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Callable Bonds & Swaptions - Chapter 3 - YouTube
Channel: DNA Training & Consulting
[1]
this is the beginning of part 2 of our
[4]
module on callable bonds and swaptions
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which discusses the mechanics pricing
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and certain applications of callable
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bonds and swaptions from a trader and
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end user perspective we remind you that
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this part 2 contains chapters 3 & 4 of
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the module with chapter 3 reviewing a
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standard pricing model for swaptions
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and chapter 4 introducing a standard
[36]
pricing model for callable bonds this
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chapter 3 reviews the standard pricing
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models for swaptions focusing mostly on
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the European alternative worksheet
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swaption flat curve appears now in front
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of you and begins like any worksheet for
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an interest rate derivative with a LIBOR
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spot and forward curve appearing in
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column C here as you can see the curve
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is assumed to be completely flat at 6%
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the swaps principal at the beginning of
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each period appears in this column I
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giving you the flexibility to input
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amortize ink or accreting notional if
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you need to further on the right sells
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l5 to l8 reveal the principal remaining
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inputs for the swaption including its
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strike and it's tenor as well as the
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tenor of the underlying swap and it's
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assumed forward volatility
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CEL l12 derives the rate for an interest
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rate swap starting on the expiration
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date of the option ie a forward start
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swap and having a maturity equal to the
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maturity of the swapped underlying the
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swaption five years this is key to the
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pricing of the swaption since it is the
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forward swap level implied for the
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exercise date of the option rather than
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the spot level that is relevant to the
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pricing sells l 13 and 14 finally
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calculate the premium for a European
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payer and receiver swaption respectively
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by means of a closed form formula
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largely identical to the one derived
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under the famous black model for pricing
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caps and floors each premium is
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expressed as a percentage of notional
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and is paid of course up front with the
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curve completely flat that is as it is
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on this worksheet it comes as no
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surprise to find the receiver and the
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payer equal in value in the put called
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parity ensures that things could not be
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otherwise
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we asked you to illustrate the potential
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arbitrage if the payer was worth four
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percent up front while the receiver was
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worth only 3.75 percent up front click
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pause while you reflect on this
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hopefully you have figured out that the
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correct strategy involves first selling
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the pair for 4% upfront second buying
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the receiver for 375 upfront and third
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very importantly offsetting this
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position by entering a five-year pay
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fixed swap starting in two years this
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has put 25 basis points up front in the
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pocket of the arbitrage or the
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difference between the upfront premiums
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and leaves him completely hedged
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regardless of the future level of swap
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rates as we show next if in two years
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time the swap rate is below the strike
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the arbitrator exercises the receiver
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swaption and is perfectly hedged the
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pair of course would expire worthless
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alternatively if in two years time the
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swap rate is above the strike bank one
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exercises the payer swaption but the
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arbitrage er is still quite visibly
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perfectly hedged
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the table appearing now summarizes the
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arbitrage strategies available when pair
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receiver parity fails in either
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direction if the payer is more expensive
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than the receiver the arbitrage or sells
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the pair buys the receiver and pays
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fixed under the forward start swap which
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earns him an upfront profit equal to the
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payer premium minus the receiver premium
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and leaves him fully hedged if the
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receiver is more expensive than the pair
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the arbitrage or sells the receiver buys
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the pair and receives fixed under the
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forward start swap which earns him
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enough front profit equal to the
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receiver premium minus the payer premium
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and again leaves him fully hedged
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if we now extend the tenor from two
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years to five and leave all else
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unchanged ie if we now price five by
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five swaptions we should not be
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surprised at all to find both premiums
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increasing but nonetheless remaining
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equal in price like the majority of
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European options especially ones that
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are at the money an increase in tenor
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generally increases the options value we
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discussed some interesting exceptions to
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this rule in our module on FX options
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which you can review if you are
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interested now we hold the option tenner
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at five years but shorten the underlying
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swap tenor to four years so we are
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pricing now five by four swaptions using
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our standard notation the price of both
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receiver and pair declines and for a
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simple reason a four-year swap has a
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lower D vo one then a five-year swap so
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by moving from a five by five to a five
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by four we kept constant the options
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tenor but replaced the underlying swap
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with one that has a lower price
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volatility so the premium necessarily
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had to fall
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indeed if we shorten the underlying
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swaps Tanner enough eg to two years only
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ie if we price a five by two the premium
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becomes even cheaper than the one for
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the two by five with which we started
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this discussion the five year option may
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be two and a half times longer than the
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original two-year swaption but its
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underlying is now a two year swap whose
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price volatility or a DV or one or
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modified duration is far less than that
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of a five-year swap so much so as to
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more than offset the longer option tenor
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we emphasize that we are still using a
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flat curve assumption and that matters
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could be quite different if we used
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another curve shape
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now we move to worksheet swaption steep
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curve where you can see in column C a
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spot LIBOR at 5% then each successive
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six month LIBOR forward lying 25 basis
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points above the preceding one note
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first that the two by five forward swap
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now lies at seven point zero five to
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zero percent if we continue pricing
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swaptions with a strike of six percent
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as before we definitely will not find
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the payer and the receiver equal in
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value any longer since the pair would
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now be significantly in the money
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against the much higher forward rate and
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the receiver equally significantly out
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of the money against the same forward
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rate indeed we see that the pair costing
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around six point four percent is worth
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some two and a half times more than the
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receiver costing a mere two point forty
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four percent an increase in vowel for
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example to forty percent would of course
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benefit both instruments and vice versa
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for a drop involved for example to
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twenty percent
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equally an upward shift in the LIBOR
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curve would as expected increase further
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the payers value since it is becoming
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even deeper in the money and reduce
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further the value of the receiver which
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becomes even deeper out of the money
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we now reset the curve to its original
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position and ask the following question
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at what strike do you think the receiver
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and the pair become equally valuable
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trial-and-error guides us slowly to a
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level somewhere above 7% and eventually
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to somewhere very close to seven point
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zero five percent a number that looks
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all too familiar presumably indeed it is
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our friend the two by five forward start
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swap and are out other friend mister
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put-call parity rears its head one more
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time insuring mathematically that the
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two instruments converge in value
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exactly when their stripes are set
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exactly equal to this forward swap level
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unfortunately the closed-form solution
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available for pricing a European
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swaption is not applicable to the
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Bermudan alternative let alone to the
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American one and this is true of both
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the constant maturity swaption version
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and the remaining maturity swaption
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version numerical solutions are required
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in these instances typically involving
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binomial or trinomial trees similar to
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ones we used in our module on FX options
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we will not be illustrating these
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techniques in this chapter on swaptions
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but we will illustrate at least the
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binomial technique in the next chapter
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for the pricing of callable bonds this
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completes this chapter 3
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