© K.Cuthbertson, D. Nitzsche1 Version 1/9/2001 FINANCIAL ENGINEERING: DERIVATIVES AND RISK...

© K.Cuthbertson, D. Nitzsche 1

Version 1/9/2001

FINANCIAL ENGINEERING:DERIVATIVES AND RISK MANAGEMENT(J. Wiley, 2001)

K. Cuthbertson and D. Nitzsche

LECTURE

T-Bond Futures


Details of Contracts and Terminology

Hedging with T-Bond Futures

Pricing T-Bond Futures

Market Timing

Topics


Details of Contracts and Terminology


Long T-bond futures positionHolder can take delivery of a long maturity T-bond at expiration, at a price F0 agreed at t=0.

SpeculationThink long rates will fall in the future, then buy a T-Bond futureIf rates do fall, then F increases and close out at profit

HedgingLock in a price today, for delivery or sale of the underlying T-Bonds

ArbitrageKeeps the cash market T-bond price (S) and the futures price,F broadly moving together

Contract and its Uses


27th July 2000s

Settlement price(CBT), Sept. delivery = ‘98-14’ (= 9814/32) =$98.4375 per $100 nominal

Contract size, z = $100,000

Face value one contract:FVF0=z(F0/100)=z ($98.4375 / $100) = $98,437.50.

Tick size = 1/32 of 1%Tick value = $31.25 per contract

US, T-Bond Futures Contract


.

Table 6.2 : US T-Bond Future (CBOT)

Contract Size $ 100,000 nominal, notional US Treasury Bond with8% coupon

Delivery Months March, June, September, DecemberQuotation Per $ 100 nominalTick Size (Value) 1/32 ($ 31.25)Last trading day 7 working days prior to last business day in expiry

monthDelivery day Any business day in delivery month (sellers choice)Settlement Any US Treasury Bond maturing at least 15 years

from the contract month (or not callable for 15 years)Margins $ 5,000 initial; $ 4,000 maintenanceTrading Hours 8 am – 2 pm – Central TimeDaily Price Limit 96/30 pnts $ 3,000


Adjusts price of actual bond to be delivered by assuming it has an 8% yield to maturity, , which then matches that of the notional bond in the futures contract. (also with maturity > 15yrs)

If the coupon on the bond actually delivered > 8% then CF > 1

If the coupon on the bond actually delivered < 8% then CF < 1

Conversion Factor, CF


CTD bond is one with smallest raw basis:

[6.1] Raw Basis = BT – FT CFT

BT = spot (“clean”) price of eligible bond for deliveryFT = settlement futures priceCFT = conversion factor of a deliverable bond.

Cheapest to Deliver


Hedgingwith

T-Bond Futures


Hedging an Existing Bond Position

S p o t P o s i t i o n : 1 s t M a y F u t u r e s : 1 s t M a y ( S e p t e m b e r d e l i v e r y )

1 0 % , 2 0 0 5 T r e a s u r y B o n d ( Y T M = 1 0 . 1 2 % ) C F o f C T D b o n d = 1 . 1 2

N o m i n a l B o n d h o l d i n g N B 0 = $ 1 m S i z e o f o n e c o n t r a c t z = $ 1 0 0 , 0 0 0

C u r r e n t P r i c e S 0 = $ 1 0 1 ( $ 1 0 1 p e r $ 1 0 0 ) P r i c e o f f u t u r e s F 0 = 1 1 0 - 1 6 ( $ 1 1 0 . 5 0 )

M a r k e t V a l u e , S p o t = T V S 0

= N B 0 ( S 0 / 1 0 0 ) = $ 1 , 0 1 0 , 0 0

D u r a t i o n D s = 6 . 9

N u m b e r o f f u t u r e s c o n t r a c t s

100

0

CTD

f

sf CF

D

D

FVF

TVSN

F a c e V a l u e F V F 0 = z F 0 = $ 1 1 0 , 5 0 0

D u r a t i o n D F = 7 . 2

T i c k v a l u e o f 1 / 3 2 e q u a l s $ 3 1 . 2 5


FINANCIAL ENGINEERING:

Outcome of the Hedge : 1st August

Spot Market (On 1st August) September Futures (On 1st August)

S1 = 98-16 ($98.5)

