ChanceBrooksAn Introduction to Derivatives and Risk Man.docx

Chance/Brooks
An Introduction to Derivatives and Risk Management, 10th ed.
Chapter 10: Forward and Futures Hedging, Spread, and Target
Strategies
The beauty of finance and speculation was that they could be
different things to different men. To some: poetry or high
drama; to others, physics, scientific and immutable; to still
others, politics or philosophy. And to still others, war.
Michael M. Thomas
Hanover Place, 1990, p. 37
© 2015 Cengage Learning. All Rights Reserved. May not be
scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
Ch. 10: *
*
Chance/Brooks
Important Concepts in Chapter 10Why firms hedgeHedging
conceptsFactors involved when constructing a hedgeHedge
ratiosExamples of foreign currency hedges, intermediate- and
long-term interest rate hedges, and stock index futures
hedgesExamples of spread and target strategies

Ch. 10: *
*
Chance/Brooks
Why Hedge?The value of the firm may not be independent of
financial decisions becauseShareholders might be unaware of
the firm’s risks.Shareholders might not be able to identify the
correct number of futures contracts necessary to hedge.
Shareholders might have higher transaction costs of
hedging than the firm.There may be tax advantages to a firm
hedging.Hedging reduces bankruptcy costs.Managers may be
reducing their own risk.Hedging may send a positive signal to
creditors.Dealers hedge their market-making activities in
derivatives.
Ch. 10: *
*

Chance/Brooks
Why Hedge? (continued)Reasons not to hedgeHedging can give
a misleading impression of the amount of risk reducedHedging
eliminates the opportunity to take advantage of favorable
market conditionsThere is no such thing as a hedge. Any hedge
is an act of taking a position that an adverse market movement
will occur. This, itself, is a form of speculation.
Ch. 10: *
*
Chance/Brooks
Hedging Concepts Short Hedge and Long HedgeShort (long)
hedge implies a short (long) position in futuresShort hedges can
occur because the hedger owns an asset and plans to sell it
later.Long hedges can occur because the hedger plans to
purchase an asset later.An anticipatory hedge is a hedge of a
transaction that is expected to occur in the future. See Table
10.1 for hedging situations.
Ch. 10: *

*
Chance/Brooks
Hedging Concepts (continued)The BasisBasis = spot price –
futures price.Hedging and the BasisP (short hedge) = ST – S0
(from spot market)
– (fT – f0) (from futures market)P (long hedge) = –ST + S0
(from spot market)
+ (fT – f0) (from futures market)If hedge is closed prior to
expiration,
P (short hedge) = St – S0 – (ft – f0)If hedge is held to
expiration, St = ST = fT = ft.
Ch. 10: *
*
Chance/Brooks
Hedging Concepts (continued)The Basis (continued)Hedging
and the Basis (continued)Example: Buy asset for $100, sell
futures for $103. Hold until expiration. Sell asset for $97, close
futures at $97. Or deliver asset and receive $103. Make $3 for

sure.Basis definitioninitial basis: b0 = S0 – f0basis at time t: bt
= St – ftbasis at expiration: bT = ST – fT = 0For a position
closed at t:P (short hedge) = St – ft – (S0 – f0) = –b0 + bt
Ch. 10: *
*
Chance/Brooks
Hedging Concepts (continued)The Basis (continued)This is the
change in the basis and illustrates the principle of basis
risk.Hedging attempts to lock in the future price of an asset
today, which will be f0 + (St – ft).A perfect hedge is practically
non-existent.Short hedges benefit from a strengthening basis.All
of this reverses for a long hedge.See Table 10.2 for hedging
profitability and the basis.
Ch. 10: *
*

Chance/Brooks
Hedging Concepts (continued)The Basis (continued)Example:
March 30. Spot gold $1,387.15. June futures $1,388.60. Buy
spot, sell futures. Note:
b0 = 1,387.15 − 1,388.60 = −1.45. If held to expiration, profit
should be change in basis or 1.45.At expiration, let ST =
$1,408.50. Sell gold in spot for $1,408.50, a profit of 21.35.
Buy back futures at $1,408.50, a profit of −19.90. Net gain
=1.45 or $145 on 100 oz. of gold.
Ch. 10: *
*
Chance/Brooks
Hedging Concepts (continued)The Basis (continued)Example:
(continued)Instead, close out prior to expiration when
St = $1,377.52 and ft = $1,378.63. Profit on spot = −9.63. Profit
on futures = 9.97. Net gain = 0.34 or $34 on 100 oz. Note that
change in basis was bt − b0 or
−1.11 − (−1.45) = 0.34.Behavior of the basis, see Figure 10.1.In
forward markets, the hedge is customized so there is no basis
risk.

