Essay --

Describe the role and functions of the PCI security standards council â€Å"The PCI Security Standards Council is an organization created by the major credit card companies in an effort to better protect credit card holder data.† (Rouse, 2012) The council was formed in response to the increase in data security breaches that not only affected customers but also credit card companies cost. With PCI Security Standards Council being a open global forum, The five founding credit card companies – American Express, Discover Financial Services, JCB International, MasterCard Worldwide and Visa Inc. – are responsible for carrying out the organization’s work. Functions of the council include coming up with a framework of specifications, measurements, and support resources to help organizations ensure the safe handling of cardholder information at every step. This is done by managing the Payment Card Industry Security Standard (PCI DSS) and the Payment Application Data Security Standard. Identify/describe key requirements for data security standards The key requirements for the Data Security ...

Accrual Swaps

ACCRUAL SWAPS AND RANGE NOTES PATRICK S. HAGAN BLOOMBERG LP 499 PARK AVENUE NEW YORK, NY 10022 [email  protected] NET 212-893-4231 Abstract. Here we present the standard methodology for pricing accrual swaps, range notes, and callable accrual swaps and range notes. Key words. range notes, time swaps, accrual notes 1. Introduction. 1. 1. Notation. In our notation today is always t = 0, and (1. 1a) D(T ) = today’s discount factor for maturity T. For any date t in the future, let Z(t; T ) be the value of $1 to be delivered at a later date T : (1. 1b) Z(t; T ) = zero coupon bond, maturity T , as seen at t. These discount factors and zero coupon bonds are the ones obtained from the currency’s swap curve. Clearly D(T ) = Z(0; T ). We use distinct notation for discount factors and zero coupon bonds to remind ourselves that discount factors D(T ) are not random; we can always obtain the current discount factors from the stripper. Zero coupon bonds Z(t; T ) are random, at least until time catches up to date t. Let (1. 2a) (1. 2b) These are de? ned via (1. 2c) D(T ) = e? T 0 f0 (T ) = today’s instantaneous forward rate for date T, f (t; T ) = instantaneous forward rate for date T , as seen at t. f0 (T 0 )dT 0 Z(t; T ) = e? T t f (t,T 0 )dT 0 . 1. 2. Accrual swaps (? xed). ?j t0 t1 t2 †¦ tj-1 tj †¦ tn-1 tn period j Coupon leg schedule Fixed coupon accrual swaps (aka time swaps) consist of a coupon leg swapped against a funding leg. Suppose that the agreed upon reference rate is, say, k month Libor. Let (1. 3) t0 < t1 < t2  ·  ·  · < tn? 1 < tn 1 Rfix Rmin Rmax L( ? ) Fig. 1. 1. Daily coupon rate be the schedule of the coupon leg, and let the nominal ? xed rate be Rf ix . Also let L(? st ) represent the k month Libor rate ? xed for the interval starting at ? st and ending at ? end (? st ) = ? t + k months. Then the coupon paid for period j is (1. 4a) where (1. 4b) and (1. 4c) ? j = #days ? st in the interval with Rmin ? L(? st ) ? Rmax . Mj ? j = cvg(tj? 1 , tj ) = day count fraction for tj? 1 to tj , Cj = ? j Rf ix ? j paid at tj , Here Mj is the total number of days in interval j, and Rmin ? L(? st ) ? Rmax is the agreed-upon accrual range. Said another way, each day ? st in the j th period contibutes the amount ? ?j Rf ix 1 if Rmin ? L(? st ) ? Rmax (1. 5) 0 otherwise Mj to the coupon paid on date tj . For a standard deal, the leg’s schedule is contructed like a standard swap schedule. The theoretical dates (aka nominal dates) are constructed monthly, quarterly, semi-annually, or annually (depending on the contract terms) backwards from the â€Å"theoretical end date. † Any odd coupon is a stub (short period) at the front, unless the contract explicitly states long ? rst, short last, or long last. The modi? ed following business day convention is used to obtain the actual dates tj from the theoretical dates. The coverage (day count fraction) is adjusted, that is, the day count fraction for period j is calculated from the actual dates tj? 1 and tj , not the theoretical dates. Also, L(? t ) is the ? xing that pertains to periods starting on date ? st , regardless of whether ? st is a good business day or not. I. e. , the rate L(? st ) set for a Friday start also pertains for the following Saturday and Sunday. Like all ? xed legs, there are many variants of these coupon legs. The only variations that do not make sense for coupon legs are â€Å"set-in-arrearsâ €  and â€Å"compounded. † There are three variants that occur relatively frequently: Floating rate accrual swaps. Minimal coupon accrual swaps. Floating rate accrual swaps are like ordinary accrual swaps except that at the start of each period, a ? ating rate is set, and this rate plus a margin is 2 used in place of the ? xed rate Rf ix . Minimal coupon accrual swaps receive one rate each day Libor sets within the range and a second, usually lower rate, when Libor sets outside the range ? j Mj ? Rf ix Rf loor if Rmin ? L(? st ) ? Rmax . otherwise (A standard accrual swap has Rf loor = 0. These deals are analyzed in Appendix B. Range notes. In the above deals, the funding leg is a standard ?oating leg plus a margin. A range note is a bond which pays the coupon leg on top of the principle repayments; there is no ? oating leg. For these deals, the counterparty’s credit-worthiness is a key concern. To determine the correct value of a range note, one needs to use an option adjusted spread (OAS) to re? ect the extra discounting re? ecting the counterparty’s credit spread, bond liquidity, etc. See section 3. Other indices. CMS and CMT accrual swaps. Accrual swaps are most commonly written using 1m, 3m, 6m, or 12m Libor for the reference rate L(? st ). However, some accrual swaps use swap or treasury rates, such as the 10y swap rate or the 10y treasury rate, for the reference rate L(? st ). These CMS or CMT accrual swaps are not analyzed here (yet). There is also no reason why the coupon cannot set on other widely published indices, such as 3m BMA rates, the FF index, or the OIN rates. These too will not be analyzed here. 2. Valuation. We value the coupon leg by replicating the payo? in terms of vanilla caps and ? oors. Consider the j th period of a coupon leg, and suppose the underlying indice is k-month Libor. Let L(? st ) be the k-month Libor rate which is ? xed for the period starting on date ? st and ending on ? end (? st ) = ? st +k months. The Libor rate will be ? xed on a date ? f ix , which is on or a few days before ? st , depending on currency. On this date, the value of the contibution from day ? st is clearly ? ? ? j Rf ix V (? f ix ; ? st ) = payo? = Z(? f ix ; tj ) Mj ? 0 if Rmin ? L(? st ) ? Rmax otherwise (2. 1) , where ? f ix the ? xing date for ? st . We value coupon j by replicating each day’s contribution in terms of vanilla caplets/? oorlets, and then summing over all days ? st in the period. Let Fdig (t; ? st , K) be the value at date t of a digital ? oorlet on the ? oating rate L(? st ) with strike K. If the Libor rate L(? st ) is at or below the strike K, the digital ? oorlet pays 1 unit of currency on the end date ? end (? st ) of the k-month interval. Otherwise the digital pays nothing. So on the ? xing date ? f ix the payo? is known to be ? 1 if L(? st ) ? K , (2. 2) Fdig (? f ix ; ? st , K) = Z(? f ix ; ? end ) 0 otherwise We can replicate the range note’s payo? for date ? st by going long and short digitals struck at Rmax and Rmin . This yields, (2. 3) (2. 4) ? j Rf ix [Fdig (? f ix ; ? st , Rmax ) ? Fdig (? f ix ; ? st , Rmin )] Mj ? ?j Rf ix 1 = Z(? f ix ; ? end ) 0 Mj 3 if Rmin ? L(? st ) ? Rmax . otherwise This is the same payo? as the range note, except that the digitals pay o? on ? end (? st ) instead of tj . 2. 1. Hedging considerations. Before ? ing the date mismatch, we note that digital ? oorlets are considered vanilla instruments. This is because they can be replicated to arbitrary accuracy by a bullish spread of ? oorlets. Let F (t, ? st , K) be the value on date t of a standard ? oorlet with strike K on the ? oating + rate L(? st ). This ? oorlet pays ? [K ? L(? st )] on the end date ? end (? st ) of the k-m onth interval. So on the ? xing date, the payo? is known to be (2. 5a) F (? f ix ; ? st , K) = ? [K ? L(? st )] Z(? f ix ; ? end ). + Here, ? is the day count fraction of the period ? st to ? end , (2. 5b) ? = cvg(? st , ? end ). 1 ? oors struck at K + 1 ? nd short the same number struck 2 The bullish spread is constructed by going long at K ? 1 ?. This yields the payo? 2 (2. 6) which goes to the digital payo? as ? > 0. Clearly the value of the digital ? oorlet is the limit as ? > 0 of (2. 7a) Fcen (t; ? st , K, ? ) = ? 1  © F (t; ? st , K + 1 ? ) ? F (t; ? st , K ? 1 ? ) . 2 2 ? 1  © F (? f ix ; ? st , K + 1 ? ) ? F (? f ix ; ? st , K ? 1 ? ) 2 2 ? ? ? ? 1 ? 1 = Z(? f ix ; ? end ) K + 1 ? ? L(? st ) 2 ? ? ? 0 if K ? 1 ? < L(? st ) < K + 1 ? , 2 2 if K + 1 ? < L(? st ) 2 if L(? st ) < K ? 1 ? 2 Thus the bullish spread, and its limit, the digitial ? orlet, are directly determined by the market prices of vanilla ? oors on L(? st ). Digital ? oorlets may develop an unbounded ? - risk as the ? xing date is approached. To avoid this di? culty, most ? rms book, price, and hedge digital options as bullish ? oorlet spreads. I. e. , they book and hedge the digital ? oorlet as if it were the spread in eq. 2. 7a with ? set to 5bps or 10bps, depending on the aggressiveness of the ? rm. Alternatively, some banks choose to super-replicate or sub-replicate the digital, by booking and hedging it as (2. 