Value of spot position TVS1 = (S1/100)B

= (98.5/100)$1m

= $985,000

F0 = 110-16 ($110.5)

F1 = 108-16 ($108.5 per $100)

Loss on spot position = ((S0 – S1)/ 100) B

= (101 – 98.5)/100)$1m = $25,000

Gain on short futures

= Nf z (F0 – F1)/100

= 10 ($100,000) (2/100) = $20,000

or

= (2 x 32) ticks x $31.25 x 10 = $20,000

NET RESULT

Net loss

Hedged = $20,000 – $25,000 = $5,000

Unhedged = $25,000


the hedge period (eg. 3 months from May to August) may not correspond to the maturity of the futures contract (eg. September contract)

the exact bond to be delivered in the futures contract is not known, neither is the precise delivery date

all of the methods for calculating the relative price response in the spot and futures markets are subject to some error, in part because it is difficult to ascertain the CTD bond (and hence its duration)

shifts in the yield curve may not be parallel, so we cannot always assume, ys = yF.

Risks in the Hedge


Position Day :The short notifies the Clearing House of intention to deliver, two business days later

Notice of Intention Day :The Clearing House assigns a trader who is long to accept delivery. The short is now obligated to deliver the next business day.

Delivery DayBonds are delivered (with the last possible delivery day being the business day prior to the last 7 days in the delivery month).

Wild Card Play


On any 'position day'.If the spot price of bonds falls between 3pm and 5pm, the short buys the “low price” CTD bond in the cash market and issues a notice of intention to deliver knowing that upon delivery she will receive the “high” futures settlement price determined as of 3pm that day.

However, if the spot (bond) price does not decline, she can wait until the next day and repeat this strategy (until the final business day before the final delivery day in the month).

the short has an implicit option that is exercisable during the delivery month, while the long has increased risk because she does not know the exact price (ie. value) of the bond that will be delivered.

Wild Card Play by the ‘short’

© K.Cuthbertson, D. Nitzsche

Figure 6.2 : Pricing a T-bond future

Deliverable bond is 10% coupon which matures 15th February 2020. Deliverable bond pays semi-annual coupons of $(10/2) on 15th Feb. and 15th Aug.

C/2C/2 C/2

15th Feb. 1999

1st July 1999 (= t)

15th Aug. 1999

Buy Spot BondAIt = (136/181)(10/2) = 3.76

11th Sept.1999 (= T)

15th Feb.2000

Delivery of Bond in FuturesAIT = (27/184)(10/2) = 0.73

Arbitrage Period = 72 days

181 days 184 days

136 days 45 27 157 days


Zero Coupon Bond[13c] F = S exp( r (T-t) )

Coupon Paying BondSynthetic bond future [6.14] Net cost (at T) of ‘carry’ in cash market = (Ster(T-t) – FVCT)

FVCT applies to the coupon payments which occur between t = 1st of July and T = 11th of September

[6.15] Invoice Price of Futures (at T): IPF = Ft CFt + AIT

Actual futures contract and the synthetic futures both deliver one bond at T then their cost must be equal, otherwise riskless arbitrage profits would be possible. Equating [6.14] and [6.15] :

[6.16a] Ft (CFt) + AIt = Ster(T-t) – FVCT [6.16b] F = (1/CFt) (Ster(T-t) – FVCT - AIT)



[6.18] St = Bt + AIt

[6.19b] St = $130 + $4.14 = $134.14

Net cost of carry in the cash market at T, is :

[6.20] (Ster(T-t) – FVCT) = $134.14 e0.03(58/365) – $5.022 = $129.76

AIT because of the next coupon payment on 15th of Feb 2000 is :

[6.21a] AIT = (27/184) (10/2) = 0.73

[6.21b] F = (1/CFt) (Ster(T-t) – FVCT - AIT) = (1/1.22) ($129.76 – $0.73) = $105.76



Market Timing


Sell futures if you expect a rise in yields and therefore require a lower duration for your existing bond portfolio

Buy futures if you expect a fall in yields and therefore require an increased duration for your bond portfolio.

Market Timing


END OF SLIDES

© K.Cuthbertson, D. Nitzsche1 Version 1/9/2001 FINANCIAL ENGINEERING: DERIVATIVES AND RISK...

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