Mortgage Options:Valuation, Risk Management,
& Relative ValueSeptember 29, 2008
Andrew Lesniewski
Ellington Management Group
53 Forest Avenue
Old Greenwich, CT 06870
Mortgage Options – p. 1
Acknowledgements
Zhouhua Li
Jim Murphy
Muhammad Sattar
Mortgage Options – p. 2
Mortgage options
Mortgage options = options on TBAs
Expire 5 business days prior to the settlement of the TBA
Standard strikes = ATM, ±1/2, ±1
Quotes available from Deutsche Bank and GoldmanSachs
Mortgage Options – p. 3
Modeling mortgage options
Duration of the TBA assumed a function D (X) of therelative rate X = M − C, where C is the coupon on theTBA, and M is the current coupon.
Option value depends on the normal volatility σ of X.
D (X) calibrated to the prepayment model impliedduration.
σ is implied by the options market.
Mortgage Options – p. 4
Duration of a TBA
We parameterize the duration of a TBA
-0.04 -0.02 0.02 0.04
2
4
6
8
10
Mortgage Options – p. 5
Duration of a TBA
by means of a logistic function of the relative rate X:
D (X) = L + (U − L)1
1 + e−κ(X−∆)
The parameters L,U, κ and ∆ are determined by fitting todurations computed from the prepayment model.
Mortgage Options – p. 6
Price of a TBA
This produces the following shape of the price function of theTBA:
-0.04 -0.02 0.02 0.04
0.7
0.8
0.9
Mortgage Options – p. 7
Price of a TBA
which is explicitly given by
P (X) = µ exp
(
−(L + U) X
2
)
cosh κ∆2
cosh κ(X−∆)2
U−L
κ
This function is the key input to the option model.
The value of µ = P (0) is determined so that the distribution ofprices is consistent with the options market, and is close topar.
Mortgage Options – p. 8
Convexity of a TBA
The convexity of the TBA has the shape:
-0.04 -0.02 0.02 0.04
-200
-150
-100
-50
50
Mortgage Options – p. 9
Convexity of a TBA
and is explicitly given by the following expression:
Ω (X) = D (X)2−
κ (U − L)
4
1
cosh2 κ(X−∆)2
Mortgage Options – p. 10
Calibrating duration function parameters
Generate macro rates scenarios (say, ±25 bp, ±50, ±140,...) and compute the corresponding durations of the TBA
Fit the function D (X). For the Fannie 5.0, the calibratedparameters are:
L = −0.677
U = 9.679
∆ = 0.00344
κ = 108.624
Mortgage Options – p. 11
Calibrating duration function parameters
The graph below shows the durations of the TBAs of differentcoupons as a function of the current coupon.
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
FNMA5
FNMA5.5
FNMA6
Mortgage Options – p. 12
Calibrating duration function parameters
On this graph, the TBA duration is compared to the calibratedlogistic function D (X).
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
0.00% 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00%
FNMA5
D(X)
Mortgage Options – p. 13
Mortgage rate process
We assume that the relative rate follows a normal process
dX (t) = σdW (t) ,
X (0) = M − C,
where W (t) is a Brownian motion. The solution
X (t) = M − C + σW (t)
is implemented by a Monte Carlo simulation.
Mortgage Options – p. 14
Mortgage option price
Calls and puts struck at K and expiring at T are valuedaccording
C = Z (T ) E [max (P (X) − K, 0)]
P = Z (T ) E [max (K − P (X) , 0)]
Here, Z (T ) denotes the discount factor.
The valuation formulas do not lead to closed formexpressions, and the evaluation of the expectations requiresMonte Carlo simulations. The simulations lead to fast andaccurate results.
Mortgage Options – p. 15
Implied current coupon volatility
We can now easily calibrate the volatility σ by requiring that
the price of the TBA, and
the premium on the option
match the market values. This is done in two steps:
Choose the parameter µ = µ (σ) so that the mean of theprice distribution is the market price of the TBA.
Choose σ to match the option premium.
Currently (as of September 17), σ ≃ 265 bp.
