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- 1. Cheapest to delivery option and creditrisk in European bond future marketBack to the futureCristiana CornoStructuring, Rates & Inflation – Banca IMIEuropean Debt Crisis: from Threat to Opportunity?Venezia, 19/21 September 2012
- 2. SummaryThe bond future is literally a “standardized forward agreement in which the seller agrees to deliverphysically to the buyer a notional amount of nominal bond at a certain date versus payment of aninvoice price”.The deliverable assets are specified by the bond future contract grade and make up the deliverable basket.The peculiarity of bond future is that the seller has the choice on “which eligible bond to deliver” and“when to deliver it”. SHORT’S DILEMMAThese rights make up the STRATEGIC DELIVERY OPTIONS in the hands of the future seller.The bond future MULTIASSET NATURE makes it one of the most traded hybrid. 1
- 3. Reasons to go backI thought it could be interesting to review this topic, because in the recent past we have witnessed both: an increase of the range of tradable products; an increase in the optionality priced in bond future markets. Therefore, in the following, we will: briefly review basis terminologies and concept with particular reference to the delivery option; look at what happened recently which has affected bond future optionality; try to identify challenges and opportunities offered by the product. For avoidance of doubts, the jump was from 21st October 2015 to 26th october 1985, “Back to the future” movie 2
- 4. Agenda1 Basis basics: terminology and concepts2 Why back to the future? Optionality and new products3 Opportunities and challenges 3
- 5. Basis Basics: generalBasis is a concept common to all future/forward market. In the commodities market, the basis is thedifference between the spot and forward price. Being FORWARD PRICE = SPOT PRICE + COST OF FUNDING + COST OF STORAGEthe basis tends to equilibrate the cost of funding and storage or BASIS = -(COST OF FUNDING + COST OF STORAGE)and it is generally negative, with Forward > Spot.In bond market, cost of storage in null, plus the asset offers an income stream therefore : FORWARD PRICE = SPOT PRICE + COST OF FUNDING - INCOME STREAMwith Forward < Spot if carry is positive. BASIS = -COST OF FUNDING + INCOME STREAM = TOTAL CARRY a positively inclined yield curve carry is positive as it is the basis with Forward price < Spot price. Bydefinition BASIS GOES TO ZERO AT CONTRACT EXPIRY 4
- 6. Basis Basics: generalAnalytically, being BASIS = SPOT – FORWARD, said S = spot price and the forward price F = [ S + ai(t) ] * ( 1 + r(T-t) ) - ai(T),with ai(t) accrued interest at time t and r equals to the repo rate, we get:BASIS = S - [( S + ai(t) ) + (S + ai(t)) * r(T-t) - ai(T)] = - (S + ai(t)) * r(T-t) + ai(T) – ai(t)BASIS = -cost of funding + income coupon = total carry = daily carry * number of daysWe expect the basis to be 0 at contract expiry and to decrease with time proportionally to the daily carry,with the main factor of volatility being the repo rate. 1.4 BTPS 5 ½ 09/01/22 BASIS 1.2 SPOT PRICE - FORWARD PRICE 1 What drives the basis? 0.8 ΔBasis ΔP = − s * r * (T − t ) = − MD * r * (T − t ) ΔPs Δy 0.6 0.4 ΔBasis = −( Ps + ai (t )) * (T − t ) 0.2 Δr 0 10 12 14 18 20 24 27 31 33 35 39 41 45 47 49 53 55 59 61 63 67 69 73 75 77 81 83 87 89 91 95 97 DAYS TO EXPIRY 5
