swap:

üReceive fixed and pay fixed;

üReceive floating and pay fixed; üReceive fixed and pay floating;

üReceive floating and pay fixed.

Pricing and valuation of currency swap

Ø The pricing and valuation of currency swap are similar to that of interest rate swap:

üThe fixed rate in a currency swap is simply the swap rate calculated from the spot rates of the corresponding

currency;

üThe value of a swap is the difference of value between the two equivalent bonds.

Equity swap

Ø There are 3 types of equity swaps, and there are no “pricing” problem for the last two type.

üEquity return for fixed rate; üEquity return for floating rate;

üEquity return for another equity return.

Ø The equity leg of an equity swap can be an individual stock, a published stock index, or a custom portfolio; and the

equity leg cash flow can be with or without dividends.

Equity swap (Cont.)

Ø Notional amount is not exchanged at the beginning or the end of the swap.

Ø On settlement dates, payments are netted.

Pricing of equity swap

Ø The pricing of equity swap is similar to that of interest rate swap, we can use the same formula to calculate the fixed rate:

n

1 2 3 n

1 - D F =

D + D + D + ... + D

Valuation of equity swap

Ø Valuation of equity swap is also similar to that of interest rate swap, and equals the difference of value between the two legs of the equity swap.

üFor receive fixed rate, pay equity return swap: Vt(T) = PVFixed-rate bond – (St/St-)×NP

• St: the current equity price;

• St-: the equity price observed at the last reset date.

Valuation of equity swap (Cont.)

üFor receive floating rate, pay equity return swap: Vt(T) = PVFloating-rate bond – (St/St-)×NP

üFor receive equity (1) return, pay another equity (2) return swap:

Vt(T) = (S1,t/S1,t-)×NP– (S2,t/S2,t-)×NP

or: Vt(T) = (R1 – R2)×NP

• R: equity return after the last reset date.

Example

Ø An investor pays the stock A return and receives stock B return in a $1 million quarterly-pay swap. After one month, stock A is up 2% and stock B is down 1%. Calculate the value of the swap to the investor.

Answer:

Vt(T) = (-2%– 1%)×$1,000,000 = $30,000.

Ø Importance: ☆☆ Ø Content:

ü Pricing and valuation of currency swap; ü Pricing and valuation of equity swap.

Ø Exam tips:

ü 常考点：计算题。

Binomial Option Valuation Model (1)

Tasks:

Ø

Describe and interpret

the binomial option

valuation model;

Ø

Identify

an arbitrage opportunity involving options

Binomial Option Valuation Model

Binomial option valuation model

Ø Binomial model is based on the idea that, over the next period, some value will change to one of two possible values.

Ø To construct a binomial model, we need to know the beginning asset value ( S0 ), the size of the two possible

One-period binomial model (Cont.)

One-period binomial model (Cont.)

Ø With one-period binomial model, the value of an option on stock can be calculated as:

ü Step 1: Calculate the payoff of the option at maturity in both the up-move (C+, P+) and down-move states (C-, P-);

ü Step 2: Calculate the expected value of the option in one period as the probability-weighted average of the payoffs in each state;

ü Steps 3: Discount this expected value back to today at the

risk-free rate.

One-period binomial model (Cont.)

Ø Value of an call option:

C0 = ( πU×C+ + πD×C− )/( 1 + Rf )T

Ø Value of an put option:

P0 = ( πU×P+ + πD×P− )/( 1 + Rf )T

Example

Ø A non-dividend-paying stock is currently trading at €100. A call option on the stock has one year to mature and exercise price of €100. Assume the risk-free interest rate is 5.15% and a single-period binomial option valuation model where U = 1.35 and D = 0.74, calculate the call option value.