Ch. 10: *
*
Chance/Brooks
Hedging Concepts (continued)Some Risks of Hedgingcross
hedgingspot and futures prices occasionally move
oppositequantity risk
Ch. 10: *
*
Chance/Brooks
Hedging Concepts (continued)Contract ChoiceWhich futures
underlying asset?High correlation with spotFavorably
pricedWhich expiration?The futures with maturity closest to but
after the hedge termination date subject to the suggestion not to
be in a contract in its expiration monthSee Table 10.3 for
example of recommended contracts for T-bond hedgeConcept of

rolling the hedge forward
Ch. 10: *
*
Chance/Brooks
Hedging Concepts (continued)Contract Choice (continued)Long
or short?A critical decision! No room for mistakes.Three
methods to answer the question.
See Table 10.4.
worst case scenario method
current spot position method
anticipated future spot transaction method
Ch. 10: *
*
Chance/Brooks

Hedging Concepts (continued)Margin Requirements and
Marking to Marketlow margin requirements on futures, butcash
will be required for margin calls
Ch. 10: *
*
Chance/Brooks
Determination of the Hedge RatioHedge ratio: The number of
futures contracts to hedge a particular exposureNaïve hedge
ratioAppropriate hedge ratio should beNf = −DS/DfNote that
this ratio must be estimated.
Ch. 10: *
*
Chance/Brooks

Determination of the Hedge Ratio (continued)Minimum
Variance Hedge RatioProfit from short hedge:P = DS +
DfNfVariance of profit from short hedge:sP2 = sDS2 + sDf2Nf2
+ 2sDSDfNfThe optimal (variance minimizing) hedge ratio is
Nf = −sDSDf/sDf2This is the beta from a regression of spot
price change on futures price change.
Ch. 10: *
*
Chance/Brooks
Determination of the Hedge Ratio (continued)Minimum
Variance Hedge Ratio (continued)Hedging effectiveness is e* =
(risk of unhedged position − risk of hedged position)/risk of
unhedged positionThis is coefficient of determination from
regression.
Ch. 10: *
*

Chance/Brooks
Determination of the Hedge Ratio (continued)Price Sensitivity
Hedge RatioThis applies to hedges of interest sensitive
securities.First we introduce the concept of duration. We start
with a bond priced at B:
where CPt is the cash payment at time t and yB is the yield, or
discount rate.
Ch. 10: *
*
Chance/Brooks
Hedge Ratio (continuation)An approximation to the change in
price for a yield change is
with DURB being the bond’s duration, which is a weighted-
average of the times to each cash payment date on the bond, and
many weaknesses but is widely used as a measure of the

sensitivity of a bond’s price to its yield.Modified duration (MD)
measures the bond percentage price change for a given change
in yield.
Ch. 10: *
*
Chance/Brooks
Hedge Ratio (continuation)The hedge ratio is as follows:
Where MDB » −(DB/B) /DyB and
MDf » −(Df/f) /DyfNote the concepts of implied yield and
implied duration of a futures. Also, technically, the hedge ratio
will change continuously like an option’s delta and, like delta,
it will not capture the risk of large moves.
Ch. 10: *
*

Chance/Brooks
Hedge Ratio (continued)Alternatively, Nf = −(Yield
beta)PVBPB/PVBPf
where Yield beta is the beta from a regression of spot bond
yield on futures yield and
PVBPB, PVBPf is the present value of a basis point change in
the bond and futures prices.
Ch. 10: *
*
Chance/Brooks
Determination of the Hedge Ratio (continued)Stock Index
Futures HedgingAppropriate hedge ratio isNf =
−(bS/bf)(S/f)where bS is the beta from the CAPM and bf is the
beta of the futures, often assumed to be 1.
Ch. 10: *

*
Chance/Brooks
Hedging StrategiesLong Hedge With Foreign Currency
FuturesAmerican firm planning to buy foreign inventory and
will pay in foreign currency.See Table 10.5.Short Hedge With
Foreign Currency ForwardsBritish subsidiary of American firm
will convert pounds to dollars.See Table 10.6.
Ch. 10: *
*
Chance/Brooks
Hedging Strategies (continued)Intermediate and Long-Term
Interest Rate HedgesFirst let us look at the CBOT T-note and
bond contractsT-bonds: must be a T-bond with at least 15 years
to maturity or first call dateT-note: three contracts (2-, 5-, and
10-year)A bond of any coupon can be delivered but the standard
is a 6% coupon. Adjustments, explained in Chapter 9, are made
to reflect other coupons.Price is quoted in units and 32nds,
relative to $100 par, e.g., 93 14/32 is $93.4375.Contract size is

