# Tag Info

9

In the beginning, we had a plot of yields of individual bonds against time to maturity, the crudest form of "yield curve." Years later, people began hand-drawing a smoothed line through these yields as closely as possible. Because bonds have different coupon rates, making their yields hard to compare, people tend to draw the curve through bonds trading ...

8

It's because of the settlement days you passed when you initialized the flat volatility curve. You're creating the spot, forward and flat volatilities as: boost::shared_ptr<BlackVarianceSurface> volatilitySurface( new BlackVarianceSurface(todaysDate, calendar, maturityArray, strikeArray, ...

7

Within the fixed income space, there's a lot of literature on PCA trading. The first 2-3 principal component factors (PCs) can typically explain 90-99% of the total variances in yield curve movement. It's also nice, because the first PC looks like a change in the overall level of the yield curve, the second PC looks like a slope change, while the third ...

7

I will refer to "Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit" by Damiano Brigo and Fabio Mercurio. In chapter 3 (One-factor short-rate models) they have a very nice table which lists some of the properties of instantaneous short rate models. In both of your models you know the distribution of $r_t$. The huge difference ...

6

There are two different issues at play here. One is that, of course, you want only the future cash flows to enter the calculation. This is taken care when you set the evaluation date to 6 months from today. In C++, you would say Settings::instance().evaluationDate() = today + 6*Months; I don't remember the corresponding function in QuantLibXL, but you can ...

5

The original Vasicek paper is "An equilibrium model of the term structure". If you google for it, you'll find it and you can read in his own words his motivation for developing it. In particular, what now is called the Vasicek model basically comes from applying his results to an Ornstein-Uhlenbeck model for the spot process, which he claims was proposed by ...

4

As a starting point to this, determining seasonality for a given market is as follows: i) Take several years of historical spot price time series, e.g. TTF spot prices. For year $i$ work out a yearly price $p_{yr,i}$ by taking the arithmetic average of daily spot prices. Do the same in respect of month number $j$ of the same year to get a monthly price $p_{... 4 I would look at the following metrics when quantifying "liquidity" in listed options: bid/offer spread number contracts traded and from that follows notional traded (in the option not underlying) frequency of bid/offer adjustments relative to changes in the underlying delta. frequency of liquidity added/removed on the bid and offer side even when no trades ... 3 What they are referring to is a very simplified version of the Ho-Lee model, i.e. on that assumes $$r(t)=r(0)+{\sigma}W(t)$$ where${\sigma}$is a constant (annualized StDev). For the sake of simplicity, imagine we are in discrete time and want to fit the model to observed (market) prices of bonds. We assume that$p=0.50$, i.e. the probability of interest ... 3 A few points can be noted. The CIR model is usually for a short, or instantaneous, spot rate$r_t, which is the forward rate over an infinitesimal interval. That is, \begin{align*} r_t = \lim_{\Delta \rightarrow 0}\frac{1}{\Delta}\left(\frac{1}{P(t, t+\Delta)}-1 \right), \end{align*} whereP(t, u)$is the price at time$t$of a zero-coupon bond with ... 3 VXV is a 3-month volatility index, and is currently not tradable (there are no futures on it). And since you cannot trade it, you cannot arb it. 3 This is indeed a standard result. You can convince yourself by noticing The bank account grows from 1 at$t=\tau$to$E\left[\exp(\int_\tau^T r(u)du)|\mathscr{F}_\tau\right]$at time$T$The price of a security paying$X$at time$T$discounted to$t=\tau$is then$E\left[X \exp(-\int_\tau^T r(u)du)\right|\mathscr{F}_\tau]$Hence the price of a credit risk-... 2 Hans Buehler investigated this in some detail, including in his doctoral thesis. When I tried it out some years ago, back when volatility exotics were more liquid, I found the models nearly impossible to calibrate to my satisfaction, even for the SP500 complex. I think the mathematical analogy is fair, and enjoyed Buehler's work, but in practice it won't ... 2 To say a curve is arbitrage-free, you need to pick an arbitrage path; a series of trades which, when followed, yield a net profit without creating exposure. We neglect counterparty exposure here, since you are presumably using market-neutral rates. One arbitrage is to buy a swap from your curve, and sell at the market price. This is a test of your curve ... 2 Nelson Siegel seems to be pretty standard too 2 Many term structure models-both single-factor and multifactor imply dynamics for the short-term riskless rate$r$that can be nested within the following stochastic differential equation:$dr = (\alpha + \beta r)dt + \sigma r^\gamma dZ. $These dynamics imply that the conditional mean and variance of changes in the short-term rate depend on the level of$...

