Financial Engineering Case

Cass Undergraduate Programmes

Financial Engineering, FR3200

Professor John Hatgioannides

Chair in Mathemetical Finance and Financial Engineering Director, MSc in Mathematical Trading and Finance Faculty of Finance, Cass Business School

O¢ ce: 5069

Email: J.Hatgioannides@city.ac.uk

LECTURE NOTES 3

Plain Vanilla Interest rate Swaps, Forward Rate Agreements, Caps & Floors, Option on Discount Bonds and European Swaptions

1

VALUATION OF SIMPLE INTEREST RATE CON- TINGENT CLAIMS.

1. The Valuation of Plain-Vanilla Swaps and Forward Rate Agree-

ments

I A plain-vanilla interest rate swap is an agreement whereby two parties undertake to exchange, at known dates in the future, a Öxed for a áoating set of payments (often referred to as the Öxed and áoating legs of a swap).

I Letís approach the valuation of Interest rate swaps from an engineering perspective using replication.

I The Öxed leg is made up by payments Bi

Bi = NiX i; (1)

where Ni is the notional principal of the swap outstanding at time ti; i; usually referred to as the frequency or the tenor of the swap, is the fraction of the year between the ith and the (i + 1)th payment (therefore approximately equal to 0.5 for a semi-annual or 0.25 for a quarterly swap), and X is the Öxed rate contracted at the outset to be paid by the Öxed rate payer at each payment time.

I For a plain-vanilla swap each Öxed payment Bi occurs at the end of the accrual period, i.e. at time ti+1:

I If we denote by P (0; t) the price of a discount bond maturing at time t, the present value of each Öxed payment Bi is given by:

PV (Bi) = NiX iP (0; i+1): (2) I As for the áoating leg, each payment Ai; also occurring at time ti+1; is given by

Ai = NiRi i; (3) 2

where Ri is a shorthand notation for the period spot rate (i.e. the 3-month or 6-month LIBOR rate, for a quarterly or semi-annual swap, respectively) prevailing at time ti; and covering the period from ti to ti+1 : Ri = R(ti;ti + ): Times ti and ti + are normally...