Derivative securities pricing - Appunti completi (ENG)

Consider a swap starting at a future time with payment dates are and define :

1 0

In the upper part we have the swap time and in the bottom part we have the option time. Suppose that the

() ( ≤ )

forward swap rate is at time so it’s the fixed rate that will be paid and it’s a function of the

agreement date. The value of the fixed side at time is:

−1 −1

−( −)

∑( − ) ∙ () ∙ = ∑( − ) ∙ () ∙ (, )

+1

+1 +1 +1

=0 =0

The forward swap rate plays the same role of the ordinary swap rate. We need to equate the expected

value of what we pay and the expected value of what we receive.

The value of the variable leg

Like in a floating rate bond (that should be traded around 100 so par), if $1 is paid at , the floating side is

= , = (, ). ,

worth $1 at time . Today, the value of $1 received at time is Today, the value

0

= (, ).

of $1 received at time is Consequently, the value of the floating side at time is (assuming

0 0

a notional principal of 1): )

(, − (, )

0

The value of the variable leg is equal to the discounted

=

value of the principal received at time less the

0

discounted value of the principal paid a time

(interpretation is that if I pay 1 at this implies that I

receive 1 at ). The intuition is that If one actually did

0 =

receive the principal at time this could be invested

0

at LIBOR until when it would mature as principal plus

interest.

Paying back the principal would leave the interest alone, which is the market value of the LIBOR leg. To get

the forward swap rate we equate fixed and variable legs:

−( −) −( −) )

− (, − (, )

0 0

() = =

−1 −( −)

∑ ()

( − ) ∙ +1

+1

=0

The denominator is the sum of all discount factors. If it was swap, the agreement date is today and

)

(, = 1.

therefore the first discount factor is equal to 1 so The numerator is the difference between

0

the last and the first discount factors. The first discount factor is the one at time (when the swap begins).

0

The denominator is the sum of the discount factors associated with the cash flow dates. This implies that

the first discount factor in the summation refers to the end of the first period (the first payment date ).

1

Black’s model for European swaptions

When valuing European swap options, it is usual to assume that the swap rate is lognormal. Consider a

-year

swaption which gives the right to pay the fixed rate (and receive LIBOR) on an swap starting at

.

time The payoff on each swap payment date is:

max( − , 0)

.

Where is principal, is payment frequency and is market swap rate at time The cash flows are

received times per year for the years of the life of the swap. Suppose that the swap payment dates are

− , measured in years from today. Each cash flow is the payoff from a call option on the swap rate

1

with strike price . A cap is a portfolio of (different) options on interest rates, a swaption is a single

option on the swap rate with repeated (constant) payoffs. The value of a swaption where the holder has

the right to pay is:

0 2

ln ( ) + /2

[

) ) )]

∑ (0, ( − ( ℎ = = − √

0 1 2 1 2 1

√

=1

Where is the forward swap rate (that you expect to see when the option is over and the swap start, in

0

some exercises forward rate is equal to spot rate because the yield curve is flat), is the forward swap rate

volatility and is the time from today until the ith swap payment.

Annuity

Let’s redefine the discount factor term as follows (with payments times per year for the years of the

swap) where stands for annuity:

1 1 −

)

= ∑ (0, = ∑

=1 =1

Then the value of the swaption becomes: ) )]

∙ [ ( − (

0 1 2

Remember that we always have two maturities:

1. Maturity of the option: used in the parameters;

2. Maturity of the payments of the swap: used to discount.

Rate conversion appendix

•

= ln (1 + ) = ( − 1);

From continuous to discrete compounding: while

•

= (1 + ) − 1;

From discrete to annual discrete compounding: 1

1

• )

= + − 1] .

[(1

From annual to compounding:

1

The general rule for IRDS is consistency between forward (caplet/forward swap rate) and strike. Remember

that forward swap rate is different from forward rates (caps for example) and has a specific formula. To get

= (1 + )

the equivalent discount rates remember: .

How to solve an exercise on swaption

What is the value of the European swaption that gives the holder the right to enter into three years annual

= 1) = 4).

( swap in four years ( The fixed rate is 5% ( , annual compounding by definition if not

specified) and is paid, the LIBOR is received. The swap principal is $10 mln, the yield curve is flat at 5% p.a.

with annual compounding and the volatility of the swap rate is 20%. Here we can immediately say that the

forward swap rate is 5% like the spot because the yield curve is flat and we have annual compounded.