$100,000 face value so price is $93,437.50
Ch. 10: *
*
Chance/Brooks
Hedging Strategies (continued)Intermediate and Long-Term
Interest Rate Hedges (continued)Hedging a Long Position in a
Government BondSee Table 10.7 for example.Anticipatory
Hedge of a Future Purchase of a Treasury NoteSee Table 10.8
for example.Hedging a Corporate Bond IssueSee Table 10.9 for
example.
Ch. 10: *
*
Chance/Brooks
Hedging Strategies (continued)Stock Market HedgesFirst look

at the contractsWe primarily shall use the S&P 500 futures. Its
price is determined by multiplying the quoted price by $250,
e.g., if the futures is at 1300, the price is 1300($250) =
$325,000Stock Portfolio HedgeSee Table 10.10 for
example.Anticipatory Hedge of a TakeoverSee Table 10.11 for
example.
Ch. 10: *
*
Chance/Brooks
Spread StrategiesIntramarket SpreadsBased on changes in the
difference in carry costsSee Figure 10.2 for
illustration.Treasury Bond Futures SpreadsSee Figure 10.3 and
Figure 10.4 for illustration the relationship between changes in
spreads and interest rates.See Table 10.12 for calculation of
Tbond futures spread profits.See Figure 10.5 for illustration of
stock index spreads
Ch. 10: *
Chance/Brooks

Intermarket Spread StrategiesIntermarket spread strategies
involve two futures contracts on different underlying
instrumentsIntermarket spread strategies tend to be more risky
than intramarket spreads because there is both the change in
spreads and the change in underlying instrumentsNOB denotes
notes over bondsIntermarket spread strategies could also
involve various equity markets
Ch. 10: *
Chance/Brooks
Target Strategies: BondsTarget Duration with Bond
FuturesNumber of futures needed to change modified duration
Goal is to move the modified duration from its current value to
a new target valueSee Table 10.13 for illustration.
Ch. 10: *
Chance/Brooks
Target Strategies: EquitiesAlpha CaptureNumber of futures to

hedge systematic risk
Goal is to move the eliminate systematic riskSee Table 10.14
for illustration.Target Beta (see Table 10.15 for illustration.)
Ch. 10: *
Chance/Brooks
Target Strategies: Equities (continued)Tactical Asset
AllocationStrategic asset allocation – long run target weights
for each asset classTactical asset allocation – short run
deviations in weights for each asset classSee Table 10.16 for
illustration.
Ch. 10: *
Chance/Brooks
SummaryTable 10.17 recaps the types of hedge situations, the
nature of the risk and how to hedge the risk

Ch. 10: *
*
Chance/Brooks
(Return to text slide)
Ch. 10: *
Chance/Brooks
Ch. 10: *
Chance/Brooks

Ch. 10: *
Chance/Brooks
Ch. 10: *
Chance/Brooks
Ch. 10: *
Chance/Brooks

(To Continue)
(Return to text slide 31)
Ch. 10: *
Chance/Brooks
(To Previous Slide)
Ch. 10: *
Chance/Brooks
Ch. 10: *
å

=
+
=
T
1
t
t
B
t
)
y
(1
CP
B
B
B
y
1
y)
(
DUR
B
B
+
D
-
=
D
÷
ø
ö
ç
è
æ
÷
÷

ø
ö
ç
ç
è
æ
-
=
f
B
D
M
D
M
N
f
B
*
f
÷
ø
ö
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
-
=
f

B
D
M
D
M
-
D
M
N
f
B
T
*
f
÷
ø
ö
ç
è
æ
b
-
=
f
S
N
S
*
f
(
)
÷
ø
ö
ç
è

æ
b
-
b
-
=
f
S
N
S
T
*
f
Chance/Brooks
Chapter 8: Principles of Pricing Forwards, Futures, and Options
on Futures
Even if we didn’t believe it for a second, there’s an undeniable
adrenaline jab that comes from someone telling you that you’re
going to make five hundred million dollars.
Doyne Farmer
Quoted in The Predictors, 1999, Page 119.
Ch. 8: *

*
Chance/Brooks
Important Concepts in Chapter 8Price and value of forward and
futures contractsRelationship between forward and futures
pricesDetermination of the spot price of an assetCarry arbitrage
model for theoretical fair priceContango, backwardation, and
convenience yieldFutures prices and risk premiumsPricing
options on futures
Ch. 8: *
*
Key AssumptionsForward contracts are not subject to margin
requirementsForward contracts are not centrally clearedForward
contracts are not otherwise guaranteed by a third party. For
forward contracts, the risk of default is so small as to be
irrelevant.
Chance/Brooks
Ch. 8: *

Chance/Brooks
Generic Carry ArbitrageThe Concept of Price Versus
ValueNormally in an efficient market, price = value.For a
futures or forward, price is the contracted rate of future
purchase. Value is something different.At the beginning of a
contract, value = 0 for both futures and forwards.
NotationVt(0,T), F(0,T), vt(T), ft(T) are values and prices of
forward and futures contracts created at time 0 and expiring at
time T.
Ch. 8: *
*
Chance/Brooks
Generic Carry Arbitrage (continued)The Value of a Forward
ContractForward price at expiration: F(T,T) = ST. That is, the
price of an expiring forward contract is the spot price.Value of
forward contract at expiration:VT(0,T) = ST – F(0,T).An
expiring forward contract allows you to buy the asset, worth ST,
at the forward price F(0,T). The value to the short party is (–1)
times this.

