Serving the Quantitative Finance Community

 
User avatar
Interestrate
Topic Author
Posts: 0
Joined: February 24th, 2011, 3:55 pm

Calibration Hull-White model

March 31st, 2011, 7:41 pm

I'm trying to calibrate the 1factor Hull-White model to the current swapcurve. From that I've found the short-rates at future maturities:dr = (theta(t) - a*r)*dt + sigma * dzThen I've found bond prices at time t (2 to 10 years):P(t,T) = A(t,T)e-B(t ,T )r(t )Where I've used the different short-rates as r(t). The closed-form formula for discount bond options:c = P(0, s)N(h)- XP(0, T)N(h-sigma )But this is the optionprice at time 0, where P(0,s) and P(0,T) also are the time 0 values. If I use the bondprices at time zero, what is the point with the bondprices at future maturities ? I thought that the point with the model was to incorporate the termstructure ? Is the next step then to calculate the time 0 optionprices and compare them with optionprices in the market ? Anyone ?
 
User avatar
list
Posts: 0
Joined: October 26th, 2005, 2:08 pm

Calibration Hull-White model

March 31st, 2011, 11:03 pm

In this general form it looks like at any t < T interest rate r can be negative with positive probability. It implies that with positive probability P ( t , T ) > 1 that can be somewhat confused one who ca not imagine such scenarios .
 
User avatar
bearish
Posts: 5186
Joined: February 3rd, 2011, 2:19 pm

Calibration Hull-White model

April 1st, 2011, 12:18 am

@list -- I admire your ability to take an ill-posed question and provide an answer that can only contribute further confusion. This is the first time you have come across Gaussian interest rate models?As for the OP, your question is not clear. At all. What are you actually trying to do?
 
User avatar
list
Posts: 0
Joined: October 26th, 2005, 2:08 pm

Calibration Hull-White model

April 1st, 2011, 5:12 am

QuoteOriginally posted by: bearish@list -- I admire your ability to take an ill-posed question and provide an answer that can only contribute further confusion. This is the first time you have come across Gaussian interest rate models?As for the OP, your question is not clear. At all. What are you actually trying to do?Sorry, I do not interpret 'OP'. Concerning my remark about r ( t ) we can write the exact solution though it looks similar to Gaussian process which takes values from - infinity to + infinity regardless of the drift. It seems that negative r is quite difficult to observed.
 
User avatar
Interestrate
Topic Author
Posts: 0
Joined: February 24th, 2011, 3:55 pm

Calibration Hull-White model

April 1st, 2011, 6:13 am

I'm trying to calibrate the model to market prices. I've calculated the shortrate for the swaprate-maturities, and I've got market prices for caplets. My plan is to minimize the squared differance between the obesrved prices and the prices fomr the HW model. But how do I find the prices for caplets in the HW model ? Do I calculate todays value of the caplets, or future values ?
Last edited by Interestrate on March 31st, 2011, 10:00 pm, edited 1 time in total.
 
User avatar
bearish
Posts: 5186
Joined: February 3rd, 2011, 2:19 pm

Calibration Hull-White model

April 1st, 2011, 3:40 pm

If you want to calibrate/test the model with today's market caplet prices you need to compare them to today's model caplet prices. You find these by computing the risk neutral expected discounted caplet pay-off based on the model parameters.OP = original poster. And yes, Gaussian models have always had an issue with negative interest rates. People still use them at times, largely because they are extremely analytically tractable and perhaps justifiable as a local approximation to a more reasonable model. Or just out of sheer laziness.
 
User avatar
list
Posts: 0
Joined: October 26th, 2005, 2:08 pm

Calibration Hull-White model

April 1st, 2011, 4:20 pm

QuoteOriginally posted by: bearishIf you want to calibrate/test the model with today's market caplet prices you need to compare them to today's model caplet prices. You find these by computing the risk neutral expected discounted caplet pay-off based on the model parameters.OP = original poster. And yes, Gaussian models have always had an issue with negative interest rates. People still use them at times, largely because they are extremely analytically tractable and perhaps justifiable as a local approximation to a more reasonable model. Or just out of sheer laziness.One of the possible scenarios after calibration can be sound as following. Parameter of the model are adjusted and we have say 95% 'consistency' between historical data and our model within a period and say 7% that at some t within this period r < 0. Then what?? Such point suggests calculating the probability of the event when the model does not make sense and include it in disclaimer of the product when before getting money from the clients.I recall a joke that somewhat related to "extremely analytically tractable and perhaps justifiable". One walking around a street lamp searching something. Other one approaches and ask "What are you doing right here?" First answered, I looking for lost keys. Second and where you lost them? First turned and pointed at a dark place somewhere behind. and why are you looking here? Here is much lighter.