- 7. Basis Basics: bond future Compared to other futures, bond future are more complicated, because the underlying asset is a theoretical government bond with a fixed specific coupon (6% in europewide) and range of maturity (8.5-11 for Btp future, 8.5-10.5 for bund, 8.5-13 for Gilt). The contract is settled by physical delivery and permits the delivery, by the short, of any coupon security, provided it meets the deliverability criteria (usually remaining maturity, amount outstanding, date of issue called “contract grade” or “contract specifications”). The eligible securities are said to be in the deliverable basket. This right make up the “quality/switch option”. In the European bond futures markets the delivery period is just one day therefore the “timing option”, related to “when to deliver” are irrelevant for the European market. In the Us, due to the different delivery process, other than the “quality” option, the set of options includes: the “month end option” (future’s last trade date is seven business days before last delivery date); the “carry option” (one entire month for delivery) and the “wild card” option (time window between future settlement price establishment and end of notification time for short investor willing to deliver). 6
- 8. European less hybrid than Us peersIn terms of their relative importance the “switch/quality” option is, by far, the most valuable followed by the“month end option”, while the timing options have shown small importance at all. (Burghardt, “The treasurybond basis”).Literature shows that the value of the shorts option is highest when rates nears notional coupon level or theinflection point, where there is high probability of switch between high and low duration bonds.Below we summarize the set of options that the short of futures owns in the different markets. EUREX LIFFE CBT Quality option Y Y Y Increase in End of the month option N N Y value Carry option N Y Y Wild card Option Y Y Y Increase in quantity 7
- 9. Basis Basics: bond future The multi asset nature of the future had the objective to make demand for bonds for the purpose of delivery less concentrated in order to avoid overpricing and squeeze. To make the deliverable security economically equal to deliver the CBOT decided to adopt, when introducing the T-bond future (1977), the conversion factor invoicing system. The aim of the conversion factor system (CFS) was to adjust the invoice price paid by the long upon delivery to the characteristics of the bond being delivered, make them economically equivalent into delivery and close to their market prices. Upon delivery of bond (i) the long future pays the short an invoice amount equal to: INVOICE PRICE(i) = CF(i) * F + AI(i) where F is the future settlement price, CF(i) and AI(i), the conversion factor and the accrued interest at delivery of bond i. 8
- 10. Basis Basics: conversion factor The conversion factor is a fixed amount defined by the exchange for each bond and each future expiry and it is approximately equal to the forward price of each bond at delivery for which the yield to maturity is equal to 6% or the notional coupon ( in next slide easy calculation on bbg). For construction the CF depends only on the cash flow structure of each bond at delivery. It does not depend on market conditions. Its aim is to compensate the long investor into delivery for the different bond structure in term of coupon and maturity, with respect to the notional underlying bond. It partially does its job: for example, for a bond with coupon higher than 6% will be higher than 1 and, vice versa, this effect will be greater depending on the bond maturity. As we will see all the CTD problem can be referred to a conversion factor fault. 9
- 11. Basis Basics: approximate conversion factor With Bloomberg function Yas it is possible to calculate the bond forward price at delivery with 6% yield to maturity: price equals conversion factor 10
- 12. Basis Basics: bond future pricing Ideally, at delivery, to make the short future investor indifferent between delivering any of the eligible bonds, the future invoice price CF(i)*F+ AI(i) of each bond should equal its purchase price in the market or S(i)+AI(i). Unfortunately in the current system, bonds will be equivalent at delivery: CF (i)* F = S(i) for each i only at flat 6% yield curve, where CF(i)=S(i) for each i, with future = 100. Each time we move away from this ideal condition, the CFS it is not able to equalize differences in bonds % and we will have one or more cheapest to deliver (CTDs) bonds. In all these cases at delivery, for non arbitrage argument*, we will have: S(i) >= CF(i)* F for each i.* Non arbitrage argument1: If S(i) < CF*F then the short can buy i and deliver it into the future by making profit: CF*F-S(i) > 0, but reverse can not be done 11