Answer:

Ø S+ = US₀ = 1.35×100 = 135; S– = DS₀ = 0.74×100 = 74;

Ø C+ _{= Max(0, S}+ _{– X) = 35; C}– _{= Max(0, S}- _{– X) = 0;}

Ø π_U = (1 + R_f - D)/(U - D)

= (1 + 5.15% - 0.74)/(1.35 - 0.74) = 0.511; Ø π_D = 1 – π_U = 0.489;

Ø Value of the call option:

C0 = ( πU×C+ + πD×C− )/( 1 + Rf )T

= (35×0.511 + 0.489×0)/1.0515 = 17.01

Arbitrage opportunity involving options

Ø If the option market price is different from the calculated price from the binomial valuation model, an arbitrage opportunity exist:

üIf market price > calculated price, sell the option and buy h shares of the stock for each option we sold;

• h: hedge ratio, or delta,

üIf market price < calculated price, buy the option and sell h shares of the stock for each option we bought.



Example

Ø A non-dividend-paying stock is currently trading at €100. A call option on the stock has one year to mature and exercise price of €100. Assume the risk-free interest rate is 5.15% and a single-period binomial option valuation model where u = 1.35 and d = 0.74.

üCalculate the hedge ratio.

üDescribe the arbitrage opportunity if the market value of the call option is €12.

Answer:

Ø S+ = US₀ = 135; S- = DS₀ = 74;

Ø C+ _{= Max(0,S}+ _{– X) = 35; C}- _{= Max(0,S}- _{– X) = 0;}

Ø Hedge ratio: h = ( C+ - C− )/( S+ - S- ) = 35/61 = 0.574;

Ø Because the market price of option (€12) is lower than the calculated arbitrage-free price (€17.01, calculated in previous example), an arbitrage profit can be earned by buy the call option and sell 0.574 share of the stock for each option we bought.

Ø Importance: ☆☆ Ø Content:

ü Binomial option valuation model and its components; ü Arbitrage opportunities involving option;

ü Hedge ratio. Ø Exam tips:

ü _{常考点：one-period option value的计算, hedge ratio 的计}

算；本任务也是option定价基本方法的介绍，对后面的 学习很重要。

Binomial Option Valuation Model (2)

Tasks:

Ø

Calculate

the no-arbitrage values of European and

American options using a two-period binomial model;

Ø

Calculate and interpret

the value of an interest rate

Two-period binomial model for European option

Ø Using the two-period binomial model to value an option is similar, but with more steps:

üStep 1: calculate the three possible values of stock at T=2; • S++ = UUS₀; S+− = S−+ = UDS₀ ; S−− = DDS₀

üStep 2: calculate the payoff of the option at T=2; • C++ = Max (0, S++ − X ); P++ = Max (0, X − S++ );

• C+− _{= C}−+ _{= Max (0, S}+− _{− X); P}+− _{= P}−+ _{= Max (0, X − S}+−_);

• C−− = Max (0, S−− − X ); P−− = Max (0, X − S−− ).

Two-period binomial model for European option (Cont.)

ü Step 3: calculate the option value at T=1 (C+ or P+, C- or P-)

by discounting the expected payoff at T=2 back one period at risk-free rate;

• C+ = ( π_U×C++ + π_D×C+− )/( 1 + R_f )T

• C− = ( π_U×C+− + π_D×C−− )/( 1 + R_f )T

• P+ = ( π_U×P++ + π_D×P+− )/( 1 + R_f )T

• P− _{= (} π

U×P+− + πD×P−− )/( 1 + Rf )T

Two-period binomial model for European option (Cont.)

ü Steps 4: calculate the option value at T=0 (C₀ or P₀) by discounting the expected option value at T=1 back one period at risk-free rate.

• C = ( πU×C+ + πD×C− )/( 1 + Rf )T

• P = ( πU×P+ + πD×P− )/( 1 + Rf )T

Two-period binomial model for European option (Cont.)

Example

ØYou observe a €50 price for a non-dividend-paying stock. An European-style call option has two years to mature and an the exercise price of €50. Assume the risk-free rate is 5%, U = 1.356 and D = 0.744.

üCalculate the current call option value. üCalculate the current put option value.

Answer:

Ø Step 1: S++ = 91.94; S+− = S−+ = 50.44; S−− = 27.68;

Ø Step 2:

üC++ = 41.95; P++ = 0;

üC+− = C−+ = 0.44; P+− = P−+ = 0;

üC−− = 0; P−− = 22.32.