Ch. 8: *
*
Chance/Brooks
Generic Carry Arbitrage (continued)The Value of a Forward
Contract (continued)The Value of a Forward Contract Prior to
ExpirationA: Go long forward contract at price F(0,T) at time
0.B: At time t go long the asset and take out a loan promising to
pay F(0,T) at T
At time T, A and B are worth the same, ST – F(0,T). Thus, they
must both be worth the same prior to T.
So Vt(0,T) = St – F(0,T)(1 + r)–(T–t)
See Table 8.1.Example: Go long 45 day contract at F(0,T) =
$100. Risk-free rate = 0.10. 20 days later, the spot price is
$102. The value of the forward contract is 102 – 100(1.10)–
25/365 = 2.65.
Ch. 8: *
*

Chance/Brooks
Generic Carry Arbitrage (continued)The Value of a Futures
ContractFutures price at expiration: fT(T) = ST.Value during
the trading day but before being marked to market:vt(T) = ft(T)
– ft–1(T).Value immediately after being marked to market:
vt(T) = 0.
Ch. 8: *
*
Chance/Brooks
Generic Carry Arbitrage (continued)Forward Versus Futures
PricesForward and futures prices will be equalOne day prior to
expirationMore than one day prior to expiration if
Interest rates are certain
Futures prices and interest rates are uncorrelatedFutures prices
will exceed forward prices if futures prices are positively
correlated with interest rates.Default risk can also affect the
difference between futures and forward prices.
Ch. 8: *

*
Chance/Brooks
Carry Arbitrage: EquitiesForward and Futures Pricing When the
Underlying Generates Cash FlowsFor example, dividends on a
stock or indexAssume one dividend DT paid at expiration.Buy
stock, sell futures guarantees at expiration that you will have
DT + f0(T). Present value of this must equal S0, using risk-free
rate. Thus,
f0(T) = S0(1 + r)T – DT.For multiple dividends, let DT be
compound future value of dividends. See Figure 8.1 for two
dividends.Dividends reduce the cost of carry.If D0 represents
the present value of the dividends, the model becomes
f0(T) = (S0 – D0)(1 + r)T.
Ch. 8: *
*
Chance/Brooks
Carry Arbitrage: Equities (continued)Forward and Futures
Pricing When the Underlying Generates Cash Flows
(continued)For dividends paid at a continuously compounded

rate of dc,
Example: S0 = 50, rc = 0.08, dc = 0.06, expiration in 60 days (T
= 60/365 = 0.164).f0(T) = 50e(0.08 – 0.06)(0.164) = 50.16.
Ch. 8: *
*
Chance/Brooks
Carry Arbitrage: Equities (continued)Valuation of Equity
Forward ContractsWhen there are dividends, to determine the
value of a forward contract during its life Vt(0,T) = St – Dt,T –
F(0,T)(1 + r)–(T–t)where Dt,T is the value at time t of the
future dividends to time TOr if dividends are continuous,
Ch. 8: *
*

Chance/Brooks
Carry Arbitrage: CurrenciesPricing Foreign Currency Forward
and Futures Contracts: Interest Rate ParityInterest Rate Parity:
the relationship between futures or forward and spot exchange
rates. Same as carry arbitrage model in other forward and
futures markets.Proves that one cannot convert a currency to
another currency, sell a futures, earn the foreign risk-free rate,
and convert back without risk, earning a rate higher than the
domestic rate.
Ch. 8: *
*
Chance/Brooks
Carry Arbitrage: Currencies (continued)Pricing Foreign
Currency Forward and Futures Contracts: Interest Rate Parity
(continued)S0 = spot rate in domestic currency per foreign
currency. Foreign rate is r. Holding period is T. Domestic rate
is r.Take S0(1 + r)–T units of domestic currency and buy (1 +

r)–T units of foreign currency.Sell forward contract to deliver
one unit of foreign currency at T at price F(0,T).Hold foreign
currency and earn rate r. At T you will have one unit of the
foreign currency.Deliver foreign currency and receive F(0,T)
units of domestic currency.
Ch. 8: *
*
Chance/Brooks
(continued)So an investment of S0(1 + r)–T units of domestic
currency grows to F (0,T) units of domestic currency with no
risk. Return should be r. Therefore
F(0,T) = S0(1 + r)–T(1 + r)TThis is called interest rate
parity.Sometimes written as
Ch. 8: *

