Dear all,I am going to do my master thesis next year and would like to focused on Fixed Interest. Can anyone tell me what would be a reasonable topic in Fixed Interest ? Any interesting one ?Thank you so much.Nick

I think you mean "Fixed Income."I think the municipal bond yield curve in the US is an interesting topic that does not have a good literature.

a) Comparison of (US treasury) Bond options and swaptions ... when is one cheap/expensive relative to the other ... what are these options saying about the spread ... if we compare the smile/skews,are there opportunities away from the money?b) Bond options are often priced in terms of price vol (if the forward price of the bond is F, then dF = sigma*F*dW, but it is more natural to model rates, not prices. Since we know the relation F(y) between the forward price of the bond and the forward yield y, we could model the bond yield directly as dy = (drift)dt + a*y*dW(where teh drift term is needed to keep the model arb free), or as a CIR-type model dy = (drift)dt + a*(y^0.5)*dW,or some other model. The questions are i) when translated into forward prices, the model becomes dF = A(F)dW ... what is the relation of A(F) with the underlying yield model; ii) how does the volatility smile of in the bond yield models compare with the smile in terms of the yield vols?c) the perennial favorite (but deceptively nasty): when I strip the yield curve, I need to choose an interpolation method for interpolating the forward rates ... what interpolaiton scheme should I use?d) A popular FI model is the BGM model, which basically has the k-th caplet rate Rk evolving according to dRk = (drift)dt + sigk*Rk*dWkwhere (drift) is determined by making the model arbitrage free; the different dWk's are highly correlated, and in fact, are usually generated by one or two independent Brownian motions. If we examined the NORMAL BGM model, dRk = (drift)dt + ak*Rk*dWk,this must be a special case of the Hull-White model (with the same number of independent factors as the BGM model has ... usually one or two). What are the formulas for the connection?e) A major drawback to BDT and BK is that there is no closed form formula for zero coupon bonds ... this means in any finited difference (or "tree") implementation, one needs to work backwards from the final pay date, not the final exercise date. (I.e, on a one year option on a 20 year swap, one needs to start the grid at year 21, and work back to zero, instead of starting at year 1, and working back to zero). Can one derive a very, very accurate (it won't be perfect) closed form formula for the zero coupon bond prices?The BK model is dX = (theta(t) - kappa(t)X)dt + sigma(t)dWwhere X = log r,wher r is the short rate and we are using the money market numeraire: V(t,x) = E{exp{-int[t to T] r(X(t')) dt' * V(T,X(T)) | given X(t)=x} for any T>tUsually, once kappa(t) and sigma(t) are set, theta(t) is chosen to match today's discount curve: D(T) = Z(0;T) = E{exp{-int[0 to T] r(X(t')) dt' * 1 | given X(0)=logr(0)}kappa(t) and sigma(t) are usually set by "calibrating" the model to current swaption/caplet prices.IN the typical scaling of FI, sigma is O(epsilon), and all other quantities are O(1).f) It is often stated (erroneously) that Monte Carlo methods have an advantage over finite difference/finite element algorithms in high dimensions, because the error decays like O(1/sqrt(n)) as n, the number of points, gets large, while a k-th order finite difference/finite element scheme supposedly goes like n^(-k/d) ... yet one can construct finite element grids converge like logn/n ... (see Prof. Peter Monk, Math Dept, Univ of Delaware). Can one develop these deterministic finite-elements methods to replace MC and its associated noise?g) etc.