Answer (Cont.):

Answer (Cont.):

Answer (Cont.):

Ø Recall the put-call parity: S0 + P0 = C0 + PV(X)

Ø The put option value can be computed simply by applying put call parity:

P0 = C0 + PV(X) – S0

= 9.71 + (50/1.052) – 50

= 5.06

Two-period binomial model for American option

Ø Non dividend-paying American call options on stock will not

be exercised early because the value of a call option will be greater than its exercise value.

üWorth more alive than dead.

Two-period binomial model for American option (Cont.)

Ø Deep in-the-money put option or call option on

dividend-paying stock may benefit from early exercise:

üFor deep in-the-money put option, the exercise value can

be invested at the risk-free rate, and earn interest that exceed the time value of the put;

üFor call option on dividend-paying stock, the stock price

falls at ex-dividend date, and it may be valuable to exercise the option before the price falling.

Two-period binomial model for American option (Cont.)

Ø When value the American options that may be exercised

early, we need to determine if the option will be exercised at each node:

üIf the exercise value is greater than the calculated

arbitrage-free value, early exercise is valuable;

üUse the higher between exercise value and the calculated

price at each node.

Example

Ø A non-dividend-paying stock is currently trading at $72, a put option on this stock has a exercise price of $75 and a maturity of 2 years. Suppose the interest rate is 3%, U = 1.356 and D = 0.541, πU = 0.6 and πD = 0.4.

üCalculate the put option value if it is European-style. üCalculate the put option value if it is American-style.

Answer:

Answer:

Binomial interest rate tree

Ø A interest rate model that assumes interest rates at any point of time (node) have an equal probability of taking one of two possible values in the next period, an upper path (U) and a lower path (L).

üThe interest rates at each node are one-period forward

rates corresponding to the nodal period.

Binomial interest rate tree (Cont.)

Ø E.g.: interest rate i_2,LU at node 2 is the rate that will occur if initial interest i0 at node 0 follows the lower path to node 1,

and then follows the upper path to node 2.

i0

Binomial valuation model for interest rate option

Ø The valuation of interest rate option is similar to that of stock option, except that the payoff at maturity is different: üCall payoff = Max(0, underlying rate – exercise rate)×NP üPut payoff = Max(0, exercise rate – underlying rate)×NP

Example

Ø A European call option on the one year interest rate (the underlying) has an exercise rate of 3.25% with maturity of two years and notional amount of $1,000,000. Calculate the call option value with the following interest rate tree.

3.05%

Answer:

Ø C++ = (3.97% – 3.25%)×1,000,000 = 7,200;

Ø C+- _{= C}-- _{= 0;}

Ø C+ = (0.5×7,200 + 0.5×0)/1.0391 = 3,465;

Ø C− = 0;

Ø C₀ = (0.5×3,465 + 0.5×0)/1.0305 = 1,681.

Answer (Cont.):

Ø Importance: ☆☆☆ Ø Content:

ü Calculation of European option value; ü Calculation of American option value;

ü Calculation of interest rate option value. Ø Exam tips:

ü _{常考点：计算题。}

Black-Scholes-Merton (BSM) model

Tasks:

Ø

Identify

assumptions and

interpret

the components

of the BSM option valuation model;

Ø

Describe

how the BSM model is used to value

European options on equities and currencies;

Ø

Describe

how the Black model is used to value

Black-Scholes-Merton Model

Assumptions of the BSM model

Ø The options are European-style;

Ø The underlying asset price follows a geometric Brownian motion, and moves smoothly from value to value;

Ø The (continuously compounded) yield of the underlying asset is constant and known;

Ø The continuously compounded risk-free interest rate is

Assumptions of the BSM model (Cont.)

Ø The volatility of the underlying asset return is constant and known;

Ø The market is frictionless:

üNo transaction costs, no taxes, no regulatory constraints;

üNo-arbitrage opportunities in the market;

üThe underlying asset is highly liquid, and continuous trading is available;

üShort selling of the underlying asset is permitted.

