Mortgage Options: Valuation, Risk Management, & Relative Value

Mortgage Options:Valuation, Risk Management,

& Relative ValueSeptember 29, 2008

Andrew Lesniewski

[email protected]

Ellington Management Group

53 Forest Avenue

Old Greenwich, CT 06870

Mortgage Options – p. 1

Acknowledgements

Zhouhua Li

Jim Murphy

Muhammad Sattar


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


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.


Duration of a TBA

We parameterize the duration of a TBA

-0.04 -0.02 0.02 0.04

2

4

6

8

10


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.


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


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.


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


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


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



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



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 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 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.


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.



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+



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


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


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.


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

...


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


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


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


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


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


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


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


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


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


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: Valuation, Risk Management, & Relative Value

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