Evaluation of interest rates options

Transcription

1 Evaluation of interest rates options Nicola Barraco Finanzmarktmodelle in der Lebensversicherung In collaboration with Generali Deutschland June 17, /16

2 Overview 1 Introduction Zero coupon bonds Coupon bonds 2 Forward rate agreement Interest rate swaps Caps and floors 2/16

3 Zero coupon bonds B(t, T )isthevalue@timet of a zero coupon bond (ZCB) with maturity T > t; z(t, T ) is the zero rate, if it satisfies one of the following equations: B(t, T )= 1 (1+z(t,T )) (T t) if (T t) > 1(e ectiveform) B(t, T )= 1 1+(T t)z(t,t ) if (T t) 6 1 (nominal form) 3/16

4 Yield to maturity Definition The yield to maturity (YTM) is the internal rate of return earned by an investor who buys the bond today at the market price, assuming that the bond will be held until maturity, and that all coupon and principal payments will be made on schedule The YTM satisfies the following equation: p(t) = Xn 1 i=1 c i (1 + i YTM ) T i t + N (1 + i YTM (t)) Tn t 4/16

5 Fixed coupon bonds The contract is specified by: time steps t < T 1 <...<T n T i T i 1 = a nominal value N a sequence of deterministic coupons c 1,...,c n, often c i = K N The price is given by the sum of the discounted CFs, i.e. p(t) = K nx i=1 B(t, T i )+B(t, T n ) N 5/16

6 Floating rate notes If the coupon payment is reset for every coupon period, then it holds: c i = 1 B(T i 1, T i ) By using the present value operator and by summing up over all time steps, it follows: p(t) =N B(t, T n )+ nx i=1 1 B(t, T i 1 ) B(t, T i ) = N B(t, T 1 ) 6/16

7 Overview 1 Introduction Zero coupon bonds Coupon bonds 2 Forward rate agreement Interest rate swaps Caps and floors 7/16

8 Forward rate agreement Definition A forward rate agreement (FRA) is an agreement to enter a loan at time T 1, but after fixing the lending/borrowing rate at time t < T 1.Thus,attimeT 2 > T 1 a fixed payment is exchanged against a floating payment based on the spot rate y(t 1, T 2 ) The 2 is given by N (T 2 T 1 ) [K y s (T 1, T 2 )]. In order to achieve a fair contract, following inequality has to hold: K = apple 1 B(t, T1 ) (T 2 T 1 ) B(t, T 2 ) 1 = F (t, T 1, T 2 ) 8/16

9 Forward rate agreement By considering a continuous time model, it holds: leading to: f (t, T )=F (t, T, T + t) ln(b(t, T B(t, T )=e R T t f (t,u)du. Question t = 0, T 1 =1/2, T 2 = 1, given B(0, 1/2) = 0.98, B(0, 1) = 0.95, on the market we observe F M =0.08; how to set up an arbitrage strategy? 9/16

10 Interest rate swap Definition A payer interest rate swap is based on the exchange of a payment stream at a fixed rate of interest for a payment stream at a floating rate, i.e.: at the coupon dates T 1,...,T n the holder of the contract: pays a fixed K N recieves the floating spot rate y s (T i 1, T i ) N The t-time value of the CF is thus given by N B(t, T i 1 ) B(t, T i ) K B(t, T i ) 10 / 16

11 Interest rate swap The total value of the swap at time t is thus: p (t) =N B(t, T 1 ) B(t, T n ) K For a reciever interest rate swap it holds r (t) = p (t). nx i=1 If the contract is fair, following rate has to be used: K = B(t, T 1) B(t, T n ) P n i=1 B(t, T i) B(t, T i ). 11 / 16

12 Caps and floors Definition A cap is a contract whose payo at every point in time T i is the face value multiplied by the positive di erence between the variable interest rate and the base level, i.e. V [r L (T i 1 ) L] +. It ensures protection against rising interest rates. Definition A floor is a contract whose payo at every time point T i is given by V [L r L (T i 1 )] +. It o ers protection against decreasing interest rates. 12 / 16

13 Caps and floors Proposition Given a set of equidistant time points of a cap or a floor with base level L and face value V with respect to the nominal interest rate r L, for the arbitrage free price of a cap it holds: apple V B(t, T 1 ) B(t, T n ) L nx B(t, T i ) 6 i=2 6 cap[r L, L, V, T ] 6 V [B(t, T 1 ) B(t, T n )] 13 / 16

14 Caps-floors parity Similarly to the put-call parity the following proposition holds. Proposition C T P T = S T K, Let r L be a nominal variable interest rate, L abaselevel,v the face value, then it holds: cap[r L, L, V, T ]=floor[r L, L, V, T ]+swap[r L, L, V, T ]. 14 / 16

15 Black s formula It is market practice to price a cap/floor according to Black s formula. For t < T 1,forthei-th cap payment it leads to: cap i [r L, L, V, T ]=V B(t, T i ) F (t, T i 1, T i ) (d 1 ) L (d 2 ), where it holds: d 1/2 = log F (t,ti 1,T i ) L ± 1 2 (t)(t i 1 t) (t) p T i 1 t 15 / 16

16 References Damir Filipovic (2009) Term-Structure Models - A Graduate Course Springer Finance. An Chen (2016) Lecture Notes 16 / 16