By Daragh McInerney, Tomasz Zastawniak
This quantity within the gaining knowledge of Mathematical Finance sequence moves simply the suitable stability among mathematical rigour and functional program. latest books at the demanding topic of stochastic rate of interest versions are frequently too complicated for Master's scholars or fail to incorporate functional examples. Stochastic rates of interest covers useful subject matters akin to calibration, numerical implementation and version boundaries intimately. The authors supply a number of routines and thoroughly selected examples to assist scholars collect the mandatory talents to accommodate rate of interest modelling in a real-world atmosphere. moreover, the book's web site at www.cambridge.org/9781107002579 offers options to all the workouts in addition to the pc code (and linked spreadsheets) for all numerical paintings, which permits scholars to ensure the implications.
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Additional resources for Stochastic Interest Rates
T n−1 . The payer swaption payoﬀ is PSwpt0,n (T 0 ) = (PS(T 0 ))+ , and its value at time t is given by PSwpt0,n (t) = B(t)EQ (PS(T 0 ))+ Ft . B(T 0 ) The underlying payer swap can be expressed in a number of diﬀerent ways. 6, we have the price of the payer swap at the ﬁrst reset date T 0 (the swaption expiry) given by n PS(T 0 ) = B(T 0 , T i )τi (F(T 0 ; T i−1 , T i ) − K) i=1 n = 1 − B(T 0 , T n ) − K τi B(T 0 , T i ). i=1 Hence the swaption can be thought of as a put option with strike 1 written on a coupon-bearing bond with coupon rate K.
In addition to the bond price, it is also convenient to have the Hull–White short-rate process that gives an exact ﬁt to the term structure at time 0. 12) we can see that s f (0, s) − e−α(s−t) f (0, t) = s θ(u)e−α(s−u) du − σ(u)2 D(u, s)e−α(s−u) du 0 t t + σ(u)2 D(u, t)e−α(s−u) du. 9) becomes s r(s) = (r(t) − f (0, t)) e−α(s−t) + f (0, s) + t − −α(s−u) σ(u) D(u, t)e 2 σ(u)2 D(u, s)e−α(s−u) du 0 s σ(u)e−α(s−u) dW(u). 13). 14). 5). Bond option formula Consider a call and a put option with strike K and expiry S written on a zero-coupon bond with maturity T > S .
The ith caplet is a European option with expiry T i written on the spot LIBOR rate L(T i−1 , T i ) with payoﬀ τi (L(T i−1 , T i ) − K)+ . 23) 34 Vanilla interest rate options and forward measure The payoﬀ of the caplet becomes known at time T i−1 (when the LIBOR rate ﬁxes), and it pays at time T i . In the following we show that an interest rate caplet is equivalent to a put option on a zero-coupon bond. 23) at time T i is an FTi−1 -measurable random variable, and is therefore equivalent to the payoﬀ τi B(T i−1 , T i )(L(T i−1 , T i ) − K)+ at time T i−1 .