2

Here is a general proof for all parameters in an open domain. $$dr = adt+bdW:=r\big(k(\theta-x)+\frac12\sigma^2\big)dt+\sigma rdW.$$ Let $$u(r(s),s):=e^{-\int_t^sr}B(r(s),s,T)=:\phi(s) B.$$ Then $$u(r(t),t)=\mathbf E\big[u(r(s),s)\big|r(t)\big],\, \forall t<s. \tag{1}$$ So, by Ito's Lemma, \begin{align} du(r(s),s) &= Bd\phi +\phi dB \\ &= \phi \...

2

The SVI is simply a function (empirically fit to the data) which given a maturity and a strike price K, computes a BS implied volatility $\sigma$. Once you have that implied volatility you can plug it into a Black Scholes routine which can compute the BS price and the Black Scholes greeks. Note that if an option is actually traded with that strike and ...

2

By the looks of it your table is cumulative PD. You can use the argument: $$P(\text{Default by end year X}) = P(\text{Def. by end year X-1}) + P(\text{Not def. by end year X-1})P(\text{Def. in year X})$$ So that, $$P(\text{Def. in year X}) = \frac{P(\text{Def. by end year X})-P(\text{Def. by end year X-1}) }{1-P(\text{Def. by end year X-1})}$$

2

You can understand convexity by working out a simple example numerically yourself. Consider two bond portfolios: P1= consists of a 6 year zero coupon bond. P2= half in a 2 year ZCB, half in a 10 year ZCB. By construction the two portfolios have same duration but different convexity. Now analyze what happens to price in 4 cases: small increase in yield, ...

2

Think of a zero coupon bond - the pv_zero (t years) $= \frac{\rm{pmt}}{(1+r)^t}$ As t increases the compounding effect of that discount increases (the larger the price change) As for rate vol - convexity brings about a couple of preferable properties - as rates decrease (rates rally), bond a and bond b which have = duration, but bond a high higher ...

2

It should be time dependent and set to the spot forward rate $= -\frac{\partial}{\partial t} \ln(\text{discount}(t))$ when simulating in continuous time. When discretizing the simulation use the forward rate $= -\frac{\ln(\text{discount}(t_{i+1})) - \ln(\text{discount}(t_{i}))}{t_{i+1} - t_{i}}$ from one time pillar $t_i$ to the next time pillar $t_{i+1}$.

2

In my understanding, the mortgage prepayment option, at any point in time, is a function of the value of the mortgage from that point in time forward. This value, in turn, is a function of the future evolution of the interest rates and any optimal decision taken by the mortgagor along that path and all paths that evolve from any future 'branch'. So in ...

1

Black-Scholes does not really require a constant interest rate. For a european option with maturity $T$ the only rate involved is the zero coupon rate for maturity $T$. The theory behind this comes from working under the $T$-forward measure (the risk neutral measure associated with the zero coupon bond as numeraire). The only subtelty is that the model ...

1

In the typical HJM model, during a single time step forward rates evolve according to their individual volatilities and according to pairwise correlations which can be specified. That arrangement is arbitrage free within the time step. In addition, different time steps are independently generated. This latter feature ensures that the model does not ...

1

Conceptually, the term-structure of interest rates (spot rate curve) is calculated with zero coupon treasuries. Nelson-Siegel will accomplish this. The yield curve is calculated with treasury Bonds, which pay coupons. While this is a good benchmark for examining the current state of interest rates, you will not gain much by comparing corporate bonds to this ...

1

We shall prove this by contradiction. Let $\theta=0$ and $\sigma=0$. $X_t=X_0e^{-kt}$ and $$B(0,t)=\exp\Big(-\int_0^te^{X_0e^{-ks}}ds\Big).$$ Suppose the contrary that $B(0,t)$ is affine. We should have $$B(0,t)=\exp{\left(A(0,t)-C(0,t)e^{X_0}\right)}\;\;\ \forall (t,X_0), \tag{1}$$ Differentiate the logarithm of Equation (1) with respect to $t$ side, \$...

1

On many occasions may the ATM volatility term structure implied from option prices exhibit non monotonicity. You could actually turn the question on its head and ask yourself why should it be monotonic? Does this reflect expectations & uncertainty about interest rates (exposure to rho?), event driven concerns about the underlying, or something else? ...

1

A PCA explains the variation in data. A slope PC is usually identified by the pattern of the signs of the loadings. If the loadings of short term contracts have the same sign which is different from the sign of the loading of longer term contracts then such a PC is identified as slope PC. It means that if this PC goes up or down it affects short term ...

1

Conventional wisdom would have it that the system would be arbitrage free if and only if: All the implied spot and forward rates on each curve are non-negative (I.e implied discount factors are monotonic non-increasing wrt maturity) All the implied spot and forward rates on the 3M curve are greater than or equal to the corresponding rates on the OIS curve (...

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