FAS 133 Option Fair Value Hedges: Financial-Engineering and Financial-Accounting Perspectives
July 2001

Not The FAS 133 Cash Flow Hedge: Financial-Engineering, Finance, and Financial-Accounting Perspectives
March 2001

Or “FAS 133-Induced Earnings Volatility, the Time-Value Option Swap-related ‘Cost’ of FAS-133 and a Proposed Amendment to the Rule
October 2000

James N. Bodurtha, Jr.
Georgetown
November 29, 2001

FAS – 133 Derivative Hedges
“Short-Cut” Rule
“Hedge Effectiveness” Tests for Forwards, Futures, …
”Fair-Value” balance sheet (market benchmarks)

Options
Generally, Intrinsic Value assigned to hedge
Option Value = Intrinsic Value + Time Value

Fair-value hedges
Time value changes assigned to earnings

Cash Flow hedges
Cash flow of exposure and hedge to earnings
Time value changes assigned to OCI (equity)

Scenarios

Short Unrecognized Firm Commitment Forward Hedge (note FAS-52)

		Currency Up		Currency Down
	Account/Entries	Debit	Credit	Debit	Credit
Current		None		None
Quarter 1	Forward receivable - B	15.20			13.20
	Gain/loss on hedge activity - I		15.20	13.20
	Gain/loss on hedge activity – I	15.20			13.20
	Firm commitment – B		15.20	13.20
Quarter 2	Forward receivable – B	17.50			11.40
	Gain/loss on hedge activity - I		17.50	11.40
	Gain/loss on hedge activity – I	17.50			11.40
	Firm commitment – B		17.50	11.40
	S/T Equipment – B	100.00		100.00
	Firm Commitment – B	32.70			24.60
	Forward receivable – B		32.70	24.60
	Cash – B		100.00		100.00
	Cost of goods sold – I	100.00		100.00
	S/T Equipment – B		100.00		100.00
Accounts are identified as I-income statement and B-balance statement..

Short Unrecognized Firm Commitment Option Hedge

		Currency Up		Currency Down
	Account	Debit	Credit	Debit	Credit
Current	Call purchase - B	$5.20		$5.20
	Cash - B		$5.20		$5.20
Quarter 1	Call intrinsic value - B	15.20			0.00
	Gain/loss on hedge activity - I		15.20	0.00
	Gain/loss on hedge activity - I	15.20			0.00
	Firm commitment - B		15.20	0.00
	Call time value - B		5.40		5.20
	Loss on hedging - I	5.40		5.20
Quarter 2	Call intrinsic value - B	17.50			0.00
	Gain/loss on hedge activity - I		17.50	0.00
	Gain/loss on hedge activity - I	17.50			0.00
	Firm commitment - B		17.50	0.00
	Call time value - B	0.20			0.00
	Gain on hedging - I		0.20	0.00
	Cash - B	32.70			0.00
	Call receivable - B		32.70	0.00
	S/T Equipment - B	100.00		75.40
	Firm Commitment - B	32.70			0.00
	Call receivable - B		32.70	0.00
	Cash - B		100.00		75.40
	Cost of goods sold - I	100.00		75.40
	S/T Equipment - B		100.00		75.40

The key differences between the forward and option fair value hedges are three:

1) The additional time value entries are necessary to expense the initial “at-the-money” option premium, and the associated time value changes.

2) The declining Euro value leads to a much lower cost of goods sold.

3) In closing the first quarter, the call time value goes negative and the charge to income is greater than the initial option premium. In closing the second quarter, this negative time value provides a hedging gain. This phenomenon can only occur for a call option close to maturity on relatively high yielding underlying (high interest rate foreign currency.)

4) With the forward hedge, the cost of goods sold was fixed, independent of changes in currency values. In contrast, the option hedge leads to different outcomes depending on changes in currency values.

Concern is earnings (time-value) variability induced by the option hedge that will be held to maturity.

Figure 1

Spot Price Process Evolution
0	0.125	0.25	0.375	0.5	0.625	0.75
r=5%						Su^6= 152.85
y=5%					Su^5= 142.41
s=20%				Su^4= 132.69		Su^5d= 132.69
t=0.75			Su^3 =123.63		Su^4d= 123.63
h=0.125		Su^2= 115.19		Su^3d= 115.19		Su^4d^2= 115.19
	S^u= 107.33		Su^2d= 107.33		Su^3d^2= 107.33
S0= 100.00		Sud= 100.00		Su^2d^2= 100.00		Su^3d^3= 100.00
	S^d= 93.17		Sud^2= 93.17		Su^2d^3= 93.17
		Sd^2= 86.81		Sud^3= 86.81		Su^2d^4= 86.81
			Sd^3= 80.89		Sud^4= 80.89
u=1.073	=exp(s*sqrt(h))			Sd^4= 75.36		Sud^5= 75.36
d=0.932	=exp(-s*sqrt(h))				Sd^5= 70.22
Su^md^n=S0*u^m*d^n						Sd^6= 65.43