BSM model

Ø Formulas for the BSM model are:

üS₀: underlying asset price; X: strike price;

üT: time to maturity; σ: volatility of underlying asset return; üR_f: continuously compounded risk-free rate;

üN(x): the standard normal cumulative distribution function.

Interpretation of BSM model

Ø According to BSM model, the option value can be regarded as the present value of expected payoff at expiration:

Ø Call option can be regarded as leveraged stock investment; put option can be regarded as combination of long bond and short stock;

Ø N(d₂) is the risk-neutral probability that a call option will be exercised at expiration; N(-d2) is the risk-neutral probability

that a put option will be exercised at expiration.

Implied volatility

Ø The volatility (standard deviation) of underlying asset return

“implied” in the option market price.

üFour inputs in fives for BSM model is observable (S, X, T,

Rf), and if the option market price is available, the

volatility can be calculated with the BSM model;

üImplied volatility can be used as a mechanism to quote

option price.

• Only one quotation is needed for different options.

BSM model with carrying benefits or costs

Ø Carry benefits include dividends for stock, foreign interest rates for currency, and coupon payments for bond;

Ø Carry costs include storage cost, insurance costs, etc. üCarry costs can be treated as negative carry benefits.



BSM model with carrying benefits or costs (Cont.)

Black model for European options on futures

Ø Ignore margin requirements and marking to market for futures, the Black model is the BSM model after

Black model for interest rate option

Ø The underlying of interest rate option is a FRA:

üAssume a “M×N” FRA:

Ø The formula for call option is as follows: M

Swaption

Ø An option with underlying of interest rate swap.

üGive the option holder the right to enter into an interest rate swap at a future data.

M

Option expires Swap maturesN

0

Swaption (Cont.)

Ø Payer swaption: the right to enter into a interest rate swap as fixed-rate payer.

üThe swaption holder expects the interest rate to increase. Ø Receiver swaption: the right to enter into a interest rate

swap as fixed-rate receiver.

üThe swaption holder expects the interest rate to decrease.

Ø Importance: ☆☆☆ Ø Content:

ü Assumption and interpretation of BSM model;

ü BSM models for options on futures, interest rate option;

ü Swaption. Ø Exam tips:

ü _{常考点：BSM model的interpretation和变形，概念题。}

Option Greeks

Tasks:

Ø

Interpret

each of the option Greeks;

Option Greeks

Option Greeks

Ø The sensitivity factors of the European option price against the input factors of BSM models, including:

üDelta (Δ): option price vs. underlying price (S); üGamma (Γ): delta vs. underlying price (S);

üTheta (θ): option price vs. time passage (t);

üVega (Λ): option price vs. underlying price volatility (σ);

Delta

Ø Delta (Δ): the sensitivity of the option price against the

underlying asset price.

Delta (Cont.)

Ø Delta for call option and put option:

üFor non-dividend paying stock (δ = 0):

Ø Note: delta is also the hedge ratio (h).

Delta (Cont.)

Ø Delta_call increases from 0 to 1 as stock price increases. üDeep out-of-the-money, Delta_call → 0;

üDeep in-the-money, Delta_call → 1.

1

üDeep out-of-the-money,

Deltaput → 0;

üDeep in-the-money,

Deltaput → -1.

Delta (Cont.)

Ø When the call/put option gets closer to maturity, the delta

will drift either toward 0 if it is out-of-the-money or drift toward 1/-1 if it is in-the-money.

Delta hedge

Ø Delta-neutral portfolio: combine the underlying assets with

the options so that the value of the portfolio does not change with variation of the price of underlying assets.

ü Number of options needed to delta hedge

= - number of shares hedged / delta of option

• Long stocks, short call options (positive delta); • Long stocks, long put options (negative delta).

Example

Ø We have a short position in put options on 10,000 shares of

stock. Deltacall = 0.532 and Deltaput = –0.419. Assume each

stock option contract is for one share of stock.