- 13. Basis Basics: bond future pricing Given that at delivery S(i) >= CF*F for each bond i, the short ‘s profit will be <=0 and he will try to maximize his PROFIT function: PROFIT% = (INVOICE PRICE – PURCHASE PRICE)/(PURCHASE PRICE) F * CFi + AI i ,del − (S i + AI i ,del ) F * CFi − S i F * CFi Π= = = −1 (S i + AI i ,del ) Si Si by delivering the bond i with the lowest converted price S(i)/CF(i): At delivery the CTD is defined as the bond i: ⎡ S (i ) ⎤ ctd = i / min i ⎢ ⎥ ⎣ CF (i ) ⎦ And the future price at delivery is equal to: S (ctd ) Fdelivery = CF (ctd ) 12
- 14. Basis Basics: bond future pricing During the life of the contract, the future price will be: ⎡ Fwd t (Ctd ) ⎤ Ft ≤ ⎢ ⎥ ⎣ CF (Ctd ) ⎦ Future price will be lower than the forward converted price of the cheapest to deliver to compensate the fact that the CTD bond could change. The CTD will be the bond which maximize the short seller profit. The difference between current future price and lower forward converted price is the value of the delivery options in the hands of the short investor. Fwd t (ctd ) delivery option value (DOV) = α = − Ft CF (ctd ) 13
- 15. Basis Basics: delivery option in chart Here, we plot the converted price/yield relationship for different level of yield. The future price will mymic the worst performing asset (lower converted forward price). The distance, at time t, between the future price and the lower converted price of the bonds in the deliverable basket is the value of the delivery option. Illustration inspired by Burghardt The value of the delivery option is a function of the probability of switch (greater near switch point) and also of the relative payoff in case of switch (homogeneity of the basket, graphically rappresented by the slope of the price/yield relationship). 14
- 16. Basis Basics: gross basis The gross basis in bond future is defined as: Basis (i ) = S (i ) − CF (i ) * F and it represents the difference between the spot price and the future implied forward price for bond i. The basis can be decomposed (by adding and subtracting the forward price Fwd(i)) into: Basis (i ) = S (i ) − Fwd (i ) + Fwd (i ) − CF (i ) * F Basis (i ) = Carry (i ) + Netbasis (i ) At delivery, basis will converge to the net basis. Basisdelivery (i ) = S (i ) − CFi * Fsettlement and it will be zero for the CTD bond and equal to the difference between the spot price and the converted price for each other bond, as to say a measure of the expensiveness to deliver it. 15
- 17. Basis Basics: net basis Net basis or basis net of carry BNOC is defined as basis net of carry or, as seen before: BNOC (i ) = S (i ) − CF (i ) * F − [S (i ) − Fwd (i )] BNOC (i ) = Fwd (i ) − CFi * F Since, as we have seen, the delivery option value has been defined as: Fwd t (ctd ) delivery option value (DOV) = α = − Ft CF (ctd ) the BNOC is an approximation of the delivery option value. netbasis( BNOC ) = Fwd t (ctd ) − Ft * CF (ctd ) = α * CF (ctd ) It is a pure option value only for the cheapest to deliver bond; for the other deliverable it is a mix of delivery option value and distance of the bond forward price from the CTD forward (a measure of expensiveness). 16
