New answers tagged

1

You can simply use Ito's lemma under the risk neutral measure $Q$.For the log-bond price $p(t,T)$ this gives $$dp(t,T)=(A_t(t,T)-B_t(t,T)r_t)dt-B(t,T)dr_t$$ $$=[A_t(t,T)-(B_t(t,T)+B(t,T)a)r_t]dt-B(t,T)\sigma dW_t$$ Here $A_t(t,T)$ and $B_t(t,T)$ are partial derivatives wrt $t$ and $W_t$ is Wiener process under $Q$.


2

Try $f_0=-0.9975, f_1=2.9975, f_2=-3, f_3=1$. This should have 3i IRRs, namely -5%, 0 and 5% with the desired behavior between about -3% and +3%.


1

There is no closed-form solution, but solving for $r^\star$ such that $$f(r^\star) = \tilde{c}^{-1} $$ should be fast and safe with a standard single dimension solver, bisection or Newton-Raphson, as function $f$ is monotonically decreasing ($B_i$'s and $\tilde{c}_i$ are positive), $$f(x) = \sum_{i=1}^n \tilde{c}_i {\rm e}^{A_i-B_ix}, $$ its derivative is ...


1

Given the non-linear nature of the constrained optimization problem ie. $exp(A(T0,Ti)-B(T0,Ti)*r)$, you will need to employ numerical solvers. The authors of the document used Simulated Annealing (shown in Appendix B) for fast convergence. They note that it could take up to 10 seconds to solve a 10-dimensional parameter space.


0

The forwards and the spot rates will be decreasing, that is correct.


3

I feel that it depends on who's writing the research, and on their personal idiosyncratic preferences. For example, picking a random credit research piece, we see Asset-swap spreads in the secondary market widened modestly by 1.3bp and another random piece from the same team: Asset-swap spreads widened significantly at the beginning of March across ...


3

Let me add a couple of points. Question 1: in my experience, ASW spread always refers to the spread between a particular Bond and the IRS of the same currency. Most commonly, this would be a spread between government bond and the corresponding IRS. In a par-par ASW, you trade a fixed notional (say 500 million USD), whereby you swap the fixed cash-flow on the ...


1

I guess by "research desk" you mean a division at e.g. an investment bank and not academic research. To give others an impression of the credit market, practioners do not speak about one individual bond spread, they speak more generally about spreadS, meaning e.g. the EUR IG corporate bond market as a whole. The (credit) spread is a risk ...


3

Simple example: euro based investor wants to buy a USTreasury, currency hedged back into Euro. Investor executes the following 2 trades at t=0: purchase Treasuries for next day settle. Assume usd12mm purchase price. execute fx swap with cashflows at t=0 : receive usd12mm/pay €10mm and cashflow at t=1yr : pay usd12.0mm/ Rec €9.9mm. (I used spot =1.20 and ...


4

Question 1 I often read by research desk that ASW-spread have widened or tighten without concrete reference to the bond. As said above, the asset swap spread depends on the credit quality of the swapped bond. So is there a market standard for this? If people talk about USD, do they mean swapped Treasuries, in EUR swapped bunds? Which research papers, credit ...


0

I found the mistake... It was in the fwd rates, not in the DCF model. Thank you.


1

The futures/forward convexity adjustment for non-interest rate futures only tends to matter for futures with maturities greater than a year (which tend to be part of bespoke structures and not traded in size on screen). You can get a closed form solution in a GBM-Ho-Lee hybrid model if you don't mind grinding though some partial differential equation work ...


2

The futures/forward convexity adjustment comes from the covariance between rates and the index. For a future/forward that settles on an index $I_T$ on expiry $T$ the future price is $F_{\text{fut}} = \mathbb{E}^{\mathbb{P}}\left[I_T \right]$, where $\mathbb{P}$ is the risk neutral measure, and the forward price is $F_{\text{fwd}} = \mathbb{E}^{\mathbb{Q}^T}\...


3

I believe there's a Bloomberg manual in the help function which explicitly shows these calculations in a spreadsheet, very useful:


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