*
Chance/Brooks
(continued)Example (from a European perspective): S0 =
€1.0304.
U. S. rate is 5.84%. Euro rate is 3.59%. Time to expiration is
90/365 = 0.2466.F(0,T) = €1.0304(1.0584)–
0.2466(1.0359)0.2466 = €1.025If forward rate is actually €1.03,
then it is overpriced.Buy (1.0584)–0.2466 = $0.9861 for
0.9861(€1.0304) = €1.0161. Sell one forward contract at
€1.03.Earn 5.84% on $0.9861. This grows to $1.At expiration,
deliver $1 and receive €1.03.Return is (1.03/1.0161)365/90 – 1
= 0.0566 (> 0.0359)This transaction is called covered interest
arbitrage.
Ch. 8: *
*
Chance/Brooks

+ r)–THere, the spot rate is being quoted in units of the foreign
currency.Note that the forward discount/premium has nothing to
do with expectations of future exchange rates.Difference
between domestic and foreign rate is analogous to difference
between risk-free rate and dividend yield on stock index
futures.
Ch. 8: *
*
Chance/Brooks
Pricing Models and Risk PremiumsSpot Prices, Risk Premiums,
and the Carry Arbitrage for Generic AssetsFirst assume no
uncertainty of future price. Let s be the cost of storing an asset
and i be the interest rate for the period of time the asset is
owned. ThenS0 = ST – s – iS0If we now allow uncertainty but
assume people are risk neutral, we haveS0 = E(ST) – s – iS0If
we now allow people to be risk averse, they require a risk
. NowS0 = E(ST) – s – iS0 –
Ch. 8: *

*
Chance/Brooks
Pricing Models and Risk PremiumsSpot Prices, Risk Premiums,
and the Carry Arbitrage for Generic Assets (continued)Let us
define iS0 as the net interest, which is the interest foregone
minus any cash received.Define s + iS0 as the cost of
meaningful concept only for storable assets
Ch. 8: *
*
Chance/Brooks
Pricing Models and Risk PremiumsThe Theoretical Fair Price
(Forward/Futures Pricing Revisited)Do the followingBuy asset
in spot market, paying S0; sell futures contract at price f0(T);
store and incur costs.At expiration, make delivery. Profit:
P = f0(T) – S0 – qThis must be zero to avoid arbitrage; thus,
f0(T) = S0 + qSee Figure 8.2.Note how arbitrage and quasi-
arbitrage make this hold.

Ch. 8: *
*
Chance/Brooks
Pricing Models and Risk PremiumsForward/Futures Pricing
Revisited (continued)See Figure 8.3 for an illustration of the
determination of futures prices.Contango is f0(T) > S0. See
Table 8.2.When f0(T) < S0, convenience yield is c , an
additional return from holding asset when in short supply or a
non-pecuniary return. Market is said to be at less than full carry
and in backwardation or inverted.
See Table 8.3. Market can be both backwardation and contango.
See Table 8.4.
Ch. 8: *
*
Chance/Brooks

Pricing Models and Risk PremiumsFutures Prices and Risk
PremiaThe no risk-premium hypothesisMarket consists of only
speculators.f0(T) = E(ST). See Figure 8.4.The risk-premium
hypothesisE(fT(T)) > f0(T).When hedgers go short futures, they
transfer risk premium to speculators who go long futures.E(ST)
= f0(T) + E(f). See Figure 8.5.Normal contango: E(ST) <
f0(T)Normal backwardation: f0(T) < E(ST)
Ch. 8: *
*
Chance/Brooks
Put-Call-Forward/Futures ParityCan construct synthetic futures
with options.See Table 8.5.Put-call-forward/futures
parityPe(S0,T,X) = Ce(S0,T,X) + [X – f0(T)](1 + r)–
TNumerical example using S&P 500. On May 14, S&P 500 at
1337.80 and June futures at 1339.30. June 1340 call at 40 and
put at 39. Expiration of June 18 so
T = 35/365 = 0.0959. Risk-free rate at 4.56%.
Ch. 8: *

*
Chance/Brooks
Put-Call-Forward/Futures Parity (continued)So Pe(S0,T,X) =
39Ce(S0,T,X) + [X – f0(T)](1 + r)–T = 40 + (1340 –
1339.30)(1.0456)–0.0959 = 40.70.Buy put and futures for 39,
sell call and bond for 40.70 and net 1.70 profit at no risk.
Transaction costs would have to be considered.
Ch. 8: *
*
Chance/Brooks
Pricing Options on FuturesThe Intrinsic Value of an American
Option on FuturesMinimum value of American call on
futuresCa(f0(T),T,X) ³ Max[0, f0(T) – X]Minimum value of
American put on futuresPa(f0(T),T,X) ³ Max[0, X –
f0(T)]Difference between option price and intrinsic value is
time value.