- 18. Basis Basics: gross and net basis in numbers The NET BASIS is the pure option value only for the CTD bond only, for all other bonds it is a mix of delivery option and amount by which the issue is expensive to deliver. Spot Price Yield Fwd Price Fwd Yld CF Fwd/CF Gross Basis Carry=S‐Fwd Net Basis Net basis at delivery CTD Basket #NAME? #VALUE! NA NA BTPS 5 1/2 09/01/22 103.4 5.1249 102.101 5.29 96.96880 105.29259 1.68 1.30 0.380687 0.329143 BTPS 4 3/4 08/01/23 97.1 5.1646 96.000 5.31 90.90224 105.60789 1.74 1.10 0.643486 0.595166 BTPS 5 03/01/22 100.3 5.0208 99.121 5.18 93.57107 105.93164 2.14 1.18 0.965312 0.915574 BTPS 4 3/4 09/01/21 99.36 4.8968 98.242 5.06 92.15330 106.60693 2.69 1.12 1.572989 1.524004 BTPS 3 3/4 08/01/21 92.44 4.8612 91.577 5.02 85.55955 107.03342 2.69 0.86 1.825343 1.779863 BTPS 5 1/2 11/01/22 102.96 5.1859 101.673 5.35 96.87481 104.95316 1.34 1.29 0.051494 0.000000 0.3 BTP SEP22 NET BASIS IN UKZ2 (CTD) 0.25 0.2 0.15 Net basis at delivery = 0 for ctd bond 0.1 and equal to = S – cf *F for others. Can 0.05 be thought as the cost of the option to 0 exchange the bond for the CTD. 6/4/2012 6/11/2012 6/18/2012 6/25/2012 7/2/2012 7/9/2012 7/16/2012 7/23/2012 7/30/2012 8/6/2012 8/13/2012 8/20/2012 8/27/2012 -0.05 -0.1 (payout of a call on S with strike CF*F) -0.15 -0.2 17
- 19. Basis Basics: net basis and IRR As we have seen, during the life of the contract, the CTD bond will maximize the short profit: F * CFi + AI i ,del − (Fwd i + AI i ,del ) F * CFi − Fwd i − netbasis Π= = = (S i + AI i ,t ) S i + AI i ,t Si + AI i ,t minimizing a sort of % net basis. If we define the implied repo rate (IRR) as the rate of return of a cash & carry strategy with delivery of the bond into the future, then the CTD is the bond which minimizes the difference between the implied repo rate (IRR) and the actual repo rate: The IRR rate will be lower than the ⎛ Invoiceprice − Purchaseprice ⎞ 360 IRR = ⎜ ⎜ ⎟* ⎟ gg corresponding actual repo rate to take in ⎝ Purchaseprice ⎠ account possible change in CTD bond, in the same way as the future price is lower It is possible to show that: than the converted forward. ⎡ Fwd t (ctd ) − Ft * CF (ctd ) ⎤ Min (ActualRepo - IRR) = Min ⎢ ⎥ ⎣ Ps + Ai ⎦ To identify the CTD, minimize the difference between IRR and actualr repo is equivalent to minimize a sort of %net basis. 18
- 20. Basis Basics: conversion factor bias and ctd problem If the conversion factor invoicing system were working properly, all the bonds in the basket would be equally economic to deliver (S(i) = CF(i)*F and future price would equal 100). Unfortunately this is true only when: the yield of curve is flat and equal to the notional coupon (6%) Any time we are away from this ideal situation, we will have one or more cheapest to deliver securities. All the cheapest to deliver optionality derives from a fault* in the invoicing system and the CTD phenomenon can be traced mainly to a bias associated with the mathematics of the conversion factor.*To overcome the misfunctionality of the conversion factor system in 2006 (Oviedo, “Improving the design of Treasury-Bond future contract”) anew system has been proposed in literature TRUE NOTIONAL BOND SYSTEM, which would makes all the deliverable bonds equal for any levelof flat curve, while in the CFS this is achieved only at a specific level of yield equal to the notional coupon. 19
- 21. Basis Basics: conversion factor bias and CTD problem The CTD bond at expiry will minimize the ratio S(i)/CF(i). Since CF(i) is approximately the price at delivery of bond i on a flat yield curve at 6%, we can rewrite the ratio: S i ( y ) Si ( y ) ⎡ Si (6%) − ( y − 6%) * MDi + 0.5 * ( y − 6%) 2 * CVXTYi ⎤ Taylor approximation formula = =⎢ ⎥ for bond price. CFi Si (6%) ⎣ Si (6%) ⎦ Therefore which bond will be deliverable, depends on their relative sensitivities to yield change as expressed by modified duration and convexity. ⎛ Si ( y ) ⎞ ⎛ − ( y − 6%) * MDi + 0.5 * ( y − 6%) 2 * CVXTYi ⎞ ⎜ CF ⎟ = min⎜ min⎜ ⎟ ⎜ ⎟ ⎟ ⎝ i ⎠ ⎝ S i (6%) ⎠ In general, when yield are below 6% the cheapest to deliver will be the lowest duration bond and vice versa when yields are above 6% the cheapest to delver bond will be the higher duration bond. For similar duration bond the cheapest to deliver bond will be the less convex bond. 20