Figure 2

Call Option Value Process Evolution
0	0.125	0.25	0.375	0.5	0.625	0.75
p=48.23%	=(e^-(r-y)*t-d)/(u-d)					Cu^6= 52.85	=Max(Su^6-X,0)
Cu^nd^m=e^-r*t*(p*Cu^n+1d^m+(1-p)*Cu^nd^m+1)					Cu^5= 42.15
				Cu^4= 32.28		Cu^5d= 32.69	=Max(Su^5d-X,0)
			Cu^3= 23.19		Cu^4d= 23.48
		Cu^2= 15.74		Cu^3d= 15.00		Cu^4d^2= 15.19	=Max(Su^4d^2-X,0)
	Cu= 10.20		Cu^2d= 8.99		Cu^3d^2= 7.28
C0=6.377		Cud= 5.17		Cu^2d^2= 3.49		Cu^3d^3= 0.00	=Max(Su^3d^3-X,0)
	Cd= 2.89		Cud^2= 1.67		Cu^2d^3= 0.00
		Cd^2= 0.80		Cud^2= 0.00		Cu^2d^4= 0.00	=Max(Su^2d^4-X,0)
			Cd^3= 0.00		Cud^4= 0.00
				Cd^4= 0.00		Cud^5= 0.00	=Max(Sud^5-X,0)
					Cd^5= 0.00
						Cd^6= 0.00	=Max(Sd^6-X,0)

Figure 3

Option Time Value (T) and Intrinsic Value (I) Process Evolution
0	0.125	0.25	0.375	0.5	0.625	0.75
Tu^md^n=Cu^md^n-Max(Su^md^n-X,0)						0.00 =Tu^6
Iu^md^n=Max(Su^md^n-X,0)					-0.26 =Tu^5	52.85 =Iu^6
				-0.41 =Tu^4	42.41 =Iu^5	0.00 =Tu^5d
			-0.44 =Tu^3	32.69 =Iu^4	-0.15 =Tu^4d	32.69 =Iu^5d
		0.55 =Tu^2	23.63 =Iu^3	-0.19 =Tu^3d	23.63 =Iu^4d	0.00 =Tu^4d^2
	2.88 =Tu	15.19 =Iu^2	1.66 =Tu^2d	15.19 =Iu^3d	-0.05 =Tu^3d^2	15.19 =Iu^4d^2
6.377 =T0	7.33 =Iu	5.17 =Tud	7.33 =Iu^2d	3.49 =Tu^2d^2	7.33 =Iu^3d^2	0.00 =Tu^3d^3
0.00 =I0	2.89 =Td	0.00 =Iud	1.67 =Tud^2	0.00 =Iu^2d^2	0.00 =Tu^2d^3	0.00 =Iu^3d^3
	0.00 =Id	0.80 =Td^2	0.00 =Iud^2	0.00 =Tud^3	0.00 =Iu^2d^3	0.00 =Tu^2d^4
		0.00 =Id^2	0.00 =Td^3	0.00 =Iud^3	0.00 =Tud^4	0.00 =Iu^2d^4
			0.00 =Id^3	0.00 =Td^4	0.00 =Iud^4	0.00 =Tud^5
				0.00 =Id^4	0.00 =Td^5	0.00 =Iud^5
					0.00 =Id^5	0.00 =Td^6
						0.00 =Id^6

Figure 4

Risk-Neutral Probabilities of Occurance and Associated Changes in Option Time Value
(that would be booked to quarterly earnings under FAS 133)
	0.25		0.5		0.75
					1.26% * 0.41=Tu^6-Tu^4
					5.41% * 0.19=Tu^5d-Tu^3d
			5.41% * -0.95=Tu^4-Tu^2		8.70% * -3.49=Tu^4d^2-Tu^2d^2
			11.62% * -5.36=Tu^3d-Tud		6.23% * 0.00=Tu^3d^3-Tud^3
	23.26% * -5.83=Tu^2-T0		6.23% * 2.69=Tu^2d^2-Td^2		1.67% * 0.00=Tu^2d^4-Td^4
					2.70% * 0.41=Tu^5d-Tu^4
					11.60% * 0.19=Tu^4d^2-Tu^3d
			11.62% * -0.74=Tu^3d-Tu^2		18.68% * -3.49=Tu^3d^3-Tu^2d^2
			24.94% * -1.68=Tu^2d^2-Tud		13.37% * 0.00=Tu^2d^4-Tud^3
	49.94% * -1.21=Tud-T0		13.38% * -0.80=Tud^3-Td^2		3.59% * 0.00=Tud^5-Td^4
					1.45% * 0.41=Tu^4d^2-Tu^4
					6.23% * 0.19=Tu^3d^3-Tu^3d
			6.23% * 2.94=Tu^2d^2-Tu^2		10.02% * -3.49=Tu^2d^4-Tu^2d^2
			13.38% * -5.17=Tud^3-Tud		7.17% * 0.00=Tud^5-Tud^3
	26.80% * -5.58=Td^2-T0		7.18% * -0.80=Td^4-Td^2		1.92% * 0.00=Td^6-Td^4
Expected Values =		-3.454		-1.684		-1.240
Fair value = Discounted Expected Values =		-3.411		-1.642		-1.194
Fair value = Discounted Expected Time Value Changes = -6.247

