Quantitative Finance Collector is a blog on Quantitative finance analysis, methods in mathematical finance focusing on derivative pricing, quantitative trading and quantitative risk management.
Sep
6
The modelling philosophy for term-structure models is somewhat different to the modelling philosophy for equity models. In the latter case, stock price dynamics are usually specified under the physical probability measure, P, before their dynamics under an EMM, Q, are determined. For example, in the binomial Black-Scholes framework a unique Q is easily determined after the P-dynamics of the stock-price are given. Moreover, it is easy to check that the model does not allow any arbitrage: we just need d < R < u.
In contrast, with term-structure models we often assume that zero-coupon bonds of every maturity exists and it is not always easy to directly specify their P-dynamics in an arbitrage-free manner that it is economically satisfactory. For example, in a T-period binomial model there are O(T) zero-coupon bond prices that we need to specify at each node. Checking that the model is arbitrage-free and that bond price processes have suitable properties (e.g. implied interest rates are always non-negative) can be a cumbersome task. As a result, we usually work with term structure models where we directly specify an EMM, Q, and price all securities using this EMM. By construction, such a model is arbitrage free. Moreover, by leaving some parameters initially unspecified (e.g. short-rate values at nodes or Q-probabilities along branches in a lattice model) we can then calibrate them so that security prices in the model coincide with security prices observed in the market.
In the lecture notes of Term Structure Models-Spring 2005 professor Martin Haugh introduces how to price a Bermudan swaption with term structure lattice, precisely speaking, binomial tree, there he cailibrates both Ho-Lee and Black Derman Toy Model and use the calibrated interested rate model to price a Bermudan swaption as an example.
In contrast, with term-structure models we often assume that zero-coupon bonds of every maturity exists and it is not always easy to directly specify their P-dynamics in an arbitrage-free manner that it is economically satisfactory. For example, in a T-period binomial model there are O(T) zero-coupon bond prices that we need to specify at each node. Checking that the model is arbitrage-free and that bond price processes have suitable properties (e.g. implied interest rates are always non-negative) can be a cumbersome task. As a result, we usually work with term structure models where we directly specify an EMM, Q, and price all securities using this EMM. By construction, such a model is arbitrage free. Moreover, by leaving some parameters initially unspecified (e.g. short-rate values at nodes or Q-probabilities along branches in a lattice model) we can then calibrate them so that security prices in the model coincide with security prices observed in the market.
In the lecture notes of Term Structure Models-Spring 2005 professor Martin Haugh introduces how to price a Bermudan swaption with term structure lattice, precisely speaking, binomial tree, there he cailibrates both Ho-Lee and Black Derman Toy Model and use the calibrated interested rate model to price a Bermudan swaption as an example.
Aug
28
A swaption is an over-the-counter derivative on a swap. Normally, the underlying swap is a vanilla interest rate swap. Nevertheless, "swaption" could be applied to relate to a derivative about whatever kind of swap.
Swaptions could be European, American, or even Bermudan type. They can be physically settled, in which case a derivative is really participated into at exercise date. They can be cash settled as well, in which example the market price of the underlying swap is cleared at maturity.
it is frequently more handy to address in terms of two common kinds of swaption:
A payer swaption is a call option on a pay-fixed swap, the swaption holder has the right to pay fixed rate on a swap.
A receiver swaption is a call option on a receive fixed swap, the swaption holder has the right to receive fixed rate on a swap.
Swaptions could be European, American, or even Bermudan type. They can be physically settled, in which case a derivative is really participated into at exercise date. They can be cash settled as well, in which example the market price of the underlying swap is cleared at maturity.
it is frequently more handy to address in terms of two common kinds of swaption:
A payer swaption is a call option on a pay-fixed swap, the swaption holder has the right to pay fixed rate on a swap.
A receiver swaption is a call option on a receive fixed swap, the swaption holder has the right to receive fixed rate on a swap.
Jul
29
Calcualtes the price of a receiver swaption (bp).
http://www.vbnumericalmethods.com/finance/
http://www.vbnumericalmethods.com/finance/