üCalculate the numbers of stock needed to delta hedge

assuming the hedging instrument is stock.

üCalculate the numbers of call option needed to delta hedge

assuming the hedging instrument is call option.

Answer:

Ø

üSell 4190 shares of stock.

Ø

üSell 7876 call options.

stock put short put

N = -N Delta  10000 0.419  4190

      



  

call call put put

Gamma

Ø Gamma (Γ): the sensitivity of the option’s delta against the

underlying asset price;

Ø Gamma for a call and put option with identical features are

the same.

ü Long call/put will have a positive gamma;

ü Gamma is largest when the option is at-the-money; ü If the option is deep in- or out-of-the-money, gamma

Gamma (Cont.)

Ø Gamma approximates the estimation error with delta for

option price.

üOption price with respect to stock price is non-linear, but

delta only measures the linear relationship.

Ø Gamma risk: stock prices often jump rather than move

continuously and smoothly in reality, holding a delta-neutral portfolio will have risk against stock price too.

üIf the BSM model assumptions hold, then we would have

no gamma risk.

Theta

Ø Theta (θ): the sensitivity of the option price against time

passage (t).

üTheta is usually negative for both call and put option;

• With excerption to deep in-the-money put option.

üIt is also termed the “time decay” of options.

Vega

Ø Vega (Λ): the sensitivity of option price against the volatility

of the underlying asset price.

üVega for call option is equal to Vega for put option with

identical features, and both are positive;

üVega is high when options are at or near the money.

Rho

Ø Rho (ρ): the sensitivity of option price against the risk-free

rate.

üRho is positive for call option; üRho is negative for put option.

Ø Importance: ☆☆☆ Ø Content:

ü Option Greeks; ü Delta hedge.

Ø Exam tips:

ü 常考点1：Greeks, 定性考察，概念和正负； ü _{常考点2：Delta hedge，计算题。}

Changing Risk Exposure

Tasks:

Ø

Describe

how interest rate, currency, and equity

swaps, futures, and forwards can be used to modify

risk and return;

Ø

Describe

how to replicate an asset by using options

Changing Risk Exposures

Interest rate swap

Ø Interest rate swap can be used to modified the duration of a

fixed-income portfolio.

üSwap value for fixed-rate receiver

= value of fixed-rate bond – value of floating-rate bond

üDuration of fixed-rate receiver swap

Interest rate futures

Ø Interest rate futures are exchange-traded derivatives and

are free of default risk.

üSometimes referred to as bond futures because the

underlying asset is often a bond.

Ø Interest rate futures can be used to modify duration of fixed

income portfolio.

üLong interest rate futures will increase portfolio duration;

üShort interest rate futures will decrease portfolio duration.

Currency swap

Ø Currency swap are usually used by companies to reduce

their funding costs.

Ø A currency swap is different from an interest rate swap in

two ways:

üThe interest rates are associated with different currencies; üThe notional value may be exchanged at the initiation and

expiration of the swap.

Currency forward/futures

Ø Currency forward/futures can be used to manage foreign

exchange rate risk.

üE.g., a Chinese company have a Euro liability due in 6

months and is exposed to foreign exchange rate risk. The company can hedge the liability by long Euro

forward/futures contract.

Equity swap

Ø An equity swap can exchange the return of an equity asset to

the return of another asset.

üE.g., an interest rate, or another equity asset.

Ø Equity swap can be used to modify exposure to equity

market temporarily without actually disposing the equity portfolio.

Stock index futures

Ø Stock index futures can be used to modify the exposure to

equity market.

üLong stock index futures will increase the exposure; üShort stock index futures will decrease the exposure.

Synthetic asset with options

Ø Long asset = long call + short put (S = c – p)

Ø Short asset = short call + long put (-S = -c + p)

üAsset price = exercise price;

üThe call option and put option have identical features.

