Appunti completo del corso Strumenti derivati

THE FORWARD SWAP RATE

Underlying asset = swap. We look at the fwd of the underlying asset bc we’re consider the possibility of entering in a

swap in a future point in time.

t = starts of the OPTION

T = MATURITY OF THE OPTION

= starts of the SWAP

= MATURITY OF THE SWAP



, multiple payments of the SWAP.

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

− =

Suppose that the forward swap rate is at time t (t≤ T) 

() s is a function of time because it depends on the date

of the quote. ( )

The value of the fixed side at time t is ∑ =∑

( ) ( )

− ∙ () ∙ − ∙ () ∙ (, )

Fixed leg = is a stream of cash flows to be discounted from T until today (t).

− = 1, −

If we have yearly payments: if I have semi-annual payments I have to divide the rate for 2, ecc.

→ time adjustment.

THE VALUE OF THE VARIABLE LEG

Like in a floating rate bond, if $1 is paid at , the floating side is worth $1 at time T =

Today, t, the value of $1 received at time T = , is

(, )

Today, t, the value of $1 received at time T = , is

(, )

Consequently, the value of the floating side at time t is (assuming a notional principal of 1) )

(, − (, )

The value of the floating leg is the first discount factor (from today to t0) – second discount factor from today to tn.

 The value of the variable leg is equal to the discounted value of the principal received at time T = less the

discounted value of the principal paid a time (interpretation: if I pay 1 at this implies that I receive 1 at )

183

 Intuition: If one actually did receive the principal at time T = this could be invested at LIBOR until TN 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

THE FORWARD SWAP RATE

 The value of the variable side at time t is )

(, − (, )

 ( ) ( )

Equating the value of the fixed and variable leg ∑ ( ) (

− ∙ () ∙ − ) = −

( ) ( ) ( )

, ( , )

 Which implies () = =

( )

( )

∑ ( )



() fwd S.R.



() ANNUITY = sequence of cash flow that will paid in future point in time. The value of the annuity is to take

the discount value of future cash flows. 

ℎ numerator



Summation between all discount factors denominator 

It’s a general formula that’s appied to swap start in future point in time forward swap.

But can also be applied to spot swap.

We see that this is a fwd swap bc T0 is set in a point in time that’s after than tk = today.

( )

 =

If instead T0 = t

Go to Excel file Class10_Swaps&ForwardSwap.xls

If we move from T0 to t, we have a forward swap, not the swap.

The same formula works for Swap that starts today or for a swap that will start in a future point in time.

A CLARIFICATION ABOUT THE TIMING OF DISCOUNT FACTORS

 The numerator is the difference between the last and the first discount factors

The first discount factor is the one at time (i.e., when the swap begins)



 The denominator is the sum of the discount factors associated with the cash flow dates  annuity.

This implies that the first discount factor in the summation refers to the end of the first period (i.e., the

 first payment date at time )

T-T0 = opt

FIRST DISCOUNT FACTOR =from now to the beginning of the swap (T_0) or the end of option (T)

SECOND DISCOUNT FACTOR = FROM NOW UNTIL THE MATUIRTY OF THE UNDERLYING SWAP (T_N) 184

SWAPTIONS AND BOND OPTIONS

An interest rate swap can be regarded as an agreement to exchange a fixed-rate bond for a floating-rate bond. At the start of a swap,

the value of the floating-rate bond always equals the principal amount of the swap. A swaption can therefore be regarded as an

option to exchange a fixed-rate bond for the principal amount of the swap—that is, a type of bond option. If a swaption gives the

holder the right to pay fixed and receive floating, it is a put option on the fixed-rate bond with strike price equal to the principal. If

a swaption gives the holder the right to pay floating and receive fixed, it is a call option on the fixed-rate bond with a strike price

equal to the principal. 

Once we enter in the contract, we’re obliged to stay in the contract forward swap.

DIFFERENCE BETWEEN THE FORWARD SWAP AND SWAPTIONS: with forward swap we have are obliged; in

swaption contract we can decide.

The behaviour over time of the mkt rate, in a cap contract = it’s a ptf of options os we have multiple option and

every time there is a reset and we assess if whether or not the option will be exercised or not because we have

multiple options. So in tk we can see if the strike is below mkt therefore we exercise the call; if the mkt is below the

strike rate, consequently we’re not going to exercise.