Ch. 8: *
*
Chance/Brooks
Pricing Options on Futures (continued)The Lower Bound of a
European Option on FuturesFor calls, construct two portfolios.
See Table 8.6.Portfolio A dominates Portfolio B
soCe(f0(T),T,X) ³ Max{0, [f0(T) – X](1 + r)–T}Note that lower
bound can be less than intrinsic value even for calls.For puts,
see Table 8.7.Portfolio A dominates Portfolio B
soPe(f0(T),T,X) ³ Max{0, [X – f0(T)](1 + r)–T}
Ch. 8: *
*
Chance/Brooks
Pricing Options on Futures (continued)Put-Call Parity of

Options on FuturesConstruct two portfolios, A and B.See Table
8.8.The portfolios produce equivalent results. Therefore they
must have equivalent current values. Thus,Pe(f0(T),T,X) =
Ce(f0(T),T,X) + [X – f0(T)](1 + r)–T.Compare to put-call parity
for options on spot:Pe(S0,T,X) = Ce(S0,T,X) – S0 + X(1 + r)–
T.If options on spot and options on futures expire at same time,
their values are equal, implying
f0(T) = S0(1 + r)T, which we obtained earlier (no cash flows).
Ch. 8: *
*
Chance/Brooks
Pricing Options on Futures (continued)Early Exercise of Call
and Put Options on FuturesDeep in-the-money call may be
exercised early becausebehaves almost identically to
futuresexercise frees up funds tied up in option but requires no
funds to establish futuresminimum value of European futures
call is less than value if it could be exercisedSee Figure
8.6.Similar arguments hold for putsCompare to the arguments
for early exercise of call and put options on spot.
Ch. 8: *

*
Chance/Brooks
Pricing Options on Futures (continued)Options on Futures
Pricing ModelsBlack model for pricing European options on
futures
Ch. 8: *
*
Chance/Brooks
Pricing Options on Futures (continued)Options on Futures
Pricing Models (continued)Note that with the same expiration
for options on spot as options on futures, this formula gives the
same price.ExampleSee Table 8.9.Software for Black-Scholes-
Merton can be used by inserting futures price instead of spot
price and risk-free rate for dividend yield. Note why this
works.For puts

Ch. 8: *
*
Chance/Brooks
See Table 8.10 for a summary of equations.
See Figure 8.7 for linkage between forwards/futures, underlying
asset and risk-free bond.
Summary
Ch. 8: *
*
Chance/Brooks

Ch. 8: *
Chance/Brooks
Ch. 8: *
Chance/Brooks
Ch. 8: *
Chance/Brooks

Ch. 8: *
Chance/Brooks
Ch. 8: *
)T
(r
0
c
e
S
T)
f(0,
c
d
-
=
)
(
)
(
t
)
,
0
(

)
,
0
(
V
t
T
r
t
T
t
c
c
e
T
F
e
S
T
-
-
-
-
-
=
d
(
)
T
d
d
T
T
/2
(T)/X)
ln(f

d
where
)]
XN(d
)
(T)N(d
[f
e
C
1
2
2
0
1
2
1
0
T
r
c
s
s
s
-
=
+
=
-
=
-
)]
N(d
[1
(T)e
f
)]

N(d
[1
Xe
P
1
T
r
0
2
T
r
c
c
-
-
-
=
-
-
Chance/Brooks
Ch. 7: *
Chapter 7: Advanced Option Strategies
Read every book by traders to study where they lost money.
You will learn nothing relevant from their profits (the markets
adjust). You will learn from their losses.
Nassim Taleb
Derivatives Strategy, April, 1997, p. 25

*
Chance/Brooks
Ch. 7: *
Important Concepts in Chapter 7Profit equations and graphs for
option spread strategies, including money spreads, collars,
calendar spreads and ratio spreadsProfit equations and graphs
for option combination strategies including straddles and box
spreads
*
Chance/Brooks
Ch. 7: *
Option Spreads: Basic ConceptsDefinitionsspread
vertical, strike, money spread
horizontal, time, calendar spreadspread notation
June 120/125

