Question $1$

A Credit Default Swaps $($CDS$){?}^1$ is a contract where one party $($credit protection buyer$)$

pays the other one $($credit protection seller$)$ a fixed periodic coupon for the life of the

contract on a specified reference asset. The party paying the premium is effectively

buying insurance against specific credit events, such as default, bankruptcy or

failure$-$to$-$pay or debt restructuring. If such a credit event occurs, the party receiving

the premium makes a payment to the protection buyer, and the swap then terminates.

Consider now that party A wishes to get covered from a potential loss of the face value

$(V_A)$ of an asset in case of a credit event.

Hence, party A decides to purchase today $(t_0=0)$ some protection from party B that

lasts until some specified maturity date $T.$ To pay for this protection, party A makes a

regular stream of payments to party $B.$ The size of these payments is a fixed percentage

of the face value of the asset being insured and it is based on the yearly contractual

spread $w_{1y},$ which represents the percentage used to determine the payments' amount

over one year. The payments are made every $3$ months until maturity of the contract or

until a credit event occurs, whichever occurs first. Assume that the credit event occurs

as the first event of a Poisson counting process ${?}^2$ and hence default time is exponentially

distributed with parameter $.$ Denote the short rate with $r.$

The aim is to value the premium leg, i$.$e$.$ to write a mathematical expression for this

stream of payments taking into account both the appropriate discounting and the

probabilities of default events.

a$)$ Illustrate the problem with a sketch representing the various payments occurring over

the considered time period. Make sure you include the time at which the payments are

made and the size of each undiscounted payments.

b$)$ Express the discounting factor at time $t_i,$ where iin$\{0,1,$dots,$N\}.$

c$)$ Express the probability that a credit event occurs before time $t_i(P_{t_i}^D)$ and the survival

probability at time $t_i,$ i$.$e$.$ the probability that no credit event has occurred before time $t_i$

$(P_{t_i}^{ND}).$

d$)$ Using the above, write down the full expression for the premium leg.

e$)$ Using the values provided for your chosen pack, calculate the premium leg and price the CDS$.$Question $1$

unt both the appropriate discounting and the probabilities of default events.

Va $35000$ pounds

T $($months$)30$

W$1$y$.-0\mathrm{.}0165$

$\backslash$lambda $-\mathrm{.}02820$

a$)$ Illustrate the problem with a sketch representing the various payments occurring over the considered time period. Make sure you include the time at which the payments are made and the size of each undiscounted payments.

$)$ Express the discounting factor at time t$,$ where i in $\{0,1,...,$ N$\}.$ b

c$)$ Express the probability that a credit event occurs before time t$($PD$)$ and the survival probability at time t$,$ i$.$e$.$ the probability that no credit event has occurred before time ti

$($PND$).$

d$)$ Using the above, write down the full expression for the premium leg.

e$)$ Using the values provided for your chosen pack, calculate the premium leg and price the CDS$.$