[1] Pricing and Hedging Motivation

Transcription

1 [] Prcng and Hedgng Motvaton Fxed-ncome products can pay ether Fxed cash-flows (e.g., fxed-rate Treasury coupon bond) Random cash-flows: depend on the future evoluton of nterest rates (e.g., floatng rate note) or other varables (prepayment rate on a mortgage pool) Objectve for ths lecture Hedge the value of a portfolo of fxed cash-flows Valuaton and hedgng of random cash-flow s a somewhat more complex task Leave t for later

2 Prcng and Hedgng Notaton B(t,T) : prce at date t of a unt dscount bond payng off $ at date T («dscount factor») R a (t,θ) : zero coupon rate or pure dscount rate, or yeld-to-maturty on a zero-coupon bond wth maturty date t + θ B(t,t + ") = (+ R a (t,")) " R(t,θ) : contnuously compounded pure dscount rate wth maturty t + θ: B(t,t + ") = exp (#" $ R(t,") ) Equvalently, R(t,") = # ln( B(t,t + ")) "

3 The value at date t of a bond payng cash-flows F() s gven by: m " = V (t) = F B(t,t + ) = Prcng and Hedgng Prcng Certan Cash-Flows m " = Example: $00 bond wth a 5 coupon F [ + R a (t,)] Therefore, the value s a functon of tme and nterest rates Value changes as nterest rates fluctuate

4 Example Assume today a flat structure of nterest rates R a (0,θ) = 0 for all θ Bond wth 0 years maturty, coupon rate = 0 Prce: $00 If the term structure shfts up to 2 (parallel shft) Bond prce : $88.7 Captal loss: $.3, or.3 Implcatons Prcng and Hedgng Interest Rate Rsk Hedgng nterest rate rsk s economcally mportant Hedgng nterest rate rsk s a complex task: 0 rsk factors n ths example!

5 Basc prncple: attempt to reduce as much as possble the dmensonalty of the problem Frst step: duraton hedgng Consder only one rsk factor Assume a flat yeld curve Assume only small changes n the rsk factor Beyond duraton Relax the assumpton of small nterest rate changes Relax the assumpton of a flat yeld curve Relax the assumpton of parallel shfts Prcng and Hedgng Hedgng Prncples

6 Use a proxy for the term structure: the yeld to maturty of the bond It s an average of the whole term structure If the term structure s flat, t s the term structure We wll study the senstvty of the prce of the bond to changes n yeld: Change n TS means change n yeld Prce of the bond: (actually y/2) V = m " = Duraton Hedgng Duraton F ( + y)

7 Interest rate rsk Rates change from y to y+dy dv V Duraton Hedgng Senstvty dy s a small varaton, say bass pont (e.g., from 5 to 5.0) Change n bond value dv followng change n rate value dy dv = V ( y + dy)! V ( y) For small changes, can be approxmated by Relatve varaton " V '(y) V (y) dv! V '( y) dy dy = Sens # dy

8 The relatve senstvty, denoted as Sens, s the partal dervatve of the bond prce wth respect to yeld, dvded by the bond prce Formally Sens = V '(y) V (y) = Duraton Hedgng Duraton " + y m # = V (y) F + y ( ) In plan Englsh: tells you how much relatve change n prce follows a gven small change n yeld mpact It s always a negatve number Bond prce goes down when yeld goes up

9 Duraton Hedgng Termnology The opposte of the senstvty Sens s referred to as «Modfed Duraton» The absolute senstvty V (y) = Sens x V(y) s referred to as «$ Duraton» Example: Bond wth 0 year maturty Coupon rate: 6 Quoted at 5 yeld or equvalently $07.72 prce The $ Duraton of ths bond s and the modfed duraton s Interpretaton Rate goes up by 0. (0 bass ponts) Absolute P&L: x.0. = $ Relatve P&L: 7.52x0. = 0.752