The discounting years that we have to use are 5, 6 and 7. The maturity of the option is 4 years (). We need

to calculate the annuity (discrete compounding in this case):

1 1 1

= + + = 2.2404

5 6 7

1.05 1.05 1.05

If we were having continuously compounding yield curve, the formula would have been:

1 −

= ∑

=1

Now we calculate the parameters with the option maturity:

2

0.05 0.2

ln ( )+ 4

2

0.05

= = 0.2000 = 0.2000 − 0.2√4 = −0.2000

1 2

0.2√4

The swaption value is: ) )]

10,000,000 ∙ 2.2404[0.05( − 0.05( = 177,575$

1 2

117,575

= 1.776% 177.6

10

If the yield curve is flat, the spot rate starting in four years for one year is equal to the spot. If the yield

curve is positively inclined, the spot rate starting in four years for one year (forward rate between 4 and 5

years) is calculated as: −

2 2 1 1

= −

2 1

As soon as you have a positively inclined yield curve the forward will be higher than any spot rate linked to

it (the one year forward is higher than the 2 years spot, the two years forward is higher than the 3 years

spot, etc.) so we have a more expensive swaption (and the other way round for the negative inclination).

Second exercise

LIBOR is 6% continuously compounded, volatility is 20% and the notional is 100 mln. What is the price of a

= 2)

swaption with 3 years maturity, semestral payments ( and starting date in 5 years (option maturity)?

Here LIBOR is continuously compounded so we need to transform it into semi-annual compounding for

swap payments (but we use it as it is for the DF):

0.06

= ( − 1) = 2 ( − 1) = 6.09%

2

This is our forward swap rate because the yield curve is flat. For the annuity, we have (DF):

0

1 1

−0.06∙ −0.06∙5.5 −0.06∙6 −0.06∙8

( )

= ∑ = + + ⋯ + ∙ = 2.0036

2 2

=1

Of course, the result is the same if we convert 6% to discrete compounding and we apply the formula:

1 1 1 1

= + + ⋯+ = 2.0036

[ ]

6∙2 8∙2

5.5∙2

2 6.09% 6.09%

6.09% (1 + ) (1 + )

(1 + ) 2 2

2

The equivalent discount factor (they are equal to the continuously compounding case) is computed as

= (1 + ) . You can get the forward swap rate also by:

− 0.7408 − 0.6188

5 8

= = = 6.09%

0 2.0036

Where is related to the beginning of the swap/end of the option and is related to the end of the

5 8

6.2%,

swap. Suppose that the strike rate is we can calculate our swaption:

2

0.0609 0.2

ln ( ) + 5

0.062 2

= = 0.183911 = 0.183911 − 0.2√5 = −0.2633

1 2

0.2√5

The value of the swaption for who pays and receives is:

0

) )]

100,000,000 ∙ 2.0036[0.0609( − 0.062( = 2.07

1 2

The value of the swaption for who pays and receives is:

0

) )]

100,000,000 ∙ 2.0036[0.062(− − 0.0609(− = 2.29

2 1

Third exercise

= 7.6%, = 4, = 5,

Suppose that volatility is 25%, payments are annual, the principal is 1 mln and

LIBOR is 8% p.a. annual compounding. What is the price of a put swaption? Let’s start with the annuity:

1 1 1 1 1

= + + + + = 2.9348

5 6 7 8 9

1.08 1.08 1.08 1.08 1.08

0.252

0.08

ln( )+ 4

0.076 2

= = 0.3526 = 0.3526 − 0.25√4 = −0.1474.

Then we have: Swaption value is:

1 2

0.25√4 ) )]

1,000,000 ∙ 2.9348[0.076(− − 0.08(− = 39,553.09

2 1

= 6.2%, = 1, = 3,

What if the yield curve is not flat? Suppose that volatility is 20% and the principal

is 100. For the LIBOR we have the following table:

Payment Year Flat rates Non-flat rates

1 2 6% 7%

2 3 6% 8%

3 4 6% 9%

Now we calculate the continuous discount factors for each payment but using the years time. Thanks to

this we can calculate the annuity:

1

•