SERVING THE QUANTITATIVE FINANCE COMMUNITY

matnos
Topic Author
Posts: 2
Joined: September 17th, 2015, 8:31 am

### Free-boundary SABR (Antonov, Konikov, Spector)

Hi, due to current negative rates in economy, there were couple of papers from Antonov, Konikov and Spector who expanded the SABR model, and came with option pricing formula. Eg. in this paper they used 2 of their models: - Normal free boundary for geneneral correlation case and Beta = 0, which I implemented. The problem is, with rates below F, where the prices do not behave correctly, after F everything is OK. (see picture what I think) Example - Free boundary SABR (beta = [0,0.5) and rho = 0) with zero correlationHerethe first sqrt(KF0) would lead to 0 price at K or F = 0.Is there something I did overlook? Can you please bring a bit more of explanation in case I am wrong?Thank youmatnos

Alan
Posts: 10033
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Free-boundary SABR (Antonov, Konikov, Spector)

If the dotted line is supposed to be a plot $E(F_T-K)^+$ under the normal SABR model with zero correlation and free crossing of F=0, it looks reasonable to me.After all, one would expect the answer to tend to $F_0 - K$ for $K \rightarrow -\infty$ and yours does. Looking at (5) in the link,the obvious suggestion is that you have plotted (5) literally, which is the 'time value' not the option value. So just add back $(F_0 - K)^+$ tothe r.h.s. of (5) and I will guess everything then agrees.
Last edited by Alan on December 14th, 2015, 11:00 pm, edited 1 time in total.

matnos
Topic Author
Posts: 2
Joined: September 17th, 2015, 8:31 am

### Free-boundary SABR (Antonov, Konikov, Spector)

Thank You Alan!I really missed that point.. and was so sure, because K>F0 was ok, so I focused on analyzing behavior of other parts.Everything is ok now, thank you for senior member look.matnos

mtsm
Posts: 352
Joined: July 28th, 2010, 1:40 pm

### Free-boundary SABR (Antonov, Konikov, Spector)

Let's see if you are still interested in this tomorrow...

Lapsilago
Posts: 259
Joined: October 15th, 2004, 7:36 am
Location: Germany

### Free-boundary SABR (Antonov, Konikov, Spector)

You may want to take a look...http://papers.ssrn.com/sol3/papers.cfm? ... 647344This has implied vol Hagan expansion for free boundary SABR as well as the recent Hagan approach for NoArbSABR with PDE.Best,Lapsi

b23
Posts: 1
Joined: February 22nd, 2016, 3:13 pm

### Free-boundary SABR (Antonov, Konikov, Spector)

Hello everyone! I'm a student and I'm writing my master thesis on the subject of negative rates, and I would like to implement the mixed SABR (Antonov et al 2015 - http://papers.ssrn.com/sol3/papers.cfm? ... id=2653682). I would like to ask if you know if I have any chance to find on the web the matlab code (or the one of the Free boundary with zero/non-zero correlation). I'm new on matlab and it would really really help.In addition, there is one thing that I do not understand. The authors write: "we can define our model as a mixture of zero-correlation and a normal SABR SABR. Assume the forward rate Ft can be written asF = χ F(1) + (1-χ) (2),where? F (1) follows a zero-correlation Free SABR model with parameters (α, β, 0, γ)? F (2) follows a normal Free SABR model with parameters (α, 0, ρ, γ)? χ is a random variable taking value 1 with probability p and 0 with probability 1-p and independent of Both SABR processes.Clearly, this model is arbitrage-free and permits negative rates.Moreover, both of these component models have closed-form solutions for option values, and, consequently, so does the mixture model."but they do not show the final formula (closed) for pricing.I can't really understand and I don't know how to proceed!Thank you very much and sorry for my english!!

Wellwn
Posts: 21
Joined: June 2nd, 2014, 6:58 am

### Re: Free-boundary SABR (Antonov, Konikov, Spector)

@b23 Hi, did you come up with a solution?