Hello,I have five month to complete a MSc Thesis. I am interested in swaption/stochastic vol/SABR related topics. My math level for stochastic calculus is beginner to medium and I can understand most derivations in Shreve Stochastic Calculus in Finance II. Also have learned basic derivations for short rate/HJM/LMM/local volatility/SABR. I can code in VBA and have learned some C++ (have installed QuantLib on my PC)The issue is that there are huge amount of local vol/stochastic vol/LMM/SABR papers available (a lot of them are quite deep to me and I need to make some effort to understand). So I need suggestions about specific issues which are relatively new and may have interest from the practioners.thanks in advance!

Well, here is a threadMaybe you two should correspond. Another thought -- probably too ambitious for 5 months, but you might try to extend the SABR small-time asymptotic expansion to third/fourth order. (see Paulot)

Thanks Alan. I suppose differential geometry and Riemman manifold are prerequisite to this topic?

Yeah, geodesics on curved manifolds, etc: you'd have to get up to speed on that. (see Paulot's refs).It's always possible there is some brute force way of doing it -- it's just a small-time expansion of asolution to a relatively simple pde f_t = y^2 (s^2b f_ss + f_yy + 2 rho s^b f_sy) -- without the dif geom. machinery. I do know a double expansion for the option implied volatility in t and moneyness can be done brute forceand automated for b=1. You might do that, instead. But to do the t expansion, exact in the moneyness and for all b, which is more interesting, probably requires the dif geom.

Just out of interest: what is the need/rationale to model this on manifolds?

You need the small-time behavior of the transition probability p(t,x,y).Due to Varadhan, this can be written p(t,x,y) ~ exp{-d^2(x,y)/(2 t)} where d is the geodesic distance from x to yon a curved surface with metric g given by the inverse of the diffusion coef matrix. This leads to a verysimple formula for the leading asymptotic smile. Then, you want to develop corrections to that.See my notes, the last talk here

QuoteOriginally posted by: AlanYou need the small-time behavior of the transition probability p(t,x,y).Due to Varadhan, this can be written p(t,x,y) ~ exp{-d^2(x,y)/(2 t)} where d is the geodesic distance from x to yon a curved surface with metric g given by the inverse of the diffusion coef matrix. This leads to a verysimple formula for the leading asymptotic smile. Then, you want to develop corrections to that.See my notes, the last talk hereout of curiosity: are all relevant for finance models spaces of negative curvature?

QuoteOriginally posted by: frolloosQuoteOriginally posted by: AlanYou need the small-time behavior of the transition probability p(t,x,y).Due to Varadhan, this can be written p(t,x,y) ~ exp{-d^2(x,y)/(2 t)} where d is the geodesic distance from x to yon a curved surface with metric g given by the inverse of the diffusion coef matrix. This leads to a verysimple formula for the leading asymptotic smile. Then, you want to develop corrections to that.See my notes, the last talk hereout of curiosity: are all relevant for finance models spaces of negative curvature?No. For the SV(p) models discussed in the talk I cited, the Gaussian curvature takes the same sign as p - 3/2. Good question, though. I work this out and show some nice related graphics in my forthcoming Vol. II

Heston belongs to the Affine Term Structure model and has closed form solution.SABR is not affine in general, depending on beta and is restrictive in certain aspect.Equity option/vol smile is better modeled nowadays. Term structure of bond yield is difficult to model/predict as far as I know in academia.