June/July 120long or short
long, buying, debit spread
short, selling, credit spread
*
Chance/Brooks
Ch. 7: *
Option Spreads: Basic Concepts (continued)Why Investors Use
Option SpreadsRisk reductionTo lower the cost of a long
positionTypes of spreadsbull spreadbear spreadtime spread is
based on volatility
*
Chance/Brooks
Ch. 7: *
Option Spreads: Basic Concepts (continued)NotationFor money

spreadsX1 < X2 < X3C1, C2, C3N1, N2, N3For time spreadsT1
< T2C1, C2N1, N2See Table 7.1 for DCRB option data
*
Chance/Brooks
Ch. 7: *
Money SpreadsBull SpreadsBuy call with strike X1, sell call
with strike X2. Let N1 = 1, N2 = –1Profit equation: P = Max(0,
ST – X1) – C1
– Max(0, ST – X2) + C2P = –C1 + C2 if ST £ X1 < X2P = ST –
X1 – C1 + C2 if X1 < ST £ X2P = X2 – X1 – C1 + C2 if X1 <
X2 < STSee Figure 7.1 for DCRB June 125/130,
C1 = $13.50, C2 = $11.35.Maximum profit = X2 – X1 – C1 +
C2,
Minimum = – C1 + C2Breakeven: ST* = X1 + C1 – C2
*

Chance/Brooks
Ch. 7: *
Money Spreads (continued)Bull Spreads (continued)For
different holding periods, compute profit for range of stock
prices at T1, T2, and T using Black-Scholes-Merton model. See
Figure 7.2.Note how time value decay affects profit for given
holding period.Early exercise not a problem.
*
Chance/Brooks
Ch. 7: *
Money Spreads (continued)Bear SpreadsBuy put with strike X2,
sell put with strike X1. Let
N1 = –1, N2 = 1Profit equation: P = –Max(0, X1 – ST) + P1
+ Max(0, X2 – ST) – P2P = X2 – X1 + P1 – P2 if ST £ X1 <
X2P = P1 + X2 – ST – P2 if X1 < ST < X2P = P1 – P2 if X1 <
X2 £ STSee Figure 7.3 for DCRB June 130/125,
P1 = $11.50, P2 = $14.25.Maximum profit = X2 – X1 + P1 – P2.

Minimum = P1 – P2.Breakeven: ST* = X2 + P1 – P2.
*
Chance/Brooks
Ch. 7: *
Money Spreads (continued)Bear Spreads (continued)For
different holding periods, compute profit for range of stock
prices at T1, T2, and T using Black-Scholes-Merton model. See
Figure 7.4.Note how time value decay affects profit for given
holding period.Note early exercise problem.A Note About Put
Money SpreadsCan construct call bear and put bull spreads.
*
Chance/Brooks
Ch. 7: *
Money Spreads (continued)CollarsBuy stock, buy put with

strike X1, sell call with strike X2. NS = 1, NP = 1, NC = –
1.Profit equation: P = ST – S0 + Max(0, X1 – ST) – P1 –
Max(0, ST – X2) + C2P = X1 – S0 – P1 + C2 if ST £ X1 < X2P
= ST – S0 – P1 + C2 if X1 < ST < X2P = X2 – S0 – P1 + C2 if
X1 < X2 £ STA common type of collar is what is often referred
to as a zero-cost collar. The call strike is set such that the call
premium offsets the put premium so that there is no initial
outlay for the options.
*
Chance/Brooks
Ch. 7: *
Money Spreads (continued)Collars (continued)See Figure 7.5
for DCRB July 120/136.165,
P1 = $13.65, C2 = $13.65. That is, a call strike of 136.165
generates the same premium as a put with strike of 120. This
result can be obtained only by using an option pricing model
and plugging in exercise prices until you find the one that
makes the call premium the same as the put premium. This will
nearly always require the use of OTC options.Maximum profit =
X2 – S0. Minimum = X1 – S0.Breakeven: ST* = S0.

*
Chance/Brooks
Ch. 7: *
Money Spreads (continued)Collars (continued)The collar is a lot
like a bull spread
(compare Figure 7.5 to Figure 7.1).The collar payoff exceeds
the bull spread payoff by the difference between X1 and the
interest on X1.Thus, the collar is equivalent to a bull spread
plus a risk-free bond paying X1 at expiration.For different
holding periods, compute profit for range of stock prices at T1,
T2, and T using Black-Scholes-Merton model. See Figure 7.6.
*
Chance/Brooks
Ch. 7: *
Money Spreads (continued)Butterfly SpreadsBuy call with
strike X1, buy call with strike X3, sell two calls with strike X2.

Let N1 = 1, N2 = –2, N3 = 1.Profit equation: P = Max(0, ST –
X1) – C1
– 2Max(0, ST – X2) + 2C2 + Max(0, ST – X3) – C3P = –C1 +
2C2 – C3 if ST £ X1 < X2 < X3P = ST – X1 – C1 + 2C2 – C3 if
X1 < ST £ X2 < X3P = –ST +2X2 – X1 – C1 + 2C2 – C3
if X1 < X2 < ST £ X3P = –X1 + 2X2 – X3
– C1 + 2C2 – C3
if X1 < X2 < X3 < STSee Figure 7.7 for
DCRB July 120/125/130, C1 = $16.00, C2 = $13.50, C3 =
$11.35.
*
Chance/Brooks
Ch. 7: *
Money Spreads (continued)Butterfly Spreads
(continued)Maximum profit = X2 – X1 – C1 + 2C2 – C3,
minimum = –C1 + 2C2 – C3Breakeven: ST* = X1 + C1 – 2C2 +
C3 and
ST* = 2X2 – X1 – C1 + 2C2 – C3For different holding periods,
compute profit for range of stock prices at T1, T2, and T using
Black-Scholes-Merton model. See Figure 7.8.Note how time