7b) or (2. 7c) Fsub (t; ? st , K, ? ) = 1 {F (t; ? st , K) ? F (t; ? st , K ? ?)} Fsup (t; ? st , K, ? ) = 1 {F (t; ? st , K + ? ) ? F (t; ? st , K)} depending on which side they own. One should price accrual swaps in accordance with a desk’s policy for using central- or super- and sub-replicating payo? s for other digital caplets and ? oorlets. 2. 2. Handling the date mismatch. We re-write the coupon leg’s contribution from day ? st as ? ?j Rf ix Z(? f ix ; tj ) ? V (? f ix ; ? st ) = Z(? f ix ; ? end ) Mj Z(? f ix ; ? end ) ? 0 4 (2. 8) if Rmin ? L(? st ) ? Rmax otherwise . f(t,T) L(? ) tj-1 ? tj ? end T Fig. 2. 1. Date mismatch is corrected assuming only parallel shifts in the forward curve The ratio Z(? ix ; tj )/Z(? f ix ; ? end ) is the manifestation of the date mismatch. To handle this mismatch, we approximate the ratio by assuming that the yield curve makes only parallel shifts over the relevent interval. See ?gure 2. 1. So suppose we are at date t0 . Then we assume that (2. 9a) Z(? f ix ; tj ) Z(t0 ; tj ) ? [L(? st )? Lf (t0 ,? st )](tj en d ) = e Z(? f ix ; ? end ) Z(t0 ; ? end ) Z(t0 ; tj ) = {1 + [L(? st ) ? Lf (t0 , ? st )](? end ? tj ) +  ·  ·  · } . Z(t0 ; ? end ) Z(t0 ; ? st ) ? Z(t0 ; ? end ) + bs(? st ), ? Z(t0 ; ? end ) Here (2. 9b) Lf (t0 , ? st ) ? is the forward rate for the k-month period starting at ? t , as seen at the current date t0 , bs(? st ) is the ? oating rate’s basis spread, and (2. 9c) ? = cvg(? st , ? end ), is the day count fraction for ? st to ? end . Since L(? st ) = Lf (? f ix , ? st ) represents the ? oating rate which is actually ? xed on the ? xing date ? ex , 2. 9a just assumes that any change in the yield curve between tj and ? end is the same as the change Lf (? f ix , ? st ) ? Lf (t0 , ? st ) in the reference rate between the observation date t0 , and the ? xing date ? f ix . See ? gure 2. 1. We actually use a slightly di? erent approximation, (2. 10a) where (2. 10b) ? = ? end ? tj . ? end ? ? st Z(? ix ; tj ) Z(t0 ; tj ) 1 + L(? st ) ? Z(? f ix ; ? end ) Z(t0 ; ? end ) 1 + Lf (t0 , ? st ) We prefer this approximation because it is the only linear approximation which accounts for the day count basis correctly, is exact for both ? st = tj? 1 and ? st = tj , and is centerred around the current forward value for the range note. 5 Rfix Rmin L0 Rmax L(? ) Fig. 2. 2. E? ective contribution from a single day ? , after accounting for the date mis-match. With this approximation, the payo? from day ? st is ? 1 + L(? st ) (2. 11a) V (? f ix ; ? ) = A(t0 , ? st )Z(? f ix ; ? end ) 0 as seen at date t0 . Here the e? ctive notional is (2. 11b) A(t0 , ? st ) = if Rmin ? L(? st ) ? Rmax otherwise 1 ? j Rf ix Z(t0 ; tj ) . Mj Z(t0 ; ? end ) 1 + Lf (t0 , ? st ) We can replicate this digital-linear-digital payo? by using a combination of two digital ? oorlets and two standard ? oorlets. Consider the combination (2. 12) V (t; ? st ) ? A(t0 , ? st ) {(1 + Rmax )Fdig (t, ? st ; Rmax ) ? (1 + ? Rmin )Fdig (t, ? st ; Rmin ) F (t, ? st ; Rmax ) + ? F (t, ? st ; Rmi n ). Setting t to the ? xing date ? f ix shows that this combination matches the contribution from day ? st in eq. 2. 11a. Therefore, this formula gives the value of the contribution of day ? t for all earlier dates t0 ? t ? ? f ix as well. Alternatively, one can replicate the payo? as close as one wishes by going long and short ? oorlet spreads centerred around Rmax and Rmin . Consider the portfolio (2. 13a) A(t0 , ? st )  © ? V (t; ? st , ? ) = a1 (? st )F (t; ? st , Rmax + 1 ? ) ? a2 (? st )F (t; ? st , Rmax ? 1 ? ) 2 2 ? 1 ? a3 (? st )F (t; ? st , Rmin + 2 ? )+ a4 (? st )F (t; ? st , Rmin ? 1 ? ) 2 a1 (? st ) = 1 + (Rmax ? 1 ? ), 2 a3 (? st ) = 1 + (Rmin ? 1 ? ), 2 ? ? a2 (? st ) = 1 + (Rmax + 1 ? ), 2 a4 (? st ) = 1 + (Rmin + 1 ? ). 2 with (2. 13b) (2. 13c) Setting t to ? ix yields (2. 14) ? V = A(t0 , ? st )Z(? f ix ; ? end ) 0 if L(? st ) < Rmin ? 1 ? 2 1 + L(? st ) if Rmin + 1 ? < L(? st ) < Rmax ? 1 ? , 2 2 ? 0 if Rmax + 1 ? < L(? st ) 2 6 with linear ramps between Rmin ? 1 ? < L(? st ) < Rmin + 1 ? and Rmax ? 1 ? < L(? st ) < Rmax + 1 ?. As 2 2 2 2 above, most banks would choose to use the ? oorlet spreads (with ? being 5bps or 10bps) instead of using the more troublesome digitals. For a bank insisting on using exact digital options, one can take ? to be 0. 5bps to replicate the digital accurately.. We now just need to sum over all days ? t in period j and all periods j in the coupon leg, (2. 15) Vcpn (t) = n X This formula replicates the value of the range note in terms of vanilla ? oorlets. These ? oorlet prices should be obtained directly from the marketplace using market quotes for the implied volatilities at the relevent strikes. Of course the centerred spreads could be replaced by super-replicating or sub-replicating ? oorlet spreads, bringing the pricing in line with the bank’s policies. Finally, we need to value the funding leg of the accrual swap. For most accrual swaps, the funding leg ? ? pays ? oating plus a margin. Let th e funding leg dates be t0 , t1 , . . , tn . Then the funding leg payments are (2. 16) f ? ? cvg(ti? 1 , ti )[Ri lt + mi ]  ¤ A(t0 , ? st )  ©? 1 + (Rmax ? 1 ? ) F (t; ? st , Rmax + 1 ? ) 2 2 j=1 ? st =tj? 1 +1 ?  ¤ ? 1 + (Rmax + 1 ? ) F (t; ? st , Rmax ? 1 ? ) 2 2 ?  ¤ ? 1 + (Rmin ? 1 ? ) F (t; ? st , Rmin + 1 ? ) 2 2 ?  ¤ ? + 1 + (Rmin + 1 ? ) F (t; ? st , Rmin ? 1 ? ) . 2 2 tj X ? paid at ti , i = 1, 2, †¦ , n, ? f ? ? where Ri lt is the ? oating rate’s ? xing for the period ti? 1 < t < ti , and the mi is the margin. The value of the funding leg is just n ? X i=1 (2. 17a) Vf und (t) = ? ? ? cvg(ti? 1 , ti )(ri + mi )Z(t; ti ), ? ? where, by de? ition, ri is the forward value of the ? oating rate for period ti? 1 < t < ti : (2. 17b) ri = ? ? Z(t; ti? 1 ) ? Z(t; ti ) true + bs0 . + bs0 = ri i i ? ? ? cvg(ti? 1 , ti )Z(t; ti ) true is the true (cash) rate. This sum Here bs0 is the basis spread for the funding leg’s ? oating rate, and ri i collapses t o n ? X i=1 (2. 18a) Vf und (t) = Z(t; t0 ) ? Z(t; tn ) + ? ? ? ? cvg(ti? 1 , ti )(bs0 +mi )Z(t; ti ). i If we include only the funding leg payments for i = i0 to n, the value is ? (2. 18b) ? Vf und (t) = Z(t; ti0 ? 1 ) ? Z(t; tn ) + ? n ? X ? ? ? cvg(ti? 1 , ti )(bs0 +mi )Z(t; ti ). i i=i0 2. 2. 1. Pricing notes. Caplet/? oorlet prices are normally quoted in terms of Black vols. Suppose that on date t, a ? oorlet with ? xing date tf ix , start date ? st , end date ? end , and strike K has an implied vol of ? imp (K) ? ? imp (? st , K). Then its market price is (2. 19a) F (t, ? st , K) = ? Z(t; ? end ) {KN (d1 ) ? L(t, ? )N (d2 )} , 7 where (2. 19b) Here (2. 19c) d1,2 = log K/L(t, ? st )  ± 1 ? 2 (K)(tf ix ? t) 2 imp , v ? imp (K) tf ix ? t Z(t; ? st ) ? Z(t; ? end ) + bs(? st ) ? Z(t; ? end ) L(t, ? st ) = is ? oorlet’s forward rate as seen at date t. Today’s ? oorlet value is simply (2. 20a) where (2. 20b) d1,2 = log K/L0 (? st )  ± 1 ? (K)tf ix 2 imp , v ? imp (K) tf ix D(? st ) ? D(? end ) + bs(? st ). ?D(? end ) ? j Rf ix D(tj ) 1 . Mj D(? end ) 1 + L0 (? st ) F (0, ? st , K) = ? D(? end ) {KN (d1 ) ? L0 (? )N (d2 )} , and where today’s forward Libor rate is (2. 20c) L0 (? st ) = To obtain today’s price of the accrual swap, note that the e? ective notional for period j is (2. 21) A(0, ? st ) = as seem today. See 2. 11b. Putting this together with 2. 13a shows that today’s price is Vcpn (0) ? Vf und (0), where (2. 22a) Vcpn (0) = n X ? j Rf ix D(tj ) j=1 Mj  ¤ ?  ¤ ? 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? [1 + L0 (? st )] ? t =tj? 1 +1  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? , ? [1 + L0 (? st )] tj X n ? X i=1 (2. 22b) Vf und (0) = D(t0 ) ? D(tn ) + ? ? ? ? cvg(ti? 1 , ti )(bs0 +mi )D(ti ). i Here B? are Black’s formula at strikes around the boundaries: (2. 22c) (2. 22d) with (2. 22e) K1,2 = Rmax  ± 1 ? , 2 K3,4 = Rmin  ± 1 ?. 2 B? (? st ) = K? N (d? ) ? L0 (? st )N (d? ) 1 2 d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix Calculating the sum of each day’s contribution is very tedious. Normally, one calculates each day’s contribution for the current period and two or three months afterward. After that, one usually replaces the sum over dates ? with an integral, and samples the contribution from dates ? one week apart for the next year, and one month apart for subsequent years. 8 3. Callable accrual swaps. A callable accrual swap is an accrual swap in which the party paying the coupon leg has the right to cancel on any coupon date after a lock-out period expires. For example, a 10NC3 with 5 business days notice can be called on any coupon date, starting on the third anniversary, provided the appropriate notice is given 5 days before the coupon date. We will value the accrual swap from the viewpoint of the receiver, who would price the callable accrual swap as the full accrual swap (coupon leg minus funding leg) minus the Bermudan option to enter into the receiver accrual swap. So a 10NC3 cancellable quarterly accrual swap would be priced as the 10 year bullet quarterly receiver accrual swap minus the Bermudan option – with quarterly exercise dates starting in year 3 – to receive the remainder of the coupon leg and pay the remainder of the funding leg. Accordingly, here we price Bermudan options into receiver accrual swaps. Bermudan options on payer accrual swaps can be priced similarly. There are two key requirements in pricing Bermudan accrual swaps. First, as Rmin decreases and Rmax increases, the value of the Bermudan accrual swap should reduce to the value of an ordinary Bermudan swaption with strike Rf ix . Besides the obvious theoretical appeal, meeting this requirement allows one to hedge the callability of the accrual swap by selling an o? setting Bermudan swaption. This criterion requires using the same the interest rate model and calibration method for Bermudan accrual notes as would be used for Bermudan swaptions. Following standard practice, one would calibrate the Bermudan accrual note to the â€Å"diagonal swaptions† struck at the accrual note’s â€Å"e? ective strikes. † For example, a 10NC3 accrual swap which is callable quarterly starting in year 3 would be calibrated to the 3 into 7, the 3. 