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Derivatives and RiskManagement
(Adapted from Prof. Dani Salazars Slides)
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Financial risks
Price risk
Interest rate risk
Credit / Default risk
Foreign currency risk
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What is a financial derivative?
A derivative is an instrumentwhose value depends on thevalues of other more basic
underlying variables.
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Uses of derivatives
Hedge
Arbitrage
Speculate
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Basic forms of derivatives
Commitments
Forwards
Futures
Swaps
Contingents
Options
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What is a forward contract?
It is an agreement to buy or sell an assetat a certain time in the future for a certainprice
Over the counter securities
Forward contracts are popular oncurrencies and interest rates
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Futures contract
An agreement to buy or sell an asset at acertain time in the future for a certain price
Futures are exchange-traded.
Examples of underlyings of futurescontract: commodities, interest rates
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Swaps
Promise to exchange cash flows atvarious future time periods.
Cash flows maybe based on differentunderlyings such as return on equitymarkets, return on bonds markets, etc.
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Options
A call option is an option to buy a certainasset by a certain date for a certain price(the strike price)
A put option is an option to sell a certainasset by a certain date for a certain price(the strike price)
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Commitments versus contingents
A commitment contract gives the holderthe obligation to buy or sell at a certainprice
A contingent gives the holder the right tobuy or sell at a certain price. Such rightsare exercised only when it is beneficial to
the holder of the right.
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Roadmap
For each of the basic form of derivative,the following subtopics will be discussed:
Nature
Pricing
Pay-off
Valuation
Application
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FORWARDS AND FUTURES
F
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Forwards
A forward contract is an agreement tobuy (long position) or sell (short position)an asset in the future at an agreed price(delivery price) today
The asset could be a stock, a foreigncurrency, another financial instrument(e.g. bond)
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Difference between Forwards andFutures
Forward Futures
Private contract between twoparties
Traded on an exchange
Not standardized Standardized
Usually one specified deliverydate
Range of delivery dates
Settled at end of contract Settled daily
Delivery or final settlement usual Usually closed out prior tomaturity
Some credit risk Virtually no credit risk
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Price versus value
For futures or forward, price is the contracted rateof future purchase. It is the future value of the underlying.
Value is similar to profit from the forward orfutures contract. At date of contract, value of forward and futures is zero.
Why?
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Forward price
f0(T) = S0 (1+r)T
Definition:
f0(T) = Forward price at time 0.
S0 = Spot price of the underlying at time 0.
r = risk-free rate
T = Number of days before expiration
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Example
The spot price of gold is 600. The one yearinterest rate is 5%. What should be theforward price of a gold forward to bedelivered one year from now.
In our examples, S=600, T=1, and r=0.05so that
F = 600(1+0.05) = 630
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Forward price of Equity
f0(T) = (S
0 D) (1+r)T
Definition:
f0(T) = Forward price at time 0.
S0 = Spot price of the underlying stock attime 0.
D = Present value of dividends to bereceived prior to expiration of forward
r = risk-free rate
T = Number of days before expiration
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Forward price of Equity
f0(T) = (S0 D) (1+r)T
Definition:
S0 = Spot price of ABC stock is P10.
D = ABC declared cash dividends of P2 tobe paid 30 days from today. PV of P1.99
r = 30 - 60 days zero-coupon rate is 6% p.a
T = 60 days
f0(T) = (10 1.99) (1+6%)60/360
f0(T) = 8.08819
Forward price of foreign currency
f0(T) = S0 (1+r1)T /(1+r2)
T
Example:
f0(T) = P / $ (1+rP)T /(1+r$)
T
Definition:
f0(T) = Forward price at time 0.
S0 = Spot price of the underlying currency attime 0.
r = risk-free rate
T = Number of days before expiration20
Example
f0(T) = P / $ (1+rP)T /(1+r$)
T
Definition:
f0(T) = Forward price at time 0.
S0 = Spot price of peso is P 46 = $1.
rP = 30 days peso zero-coupon bond is 3%.
r$ = 30 days peso zero-coupon bond is 1%.T = 30 days
f0(T) = P 46/ $1 (1+3%)30/360 /(1+1%)30/360
f0(T) = 46.07521
Pay-off
On the date of expiration, the long paysthe forward price (F).
And receives delivery of the underlying(Stm).
Mathematically:
Pay-off = Stm F
If S > F, the long is a net receiver.
If S < F, the short is a net payor.
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Profit from aLong Forward Position
Profit
Price of Underlying
at Maturity, STK
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Profit from aShort Forward Position
Profit
Price of Underlying
at Maturity, STK
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What is the value of a forward?
Vt(T) = St - f0(T)/(1+r1)T-t
Definition
f0(T) = Forward price at time 0.
S0 = Spot price of the underlying at t ime t.
r = risk-free rate
t- T= time to expiration
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Value of forward at time 0
Vt(T) = St - f0(T)/(1+r1)T-t
Vt(T) = 600 - 630/(1+5%)1
Vt(T) = 600 600
Vt(T) = 0
Definition
f0(T) = Forward price at time 0 = 630.
S0 = Spot price of the underlying at time t = 600.
r = risk-free rate = 5%
t- T= time to expiration = 1 year
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Value of forward 6 months fromnow when spot is 610.
