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Chapter 30
Convexity, Timing, andTiming, and Quanto
Adjustments
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 1
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Forward Yields and
Forward Pr ices
We define the forward yield on a bond as the yield
calculated from the forward bond priceThere is a non-linear relation between bond yieldsand bond prices
It follows that when the forward bond price equals
the expected future bond price, the forward yielddoes not necessarily equal the expected futureyield
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 2
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Relationship Between Bond Yields and
Prices (F igure 30.1, page 694)
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 3
BondPrice
YieldY3
B1
Y1Y2
B3B2
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Convexi ty Adjustment for Bond Yields(Eqn 30.1, p. 695)
Suppose a derivative provides a payoff at time Tdependent on a bond yield,yTobserved at time T.Define:
G(yT) : price of the bond as a function of its yield
y0 : forward bond yield at time zero
sy : forward yield volatility
The expected bond price in a world that is FRN wrtP(0,T)is the forward bond price
The expected bond yield in a world that is FRN wrt
P(0,T)is
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 4
)(
)(
2
1
0
022
0yG
yGTy y
sYieldBondForward
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Convexity Adjustment for Swap
Rate
The expected value of the swap rate for the period Tto T+tin a world that is FRN wrtP(0,T)is
(approximately)
where G(y) defines the relationship between price
and yield for a bond lasting between Tand T+tthatpays a coupon equal to the forward swap rate
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 5
)(
)(
2
1
0
022
0yG
yGTy y
sRateSwapForward
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Example 30.1 (page 696)An instrument provides a payoff in 3 yearsequal to the 1-year zero-coupon rate
multiplied by $1000Volatility is 20%
Yield curve is flat at 10% (with annualcompounding)
The convexity adjustment is 10.9 bps so thatthe value of the instrument is 101.09/1.13=75.95
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 6
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Example 30.2 (Page 696-697)An instrument provides a payoff in 3 years =to the 3-year swap rate multiplied by $100
Payments are made annually on the swap
Volatility is 22%
Yield curve is flat at 12% (with annualcompounding)
The convexity adjustment is 36 bps so thatthe value of the instrument is 12.36/1.123=8.80
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 7
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Timing Adjustments (Equation 30.4, page698)
The expected value of a variable, V, in a world that isFRN wrtP(0,T*) is the expected value of the variable in aworld that is FRN wrtP(0,T) multiplied by
whereRis the forward interest rate between Tand T*
expressed with a compounding frequency of m, sRis thevolatility ofR,R0is the value ofR today, sVis the volatilityofF, andris the correlation betweenRandV
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 8
ssr T
mR
TTRRVVR
/1
)(exp
0
*
0
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Example 30.3 (page 698)
A derivative provides a payoff 6 years equal to thevalue of a stock index in 5 years. The interest rate is
8% with annual compounding1200 is the 5-year forward value of the stock index
This is the expected value in a world that is FRN wrtP(0,5)
To get the value in a world that is FRN wrtP(0,6) wemultiply by 1.00535
The value of the derivative is 12001.00535/(1.086)or 760.26
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 9
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Quantos(Section 30.3, page 699-702)
Quantos are derivatives where the payoff is
defined using variables measured in onecurrency and paid in another currency
Example: contract providing a payoff of
ST
K dollars ($) where S is the Nikkei stockindex (a yen number)
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Diff SwapDiff swaps are a type of quanto
A floating rate is observed in one currency
and applied to a principal in another currency
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Quanto Adjustment (page 700)
The expected value of a variable, V, in aworld that is FRN wrtPX(0,T) is its expected
value in a world that is FRN wrtPY(0,T)multiplied by exp(rVWsVsWT)
Wis the forward exchange rate (units of Yperunit ofX) and r
VWis the correlation between V
and W.
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Example 30.4 (page 700)
Current value of Nikkei index is 15,000
This gives one-year forward as 15,150.75
Suppose the volatility of the Nikkei is 20%,the volatility of the dollar-yen exchange rate is12% and the correlation between the two is0.3
The one-year forward value of the Nikkei for acontract settled in dollars is15,150.75e0.3 0.20.121or 15,260.23
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Quantos continued
When we move from the traditional riskneutral world in currency Y to the tradional
risk neutral world in currency X, the growthrate of a variable Vincreases by
rsVsS
where sV is the volatility of V, sS is thevolatility of the exchange rate (units of Y perunit of X) andr is the correlation betweenthe two
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Siegels Paradox
Options, Futures, and Other Derivatives, 9th Edition,Copyright John C. Hull 2014 15
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When is a Convexity, Timing, or
Quanto Adjustment Necessary
A convexity or timing adjustment is necessary
when interest rates are used in a nonstandardway for the purposes of defining a payoff
No adjustment is necessary for a vanilla swap,a cap, or a swap option
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