Hot answers tagged pricing
6
You have the right intuition but the approach is not quite right.
The issuer has the right to call back the bond at a pre-defined call price. So your decision criterion is "call when the value of the bond >= contractual call price". We are comparing prices in the decision rule, not the YTM of the callable bond with the coupon of the bond.
Note that ...
5
I have hardly ever seen HJM in action - I have to admit this. Short rate models and LIBOR market models are more widespread in my experience.
But let me share the following thoughts:
In my mind the drift condition is the central statement of HJM. It says that given today's term-structure and the volatilities what the only arbitrage-free drift is.
A very ...
5
Numerical methods are only approximations. So binomial trees, Monte Carlo simulations, and finite difference methods should all produce different numbers.
As for whether you should install it for your customers, that can only be answered by what you think your customers want. Do retail customers really want the potential confusion? Are they going to ...
5
Not really a quant question, but a quick search led to this from the CME: http://www.cmegroup.com/market-data/files/CME_Group_Settlement_Procedures.pdf. Unfortunately it depends on the contract, for example:
Equity Futures: For S&P and NASDAQ, the settlement price of the lead* month contract is the midpoint of the closing range determined based on pit ...
4
CMS adjustments in single curve context can be roughly explained if you consider a CMS swaplet by the fact that there is a single payment at the CMS rate at a single date and not on the whole strip of the underlying CMS tenor schedule.
So if you are trying to hedge a CMS swaplet with the corresponding swap of CMS tenor length (with correct naïve nominal ...
4
A swap does not require a model because its price can be derived from the yield curve without any assumptions about how the yield curve may move in the future.
The PFE however is an indication of by how much the swap's mark-to-market may move between now and a moment in the future. It is of course influenced by how volatile rates are. The more volatile ...
4
The HJM model should be rather called HJM framework. Because: For specific choices of the forward rate's instantaneous volatility function $\sigma(t,T)$ you obtain other models from the HJM. The HJM is a super-set.
The rate curve is calibrate via the specification of $f(0,T)$.
The drift $\alpha(t,T)$ is - as always - confined by risk neutrality, i.e., it ...
3
I had to go and dig in one of the books I worked with in my grad studies which is particularly useful for Fixed Income: Term Structure Models by Filipovic.
The Chapter 6 is dedicated to the HJM models, and the most important theorem states the equation you mentioned in $(1)$ and that the discounted bond price go as follows:
$$\frac{P(t,T)}{B(t)} = P(0,T) ...
3
You have to look at the terms and conditions on your individual bond. The way the specifications usually work is that a call will result in accrued interest being paid, effectively making up for the lost coupon. Sometimes there's even an extra penalty. A put will result in a loss of coupon in almost all cases, and so is almost always done just after a ...
3
I have a little more informations, so let me share it with you.
Even though I think that the frameworks I presented in my question are both corrects (i.e. aribtrage free), it happens to be the case that the market seems to have more "structure".
Here is a methodology that allows to retreive market quotes and which is the same as BBG (which is the best ...
3
You should use the full yield curve, discounting cash flows at specific dates using the appropriate zero-coupon interest rate. As to which yield curve, that is often a matter of convention. Generally one uses the LIBOR/swaps curve for all but the most liquid products (in which case you use the treasury curve). The curve is constructed from LIBOR/Eurodollar ...
3
You assume that interest rates are never negative, however, all kinds of strange things happened already in the recent years, e.g. the Euroswiss futures traded above 100 in August 2011, SARON is negative (http://www.six-swiss-exchange.com/indices/swiss_reference_rates/reference_rates_en.html), German short term debt (Schatz) traded with negative yields this ...
2
Let's approximate the time to maturity to be 3 years and 10 months. Assume that coupon is paid on March 6 each year. Let face value $F=100$ and coupon $c=0.07375F$. Let the discount factor be $d(0,T)=e^{−r T}$ where $r=0.06535$. The price of the bond is
$$ce^{−10/12 \bullet r}+ce^{−22/12 \bullet r}+ce^{−34/12 \bullet r}+(F+c)e^{−46/12 \bullet r}=103.24 \; ...
2
Assume you have an USD-EUR Cross Currency Swap (3M-FloatUSD+SpreadUSD vs 3M-FloatEUR+SpreadEUR) (spread on USD side is usually zero), collateralized by USD-OIS (Fed Fund)
I assume you know the USD-OIS discount curve, then you know the discount curve for USD cash flows.
I further assume that you know the USD-3M forwards collateralized w.r.t. USD-OIS (from ...
2
The question should not be about the swaption pricing formula, its well established and widely accepted and utilized every single day. The question you SHOULD be asking, however, is which underlying volatility model you are using. Its idential to you questioning the use of B-S in transforming vols -> prices, and prices -> vols in the equity world while you ...
2
In my mind you are simply right: you arrive at
$$
f(t,S) = S(t) - K e^{-r(T-t)}.
$$
Assume that $t=0$, so we are at the inception of the contract, then
$$
f(0,S) = S(0) - Ke^{-r T}.
$$
If you choose $K = S(0) e^{r T}$ then the contract value at inception is zero. This simply means that the fair price for the forward is given by $K= S(0) e^{r T}$ which is ...
2
I would say
Start with Black Scholes to look at accuracy. In particular, you have a closed formula and you know what the characteristic function for lognormal is. Running FFT and comparing FFT pricing with the closed formula will give you an idea of what are the convergence issues, what is the behaviour at the boundaries (extreme strikes) etcetera.
Then ...
1
I don't know if I understand your question correctly but the procedure how to calculate ATM option prices with publicly available implied volatility indices (like VXO) for the vol parameter can be found in the mentioned paper on pages 5-7:
How Students Can Backtest Madoff’s Claims by Michael J. Stutzer (2009)
1
Here's an answer from a purely statistical point of view:
http://www.duke.edu/~rnau/regnotes.htm#constant
And another from Cross Validated:
http://stats.stackexchange.com/questions/7948/when-is-it-ok-to-remove-the-intercept-in-lm
The lean in both cases is to include the intercept unless there is a strong theoretical reason.
A more satisfying answer would ...
1
Time-series regression is not a great method for determining betas on individual securities. Rather, the most common method used by the commercial risk model providers is called "predicted beta" or "fundamental beta." The leader in this area is Barra. The way they define the predicted beta, it appears that they include the constant in the regression.
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