Mortgage Options – p. 16
Implied current coupon volatility
Specifically, here is the September 17 snapshot of the options market(November expires):
FNCL5.0
99 − 016
K −1 −1/2 ATM +1/2 +1
σ 1 − 07 1 − 12+ 1 − 185 1 − 095 1 − 013
FNCL5.5
100 − 26+
K −1 −1/2 ATM +1/2 +1
σ 0 − 273 1 − 002 1 − 057 0 − 28+ 0 − 201
FNCL6.0
102 − 012
K −1 −1/2 ATM +1/2 +1
σ 0 − 172 0 − 212 0 − 261 0 − 16+ 0 − 08+
Mortgage Options – p. 17
Implied current coupon volatility
and the implied current coupon volatilities (in basis points) are:
FNCL5.0K −1 −1/2 ATM +1/2 +1
σ 262.5 263.4 264.2 265.7 267.4
FNCL5.5K −1 −1/2 ATM +1/2 +1
σ 263.7 265.6 267.6 270.1 272.9
FNCL6.0K −1 −1/2 ATM +1/2 +1
σ 259.5 261.6 263.5 267.3 272.1
Mortgage Options – p. 18
Volatility smile
Implied volatility exhibits smile:
250
255
260
265
270
275
-1 -0.5 0 0.5 1
FNCL5.0
FNCL5.5
FNCL6.0
Mortgage Options – p. 19
Relative value
The stochastic model allows to asses richness / cheapness ofcurrent coupon volatility:
Versus realized current coupon volatility (≃ 235 bp as ofSeptember 17).
Versus suitable swaption volatilities.
Mortgage Options – p. 20
Risk characteristics of mortgage options
Mortgage options are options on convex assets (prepaymentoptions). Their risk is a composition of optionalities.
Delta
∆ = ∆BS × ∆TBA
Gamma
Γ = ΓBS × (∆TBA)2 + ∆BS × ΓTBA
...
Mortgage Options – p. 21
Delta risk of a mortgage call
0.00%
0.50%
1.00%
1.50%
2.00%
2.50%
-500 -400 -300 -200 -100 0 100 200 300 400 500
Relative rate
DV0
1
Mortgage Options – p. 22
Gamma risk of a mortgage call
-0.000250
-0.000200
-0.000150
-0.000100
-0.000050
0.000000
0.000050
0.000100
0.000150
0.000200
0.000250
0.000300
-500 -400 -300 -200 -100 0 100 200 300 400 500
Relative rate
Gam
ma
Mortgage Options – p. 23
CC delta risk of a mortgage call
-0.0120
-0.0100
-0.0080
-0.0060
-0.0040
-0.0020
0.0000
0.0020
-500 -400 -300 -200 -100 0 100 200 300 400 500
Relative rate
CC
DV0
1
Mortgage Options – p. 24
CC gamma risk of a mortgage call
0.00000
0.00005
0.00010
0.00015
0.00020
0.00025
-500 -400 -300 -200 -100 0 100 200 300 400 500
Relative rate
CC
Gam
ma
Mortgage Options – p. 25
Swaption vol vega risk of a mortgage call
-0.0450
-0.0400
-0.0350
-0.0300
-0.0250
-0.0200
-0.0150
-0.0100
-0.0050
0.0000
-300 -200 -100 0 100 200 300 400 500
Relative rate
Vega
Mortgage Options – p. 26
Delta risk of a mortgage put
-8.00%
-7.00%
-6.00%
-5.00%
-4.00%
-3.00%
-2.00%
-1.00%
0.00%
-500 -400 -300 -200 -100 0 100 200 300 400 500
Relative rate
DV0
1
Mortgage Options – p. 27
Gamma risk of a mortgage put
-0.000100
-0.000050
0.000000
0.000050
0.000100
0.000150
0.000200
0.000250
0.000300
0.000350
0.000400
0.000450
-500 -400 -300 -200 -100 0 100 200 300 400 500
Relative rate
Gam
ma
Mortgage Options – p. 28
CC delta risk of a mortgage put
-0.0120
-0.0100
-0.0080
-0.0060
-0.0040
-0.0020
0.0000
0.0020
0.0040
-500 -400 -300 -200 -100 0 100 200 300 400 500
Relative rate
CC
DV0
1
Mortgage Options – p. 29
CC gamma risk of a mortgage put
-0.00008
-0.00007
-0.00006
-0.00005
-0.00004
-0.00003
-0.00002
-0.00001
0.00000
-500 -400 -300 -200 -100 0 100 200 300 400 500
Relative rate
CC
Gam
ma
Mortgage Options – p. 30
Swaption vol vega risk of a mortgage put
0.0000
0.0100
0.0200
0.0300
0.0400
0.0500
0.0600
0.0700
0.0800
-300 -200 -100 0 100 200 300 400 500
Relative rate
Vega
Mortgage Options – p. 31
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