25 into 6. 75, †¦ , the i 8. 75 into 1. 25, and the 9 into 1 swaptions. The strike Ref f for each of these â€Å"reference swaptions† would be chosen so that for swaption i, (3. 1) value of the ? xed leg value of all accrual swap coupons j ? i = value of the ? oating leg value of the accrual swap’s funding leg ? i This usually results in strikes Ref f that are not too far from the money. In the preceding section we showed that each coupon of the accrual swap can be written as a combination of vanilla ? oorlets, and therefore the market value of each coupon is known exactly. The second requirement is that the valuation procedure should reproduce today’s m arket value of each coupon exactly. In fact, if there is a 25% chance of exercising into the accrual swap on or before the j th exercise date, the pricing methodology should yield 25% of the vega risk of the ? oorlets that make up the j th coupon payment. E? ectively this means that the pricing methodology needs to use the correct market volatilities for ? oorlets struck at Rmin and Rmax . This is a fairly sti? requirement, since we now need to match swaptions struck at i Ref f and ? oorlets struck at Rmin and Rmax . This is why callable range notes are considered heavily skew depedent products. 3. 1. Hull-White model. Meeting these requirements would seem to require using a model that is sophisticated enough to match the ? oorlet smiles exactly, as well as the diagonal swaption volatilities. Such a model would be complex, calibration would be di? ult, and most likely the procedure would yield unstable hedges. An alternative approach is to use a much simpler model to match the diagonal swaption prices, and then use â€Å"internal adjusters† to match the ? oorlet volatilities. Here we follow this approach, using the 1 factor linear Gauss Markov (LGM) model with internal adjusters to price Bermudan options on accrual swaps. Speci ? cally, we ? nd explicit formulas for the LGM model’s prices of standard ? oorlets. This enables us to compose the accrual swap â€Å"payo? s† (the value recieved at each node in the tree if the Bermudan is exercised) as a linear combination of the vanilla ? orlets. With the payo? s known, the Bermudan can be evaluated via a standard rollback. The last step is to note that the LGM model misprices the ? oorlets that make up the accrual swap coupons, and use internal adjusters to correct this mis-pricing. Internal adjusters can be used with other models, but the mathematics is more complex. 3. 1. 1. LGM. The 1 factor LGM model is exactly the Hull-White model expressed as an HJM model. The 1 factor LGM model has a single state variable x that determines the entire yield curve at any time t. 9 This model can be summarized in three equations. The ? st is the Martingale valuation formula: At any date t and state x, the value of any deal is given by the formula, Z V (t, x) V (T, X) (3. 2a) = p(t, x; T, X) dX for any T > t. N (t, x) N (T, X) Here p(t, x; T, X) is the probability that the state variable is in state X at date T , given that it is in state x at date t. For the LGM model, the transition density is Gaussian 2 1 e? (X? x) /2[? (T ) (t)] , p(t, x; T, X) = p 2? [? (T ) ? ?(t)] (3. 2b) with a variance of ? (T ) ? ?(t). The numeraire is (3. 2c) N (t, x) = 1 h(t)x+ 1 h2 (t)? (t) 2 , e D(t) for reasons that will soon become apparent. Without loss of generality, one sets x = 0 at t = 0, and today’s variance is zero: ? (0) = 0. The ratio (3. 3a) V (t, x) ? V (t, x) ? N (t, x) is usually called the reduced value of the deal. Since N (0, 0) = 1, today’s value coincides with today’s reduced value: (3. 3b) V (0, 0) ? V (0, 0) = V (0, 0) ? . N (0, 0) So we only have to work with reduced values to get today’s prices.. De? ne Z(t, x; T ) to be the value of a zero coupon bond with maturity T , as seen at t, x. It’s value can be found by substituting 1 for V (T, X) in the Martingale valuation formula. This yields (3. 4a) 1 2 Z(t, x; T ) ? Z(t, x; T ) ? = D(T )e? (T )x? 2 h (T )? (t) . N (t, x) Since the forward rates are de? ned through (3. 4b) Z(t, x; T ) ? e? T t f (t,x;T 0 )dT 0 , ? taking ? ?T log Z shows that the forward rates are (3. 4c) f (t, x; T ) = f0 (T ) + h0 (T )x + h0 (T )h(T )? (t). This last equation captures the LGM model in a nutshell. The curves h(T ) and ? (t) are model parameters that need to be set by calibration or by a priori reasoning. The above formula shows that at any date t, the forward rate curve is given by today’s forward rate curve f0 (T ) plus x times a second curve h0 (T ), where x is a Gaussian random variable with mean zero and variance ? (t). Thus h0 (T ) determines possible shapes of the forward curve and ? (t) determines the width of the distribution of forward curves. The last term h0 (T )h(T )? (t) is a much smaller convexity correction. 10 3. 1. 2. Vanilla prices under LGM. Let L(t, x; ? st ) be the forward value of the k month Libor rate for the period ? st to ? end , as seen at t, x. Regardless of model, the forward value of the Libor rate is given by (3. 5a) where (3. 5b) ? = cvg(? st , ? end ) L(t, x; ? st ) = Z(t, x; ? st ) ? Z(t, x; ? end ) + bs(? st ) = Ltrue (t, x; ? st ) + bs(? st ), ? Z(t, x; ? end ) is the day count fraction of the interval. Here Ltrue is the forward â€Å"true rate† for the interval and bs(? ) is the Libor rate’s basis spread for the period starting at ? . Let F (t, x; ? st , K) be the value at t, x of a ? oorlet with strike K on the Libor rate L(t, x; ? st ). On the ? xing date ? f ix the payo? is (3. 6) ?  ¤+ F (? f ix , xf ix ; ? st , K) = ? K ? L(? f ix , xf ix ; ? st ) Z(? f ix , xf ix ; ? end ), where xf ix is the state variable on the ? xing date. Substituting for L(? ex , xex ; ? st ), the payo? becomes (3. 7a)  · ? + F (? f ix , xf ix ; ? st , K) Z(? f ix , xf ix ; ? st ) Z(? f ix , xf ix ; ? end ) . = 1 + ? (K ? bs(? st )) ? N (? ix , xf ix ) N (? f ix , xf ix ) Z(? f ix , xf ix ; ? end ) Knowing the value of the ? oorlet on the ? xing date, we can use the Martingale valuation formula to ? nd the value on any earlier date t: Z 2 1 F (t, x; ? st , K) F (? f ix , xf ix ; ? st , K) e? (xf ix ? x) /2[? f ix ] =q dxf ix , (3. 7b) N (t, x) N (? f ix , xf ix ) 2? [? f ix ? ?] where ? f ix = ? (? f ix ) and ? = ? (t). Substituting the zero coupon bond formula 3. 4a and the payo? 3. 7a into the integral yields (3. 8a) where log (3. 8b) ? 1,2 =  µ 1 + ? (K ? bs) 1 + ? (L ? bs)  ¤ ?  ± 1 (hend ? hst )2 ? f ix ? ?(t) 2 q , (hend ? hst ) ? f ix ? (t)  ¶ F (t, x; ? st , K) = Z(t, x; ? end ) {[1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L ? bs)]N (? 2 )} , and where L ? L(t, x; ? st ) = (3. 8c)  µ  ¶ 1 Z(t, x; ? st ) ? 1 + bs(? st ) ? Z(t, x; ? end )  ¶  µ 1 Dst (hend ? hst )x? 1 (h2 ? h2 )? end st 2 = e ? 1 + bs(? st ) ? Dend 11 is the forward Libor rate for the period ? st to ? end , as seen at t, x. Here hst = h(? st ) and hend = h(? end ). For future reference, it is convenient to split o? the zero coupon bond value Z(t, x; ? end ). So de? ne the forwarded ? oorlet value by (3. 9) Ff (t, x; ? st , K) = F (t, x; ? st , K) Z(t, x; ? end ) = [1 + ? (K ? bs)]N (? 1 ) ? [1 + ? L(t, x; ? st ) ? bs)]N (? 2 ). Equations 3. 8a and 3. 9 are just Black’s formul as for the value of a European put option on a log normal asset, provided we identify (3. 10a) (3. 10b) (3. 10c) (3. 10d) 1 + ? (L ? bs) = asset’s forward value, 1 + ? (K ? bs) = strike, ? end = settlement date, and p ? f ix ? ? (hend ? hst ) v = ? = asset volatility, tf ix ? t where tf ix ? t is the time-to-exercise. One should not confuse ? , which is the ? oorlet’s â€Å"price volatility,† with the commonly quoted â€Å"rate volatility. † 3. 1. 3. Rollback. Obtaining the value of the Bermudan is straightforward, given the explicit formulas for the ? orlets, . Suppose that the LGM model has been calibrated, so the â€Å"model parameters† h(t) and ? (t) are known. (In Appendix A we show one popular calibration method). Let the Bermudan’s noti? cation dates be tex , tex+1 , . . . , tex . Suppose that if we exercise on date tex , we receive all coupon payments for the K k0 k0 k intervals k + 1, . . . , n and recieve all funding leg payments f or intervals ik , ik + 1, . . . , n. ? The rollback works by induction. Assume that in the previous rollback steps, we have calculated the reduced value (3. 11a) V + (tex , x) k = value at tex of all remaining exercises tex , tex . . . , tex k k+1 k+2 K N (tex , x) k at each x. We show how to take one more step backwards, ? nding the value which includes the exercise tex k at the preceding exercise date: (3. 11b) V + (tex , x) k? 1 = value at tex of all remaining exercises tex , tex , tex . . . . , tex . k? 1 k k+1 k+2 K N (tex , x) k? 1 Let Pk (x)/N (tex , x) be the (reduced) value of the payo? obtained if the Bermudan is exercised at tex . k k As seen at the exercise date tex the e? ective notional for date ? st is k (3. 12a) where we recall that (3. 12b) ? = ? end (? st ) ? tj , ? end (? st ) ? ? st ? = cvg(? st , ? end (? st )). 