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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

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 ...

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

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 ...

5

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 $p_{... 5 In my understanding, the mortgage prepayment option, at any point in time, is a function of the value of the mortgage from that point in time forward. This value, in turn, is a function of the future evolution of the interest rates and any optimal decision taken by the mortgagor along that path and all paths that evolve from any future 'branch'. So in ... 5 I interpret your question to be asking about curve fitting techniques (for constructing fitted par/zero curves), since a term structure model (HW, LMM, etc.) can always be constructed to fit a given yield curve perfectly. In an institutional setting, there really hasn't been any new models being proposed, because the existing ones are all very flexible and ... 4 The SVI is simply a function (empirically fit to the data) which given a maturity and a strike price K, computes a BS implied volatility$\sigma$. Once you have that implied volatility you can plug it into a Black Scholes routine which can compute the BS price and the Black Scholes greeks. Note that if an option is actually traded with that strike and ... 4 Since contracts on physical goods have associated costs, it makes sense that the term structure curve would be upward sloping. Since there is no cost associated with delivery for the VIX and contango is considered to exist in healthy markets, is the upward slope simply accounting for the greater potential for the market to become unhealthy over longer ... 3 This is indeed a standard result. You can convince yourself by noticing The bank account grows from 1 at$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-... 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$tof 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 It really depends for what purpose you are using the model. Let’s say you are using it for valuation of some instrument. If you want the fair market value, then a) is irrelevant and you would instead calibrate to the current term structure. For hedging , one usually means hedging the market value so again b) is appropriate. The only reason to use a) is to ... 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 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 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}$. 2 Many term structure models-both single-factor and multifactor imply dynamics for the short-term riskless rate$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

Nelson Siegel seems to be pretty standard too

2

This is not an answer, but instead advice: Since you're new to quant and volatility then you should start with something other than a volatility or a rates product because those are going to be some of the most complicated products. First, you need to gain some comfort with quant stuff. Then, you can move onto more complicated topics. I'm not saying that ...

2

For what date is the chart derived? One definition for implied volatility skew is: (25 delta put implied volatility - 25 delta call implied volatility) / 50 delta. Can you test to see if this calculation for the options maturities is consistent with the values on your graph for the date in question? With respect to the vol beta, this appears to be a ...

2

Maybe you would like to take a look at Managing forward volatility and skew risk for a direct and robust relation between spot-volatility correlation/covariance and the implied vol skew in the context of (fractional) stoch vol models. Although the result is for forward start case, by letting the forward start date equal spot date it is valid for spot ...

2

By definition, the overnight rate is the rate at which banks lend to each other overnight. Overnight index swaps (OIS) allow banks to 'lock in' the cost of funding overnight for a specific term. They exchange a predetermined OIS rate for a payoff equal to the growth of the notional amount of money lent at the overnight rate for a specific term. The overnight ...

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 ...

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