10 Duraton Hedgng Duraton Defnton of Duraton D: Also known as Macaulay duraton It s a measure of average maturty Relatonshp wth senstvty and modfed duraton:

11 Tme of Cash Flow () Cash Flow F w = V! F ( + y) Duraton Hedgng Example! w Example: m = 0, c = 5.34, y = D m = # " = w! Total

12 Duraton of a zero coupon bond s Equal to maturty For a gven maturty and yeld, duraton ncreases as coupon rate Decreases For a gven coupon rate and yeld, duraton ncreases as maturty Increases For a gven maturty and coupon rate, duraton ncreases as yeld rate Decreases Duraton Hedgng Propertes of Duraton

13 Duraton Hedgng Propertes of Duraton - Lnearty Duraton of a portfolo of n bonds where w s the weght of bond n the portfolo, and: Ths s true f and only f all bonds have same yeld,.e., f yeld curve s flat If that s the case, n order to attan a gven duraton we only need two bonds

14 Prncple: mmunze the value of a bond portfolo wth respect to changes n yeld Denote by P the value of the portfolo Denote by H the value of the hedgng nstrument Hedgng nstrument may be Bond Swap Future Opton Assume a flat yeld curve Duraton Hedgng Hedgng

15 Changes n value Portfolo Hedgng nstrument dp " P'(y)dy Duraton Hedgng Hedgng dh! H '( y) dy Strategy: hold q unts of the hedgng nstrument so that dp + qdh = ( qh'(y) + P'(y) )dy = 0 Soluton q = " P'(y) H'(y) = "P #Sens P H #Sens H = "P # Dur P H # Dur H

16 Duraton Hedgng Hedgng Example: At date t, a portfolo P has a prce $328635, a 5.43 yeld and a 7.08 duraton Hedgng nstrument, a bond, has a prce $8.786, a yeld and a duraton Hedgng strategy nvolves a buyng/sellng a number of bonds q = (328635x7.08)/(8.786x5.748) = 342 If you hold the portfolo P, you want to sell 342 unts of bonds

17 Duraton Hedgng Lmts Duraton hedgng s Very smple Bult on very restrctve assumptons Assumpton : small changes n yeld The value of the portfolo could be approxmated by ts frst order Taylor expanson OK when changes n yeld are small, not OK otherwse Ths s why the hedge portfolo should be re-adjusted reasonably often Assumpton 2: the yeld curve s flat at the orgn In partcular we suppose that all bonds have the same yeld rate In other words, the nterest rate rsk s smply consdered as a rsk on the general level of nterest rates Assumpton 3: the yeld curve s flat at each pont n tme In other words, we have assumed that the yeld curve s affected only by a parallel shft

18 [2] Accountng for Larger Changes n Yeld Duraton and Interest Rate Rsk

19 Accountng for Larger Changes n Yeld Convexty Relatonshp between prce and yeld s convex: Taylor approxmaton: Relatve change "V V # V '(y) V (y) "y + 2 V"(y) V (y) ("y) 2 = Sens $ "y + Conv $ ("y) 2 2 Conv s relatve convexty,.e., the second dervatve of value wth respect to yeld dvded by value

20 Accountng for Larger Changes n Yeld Convexty and $ Convexty (Relatve) convexty s ( ) ( ) = Conv = V" y V y m " ( +)F ( + y) 2 ( + y) = V y ( ) $ Convexty = V (y) = Conv x V(y) Example (back to prevous) 0 year maturty bond, wth a 6 annual coupon rate, a 7.36 modfed duraton, a 6974 $ convexty and whch sells at par Case 2: yelds go from 6 to 8 Second order approxmaton to change n prce Fnd: (6974.(0.02)²/2) = -$3.33 Exact soluton s -$3.42 and frst order approxmaton s -$4.72

21 Accountng for Larger Changes n Yeld Propertes of Convexty - Lnearty Convexty of a portfolo of n bonds where w s the weght of bond n the portfolo, and: Ths s true f and only f all bonds have same yeld,.e., f yeld curve s flat