Vt(T) = St - f0(T)/(1+r1)T-t
Vt(T) = 610 - 630/(1+5%)6/12
Vt(T) = 610 614.82
Vt(T) = - 4.82
Definition
f0(T) = Forward price at time 0 = 630.S0 = Spot price of the underlying at time t = 610.
r = risk-free rate = 5%
t- T= time to expiration = 30 days
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SWAPS
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Nature
A swap is an agreement to exchange cashflows at specified future times according tocertain specified rules
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Examples:
Plain vanilla interest rate swap
Foreign currency swap
Equity swap
Equity for bond swaps
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Uses of an Interest Rate Swap
Converting a liability from fixed rate to floating rate floating rate to fixed rate
Converting an investment from fixed rate to floating rate floating rate to fixed rate
Valuation of an Interest Rate Swap
Portfolio of bonds = Difference betweenthe Value of a fixed-rate bond and thevalue of a floating-rate bond
Portfolio of forwards = Values ofdifferent cash flows at different periods.
FVTPL - Swaps
A speculator is expecting interest rates to go down. On
January 1, he entered in a 2 year plain-vanilla interest
rate swap to receive a fix interest rate of 6% and pay
floating equivalent to 180-day treasury rate + 3%. The
nominal principal is P5 Million. The counterparties
agreed to swap every June 30 and December 31. The
180-day treasury on January 1 is 3%.
Period 180 day
treasury rates
January 1, 2003 3%
June 30, 2003 4%
December 31, 2003 2%
FVTPL - Swaps
FVTPL - Swaps
Value of Swap
Date of inceptionPoint of view of the speculator.
Value of fixed income investmentcoupon rate: 6%floating rate: 3% + 3%
P 5 Million
Value of floating rate borrowings ( 5 Million)
Value of swaps - 0 -
FVTPL - SwapsSwapJune 30180-day treasury rate= 3%.
No payment on June 30, first swap period
SpeculatorCounter-
Party
6% X P5M [fix]
(3%+3%)X P5M [fl]
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FVTPL - SwapsSwapDecember 31180-day treasury rate= 4%.
Speculator is net payor of P25K
SpeculatorCounter-
Party
6% X P5M X 6/12 [fix]
(4%+3%)X P5M X 6/12[fl]
FVTPL - SwapsValue of SwapOn Balance Sheet Date (December 31)180-day interest rate = 2%Point of view of the speculator.
Value of fixed income investmentcoupon rate: 6%discounting rate: 2% + 3%
P 5.048 Million
Value of floating rate borrowings ( 5 Million)
Value of swaps P 0.048 Million
Therefore, the swap is a derivative assetto the speculator.
Principal 5,000,000.00
market rate 5%
coupon 6%
remaining swap period 2
PV of principal 4,759,071.98
PV of interest 289,113.62PV of fixed income 5,048,185.60
Value of Fixed Income InvestmentValue of Fixed Income InvestmentFVTPL - Swaps
SwapJune 30, Y2180-day treasury rate= 2%.
Speculator is net receiver of P25K
SpeculatorCounter-
Party
6% X P5M X 6/12[fix]
(2%+3%)X P5M X 6/12[fl]
Currency swaps
Counterparties swapped principal at thebeginning of the contract.
Interest are paid periodically based on thecurrency received and agreed uponinterest rate.
Counterparties return the principal at theend of the swap contract.
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Uses of a Currency Swap
Conversion from a liability in one currencyto a liability in another currency
Conversion from an investment in onecurrency to an investment in anothercurrency
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Valuation of Currency Swaps
Like interest rate swaps, currencyswaps can be valued either as thedifference between 2 bonds or as aportfolio of forward contracts
OPTIONS
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Options
A call is an option to buy
A put is an option to sell
A European option can be exercised onlyat the end of its life
An American option can be exercised atany time
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Option Positions
Long Call
Profit from buying one European call option:
option premium = $5, strike price = $100.
30
20
10
0-5
70 80 90 100
110 120 130
Profit ($)
Terminalstock price ($)
Profit = Spot Exercise price premium paid
Profit = premium paid
Short Call
Profit from writing one European call option: option
premium = $5, strike price = $100
-30
-20
-10
05
70 80 90 100
110 120 130
Profit ($)
Terminalstock price ($)
Profit = Exercise price + premium paid - Spot
Profit = + premium paid
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Long Put
Profit from buying a European put option: optionpremium = $7, strike price = $70
30
20
10
0
-770605040 80 90 100
Profit ($)
Terminalstock price ($)
Profit = Exercise price - Spot premium paid
Profit = premium paid
Short Put
Profit from writing a European put option: optionprice = $7, strike price = $70
-30
-20
-10
7
070
605040
80 90 100
Profit ($)Terminal
stock price ($)
Profit = Spot + premium paid Exercise price
Profit = + premium paid
Payoffs from Options
K= Strike price, ST = Price of asset atmaturity
Payoff Payoff
ST STK
K
Payoff Payoff
ST STK
K
Terminology: Moneyness :
At-the-money option
In-the-money option
Out-of-the-money option
Simple Option Pricing
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0or))1(
(
0or))1(
(
0
0
Sr
Xp
r
XSc
t
t
+
=
+
=
Factors affecting option value
Price of underlying
Strike price or exercise price
Time to maturity (life of option)
risk-free interest rate (corresponding tooption maturity; continuouslycompounded)
volatility of asset price
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References:
Hull, John. 2007. Fundamentals ofFutures and Options Markets, 6th Edition
Brooks, Robert. Don Chance. 2008.Derivatives and Risk Management Basics
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