Pricing Interest Rate Derivatives: Autumn 2 0 0 9

A U T U M N    2 0 0 9

PRICING INTEREST RATE DERIVATIVES

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Numerical Methods in Finance (Implementing Market Models)

©Finbarr Murphy 2007

Lecture Objectives  Pricing Interest Rate Derivatives  How one can use our implied BDT binomial trees to calculate

options on pure discount bound

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 Swaption values

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Agenda Page

Pricing Interest Rate Derivatives

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3

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Pricing Interest Rate Derivatives  Constructing a tree to fit the existing yield and

volatility curves allows us to price a wide variety of interest rate derivatives  And, this is the point of the work so far!

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 Using the same software, every time there is a

change in the yield curve or volatility curve, a new bi/trinomial is created  And the associated derivative values are re-

calculated

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Pricing Interest Rate Derivatives  As an example we will consider an option on a

discount bond  Such options are not actively traded but the

pricing principal is easily translated to swaptions

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 The total amount of equity derivatives

outstanding at the end of 2006 was $7,178,480,000,000  The total amount of IR and Currency derivatives

outstanding at the end of 2006 was $285,728,140,000,000  Source ISDA – Figures based on Notional Amounts 5

©Finbarr Murphy 2007

Pricing Interest Rate Derivatives  Using the following notation:  T = Option maturity  S = Underlying bond maturity  K = Strike Price  NS = No of steps to Bond Maturity  NT = No of steps to Option Maturity

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 PSi,j = Value of S-Maturity Bond at node i,j

 We start by setting some boundary conditions

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Pricing Interest Rate Derivatives  Clearly, the discount bond matures at S with a

value 1

PSN S,N S=1

j=N

PSN S,N S-1 =1 j=2

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j=1 j=0 j=-1 j=-2

j=-N

PSN S,-N S=1 i=0

i=1

i=2

i=NT

i=NS-1

i=NS

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©Finbarr Murphy 2007

Pricing Interest Rate Derivatives  In other words,

PSN S , j = 1 ∀ nodes j at N S  Now, using backward induction, we can calculate

the price of a Psi,j bond (one that matures at S) for each node

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 We only need to work back as far as node T for

European style options

[

PSi , j = 12 d i , j PSi+1, j+1 + PSi+1, j−1

]

 We have calculated all the di,j’s previously 8

©Finbarr Murphy 2007

Pricing Interest Rate Derivatives  Here are the discount values created by

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BDT90CurveFit.m, changing T from 4 to 10

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Pricing Interest Rate Derivatives  Now, when we reach the T step, we can stop

(european style) and calculate the maturity values of the call options

{

PNT , j = max 0, PSNT , j − K

}

∀ j at N T

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 When the terminal conditions are calculated, we

can work back to find the current value of the option

[

Ci , j = 12 d i , j Ci +1, j +1 + Ci +1, j −1

] 10

©Finbarr Murphy 2007

Pricing Interest Rate Derivatives  Next, we consider how we might use the fitted

BDT tree to price swaptions  A swaption gives the holder the right to pay fixed

and receive floating (“payer” option)  The option has a maturity date and the underlying

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Swap will have a maturity  This can be considered a put on a fixed coupon

bond with a strike price equal to the swap notional  The coupons are equal to half the quoted swap

rate (assuming semi-annual reset dates) 11

©Finbarr Murphy 2007

Pricing Interest Rate Derivatives  A “receiver” swaption gives the holder the right

to pay floating and receive fixed  This is equivalent to a call option on a coupon

bearing bond  The following timeline shows a 1-year option on a

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3-year swap (see C&S P251) Period of Swap

Period of Option

0

1

2

3

4

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Pricing Interest Rate Derivatives  Using the same notation as that describing an

option on a pure discount bond, we start at the end of the bond life  The terminal bond price for all nodes at i=NS is

given by

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BN S , j

coupon = 1+ 2

 Recall that the final coupon is paid at maturity

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Pricing Interest Rate Derivatives  As before we can work these terminal values

backwards to NT using discounted expectations di,j  But we must include coupon payments when

these are traversed  At NS, we can calculate the terminal option values

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as before and work these back in time to t=0

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Pricing Interest Rate Derivatives  At NS, the terminal option values are given as

payer swaption receiver swaption

{ max{0, B

} − 1}

= ∑ QNT , j max 0,1 − BNT , j j

= ∑ QN T , j

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j

NT , j

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Recommended Texts  Required/Recommended  Clewlow, L. and Strickland, C. (1996) Implementing derivative

models, 1st ed., John Wiley and Sons Ltd. — Chapter 8

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 Additional/Useful

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