value decay affects profit for given holding period.Note early
exercise problem.
*
Chance/Brooks
Ch. 7: *
Calendar SpreadsBuy call with longer time to expiration, sell
call with shorter time to expiration.Note how this strategy
cannot be held to expiration because there are two different
expirations.Profitability depends on volatility and time value
decay.Use Black-Scholes-Merton model to value options at end
of holding period if prior to expiration.See Figure 7.9.Note time
value decay. See Table 7.2 and Figure 7.10.Early exercise can
be problem.Can be constructed with puts as well.
*
Chance/Brooks

Ch. 7: *
Ratio SpreadsLong one option, short another based on deltas of
two options. Designed to be delta-neutral. Can use any two
options on same stock.Portfolio valueV = N1C1 + N2C2Set to
zero and solve for N1/N2 = –D2/D1, which is ratio of their
deltas (recall that D = N(d1) from Black-Scholes-Merton
model).Buy June 120s, sell June 125s. Delta of 120 is 0.630;
delta of 125 is 0.569. Ratio is –(0.569/0.630) = –0.903. For
example, buy 903 June 120s, sell 1,000 June 125sNote why this
works and that delta will change.Why do this? Hedging
mispriced option
*
Chance/Brooks
Ch. 7: *
StraddlesStraddle: long an equal number of puts and callsProfit
equation: P = Max(0, ST – X) – C
+ Max(0, X – ST) – P (assuming Nc = 1, Np = 1)P = ST – X – C
– P if ST ³ XP = X – ST – C – P if ST < XEither call or put will
be exercised (unless ST = X).See Figure 7.11 for DCRB June
125,
C = $13.50, P = $11.50.Breakeven: ST* = X – C – P and ST* =
– C – PSee Figure

7.12 for different holding periods. Note time value decay.
*
Chance/Brooks
Ch. 7: *
Straddles (continued)Applications of StraddlesBased on
perception of volatility greater than priced by marketA Short
Straddle Unlimited loss potentialBased on perception of
volatility less than priced by market
*
Chance/Brooks
Ch. 7: *
Box SpreadsDefinition: bull call money spread plus bear put
money spread. Risk-free payoff if options are
EuropeanConstruction:Buy call with strike X1, sell call with

strike X2Buy put with strike X2, sell put with strike X1Profit
equation: P = Max(0, ST – X1) – C1
– Max(0, ST – X2) + C2 + Max(0, X2 – ST) – P2 – Max(0, X1 –
ST) + P1P = X2 – X1 – C1 + C2 – P2 + P1 if ST £ X1 < X2P =
X2 – X1 – C1 + C2 – P2 + P1 if X1 < ST £ X2P = X2 – X1 – C1
+ C2 – P2 + P1 if X1 < X2 < ST
*
Chance/Brooks
Ch. 7: *
Box Spreads (continued)Evaluate by determining net present
value (NPV)NPV = (X2 – X1)(1 + r)–T – C1 + C2 – P2 +
P1This determines whether present value of risk-free payoff
exceeds initial value of transaction.If NPV > 0, do it. If NPV <
0, do the reverse.See Figure 7.13.Box spread is also difference
between two put-call parities.
*

Chance/Brooks
Ch. 7: *
Box Spreads (continued)Evaluate June 125/130 box spreadBuy
125 call at $13.50, sell 130 call at $11.35Buy 130 put at $14.25,
sell 125 put at $11.50Initial outlay = $4.90, $490 for 100
eachNPV = 100[(130 – 125)(1.0456)–0.0959 – 4.90]
= 7.85NPV > 0 so do itEarly exercise a problem only on short
box spreadTransaction costs high
*
Chance/Brooks
Ch. 7: *
Summary
*

Chance/Brooks
Ch. 7: *
Chance/Brooks
Ch. 7: *
Chance/Brooks
Ch. 7: *

ChanceBrooksAn Introduction to Derivatives and Risk Man.docx

Recommended

Recommended

More Related Content

Similar to ChanceBrooksAn Introduction to Derivatives and Risk Man.docx

Similar to ChanceBrooksAn Introduction to Derivatives and Risk Man.docx (20)

More from cravennichole326

More from cravennichole326 (20)

Recently uploaded

Recently uploaded (20)

ChanceBrooksAn Introduction to Derivatives and Risk Man.docx