12 A(tex , x, ? t ) = k ?j Rf ix Z(tex , x; tj ) 1 k , Mj Z(tex , x; ? end ) 1 + Lf (tex , x; ? st ) k k Reconstructing the reduced value of the payo? (see equation 2. 15) yields (3. 13a) Pk (x) = N (tex , x) k n X ? j Rf ix Z(tex , x; tj ) k Mj N (tex , x) ? k tj X j=k+1 st =tj? 1 +1 ? 1 + (Rmax ? 1 ? ) 2 Ff (tex , x; ? st , Rmax + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ? ? 1 + (Rmax + 1 ? ) 2 Ff (tex , x; ? st , Rmax ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin ? 1 ? ) 2 Ff (tex , x; ? st , Rmin + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin + 1 ? ) 2 + Ff (tex , x; ? st , Rmin ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ? n ? X ? ? Z(tex , x, tik ? 1 ) ? Z(tex , x, tn ) Z(tex , x, ti ) k k k ? ? cvg(ti? 1 , ti )(bsi +mi ) ? ex , x) ex , x) . N (tk N (tk i=i +1 k ? This payo? includes only zero coupon bonds and ? oorlets, so we can calculate this reduced payo? explicitly using the previously derived formula 3. 9. The reduced valued including the kth exercise is clearly ? ? Pk (x) V + (tex , x) V (tex , x) k k = max , at each x. (3. 13b) N (tex , x) N (tex , x) N (tex , x) k k k Using the Martingale valuation formula we can â€Å"roll di? erences, trees, convolution, or direct integration to Z V + (tex , x) 1 k? 1 (3. 3c) =p N (tex , x) 2? [? k ? ? k? 1 ] k? 1 back† to the preceding exercise date by using ? nite compute the integral V (tex , X) ? (X? x)2 /2[? k k? 1 ] k dX e N (tex , X) k at each x. Here ? k = ? (tex ) and ? k? 1 = ? (tex ). k k? 1 At this point we have moved from tex to the preceding exercise date tex . We now repeat the procedure: k k? 1 at each x we t ake the max of V + (tex , x)/N (tex , x) and the payo? Pk? 1 (x)/N (tex , x) for tex , and then k? 1 k? 1 k? 1 k? 1 use the valuation formula to roll-back to the preceding exercise date tex , etc. Eventually we work our way k? 2 througn the ? rst exercise V (tex , x). Then today’s value is found by a ? nal integration: k0 Z V (tex , X) ? X 2 /2? V (0, 0) 1 k0 k0 dX. (3. 14) V (0, 0) = =p e N (0, 0) N (tex , X) 2 k0 k0 3. 2. Using internal adjusters. The above pricing methodology satis? es the ? rst criterion: Provided we use LGM (Hull-White) to price our Bermudan swaptions, and provided we use the same calibration method for accrual swaps as for Bermudan swaptions, the above procedure will yield prices that reduce to the Bermudan prices as Rmin goes to zero and Rmax becomes large. However the LGM model yields the following formulas for today’s values of the standard ? orlets: F (0, 0; ? st , K) = D(? end ) {[1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L0 ? bs)]N (? 2 )} log  µ  ¶ 1 + ? (K ? bs)  ± 1 ? 2 tf ix 2 mod 1 + ? (L0 ? bs) . v ? mod tf ix 13 (3. 15a) where (3. 15b) ?1,2 = Here (3. 15c) L0 = Dst ? Dend + bs(? st ) ? Dend is today’s forward value for the Libor rate, and (3. 15d) q ? mod = (hend ? hst ) ? f ix /tf ix 3. 2. 1. Obtaining the market vol. Floorlets are quoted in terms of the ordinary (rate) vol. Suppose the rate vol is quoted as ? imp (K). Then today’s market price of the ? oorlet is is the asset’s log normal volatility according to the LGM model. We did not calibrate the LGM model to these ? oorlets. It is virtually certain that matching today’s market prices for the ? oorlets will require using q an implied (price) volatility ? mkt which di? ers from ? mod = (hend ? hst ) ? f ix /tf ix . (3. 16a) where (3. 16b) Fmkt (? st , K) = ? D(? end ) {KN (d1 ) ? L0 N (d2 )} d1,2 = log K/L0  ± 1 ? 2 (K)tf ix 2 imp v ? imp (K) tf ix The price vol ? mkt is the volatility that equates the LGM ? oorlet value to this market value. It is de? ned implicitly by (3. 17a) with log (3. 17b) ? 1,2 =  µ  ¶ 1 + ? (K ? bs)  ± 1 ? 2 tf ix 2 mkt 1 + ? (L0 ? bs) v ? kt tf ix [1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L0 ? bs)]N (? 2 ) = ? KN (d1 ) ? ?L0 N (d2 ), (3. 17c) d1,2 = log K/L0  ± 1 ? 2 (K)tf ix 2 imp v ? imp (K) tf ix Equivalent vol techniques can be used to ? nd the price vol ? mkt (K) which corresponds to the market-quoted implied rate vol ? imp (K) : (3. 18) ? imp (K) = 1 + 5760 ? 4 t2 ix +  ·  ·  · 1+ imp f ? mkt (K) 1 2 1 4 2 24 ? mkt tf ix + 5760 ? mkt tf ix  µ log L0 /K  ¶ 1 + ? (L0 ? bs) 1 + ? (K ? bs) 1+ 1 2 24 ? imp tf ix log If this approximation is not su? ciently accurate, we can use a single Newton step to attain any reasonable accuracy. 14 igital floorlet value ? mod ? mkt L0/K Fig. 3. 1. Unadjusted and adjusted digital payo? L/K 3. 2. 2. Adjusting the price vol. The price vol ? mkt obtained from the market price will not match the q LGM model’s price vol ? mod = (hend ? hst ) ? f ix /tf ix . This is easily remedied using an internal adjuster. All one does is multiply the model volatility with the factor needed to bring it into line with the actual market volatility, and use this factor when calculating the payo? s. Speci? cally, in calculating each payo? Pk (x)/N (tex , x) in the rollback (see eq. 3. 13a), one makes the replacement k (3. 9) (3. 20) (hend ? hst ) q q ? mkt ? f ix ? ?(tex ) =? (hend ? hst ) ? f ix ? ?(t) k ? mod q p = 1 ? ?(tex )/? (tf ix )? mkt tf ix . k With the internal adjusters, the pricing methodology now satis? es the second criteria: it agrees with all the vanilla prices that make up the range note coupons. Essentially, all the adjuster does is to slightly â€Å"sharpen up† or â€Å"smear out† the digital ? oorlet’s payo? to match today’s value at L0 /K. This results in slightly positive or negative price corrections at various values of L/K, but these corrections average out to zero when averaged over all L/K. Making this volatility adjustment is vastly superior to the other commonly used adjustment method, which is to add in a ? ctitious â€Å"exercise fee† to match today’s coupon value. Adding a fee gives a positive or negative bias to the payo? for all L/K, even far from the money, where the payo? was certain to have been correct. Meeting the second criterion forced us to go outside the model. It is possible that there is a subtle arbitrage to our pricing methodology. (There may or may not be an arbitrage free model in which extra factors – positively or negatively correlated with x – enable us to obtain exactly these ? orlet prices while leaving our Gaussian rollback una? ected). However, not matching today’s price of the underlying accrual swap would be a direct and immediate arbitrage. 15 4. Range notes and callable range notes. In an accrual swap, the coupon leg is exchanged for a funding leg, which is normally a standard Libor leg plus a margin. U nlike a bond, there is no principle at risk. The only credit risk is for the di? erence in value between the coupon leg and the ? oating leg payments; even this di? erence is usually collateralized through various inter-dealer arrangements. Since swaps are indivisible, liquidity is not an issue: they can be unwound by transferring a payment of the accrual swap’s mark-to-market value. For these reasons, there is no detectable OAS in pricing accrual swaps. A range note is an actual bond which pays the coupon leg on top of the principle repayments; there is no funding leg. For these deals, the issuer’s credit-worthiness is a key concern. One needs to use an option adjusted spread (OAS) to obtain the extra discounting re? ecting the counterparty’s credit spread and liquidity. Here we analyze bullet range notes, both uncallable and callable. The coupons Cj of these notes are set by the number of days an index (usually Libor) sets in a speci? ed range, just like accrual swaps: ? tj X ? j Rf ix 1 if Rmin ? L(? st ) ? Rmax (4. 1a) Cj = , 0 otherwise Mj ? =t +1 st j? 1 where L(? st ) is k month Libor for the interval ? st to ? end (? st ), and where ? j and Mj are the day count fraction and the total number of days in the j th coupon interval tj? 1 to tj . In addition, these range notes repay the principle on the ? nal pay date, so the (bullet) range note payments are: (4. 1b) (4. 1c) Cj 1 + Cn paid on tj , paid on tn . j = 1, 2, . . . n ? 1, For callable range notes, let the noti? ation on dates be tex for k = k0 , k0 + 1, . . . , K ? 1, K with K < n. k Assume that if the range note is called on tex , then the strike price Kk is paid on coupon date tk and the k payments Cj are cancelled for j = k + 1, . . . , n. 4. 1. Modeling option adjusted spreads. Suppose a range note is issued by issuer A. ZA (t, x; T ) to be the value of a dollar paid by the note on date T , as seen at t, x. We assume that (4. 2) ZA (t, x; T ) = Z(t, x; T ) ? (T ) , ? (t) De? ne where Z(t, x; T ) is the value according to the Libor curve, and (4. 3) ? (? ) = DA (? ) . e D(? ) Here ? is the OAS of the range note. The choice of the discount curve DA (? ) depends on what we wish the OAS to measure. If one wishes to ? nd the range note’s value relative to the issuer’s other bonds, then one should use the issuer’s discount curve for DA (? ); the OAS then measures the note’s richness or cheapness compared to the other bonds of issuer A. If one wishes to ? nd the note’s value relative to its credit risk, then the OAS calculation should use the issuer’s â€Å"risky discount curve† or for the issuer’s credit rating’s risky discount curve for DA (? ). If one wishes to ? nd the absolute OAS, then one should use the swap market’s discount curve D(? , so that ? (? ) is just e . When valuing a non-callable range note, we are just determining which OAS ? is needed to match the current price. I. e. , the OAS needed to match the market’s idiosyncratic preference or adversion of the bond. When valuing a callable range note, we are ma king a much more powerful assumption. By assuming that the same ? can be used in evaluating the calls, we are assuming that (1) the issuer would re-issue the bonds if it could do so more cheaply, and (2) on each exercise date in the future, the issuer could issue debt at the same OAS that prevails on today’s bond. 16 4. 