The path-dependent time-value evolution is noted as Tuⁿd^m-Tuⁱd^j, that indicates the time value change associated with a spot price move from Suⁱd^j to Suⁿd^m

Simple Example

Figure 5

Value of $1 Contingent on Particular Changes in Option Time Value
	0.25		0.5		0.75
					0.0121 if Tu^6 from Tu^4
					0.0521 if Tu^5d from Tu^3d
			0.0528 if Tu^4 from Tu^2		0.0838 if Tu^4d^2 from Tu^2d^2
			0.1133 if Tu^3d from Tud		0.0600 if Tu^3d^3 from Tud^3
	0.2298 if Tu^2 from T0		0.0608 if Tu^2d^2 from Td^2		0.0161 if Tu^2d^4 from Td^4
					0.0260 if Tu^5d from Tu^4
					0.1118 if Tu^4d^2 from Tu^3d
			0.1133 if Tu^3d from Tu^2		0.1799 if Tu^3d^3 from Tu^2d^2
			0.2432 if Tu^2d^2 from Tud		0.1287 if Tu^2d^4 from Tud^3
	0.4932 if Tud from T0		0.1305 if Tud^3 from Td^2		0.0345 if Tud^5 from Td^4
					0.0140 if Tu^4d^2 from Tu^4
					0.0600 if Tu^3d^3 from Tu^3d
			0.0608 if Tu^2d^2 from Tu^2		0.0966 if Tu^2d^4 from Tu^2d^2
			0.1305 if Tud^3 from Tud		0.0691 if Tud^5 from Tud^3
	0.2647 if Td^2 from T0		0.0700 if Td^4 from Td^2		0.0185 if Td^6 from Td^4
Total cost across all outcomes =		0.9876		0.9753		0.9632
(Discounted value of $1 in all outcomes)

Figure 6

Figure 7

Forward Price Contingent Cashflows of a Fixed Option Time Value Cash Flow Swap
[forward time value rebate - fixed (swap) + option (@ maturity)]
	0.25		0.5		0.75
					(-0.42 - 2.179)*1.26%
					(-0.20 - 2.179)*5.41%
			(0.98 - 2.179)*5.41%		(3.62 - 2.179)*8.70%
			(0.00 - 2.179)*11.62%		(0.00 - 2.179)*6.23%
	(5.90 - 2.179)*23.26%		(-2.76 - 2.179)*6.23%		(0.00 - 2.179)*1.67%
					(-0.42 - 2.179)*2.70%
					(-0.20 - 2.179)*11.60%
			(0.76 - 2.179)*11.62%		(3.62 - 2.179)*18.68%
			(1.72 - 2.179)*24.94%		(0.00 - 2.179)*13.37%
	(1.22 - 2.179)*49.94%		(0.82 - 2.179)*13.38%		(0.00 - 2.179)*3.59%
					(-0.42 - 2.179)*1.45%
					(-0.20 - 2.179)*6.23%
			(-3.02 - 2.179)*6.23%		(3.62 - 2.179)*10.02%
			(5.30 - 2.179)*13.38%		(0.00 - 2.179)*7.17%
	(5.65 - 2.179)*26.80%		(0.82 - 2.179)*7.18%		(0.00 - 2.179)*1.92%
RN Expected Values =		1.318		-0.453		-0.892
Discounted RN Expected Values =		1.301		-0.442		-0.859
Discounted RN Expected Value of Net Time Value Rebate less Swap Payment (2.179) = 0.000

2.c. A General Case

3)

Sequence of time value changes has the following discounted risk-neutral measure expected value:

4)

As , and set .

5)

In most cases, this total cost of time-value changes will be negative. In the case of zero interest rates, this quantity is current option time value. In cases with no underlying yield, higher rates lower this cost. With a yield on the underlying, then the sign of the cost is undetermined (though for most cases the cost remains positive.) In the usual cases that longer maturity and/or higher volatility raise option values, then the cost of option time-value changes will also rise.

“Fixing” FAS-133 Induced Option Time Value Changes

Define forward time value change

6)

7)

Time Value Swap (TVS) payment value is the root, V.