22 Accountng for Larger Changes n Yeld Duraton-Convexty Hedgng Prncple: mmunze the value of a bond portfolo wth respect to changes n yeld Denote by P the value of the portfolo Denote by H and H 2 the value of two hedgng nstruments Needs two hedgng nstrument because want to hedge one rsk factor (stll assume a flat yeld curve) up to the second order Changes n value Portfolo Hedgng nstruments dp " P'(y)dy + P''(y) dy 2 2 # dh " H '(y)dy + 2 H ''(y)dy 2 $ dh 2 " H 2 '(y)dy + 2 H 2''(y)dy 2 &

23 Accountng for Larger Changes n Yeld Duraton-Convexty Hedgng Strategy: hold q (resp. q 2 ) unts of the frst (resp. second) hedgng nstrument so that dp + q dh! dh + q2! 2 = Soluton (under the assumpton of unque dy parallel shfts) # P'( y) + qh '( y) + q2h 2'( y) = 0 "! P''( y) + qh ''( y) + q2h 2''( y) = 0 Or (under the assumpton of a unque y flat yeld curve) # q H (y)dur + q 2 H 2 (y)dur 2 = "P(y)Dur P $ q H (y)conv + q 2 H 2 (y)conv 2 = "P(y)Conv P 0

24 Accountng for a Non Flat Yeld Curve Allowng for a Term Structure Problem wth the prevous method: we have assumed a unque yeld for all nstrument,.e., we have assumed a flat yeld curve We now relax ths smplfyng assumpton and consder 3 potentally dfferent yelds y, y, y 2 On the other hand, we mantan the assumpton of parallel shfts,.e., we assume dy = dy = dy 2 We are stll lookng for q and q 2 such that dp + q dh! dh + q2! 2 = 0

25 Accountng for a Non Flat Yeld Curve Accountng for a Non Flat Yeld Curve Soluton (under the assumpton of unque dy parallel shfts) " P'(y) + q H '(y ) + q 2 H 2 '(y 2 ) = 0 # $ P''(y) + q H ''(y ) + q 2 H 2 ''(y 2 ) = 0 Or (relaxng the assumpton of a flat yeld curve) # q H (y )Sens + q 2 H 2 (y 2 )Sens 2 = "P(y)Sens P $ q H (y )Conv + q 2 H 2 (y 2 )Conv 2 = "P(y)Conv P Just replace (Macaulay) duraton by senstvty or modfed duraton n the frst equaton

26 Accountng for a Non Flat Yeld Curve Tme for an Example! Portfolo at date t Prce P = $ Yeld y = 5.43 Sens = 6.76 Conv = Hedgng nstrument Prce H = $ Yeld y = Sens = 8.83 Conv = Hedgng nstrument 2: Prce H 2 = $ Yeld y 2 = Sens 2 = Conv 2 = 0.68

27 Accountng for a Non Flat Yeld Curve Tme for an Example! Optmal quanttes q and q 2 of each hedgng nstrument are gven by # "! q q $ 8.83$ q2 $ 2.704$ = $ 6.76 $ 99.08$ q $ 0.68$ = $ Or q = -305 and q 2 = 40 2 If you hold the portfolo, you should sell 305 unts of H and buy 40 unts of H 2

28 Accountng for Non Parallel Shfts Accountng for Changes n Shape of the TS Bad news s: not only the yeld curve s not flat, but also t changes shape! Aforementoned methods do not allow to account for such deformatons Addtonal rsk factors One has to regroup dfferent rsk factors to reduce the dmensonalty of the problem: e.g., a short, medum and long maturty factors Systematc approach: factor analyss on hstorcal data has shed some lght on the dynamcs of the yeld curve 3 factors account for more than 90 of the varatons Level factor Slope factor Curvature factor