2. Non-callable range notes. Range note coupons are ? xed by Libor settings and other issuerindependent criteria. Thus the value of a range note is obtained by leaving the coupon calculations alone, and replacing the coupon’s discount factors D(tj ) with the bond-appropriate DA (tj )e tj : (4. 4a) VA (0) = n X j=1 ?j Rf ix DA (tj )e tj Mj  ¤ ?  ¤ ? 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? [1 + L0 (? st )] ? st =tj? 1 +1  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? ? [1 + L0 (? st )] +DA (tn )e tn . tj X Here the last term DA (tn )e n is the value of the notional repaid at maturity. As before, the B? are Black’s formulas, (4. 4b) B? (? st ) = Kj N (d? ) ? L0 (? st )N (d? ) 1 2 (4. 4c) d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix (4. 4d) K1,2 = Rmax  ± 1 ? , 2 K3,4 = Rmin  ± 1 ? , 2 and L0 (? ) is today’s forward rate: (4. 4e) Finally, (4. 4f) ? = ? end ? tj . ? en d ? ? st L0 (? st ) = D(? st ) ? D(? end ) ? D(? end ) 4. 3. Callable range notes. We price the callable range notes via the same Hull-White model as used to price the cancelable accrual swap. We just need to adjust the coupon discounting in the payo? function. Clearly the value of the callable range note is the value of the non-callable range note minus the value of the call: (4. 5) callable bullet Berm VA (0) = VA (0) ? VA (0). bullet Berm (0) is the today’s value of the non-callable range note in 4. 4a, and VA (0) is today’s value of Here VA the Bermudan option. This Bermudan option is valued using exactly the same rollback procedure as before, 17 except that now the payo? is (4. 6a) (4. 6b) Pk (x) = N (tex , x) k ? tj X st =tj? 1 +1 j=k+1 n X ? j Rf ix ZA (tex , x; tj ) k Mj N (tex , x) ? k 1 + (Rmax ? 1 ? ) 2 Ff (tex , x; ? st , Rmax + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ? ? + (Rmax + 1 ? ) 2 Ff (tex , x; ? st , Rmax ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin ? 1 ? ) 2 Ff (tex , x; ? st , Rmin + 1 ? ) k 2 1 + Lf (tex , x; ? st ) k 1 + (Rmin + 1 ? ) 2 + Ff (tex , x; ? st , Rmin ? 1 ? ) k 2 1 + Lf (tex , x; ? st ) k ZA (tex , x, tn ) ZA (tex , x, tk ) k k + ? Kk ex , x) N (tk N (tex , x) k Here the bond speci? c reduced zero coupon bond value is (4. 6c) ex ex 1 2 ZA (tex , x, T ) D(tex ) k k = DA (T )e (T ? tk ) e? h(T )x? 2 h (T )? k , ex , x) N (tk DA (tex ) k ? the (adjusted) forwarded ? oorlet value is Ff (tex , x; ? st , K) = [1 + ? (K ? bs)]N (? 1 ) ? [1 + ? (L(tex , x; ? t ) ? bs)]N (? 2 ) k k log (4. 6d) ? 1,2 =  µ  ¶ 1 + ? (K ? bs)  ± 1 [1 ? ?(tex )/? (tf ix )]? 2 tf ix k mkt 2 1 + ? (L ? bs) p , v 1 ? ?(tex )/? (tf ix )? mkt tf ix k  ¶ Z(tex , x; ? st ) k ? 1 + bs(? st ) Z(tex , x; ? end ) k  ¶ (hend ? hst )x? 1 (h2 ? h2 )? ex end st k ? 1 + bs(? 2 e st ) 1 = ?  µ and the forward Libor value is (4. 6e) (4. 6f) L? L (tex , x; ? st ) k  µ Dst Dend 1 = ? The only remaining issue is calibration. For range notes, we should use constant mean reversion and calibrate along the diagonal, exactly as we did for the cancelable accrual swaps. We only need to specify the strikes of the reference swaptions. A good method is to transfer the basis spreads and margin to the coupon leg, and then match the ratio of the coupon leg to the ? oating leg. For exercise on date tk , this ratio yields (4. 7a) n X ?k = ? j Rf ix DA (tj )e (tj ? tk ) Mj Kk DA (tk ) j=k+1 (?  ¤ ?  ¤ 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B1 (? st ) 2 2 ? [1 + L0 (? st )] ? st =tj? 1 +1 )  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B3 (? st ) 2 2 ? 1 + Lf (tex , x; ? st ) k tj X + DA (tn )e (tn ? tk ) Kk DA (tk ) 18 As before, the Bj are dimensionless Black formulas, (4. 7b) B? (? st ) = K? N (d? ) ? L0 (? st )N (d? ) 1 2 d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix K3,4 = Rmin  ± 1 ? , 2 (4. 7c) (4. 7d) K1,2 = Rmax  ± 1 ? , 2 and L0 (? st ) is today’s forward rate: Appendix A. Calibrating the LGM model. The are several methods of calibrating the LGM model for pricing a Bermudan swaption. The most popular method is to choose a constant mean reversion ? , and then calibrate on the diagonal European swaptions making up the Bermudan. In the LGM model, a â€Å"constant mean reversion ? † means that the model function h(t) is given by (A. 1) h(t) = 1 ? e t . ? Usually the value of ? s selected from a table of values that are known to yield the correct market prices of liquid Bermudans; It is known empirically that the needed mean reversion parameters are very, very stable, changing little from year to year. ? 1M 3M 6M 1Y 3Y 5Y 7Y 10Y 1Y -1. 00% -0. 75% -0. 50% 0. 00% 0. 25% 0. 50% 1. 00% 1. 50% 2Y -0. 50% -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 3Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 4Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 5Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 7Y -0. 25% 0. 00% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% 10Y -0. 25% 0. 0% 0. 25% 0. 50% 1. 00% 1. 25% 1. 50% 1. 75% Table A. 1 Mean reverssion ? for Bermudan swaptions. Rows are time-to-? rst exercise; columns are tenor of the longest underlying swap obtained upon exercise. With h(t) known, we only need determine ? (t) by calibrating to European swaptions. Consider a European swaption with noti? cation date tex . Suppose that if one exercises the option, one recieves a ? xed leg worth (A. 2a) Vf ix (t, x) = n X i=1 Rf ix cvg(ti? 1 , ti , dcbf ix )Z(t, x; ti ), and pays a ? oating leg worth (A. 2b) Vf lt (t, x) = Z(t, x; t0 ) ? Z(t, x; tn ) + n X i=1 cvg(ti? 1 , ti , dcbf lt ) bsi Z(t, x; ti ). 9 Here cvg(ti? 1 , ti , dcbf ix ) and cvg(ti? 1 , ti , dcbf lt ) are the day count fraction s for interval i using the ? xed leg and ? oating leg day count bases. (For simplicity, we are cheating slightly by applying the ? oating leg’s basis spread at the frequency of the ? xed leg. Mea culpa). Adjusting the basis spread for the di? erence in the day count bases (A. 3) bsnew = i cvg(ti? 1 , ti , dcbf lt ) bsi cvg(ti? 1 , ti , dcbf ix ) allows us to write the value of the swap as (A. 4) Vswap (t, x) = Vf ix (t, x) ? Vf lt (t, x) n X = (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )Z(t, x; ti ) + Z(t, x; tn ) ? Z(t, x; t0 ) i=1 Under the LGM model, today’s value of the swaption is (A. 5) 1 Vswptn (0, 0) = p 2 (tex ) Z e? xex /2? (tex ) 2 [Vswap (tex , xex )]+ dxex N (tex , xex ) Substituting the explicit formulas for the zero coupon bonds and working out the integral yields (A. 6a) n X (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )D(ti )N Vswptn (0, 0) = where y is determined implicitly via (A. 6b) y + [h(ti ) ? h(t0 )] ? ex p ? ex i=1 A A ! ! y + [h(tn ) ? h(t0 )] ? ex y p ? D(t0 )N p , +D(tn )N ? ex ? ex A ! n X 2 1 (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix )e? [h(ti )? h(t0 )]y? 2 [h(ti )? h(t0 )] ? ex i=1 +D(tn )e? [h(tn )? h(t0 )]y? [h(tn )? h(t0 )] 1 2 ? ex = D(t0 ). The values of h(t) are known for all t, so the only unknown parameter in this price is ? (tex ). One can show that the value of the swaption is an increasing function of ? (tex ), so there is exactly one ? (tex ) which matches the LGM value of the swaption to its market price. This solution is easily found via a global Newton iteration. T o price a Bermudan swaption, one typcially calibrates on the component Europeans. For, say, a 10NC3 Bermudan swaption struck at 8. 2% and callable quarterly, one would calibrate to the 3 into 7 swaption struck at 8. 2%, the 3. 25 into 6. 5 swaption struck at 8. 2%, †¦ , then 8. 75 into 1. 25 swaption struck at 8. 25%, and ? nally the 9 into 1 swaption struck at 8. 2%. Calibrating each swaption gives the value of ? (t) on the swaption’s exercise date. One generally uses piecewise linear interpolation to obtain ? (t) at dates between the exercise dates. The remaining problem is to pick the strike of the reference swaptions. A good method is to transfer the basis spreads and margin to the coupon leg, and then match the ratio of the coupon leg to the funding leg to the equivalent ratio for a swaption. For the exercise on date tk , this ratio is de? ed to be 20 n X ? j D(tj ) (A. 7a) ? k = Mj D(tk ) ? j=k+1 D(tn ) X D(ti ) + cvg(ti? 1 , ti )(bs0 +mi ) ? i D(tk ) i=1 D(tk ) n  ¤ ?  ¤ 1 + (Rmax ? 1 ? ) B1 (? st ) ? 1 + (Rmax + 1 ? ) B2 (? st ) 2 2 ? [1 + L0 (? st )] st =tj? 1 +1  ¤ ?  ¤ ? 1 + (Rmin ? 1 ? ) B3 (? st ) ? 1 + (Rmin + 1 ? ) B4 (? st ) 2 2 ? ? [1 + L0 (? st )] tj X ? where B? are Black’s formula at strikes around the boundaries: (A. 7b) B? (? st ) = ? D(? end ) {K? N (d? ) ? L0 (? st )N (d? )} 1 2 d? = 1,2 log K? /L0 (? st )  ± 1 ? 2 (K? )tf ix 2 imp v ? imp (K? ) tf ix (A. 7c) with (A. 7d) K1,2 = Rmax  ± 1 ? , 2 K3,4 = Rmin  ± 1 ?. 2 This is to be matched to the swaption whose swap starts on tk and ends on tn , with the strike Rf ix chosen so that the equivalent ratio matches the ? k de? ned above: (A. 7e) ? k = n X i=k+1 (Rf ix ? bsi ) cvg(ti? 1 , ti , dcbf ix ) D(ti ) D(tn ) + D(tk ) D(tk ) The above methodology works well for deals that are similar to bullet swaptions. For some exotics, such as amortizing deals or zero coupon callables, one may wish to choose both the tenor of the and the strike of the reference swaptions. This allows one to match the exotic deal’s duration as well as its moneyness. Appendix B. Floating rate accrual notes. 21

. Summarise the Historical Changes in Childhood Experience...