Other

Section 2.b.2 variable TVS set each quarter smoothes time-value changes and doesn’t require FAS-133.

3) Accounting perspective -

Look to held – to – maturity bonds,

principal like intrinsic value, time value could be amortized like bond discount or premium over option life.

Paritally done with DIG G20 for some cash flow hedges

Appendix – Value and expected value changes
(literally, minding p’s and q’s.)

Changes in Call Option Fair Value and Expected Value

0

0.25

Fair Values

15.74 = Cu2

9.36=Cu2-C0

6.377 = C0

5.17 = Cud

-1.21=Cud-C0

0.80 = Cd2

-5.58=Cd2-C0

Expected Values

18.23 = ECu2

9.89=ECu2-EC0

8.345 = EC0

6.42 = ECud

-1.93=ECud-EC0

1.09 = ECd2

-7.25=ECd2-EC0

Hedge Effectiveness

5.6% in u2

(% of expected value change)

59.5% in ud

30.1% in d2

Finesse to time value change in OCI won’t generally be “effective.”

Methodology – Linking objective and risk-neutral probabilities

annualized expected logarithmic change in the underlying price a-1)

Cox-Ross-Rubinstein (1976) and Rubinstein (1976) implicitly use this specification, . For this case, and . is an unbiased risk premium estimate.

objective probability a-3)

for a = 0, equals risk-neutral probability, p.

a-8)

For underlying valuation (fixed risk premium and volatility), may use either valuation measure/probabilities.

Table A-1 Spot outcomes	# of paths to outcome	Risk neutral probability of outcome	True probability of outcome
152.8	1	1.26%	1.93%
132.7	2	8.11%	10.76%
115.2	15	21.76%	25.07%
100.0	20	31.13%	31.13%
86.8	15	25.06%	21.75%
75.4	6	10.76%	8.10%
65.4	1	1.92%	1.26%
Expected Value		100	103.05
Add 5% Yield		103.05	106.98
Discounted value	@ Risk neutral rate =5%	100	N/A
Discounted value	@ Risk-adjusted rate = 9%	N/A	100

4% risk premium, a, as m = 7%, s = 20%, r= 5%;
, expected return – continuously compounded risk-adjusted rate.

Option Expected Values

a-8)

B[ ] is the complementary binomial distribution; “a stands for the minimum number of upward moves the stock must make over the next n periods for the call to finish in-the-money (a will be the smallest nonnegative integer such that);”[1] and .[2]

Following Rubinstein (1976) and allowing the number of periods, n, to go to infinity (h to go to zero), we have

, a-9)

where N is the standard normal distribution, and

Figure A-1

Delta Hedge Process Evolution
0	0.125	0.25	0.375	0.5	0.625
Du^nd^m=exp(-y*h)*(Cu^n+1d^m-Cu^nd^m+1)/(Su^n+1d^m-Su^nd^m+1)					Du5=0.994
				Du4=0.988
			Du3=0.981		Du4d=0.994
		Du2=0.866		Du3d=0.988
	Du=0.692		Du2d=0.753		Du3d2=0.994
D0=0.513		Dud=0.513		Du2d2=0.511
	Dd=0.329		Dud2=0.263		Du2d3=0.000
		Dd2=0.135		Dud3=0.000
			Dd3=0.000		Dud4=0.000
				Dd4=0.000
					Dd5=0.000

Figure A-2

Option Elasticity Process Evolution
0	0.125	0.25	0.375	0.5	0.625
eu^nd^m=Du^nd^m*Su^nd^m/Cu^nd^m					eu^5=3.358
				eu^4=4.059
			eu^3=5.232		eu^4d=5.232
		eu^2=6.337		eu^3d=7.583
	eu=7.275		eu^2d=8.994		eu^3d^2=14.648
e0=8.052		eud=9.936		eu^2d^2=14.648
	ed=10.609		eud^2=14.648		#N/A
		ed^2=14.648		#N/A
			#N/A		#N/A
				#N/A
					#N/A

Figure A-3

Risk-adjusted Discount Rate Process Evolution under the True-Objective-Q Measure
	Underlying Risk premium=a=	4.00%
Risk-adjusted underlying discount rate=er=a+r=			9.00%
	ru^nd^m=ln (eu^nd^m (e^er*h-e^r*h)+e^r*h)/h
0	0.125	0.25	0.375	0.5	0.625
					ru^5=18.35%
				ru^4=21.11%
			ru^3=25.71%		ru^4d=25.71%
		ru^2=30.02%		ru3^d=34.84%
	ru=33.65%		ru^2d=40.28%		ru^3d^2=61.68%
r0=36.65%		rud=43.88%		ru^2d^2=61.68%
	rd=46.45%		rud^2=61.68%		#N/A