29 Accountng for Non Parallel Shfts Accountng for Non Parallel Shts To properly account for the changes n the yeld curve, one has to get back to pure dscount rates m " = V (t) = F()B(t,t + ) = m m " = F() [ + R a (t,)] Or, usng contnuously compounded rates V (t) = " F()B(t,t + ) = F()exp # $ R(t,) = m " = [ ]

30 Accountng for Non Parallel Shfts Nelson-Segel Model The challenge s that we are now facng m rsk factors Reduce the dmensonalty of the problem by wrtng dscount rates as a functon of 3 parameters One classc model s Nelson and Segel s & ' exp( ') ( )# R( 0, ) = * 0 + * $! + * 2 ) ( " & $ ' exp( ') ( ) ) ( ' exp( ') wth R(0,θ): pure dscount rate wth maturty θ β 0 : level factor β : slope factor β 2 : curvature factor τ : fxed scalng parameter Hedgng prncple: mmunze the portfolo wth respect to changes n the value of the 3 parameters # ( )! "

31 Accountng for Non Parallel Shfts Nelson Segel Model Mechancs of the model: changes n beta parameters mply changes n dscount rates, whch n turn mply changes n prces One may easly compute the senstvty (partal dervatve) of R(0,θ) wth respect to each parameter beta (see next slde) Very consstent wth factor analyss of nterest rates n the sense that they can be regarded as level, slope and curvature factors, respectvely

32 Accountng for Non Parallel Shfts Nelson Segel.2 Senstvty of rates béta 0 béta béta Maturty of rates

33 Accountng for Non Parallel Shfts Nelson Segel Model Let us consder at date t=0 a bond wth prce P delverng the future cash-flows F The prce s gven by P 0 = F " B(0,# ) = F " e $# R(0,# )!!!! "!!! # $ & ' ( ) * = +,, = & ' ( ) * + + = +,, = = +,, = ) (0, ) (0, 0 ) (0, ) / exp( / ) / exp( / ) / exp( R R R F e P S F e P S F e P S /. /. /.. 0 /. / Senstvtes of the bond prce wth respect to each beta parameter are

34 Accountng for Non Parallel Shfts Example At date t=0, parameters are estmated (ftted) to be Beta 0 Beta Beta 2 Scale parameter Senstvtes of 3 bonds wth respect to each beta parameter, as well as that of the portfolo nvested n the 3 bonds, are Maturty Coupon Prce S0 S S2 Bond 2 ans $ Bond 2 7 ans $ Bond 3 0 ans $ Portfolo

35 Accountng for Non Parallel Shfts Hedgng wth Nelson Segel Prncple: mmunze the value of a bond portfolo wth respect to changes n parameters of the model Denote by P the value of the portfolo Denote by H, H 2 and H 3 the value of three hedgng nstruments Needs 3 hedgng nstruments because want to hedge 3 rsk factors (up to the frst order) Can also mpose dollar neutralty constrant q 0 H 0 + q H + q 2 H 2 + q 3 H 3 + q 4 H 4 = - P (need a 4 th nstrument for that) Formally, look for q, q 2 and q 3 such that!!! "!!! # $ = = = & & & & & & & & & & & & G q G q G q P G q G q G q P G q G q G q P

36 Beyond Duraton General Comments Whatever the method used, duraton, modfed duraton, convexty and senstvty to Nelson and Segel parameters are tme-varyng quanttes Gven that ther value drectly mpact the quanttes of hedgng nstruments, hedgng strateges are dynamc strateges Re-balancement should occur to adjust the hedgng portfolo so that t reflects the current market condtons In the context of Nelson and Segel model, one may elect to partally hedge the portfolo wth respect to some beta parameters Ths s a way to speculate on changes n some factors; t s known as «sem-hedgng» strateges For example, a portfolo bond holder who antcpates a decrease n nterest rates may choose to hedge wth respect to parameters beta and beta 2 (slope and curvature factors) whle remanng voluntarly exposed to a change n the beta 0 parameter (level factor)