There is little evidence of what it was really like in the past so it is difficult for a lot of people to re-construct the life of a child, however from what I have researched, and in my own opinion I am going to summarise the historical changes in childhood experience and relate these changes to childhood development and rights. In the 19th century I think children missed out on most of their childhood as most of them took on jobs such as chimney sweepers, street sellers and farms for example. These were mainly children from poor families who were seen as extra farm hands and were exploited by receiving low pay for long hours and working in poor conditions. Families did not look at how the children were treated and the possible impact†¦show more content†¦Also from what my grandad has told me, back then they weren’t allowed to express themselves as they would get emotional and physical abuse towards them, so they just carried on with everyday life. They was wealthy families and was a lot easier from them as they had the money to attend a private school or was home-schooled. They were also encouraged to donate money and goods to the poor. Kerry Woolford Even though a legislation was drawn up and improved childhood education, healthcare and welfare, children were and even now in modern society around the world are being taken advantage of for monetary gain. Modern society now especially since 1997 have tried putting children first such as committing to meeting children’s needs. There are still problems such as school truancy, adolescents not in education or training and also child protection as there have been tragic deaths including baby P and Victoria climbie, which her death was largely responsible for the formation of every child matters inactive plus a lot of other changes in different systems, 2. Discuss how family systems are influential in child development processes and include challenges to meeting a child’s needs, such as conflicts and poor parenting practices. Family systems can influence in a child’sShow MoreRelatedTo What Extent Can Childhood Be Considered a Social Construction?1489 Words   |  6 PagesTo what extent can childhood be considered a social construction? This essay will analyse the major experiences by which childhood is constructed: one determined by the society and the other examined personally. Following this approach will be explained socially constructed childhood that asserts children’s attitudes, expectations and understandings that are defined by a certain society or culture. Furthermore various aspects of childhoods will be taken into account in relation to social, economicRead MoreThe Importance Of Teaching And Learning Processes1867 Words   |  8 Pages For educators, evaluating what is being taught and why can be a somewhat elusive and subjective task. The amount of theoretical research in regards to teaching and learning processes is immense as there is no particular ‘right answer’, allowing much room for interpretation (Elmborg, 2006). It is therefore of importance that educators understand what they are teaching and why, by deepening their knowledge on various theoretical standpoints. Furthermore, discussion of these standpoints, particularlyRead MoreScientific Method and Children4906 Words   |  20 Pagesuniversally valid; in other words, it is a natural law. The laws of nature cannot change. Every technical construction and measuring apparatus is a practical application of the laws of nature. 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For these reasons no patent officeRead MoreClient Presentation And Service Delivery10192 Words   |  41 PagesLife 6 Stages of Human Development 7 Physical Development†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦....8 Psychological Development Social Development Cognitive Development Affective Development Theories that Inform Community Services Practice 10 Attachment Theory Trauma Trauma Theory Identify Relevant Support for Clients 12 Outcomes Measurement Tools 15 ABAS-II – Adaptive Behaviour Assessment System The Griffith Mental Development Scales (GMDS) The Bayley Scales of Infant Development (Bayley-III) Sensory ProfileRead MoreCase Study148348 Words   |  594 Pages---------------------------------This edition published 2011  © Pearson Education Limited 2011 The rights of Gerry Johnson, Richard Whittington and Kevan Scholes to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Pearson Education is not responsible for the content of third party internet sites. ISBN: 978-0-273-73557-1 (printed) ISBN: 978-0-273-73552-6 (web) All rights reserved. Permission is hereby given for the material in this publicationRead MoreGsk Annual Report 2010135604 Words   |  543 Pageslive longer GlaxoSmithKline Annual Report 2010 Contents Business review P08–P57 Governance and remuneration P58–P101 Financial statements P102–P191 Shareholder information P192–P212 Business review 2010 Performance overview Research and development Pipeline summary Products, competition and intellectual property Regulation Manufacturing and supply World market GSK sales performance Segment reviews Responsible business Financial review 2010 Financial position and resources Financial reviewRead MoreExploring Corporate Strategy - Case164366 Words   |  658 Pagesresponsibility Culture Competitive strategy Strategic options: directions Corporate-level strategy International strategy Innovation and Entrepreneurship Strategic options: methods Strategy evaluation Strategic management process Organising Resourcing Managing change Strategic leadership Strategy in practice Public sector/not-for-proï ¬ t management Small business strategy ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€"  ââ€" Read MoreMonsanto: Better Living Through Genetic Engineering96204 Words   |  385 Pagessuch a course. The ‘full story’ that follows this summary gives you considerable detail about how to go about a case analysis, but for now here is a brief account. Before we start, a word about attitude – make it a real exercise. You have a set of historical facts; use a rigorous system to work out what strategies shoul d be followed. All the cases are about real companies, and one of the entertaining bits of the analysis process is to compare what you have said they should do with what they really have

ACT Format What to Expect on the ACT Exam

Students who take the ACT are really taking tests in four subject areas: mathematics, English, reading, and science. The ACT also has an optional writing test. The number of questions and time allocation varies by subject area: ACT Section Number of Questions Time Allowed English 75 45 minutes Mathematics 60 1 hour Reading 40 35 minutes Science 40 35 minutes Writing (optional) 1 essay 40 minutes The total exam time is 2 hours and 55 minutes, although the actual exam will take ten minutes longer because of a break after the math section. If you take the ACT Plus Writing, the exam is  3 hours and 35 minutes long plus the 10 minute break after the math section and a 5 minute break before you begin the essay. The ACT English Test With 75 questions to complete in 45 minutes, youll need to work quickly  to complete the English section of the ACT. Youll  be asked to answer questions about five short passages and essays. The questions  cover several different aspects of English language and writing: Production of Writing. This content area represents 29-32% of the English test. These questions will be focused on the big picture of the passage. What is the passages purpose? What is the tone? What literary strategies is the author employing? Has the text achieved its goal? Is an underlined part of the text relevant to the overall goal of the passage?Knowledge of Language. This part of the English section  focuses on issues of language use, such as style, tone, conciseness, and precision. Questions from this category account for 13-19% of the English test.Conventions of Standard English. This content area is the largest part of the English test. These questions focus on correctness in grammar, syntax, punctuation, and word usage. This content area makes up 51-56% of the English Test. The ACT Mathematics Test At 60 minutes long, the math section of the ACT is the most time-consuming part of the exam. There are 60 questions in this section, so youll have one minute per question. While a calculator is not necessary to complete the math section, you are allowed to use one of the permitted calculators, which will save you precious time during the exam. The  ACT Mathematics Test covers standard high school math concepts  before  calculus: Preparing for Higher Mathematics. This content area represents 57-60% of the math questions broken down into several sub-categories.Number and Quantity. Students must understand real and complex number systems, vectors, matrices, and expressions with integer and rational exponents. (7-10% of the Mathematics Test)Algebra. This section requires test-takers to know how to solve and graph several kinds of expressions as well as understand linear, polynomial, radical, and exponential relationships. (12-15% of the Mathematics Test)Functions. Students need to understand both the representation and application of functions. Coverage includes linear, radical, polynomial, and logarithmic functions. (12-15% of the Mathematics Test)Geometry. This section focuses on shapes and solids, and students need to be able to calculate area and volume of different objects. Test-takers must be prepared to solve for missing values in triangles, circles, and other shapes. (12-15% of the Mathematics Test)Stati stics and Probability. Students need to be able to understand and analyze distributions of data, data collection methods, and probabilities related to a data sample. (8-12% of the Mathematics Test)Integrating Essential Skills. This content area accounts for 40-43% of the questions on the math section. The questions here draw on the information covered in the Preparing for Higher Mathematics section, but students will be asked to synthesize and apply their knowledge to solve more complex problems. Subjects covered here include percentages, surface area, volume, average, median, proportional relationships, and different ways to express numbers. You may need to work through multiple steps to solve these problems. The ACT Reading Test Whereas the English Test focuses primarily on grammar and usage, the ACT reading test assesses your ability to understand, analyze, interpret, and draw conclusions from a passage. The reading part of the ACT has four sections. Three of those sections ask questions about a single passage, and the fourth asks you to answer questions related to a pair of passages. Note that these passages can be from any discipline, not just English literature. Your close-reading and critical-thinking skills are essential for the reading part of the ACT. The questions can be broken down into three categories: Key Ideas and Details. These questions require you to identify the central ideas and themes in the passage. Youll also need to understand how passages develop their ideas. Is it through sequential relationships, comparisons, or cause and effect? These questions make up 55-60% of the reading questions.Craft and Structure. With these questions, youll analyze meanings of specific words and phrases, rhetorical strategies, and narrative points of view. You might be asked about the authors purpose and perspective, or you might need to identify shifts in perspective. These questions account for 25-30% of the reading questions.Integration and Knowledge of Ideas. Questions in this category ask you to differentiate between facts and an authors opinions, and you may be asked to use evidence to make links between different texts. These questions represent 13-18% of the reading section of the exam. The ACT Science Test The ACT science test questions draw from the four common fields of high school science: biology, earth science, chemistry, and physics. However, the questions do not demand advanced knowledge in any of the subject areas. The science portion of the ACT tests your ability to interpret  graphs, analyze data, and structure an experiment,  not  your ability to memorize facts. With 40 questions and 35 minutes, youll have just over 50 seconds per question.  Calculators are not allowed on this section. The ACT science questions can be broken down into three broad categories: Data Representation. With these questions, youll need to be able to read tables and graphs, and youll be asked to draw conclusions from them. You may also be asked to work in the opposite direction and translate data into graphs. These questions account for 30-40% of the science portion of the ACT.Research Summaries. If given the description of one or more experiments, can you answer questions related to the design of the experiments and the interpretation of the experimental results? These questions represent about half of the science test (45-55% of the questions).Conflicting Viewpoints. Given a single scientific phenomena, these questions ask you to explore how different conclusions might be drawn. Issues such as incomplete data and differing premises are central to this category of question. 15-20% of the Science Test focuses on this topic area. The ACT Writing Test Few colleges require the ACT writing test, but many still recommend the essay portion of the exam. Thus, it is often  a good idea to take the ACT Plus Writing.   The optional writing portion of the ACT asks you to write a single essay in 40 minutes. Youll be provided an essay question as well as three different perspectives related to the question. Youll then craft an essay that takes a position on the topic while engaging at least one of the perspectives presented in the prompt. The essay will be scored in four areas: Ideas and Analysis. Does the essay develop meaningful ideas related to the situation presented in the prompt, and have you engaged successfully with other perspectives on the issue?Development and Support. Has your essay succeeded in backing up your ideas with a discussion of the implications, and have you backed up your main points with well chosen examples?Organization. Do your ideas flow smoothly and clearly from one to the next? Is there a clear relationship between your ideas? Have you guided your reader through your argument effectively?Language Use and Conventions. This area focuses on the nuts and bolts of proper English usage. Is your language clear, and have you used proper grammar, punctuation, and syntax? Is the style and tone engaging and appropriate? A Final Word on the ACT Format While the ACT is broken down into four distinct test subjects, realize that there is a lot of overlap between sections. Whether youre reading a literary passage or a scientific graph, youll be asked to use your analytical skills to understand the information and draw conclusions. The ACT is not an exam that requires a remarkable vocabulary and advanced calculus skills. If youve done well in high school in core subject areas, you should earn a good score on the  ACT.