Swaptions = once we decide that we exercise the contract, it’s done for the remaining time we’re going to pay all the

cash flows involved in the swap contract. Swap = multiperiod contract so we need to pay the associated cash flows

for many years. 

Mkt rate = Libor we look at it to decide if exercise or not



Swaption is not the short-term rat but it’s the swap rate given that we’re buying a swap, we want to enter in the



swap when the swap rate of the mkt is higher that the swap rate agreed in the contract so we pay at most the

swap rate.

Once the option expires we have to decide if entering or not in the swap and we enter for the entire life of the swap.

We don’t have cash flows immediately but at the end (t_1 = first cash flow for the swap, Tn+1)



How to set the swap rate for a forward swap = breakeven for the fixed rate leg it’s the swap rate that we’re



supposed to pay * discount factor (bc payments occur in future points in time) * P(t, Ti+1) delta in order to

consider when the payment occur.

Floating leg =

A floating rate bond will pay 100 par at the end of the bond if there is no default. The value of the par bond

when we set the rate we have

problems: when there are changing in rate between one coupon and the other.

The value of the floating bond at the date of the fixing is always par.

Once we have the value of the floating leg, we make it equal to the fixed leg. 185

At fixed rate setting, B = 100 always



Coupon = 5% paid by the bond; r = 7%



B = 100 par it’s below par because the cash flows are discounted at higher rate

Loosing 2% running = every year

If B has 2 years until maturity left, 4% to make up so the current value of the bond should be like 96.

B = PV(f_i)

If c = 5%; r = 3%;

 

B > 100 it’s above par I will have 100 at T so I lose on capital gain but I have higher coupons.



If c = 5%; r = 5%; paar

Z = B slightly lower than 100 because I’m earning less than I should.

 

Floating rate bond: = I set the rate the rate observed in the market is 3% 3% will be paid in . But in

the rate is no more 3% but it’s 4% that will be paid in .

The swap rate for a particular maturity at a particular time is the (mid-market) fixed rate that would be exchanged for the floating

rate in a newly issued swap with that maturity. The model usually used to value a European option on a swap assumes that the 186

underlying swap rate at the maturity of the option is lognormal. Consider a swaption where the holder has the right to pay a rate

and receive the floating rate on a swap that will last n years starting in T years. We suppose that there are m payments per year

under the swap and that the notional principal is L. Day count conventions may lead to the fixed payments under a swap being

slightly different on each payment date. For now we will ignore the effect of day count conventions and assume that each fixed

payment on the swap is the fixed rate times . Suppose that the swap rate for an n-year swap starting at time T proves to be . By

comparing the cash flows on a swap where the fixed rate is to the cash flows on a swap where the fixed rate is , it can be seen

that the payoff from the swaption consists of a series of cash flows equal to max( − , 0).

The cash flows are received m times per year for the n years of the life of the swap. Suppose that the swap payment dates are T1,

T2, …, Tmn, measured in years from today. (It is approximately true that .) Each cash flow is the payoff from a call

= +

option on with strike price .Whereas a cap is a portfolio of options on interest rates, a swaption is a single option on the swap

rate with repeated payoffs. The standard market model gives the value of a swaption where the holder has the right to pay as

∑ where and

)[ ) )]

(0, ( − ( = = − √.

√

= forward swap rate at time zero; = volatility of the forward swap rate.

∑ = discount factor for the mn payoffs.

)

(0,

A = value of a contract that pays at time the value of the swaption becomes: ) )]

(1 ≤ ≤ ), [ ( − (

∑

where .

)

= (0,

If the swaption gives the holder the right to receive a fixed rate of instead of paying it, the payoff from the swaption is

This is a put option on . The payoffs are received at times

max( − , 0). (1 ≤ ≤ ).

The standard market model gives the value of the swaptions as: ) )].

[ (− − (−

THE IMPACT OF DAY COUNT CONVENTIONS

The fixed rate for the swap underlying the swap option is expressed with a day count convention such as actual>365 or 30>360.

Suppose that and that, for the applicable day count convention, the accrual fraction corresponding to the time period

=

between and is . The formulas that have been presented are then correct with the annuity factor A being defined as:

)

= (0,

BLACK’S MODEL FOR EUROPEAN SWAPTIONS

2 scenarios: flat curve or non flat curve.

Flat curve = all fwd rates are equal to spot rates bc all rates are at the same level.

 When valuing European swap options it is usual to assume that the swap rate is lognormal  the only time that

matters is T bc it’