Synthetic options

Ø Long call = long asset + long put (c = S + p) Ø Long put = Short asset + long call (p = -S + c)

Synthetic asset with forward/futures

Ø Long asset = long futures + risk-free asset (cash)

üLong risk-free asset (synthetic cash)

= long asset + short futures

Ø Importance: ☆ Ø Content:

ü Changing risk exposure with interest rate, currency, and equity swaps, futures, and forwards;

ü Synthetic equivalencies with options, forward/futures. Ø Exam tips:

ü _{常考点：不是考试重点。}

Derivative Strategies

Tasks:

Ø

Describe, calculate and interpret

the value at

expiration, profit, maximum profit, maximum loss,

and breakeven underlying price at expiration for

Derivative Strategies

Covered call (S - C)

Ø Structure: short call option + long the underlying stock; Ø Investment objectives:

üIncome generation: earn the option premium if the option

expire worthless;

üImproving on the market: capture the time value by

constructing covered call with a in-the-money call option. • Option value = intrinsic value + time value

üTarget price realization: construct covered call with a call

Covered call (Cont.)

Ø Example: Buy stock at $39, sell call option with strike price

Covered call (Cont.)

Ø Risk of covered call:

üKeeps the downside risk of the stock position;

üGives up the upside potential of the stock position.

Covered call vs. (long asset + short forward)

Ø Recall: delta is the sensitivity of option price against

underlying price.

üDelta of a long stock position: 1; üDelta of a short stock position: -1;

üDelta of a long forward position: 1. üDelta of a short forward position: -1.

Covered call vs. (long asset + short forward)

Ø From the aspect of delta (price sensitivity against underlying

asset price), a covered call position (S – C) is equivalent to a position of long a stock and short forward for detalcall unit.

üBoth of them have delta of (1- delta_call).

Protective put (S + P)

Ø Structure: long put option + long underlying stock; Ø Investment objectives:

üProvide protection or insurance against a price decline.

Protective put (Cont.)

Ø Example: Buy stock at $41, buy put option with strike price

of $40 at $3.

üMax. profit: Unlimited üMax. loss: S₀ + P₀ - X üBreakeven: S₀ + P₀

Protective put (Cont.)

Ø Risk of protective put:

üThe put premium will reduce the portfolio return.

• Continuous purchase of protective puts may be expensive and probably suboptimal, but occasional purchase to deal with a bearish short-term outlook can be a reasonable risk-reducing activity.

Protective put vs. (long asset + short forward)

Ø From the aspect of delta (price sensitivity against underlying

asset price), a protective put position (S + P) is equivalent to a position of long a stock and short forward for detalput unit.

üBoth of them have delta of (1+ delta_put).

• Remember: detalput is negative.

Spread strategy

Ø Long an option and short another option on same

underlying asset but with different exercise price.

üBull spread: long an option and short another with a higher

exercise price;

• The underlying asset price is expected to increase.

üBear spread: long an option and short another with a lower

exercise price;

• The underlying asset price is expected to decrease.

üBoth bull spread and bear spread can be constructed with

either call or put option.

Bull call spread

Ø Long call with lower exercise price (X_L) and short call with

higher exercise price (XH).

Bull put spread

Ø Long put with lower exercise price (X_L) and short put with

higher exercise price (XH).

Bear call spread

Ø Long call with higher exercise price (X_H) and short call with

lower exercise price (XL).

Bear put spread

Ø Long put with higher exercise price (X_H) and short put with

lower exercise price (XL).

Collar

Ø Structure: long put + short call + underlying asset

üThe exercise price (X_L) of put option is typically below the

asset price (S0);

üThe exercise price (X_H) of call option is typically above the

asset price (S0).

Ø Investment objective: buy a protective put and sell a call to

offset the premium.

üZero-cost collar: the premiums for call and put are equal.

Straddle

Ø Long straddle: Long call + long put, with the same exercise

price, on the same underlying asset.

üProfit: Max(0, S_T - X) + Max(0, X- S_T) - (C₀ + P₀)

ST

0 X

P/L ü Max. profit: unlimited

ü Max. loss: C₀ + P₀

ü Breakeven: X + (C₀ + P₀) or: X - (C0 + P0)