- 22. Basis Basics: conversion factor bias and CTD problemBelow, a graphical representation of the conversion bias and CTD change with 2 bonds ofdifferent and equal duration. Before delivery the future will track the bond with lower converted forward price: the long will receive always the worst performing bond in the deliverable basket. This gives it a NEGATIVE CONVEXITY feature compared to the deliverable bonds. The price of the convexity is the delivery option value. 21
- 23. Basis Basics: conversion factor bias and CTD problem In chart below, we chart the forward converted price for Btp deliverables in Dec12 contract for simulated forward yield with the corresponding net basis at delivery to identify the possible CTD switch. Considering only parallel shift there is one switch point around 5.85% in yield (ref Aug21) 109 2.5 107 Converted prices for deliverable IKZ2 Net basis at delivery for IKZ2 105 2 8/1/2023 8/1/2023 103 9/1/2022 9/1/2022 11/1/2022FWD(i)/Cf(i) 101 11/1/2022 1.5 99 97 6% switch point 1 95 93 0.5 91 89 0 4.58 4.68 4.78 4.88 4.98 5.08 5.18 5.28 5.38 5.48 5.58 5.68 5.78 5.88 5.98 6.08 6.18 6.28 6.38 6.48 6.58 3.04 3.24 3.44 3.64 3.84 4.04 4.24 4.44 4.64 4.84 5.04 5.24 5.44 5.64 5.84 6.04 6.24 6.44 6.64 6.84 yield level 22
- 24. Basis Basics: option delivery valueThe value of the delivery option depends on the probability and by the outcomes of a CTD change (nearness to switch point and difference in slope of the 2 curves). Therefore the option value depends on: yield change slope change unanticipated new issues. Yield change. As seen, the CTD bond is related to the level of yields. It will tend to be the lowest duration bond for yield lower than 6% and vice versa. It will generally, be the least convex bond. The value of the delivery option will depend on the yield volatility and on bonds different sensitivities to yield changes (homogeneity of the basket). Yield slope. We can distinguish 2 kind of slope move with opposite effect the delivery option. “Systemic move”: generally, as yield rise curve flattens and vice versa. This kind of move has the effect to reduce the option value by reducing the switch probability. We can appreciate how this happens from chart in next slide. 23
- 25. Basis Basics: option delivery value Systemic move: as yield come down on higher duration bond, curve steepens and lower duration bond outperform, shifting its curve from LD to LD’. The switch point shift to the left decreasing the slope of the basis. Final Future Systemic slope move Switch point 2 reduce option value byInitial Future moving away switch points. Switch point 1 Due to correlation between slope and level a move in yield will reduce the change in the net basis, we would have otherwise. 24
- 26. Basis Basics: option delivery value As an example, in chart below, we plot net basis for Short Btp future dec12, at delivery in 2 hypothesis : 1. parallel shift only; 2. parallel shift and slope move using historical beta (for opportunity we choose the shortest bond as benchmark) 4 Net basis with parallel shift only 2/1/20153.5 8/1/2015 4/15/2015 6/15/2015 2.5 3 11/1/2015 3/1/2015 Net basis with slope and shift move 2/1/2015 7/15/2015 8/1/20152.5 4/15/2015 2 6/15/2015 11/1/2015 2 3/1/2015 7/15/20151.5 1.5 1 10.5 0 0.5 0.56 0.91 1.26 1.61 1.96 2.31 2.66 3.01 3.36 3.71 4.06 4.41 4.76 5.11 5.46 5.81 6.16 6.51 6.86 7.21 7.56 0 0.56 0.91 1.26 1.61 1.96 2.31 2.66 3.01 3.36 3.71 4.06 4.41 4.76 5.11 5.46 5.81 6.16 6.51 6.86 7.21 7.56 Considering parallel shift only we get 3 switch points, which get reduced to 2, if we consider shift and correlated slope move. 25