The Overcrowding of Prisons Essays - 1786 Words

According to Mark Early, president of the Prison Fellowship International, the Bureau of Justice Statistics report shows that there are 19 states that have prisons operating at 100 percent capacity and another 20 are falling right behind them. There is no wonder why the overcrowding of prisons is being discussed everywhere. Not to mention how serious this predicament is and how serious it can get. Many of the United States citizens don’t understand why this is such a serious matter due to the fact that some of them believe it’s not their problem. Therefore they just don’t care about it. Also, some Americans may say that they don’t care about what could happen to the inmates due to repercussions of overcrowding, because it’s their own†¦show more content†¦That was two years ago and the prison population have continued to grow since then, so can you imagine what it is now? Now, think of what the prisoners have to go through. Andrea Caumont sh ared some of the problems prisoners are having, because of prison overcrowding. She says that in California the United States Supreme Court told the state to reduce its prison population, because overcrowded conditions are amounting to cruel and unusual punishment. Can you imagine what must be happening for the Supreme Court to rule it as cruel and unusual punishment? Not only are they being punished cruelly, the prisoners are also losing their opportunities to participate in the prisons self-improvement programs and the vocational training, because there is less of everything to go around. Therefore all resources, even food, are being stretched to the max. (John Howard goes into more detail on this issue in his paper The Effects of Prison Overcrowding on the P.A.T.R.I.C.K Crusade website.) The prisoners are not the only ones being affected by this problem, this problem is affecting everyone. The overcrowding of prisons can date back to the 1850’s, when they imprisoned debtors, delinquent juveniles, minor misdemeanants and felons all together with no separation. The King Edwards Public School students created their own websiteShow MoreRelatedPrison Overcrowding1187 Words   |  5 PagesAbstract This paper will discuss prison overcrowding and what type of numbers have come about over the years when it comes to inmates being imprisoned. It will discuss the cost of a prisoner annually as well as the decision to add verses build when it comes to new facilities. The overcrowding in one particular prison will be touched on as well as whose responsibility it is for upkeep. It will discuss how funding plays a role in overcrowding as well as the â€Å"three strikes† rule in California andRead MorePrison Overcrowding953 Words   |  4 PagesPrison Overcrowding Sherita Bowens American Intercontinental University April 24, 2010 Abstract Prison overcrowding is one of the many different problems throughout the world that law enforcement faces. Prison overcrowding not only affects those in law enforcement, it also affects the taxpayers in the community as well. The problem of overcrowded prisons is being handled in many different ways. Some of these ways have been proven to be sufficient and successful while others have not beenRead MorePrison Overcrowding2966 Words   |  12 Pagesâ€Å"Prison Overcrowding: Using Proposals from Nevada and California to Recommend an Alternative Answer† By: Casey Apao For: Dr. Sarri CSN Fall 2010 Dedication: â€Å"I, the undersigned, Casey Apao hereby certify that without the assistance of Henry Apao this Critical Thinking Scientific Paper wouldn’t be done.† Signed , Casey Apao Disclaimer: â€Å"I, Casey Apao hereby certify that this Critical Thinking Scientific Paper is the result of my sole intellectualRead MoreOvercrowding Prisons And The Prisons Essay1785 Words   |  8 PagesAlma Gonzalez Professor Shaw SOC 474OL 11 August 2016 Overcrowding Prisons Prisons were essentially built to accommodate a number of prisoners, but over the years, it has reached over capacity. Today in the United States, there are approximately 193,468 federal inmates that consist of the Bureau of Prisons Custody, private managed facilities and other facilities. The inmates ages range from 18- 65 with the median age being in their late 30’s. This number is counting both male and female populationRead MoreArticle Report On Overcrowding Of Prison Overcrowding1074 Words   |  5 Pages Clark, Charles S. Prison Overcrowding. CQ Researcher 4 Feb. 1994: 97-120. Web. 26 Mar. 2016. This article discusses overcrowding in the United States prison system, due in part to mandatory prison sentences. Additionally, this article also discusses the challenges in managing the overpopulation of prisons and gives an objective look at solutions, to include building more prisons, to combat overcrowding. While the author does not include information about himself and his qualifications, hisRead MoreEffects Of Prison Overcrowding1345 Words   |  6 Pagesa few of the effects that prison overcrowding causes towards the inmates and the guards. I will first address the issue of violence that prison overcrowding causes. My next point will be the health of the inmates discussing both their physical and mental while in overcrowded prisons. Lastly I will discuss the physical and mental health of the correctional officers and how the job could lead to correctional officers having issues in their private life. Prison Overcrowding has become a major issueRead MoreThe Problem Of Prison Overcrowding1572 Words   |  7 Pagesthe past 30 odd years, California’s prison population has grown by 750 percent (â€Å"California’s Perpetuating Prison Crisis†). As this percentage perpetuates to make substantial gains, inmates are suffering in confinement cells, officials are negotiating over the issue, and the public is protesting to make their opinions count. The prison crisis has continued to grow over the years, causing a great uproar among all of California’s 32 state prisons. Prison overcrowding has been an increasingly vital issueRead MoreThe Overcrowding Of Prison And Massachusetts1261 Words   |  6 PagesAfter exploring options of which states had the most overcrowding in prisons, the best option to go with was California because states like Alabama and Massachusetts did not have current statistics. If anything, their statistics were from 2016 or 2013, making data harder to collect. Therefore, according to the California Department of Corrections and Rehabilitation website, the most recent report they have of the total population is as of midnight February 8, 2017. The total population is 180,885Read MoreThe Problem Of Prison Overcrowding1166 Words   |  5 PagesBlackboard #1: Realignment Prison overcrowding has always been a problem in California prisons. It has been growing over the year and has now lead the United States Supreme Court to take part in trying to find a solution to this problem. Because of this issue, Plata litigation came through and had a significant impact on the way we see community corrections. The ruling in the Plata litigation in turn lead to AB 109 or The Public Safety Realignment Act to be implemented as a solution to California’sRead MorePrison Overcrowding Essay1184 Words   |  5 Pages Prison Overcrowding Nicole Neal American Intercontinental University Abstract This research paper is to explore the impact of prison overcrowding. The United States has a, what seems to be everlasting, prison overcrowding problem. Not only does the United States have this dilemma, but also many other countries have overcrowded prisons as well. Many issues need to be addressed; ways to reduce the prison populations and how to effectively reduce prison cost without

Sociology Essay Impact of Globalization Trends Free Essays

Introduction Globalization is a widely discussed and contested topic. The process of globalization has profound impacts on the capacity of a nation to formulate its policies. It is accompanied by a seemingly endless process of change within education (Peters, 1992). We will write a custom essay sample on Sociology Essay: Impact of Globalization Trends or any similar topic only for you Order Now Globalization is one main issue that is increasingly attracting the attention of most academicians, researchers and policy makers. It has gained relevance in the context of higher education. Education is an important driver of growth and poverty reduction. Education policies have been in existence for quite some time and have played an important role in the development policy. The most recent wave of globalization is likely to have profound effects on education structures and policies across the world. What is globalization? ‘Globalisation’ is a term that describes the process of integrating societies by removing legal, political and geographical constraints (Trowler, 1998). Vulliamy (2004) describes it as a process which is rapidly integrating the world into one economic space via an increasingly networked global telecommunication system. A study by Tikly (2003), suggest globalization as an inevitable and largely irresistible phenomenon that contains opportunities and threats for national development. Globalization is therefore seen to be concerned principally with integration into global and regional markets underpinned by technologies Although internationalization is not new to education policies, the forces and tensions under the umbrella concept of globalization constitute dramatically different environment in which education institutions and policy makers operate in (Marginson, 1999). The changes to which education structures inUKand around the globe is exposed are complex and varied (Marginson, 1999). Nonetheless, the globalization concept indicates that these changes are somehow interrelated. For the purpose of this analysis, we will stress the following tendencies within the overall force of globalization: Restructuring of the economic world system due to rapid integration of the world economy resulting from a transformation to a post industrial knowledge economy and increasingly liberalized trade and commerce. Rise of network society due to technological advancements and the expansion of the internet Increasing virtual mobility of people, knowledge and capital resulting from the development of new transport facilities, expansion of the internet and increasingly world integrated community Complex cultural developments whereby we have an increasing cultural exchange and multicultural reality on the one hand of homogeneity and cultural differentiation and segregation on the other hand. Erosion of the nation state and a widening of the gap between socio-political regulation and economic activity. Such is the nature and complexities of forces associated with globalization. These forces define the social environment in which education structures and policies operate in (Green, 1999). Further, these forces condition the context in which education policies and structures have to operate and profoundly alter people’s experience of both formal and informal education (Green, 1995). For example, most institutions are transformed to become targets of corporate expansion and sites for branding. A more detailed explanation will be discussed below. Impact of globalization on education structures and policies Globalisation has profound impacts on education structures and policies. The impact is profound but also diverse, depending on the locality within the global arena. While there is often a danger of oversimplification and generalisation when dealing with globalisation, diversity has to be recognised and promoted to a certain extent. Various views have been expressed in literature with regard to the impact of contemporary globalization on the processes and structure of education worldwide. 