- 27. Basis Basics: option delivery value Unsystemic move: at constant yield a steepening of the curve (LD from LD’ higher performance of lowduration bond) shift the switch point from S to S1 decreasing the basis. Vice versa a flattening of the curve(shift of the initial curve from LD to LD’’) shifts the switch point from S to S2 increasing the basis. At constant yield on the high duration bond, the steepening reduces the basis and the flattening increases it. Pure slope movement will increase the volatitlity of switch points and the value of the delivery option 26
- 28. Basis Basics: option delivery valueThe announcement of issuance of new bonds, which will become cheapest to deliver represents a sourceof risk for basis trading. This is likely to happen, when rates are trading near the coupon rate.As an example, below, what happened at the basis of Mar20 in the June2011 contract when the issuanceof Sep21 was announced (18° February 2011). 0.6 CTD net basis in IKH2 on new issuance 0.5 0.4 0.3 0.2 Announcement 0.1 0 1/3/2011 1/5/2011 1/7/2011 1/9/2011 1/11/2011 1/13/2011 1/15/2011 1/17/2011 1/19/2011 1/21/2011 1/23/2011 1/25/2011 1/27/2011 1/29/2011 1/31/2011 2/2/2011 2/4/2011 2/6/2011 2/8/2011 2/10/2011 2/12/2011 2/14/2011 2/16/2011 2/18/2011 2/20/2011 2/22/2011 2/24/2011 2/26/2011 2/28/2011 3/2/2011 3/4/2011 3/6/2011 3/8/2011 27
- 29. Agenda1 Basis basics: terminology and concepts2 Why back to the future? Optionality and new products3 Opportunities and challenges 28
- 30. New products From the beginning of the European crisis, to encounter hedging/downloading needs the available bond futures have increased notably with: Eurex adding three new bond future on the Italian curve: short (Oct 2010), medium (Sep 2011) and long end (Sep 2009) and a long end bond future on the Oat market (Apr 2012); Meff adding a long end bond future on the Spanish government market (May 2012). Unfortunately not very successfully. Below, you can find the main features of the contract specification for the Btp futures. Undelyings Notional short-, mid- and long-term debt instruments issued by the Republic of Italy with a remaining term of 2 to 3.25 years (short-term), respectively 4.5 to 6 years (mid-term) and 8.5 to 11 years (long-term) with a notional coupon of 6 percent. Mid- and long-term debt instruments must have an original maturity of no longer than 16 years. Settlement A delivery obligation arising out of a short position may only be fulfilled by the delivery of certain debt securities issued by the Republic of Italy with a remaining term of respectively 2 to 3.25 years (short-term), 4.5 to 6 years (mid-term) and 8.5 to 11 years (long-term) and an original maturity of no longer than 16 years. Such debt securitiesDelivery must have a minimum issue amount of EUR 5 billion and a nominal fixed payment.basket freeze Starting with the contract month of June 2012, debt securities of the Republic of Italy have to possess a minimum(Jan2012) issuance volume of EUR 5 billion no later than 10 exchange days prior to the last trading day of the current front month, otherwise, they shall not be deliverable until the delivery day of the current front month contract. Last trading day Two exchange days prior to the delivery day of the relevant maturity month. End of trading for the maturing delivery month is 12:30 CET Delivery day The tenth calendar day of the respective quarterly month, if this day is an exchange day; otherwise, the exchange day immediately succeeding that day. 29
- 31. Optionality is backOptionality in the European bond future market is related only to the possible changes in the CTD. This option becomes more valuable when:I. yield and yield slope volatility increases;II. level of rates nears the notional coupon of the theoretical bond underlying;III. forward volatility increases.All of these factors are and have been in place lately:I. credit risk has increased yield volatility, while institutional intervention (ltro, non standard intervention talking) has increased slope volatility by interfering with the curve shape (a comparison of net basis for Btp and Bund future in chart1).II. Btp rates have often reached the 6% nominal coupon of the bond future, while Liffe in October 2011 has lowered the notional coupon from 6% to 4% (Gilt net basis behaviour in chart 2).III. forward price is a function of both spot price and repo rate. With the repo rate stuck the volatility of the forward rate has increased. 30
- 32. Optionality is back: credit risk and institutionalinterventionNet basis in Btp future has become more valuable due to:I. higher credit risk: increase in yield volatility and in the level of rates, reaching the 6% notional coupon of the bond future or the “inflection point”;II. institutional intervention has affected market in two ways: increasing yield slope volatility and lowering the correlation between repo and long rates thereby augmenting the forward volatility 1.4 1.2 Net basis in 10y Btp and Bund from 2009 onwards ctd_btp_net_basis 1 ctd_bund_net_basis 0.8 0.6 0.4 0.2 0 29-Nov-09 29-Nov-10 29-Nov-11 29-Jan-10 29-Jun-10 29-Jul-10 29-Jan-11 29-Jun-11 29-Jul-11 29-Jan-12 29-Jun-12 29-Jul-12 29-Sep-09 29-Oct-09 29-Dec-09 28-Feb-10 29-Apr-10 29-Mar-10 29-May-10 29-Aug-10 29-Sep-10 29-Oct-10 29-Dec-10 28-Feb-11 29-Apr-11 29-Mar-11 29-May-11 29-Aug-11 29-Sep-11 29-Oct-11 29-Dec-11 29-Feb-12 29-Apr-12 29-Mar-12 29-May-12 29-Aug-12 -0.2 -0.4 -0.6 31
- 33. Optionality is back: credit risk and institutionalinterventionForward prices are a function of spot price and repo rate: F = [ S + ai(t) ] * ( 1 + r(T-t) ) - ai(T)An increase in rates has the effect: to lower the forward price via a reduction in spot price; to increase the forward price via an increase of the funding cost. Even if, in the short term the two rates (repo and bond yield) respond to different forces, they showed a positive correlation in the past R^=0.43%. With repo stuck, independent by market forces correlation has gone down, and forward volatility has increased*. Application to Pf formula of the following Var ( xy ) = V ( x) E ( y ) 2 + V ( y ) E ( x) 2 + 2 E ( x) E ( y )Cov( x, y ) + V ( x)V ( y ) + Cov( x, y ) 2 With Cov(X,Y) going from negative to zero 32
- 34. Optionality is back: nearing inflection pointIn October 2011 Liffe has lowered the Gilt notional coupon from 6% to 4%:I. to reckon a lower long term level of interest rates;II. to give back duration to the future. The wide distance between notional coupon and the actual level of rates have made the shortest bond CTD for six consecutive rolls with a effective shortening of the future duration. 2 Net basis in 10y Gilt from Sep 2009 onwards 1.5 gilt_net_basis 1 Same ctd for 6 consecutive rolls 0.5 0 25-Nov-09 22-Nov-10 15-Nov-11 2-Jul-10 27-Jun-11 26-Jan-12 21-Jun-12 26-Jul-12 20-Oct-09 31-Dec-10 5-Feb-10 15-Mar-10 22-Apr-10 28-May-10 6-Aug-10 13-Sep-10 18-Oct-10 28-Dec-10 1-Feb-11 9-Mar-11 13-Apr-11 20-May-11 1-Aug-11 6-Sep-11 11-Oct-11 21-Dec-11 5-Mar-12 11-Apr-12 16-May-12 31-Aug-12 -0.5 -1 33
- 35. Agenda1 Basis basics: terminology and concepts2 Why back to the future? Optionality and new products3 Opportunities and challenges 34
- 36. Opportunities & challenges: how to value deliveryoptionThe first challenge is to be able to price the delivery option. At this aim 4 steps are necessary. Have a precise idea of deliverable bonds forward yield distribution and of the way to model it (1-2 factor model). Calculate for each state of the world the ctd and the implied future price. Calculate the value of the net basis for each deliverable in each state of the world. Average each state of the world net basis by its risk neutral probability. This is important both: for pricing (evaluate future richness/cheapness versus theoretical) and for hedging. If we do not consider the delivery option each time the CTD changes we have a jump in future DV01 (pictures below). 35

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