1. Direct impacts on both the curriculum and pedagogy Carnoy (1999) suggests direct impacts on both the curriculum and pedagogy. There is little evidence however to support such an assessment. Whilst attempts have been made to inject global awareness on school curricula in western industrialized countries, these have generally remained very low status add-ons. Carnoy (1999) continues to argue that whilst the direct impacts on pedagogy and curriculum are limited, the more general influences of economic restructuring and political ideologies are immense. For instance, globalization is putting considerable premium on highly skilled and flexible workers in an organisation hence increasing the demand for university education. 2. Emerging ‘bordeless’ higher education market The most visible manifestation of globalisation in the education sector is the emerging ‘bordeless’higher education market. Globalization leads to huge increases in worldwide demand for higher education through opportunities created by the internet and new communication technologies which in turn shape an environment in which providers can expand their supply of educational facilities (Breier, 2001). Universities fromAustralia, North America, Europe andEnglandare reaching out their educational provisions to the international market by actively recruiting international students through establishing branch campuses or via distance education, e-learning and other transnational activities (Breier, 2001). These increasing demands bring new providers into the market. The business of borderless education comprises various forms and developments including the emergence of corporate universities, professional association that are directly active in higher education, and media companies delivering educational programmes among others (Alao Kayode, 2005). These new providers extensively use the Internet and ICT as a delivery channel. 3. Erosion of national regulatory and policy framework Globalization is also associated with the erosion of national regulatory and policy frameworks in which institutions are embedded (Slattery, 1995). The policy framework is subject to erosion in an increasingly international environment marked by globalizing professions, liberalized market place, mobility of skilled labour, and international competition between institutions (Slattery, 1995). Most institutions acknowledge this and thus develop consortia, partnerships and networks to strengthen their position in the global arena. Schemes such as the European Credit transfer system and mobility programmes such as UMAP and SOCRATES can be developed to stimulate internationalization in higher education with respect to the various national policy frameworks (Dearden et al, 2002). There is need for an international regulatory framework that transcends the eroded national policy framework and steer to some extent the global integration of higher education system. 4. Create new and tremendously important demands and exigencies towards universities as knowledge centre’s Consequently, globalization creates new and tremendously important demands and exigencies towards universities as knowledge centers (Dearden et al, 2005). Research and development is crucial in any knowledge and information driven society. Globalization of research and development leads to a more mobile and highly competitive international market of researchers. Moreover, universities are called upon to take up responsibilities in the society, deepen democracy, act as mediators and to function as centre’s of critical debate. These higher demands placed upon them create tensions in institutions and stimulate other organizations to engage in such kind of activities. 5.Increasing demand for higher education worldwide Finally, the continuing trend of globalization is expected to increase the demand for higher education worldwide. In the developed world, the society will always ask for highly qualified and flexible workers. Modernization, economic development and demographic pressure increase the demand for higher education in most parts of the world (Blanden Machin, 2004). Governments and local institutions generally lack enough resources to deal with the increasing demand hence leaving an unmet demand to the international and virtual providers. This demand not only grows quantitatively but also becomes more diverse. The internet together with new technologies are increasingly providing new opportunities for more flexible delivery of higher education, thus increasing demand in some countries and meeting demands in others where traditional institutions have failed. These developments brought by globalization underpin the assertion that higher education will emerge as one of the booming markets in future (Blanden Machin, 2004). The need for an international regulatory framework There is a big difference in the way countries deal with private universities and transnational higher education.GreeceandIsrael, for instance, rarely recognize their diplomas and degrees (Blanden, Gregg Machin, 2005). While other countries residing in the developing world such asMalaysiarecognize their incapacity to meet the increasing demand and thus welcome foreign providers (Blanden, Gregg Machin, 2005). Principally, there is no reason to oppose a positive and open attitude towards transnational higher education and private universities. In modern policy approach, it must be recognized that private and transnational institutions are also capable of fulfilling public functions. Despite the fact that traditional higher education institutions have a specific tradition and academic culture to defend, it should be amenable to competitors from diverse backgrounds. It therefore becomes imperative to have in place international and sustainable policy framework that deals with private and transnational providers. Conclusion The globalization trends are leading to a wide spread changes that are impacting on education worldwide. Nation states acknowledge this and have developed reforms to their educational systems in response to modernizing ideas and international trends. It should be noted that globalization represents a new and distinct shift in the relationship between states and supranational forces and that its impact on education is profound in a range of ways. Whilst this analysis does not present an exhaustive listing of the impact of globalization on education, it does bring out key dynamics and highlight important areas of action for academicians and policy makers with respect to globalization. (1557 words) Reference Alao Kayode (2005), Emerging Perspectives on Educational Assessment in an Era of Postmodernism, Commissioned paper presented at 31st Annual conference on International Association for Educational Assessment. Blanden.J.P., Gregg Machin.S (2005), Educational inequality and intergenerational mobility, The economics of education in theUnited Kingdom, Princeton,PrincetonUniversitypress. Blanden.J Machin.S (2004), Educational inequality and the expansion ofUKhigher education, Scottish Journal of political economy, Vol 54, PP.230-49 Breier.M (2001), Curriculum Restructuring in Higher Education in Post-ApartheidSouth Africa,Pretoria Carnoy (1999), Education, globalization and nation state,Oxford,Oxforduniversity press Dearden.L, Emmerson.C, Frayne Meghir.C (2005), Education subsidies and school drop-out rates Dearden.L, Mcintosh.C, Myck.M Vignoles.A (2002), The returns to academic and vocational qualifications inBritain, Bulletin of economic research, Vol 54, PP. 249-75 Green.A (1999), Education and globalization in Europe andEast Asia: convergent and divergent trends, Journal of education policy, Vol 14, pp.55-71 Green.M.F (1995), Transforming British higher education: a view from across theAtlantic, Higher Education, Vol 29, pp.225-239 Marginson.S (1999), After globalization: emerging politics of education, Journal of Education Policy, Vol 14, pp.19-31. Peters M (1992), Performance and Accountability in ‘Post-industrial Society’: the crisis of the British universities, Studies in Higher Education, Vol 17, PP.123-139. Slattery, P. (1995) Curriculum development in the post modern era,New York, Garland Publishing Tikly (2003), Globalisation, knowledge economy and comparative education, vol 41, pp. 117-149 Trowler P.R (1998), Academics responding to change: new higher education frameworks and academic cultures, Buckingham, Open University Press. Vulliamy.G (2004), the impact of globalization on qualitative research in comparative and international education, journal of comparative and international education, Vol 34, pp.261-284 How to cite Sociology Essay: Impact of Globalization Trends, Essay examples

Matrix the movie Essay Example For Students

Matrix the movie Essay Here is the essay I think the uploader strung the sentences together so the paragraphs are in one long line: As the worlds technological capabilities reach increasingly impressive new heights, we are faced with new problems caused by these new technological capabilities. Along with these newfound problems, such as the now infamous Y2K, come the latest futuristic prophecies about the worlds damnation due to the new technologies. Thus is the need for, and theory behind the movie Matrix. The Matrix explores unthinkable realms of computer world omination, human cultivation, and a specious reality so profoundly using exceptional writing by brothers Andy and Larry Wachowski. This thought provoking screenplay is based on the premise that cyberspace is becoming far too much the center of our existence. Additionally the Wachowski brothers play on the idea that the line between reality and virtual reality is getting a little too thin. These ideas are so farfetched and mind-boggling that they lead one to stop for a second to question ones own sense of reality. This questioning of reality is a yproduct of the brilliant filmmaking used to captivate the audience for an incredible two and a quarter hour journey. While watching The Matrix, the pulse is quickened, the eyes are dazzled, and the brain is twisted beyond recognition several times over. Ones eyes are dazzled incessantly by numerous computer-generated special effects. Among these aesthetically pleasing scenes are the action packed martial arts scenes featuring the always lovable Keanu Reeves. Although one is predetermined to question Reeves comic book-like kung fu sequences, on must sit back and be amazed by the rothers Wachowski filming methods. The Wachowskis filming methods allow them to slow down sequences to show moves and actions normally made impossible by the laws of physics. Additional scenes depict bullets as they are slowed down to a crawl while the audience takes the vantage of the depth-defying characters dodging these bullets. The audience watches as characters leap buildings and virtually fly in hair- trigger quick movements that Hollywood was incapable of depicting just two or three years ago. Simply put the visual display is indescribable. The Matrixs visual barrage should propel the art of filmmaking into the next millennium. In addition to the exceptional writing and visual depiction, the Matrix offers commendable acting. As an action movie, the Matrix is not asked to have any dramatic performances. Its simply expected to have explosions, disastrous chases, and gun-driven violence, yet many of the actors give surprising performances. The before mentioned Keanu Reeves, gives an excellent performance as the geek-gone superhero, Neo, considering his resume` of painful performances. Additionally, Carrie-Anne Moss gave a respectable erformance as Trinity, the behind kicking beauty. Also Lawrence Fishburne, cast intelligently as Morpheus, brilliantly plays role of a futuristic Yoda trying to get Neo to realize his potential as humanitys savior. These performances coupled with a career- defining performance from Hugo Weaving, the dark suited Men in Black detective-type, add a dramatic flair to this already promising movie. In conclusion, Andy and Larry Wachowski combine ingenious writing, innovative filming, and good performances from a well-selected cast, to make The Matrix more than just a movie, but rather an experience.