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10

Note that the Ito integral of a deterministic integrand $f: \mathbb{R}_+ \rightarrow \mathbb{R}$ is normally distributed $$\int_0^t f(u) \mathrm{d}W_u \sim \mathcal{N} \left( 0, \int_0^t f^2(u) \mathrm{d}u \right).$$ In your case, we have $f(t) = e^{-\lambda t}$ and thus \int_0^t f^2(u) \mathrm{d}u = \...

7

Risk-neutrality isn't really a property of a model. Instead, it describes a certain calibration of a model (almost always represented by an SDE). We say a model has been calibrated to risk-neutral probabilities if model parameters can be inferred from traded security prices, and there's some anti-arbitrage assumption and hedging scheme available for those ...

6

Short rate models are broadly divided into equilibrium models and no-arbitrage models. The models from Vasicek, Dothan and Cox, Ingersoll and Ross are examples of equilibrium short rate models. The models from Ho-Lee, Hull-White and Black-Karasinski are no-arbitrage models. Take Vasicek and Hull-White as an example. The short rate processes are $\mathrm{d}... 5 The original Vasicek paper is "An equilibrium model of the term structure". If you google for it, you'll find it and you can read in his own words his motivation for developing it. In particular, what now is called the Vasicek model basically comes from applying his results to an Ornstein-Uhlenbeck model for the spot process, which he claims was proposed by ... 5 Note that you can understand the$\Delta$as an "operator" acting on$r$. So just act on$r$twice: $$\Delta^2 r_t = r_t - 2 r_{t-1} + r_{t-2}.$$ In fact if you write the$r$as a vector,$r = (r_1, r_2, \ldots, r_N)$, then$\Delta$is an$N\times N$matrix with elements$\Delta_{i,j} = \delta_{i,j} - \delta_{i-1,j}. The AR(2) model can be written as $$... 5 Apply the Ito product rule, noting the cov of a deterministic and stochastic term is zero:$$\begin{align} d\left(e^{at}r_t\right)&=e^{at} dr_t+r_t de^{at} \\[6pt] &=e^{at} dr_t+r_t e^{at} d(at) \\[6pt] &=e^{at} dr_t+r_t e^{at} a dt \end{align}5 Intuition is just that the bond price by definition is a convex function of the rates, and the expectation of a convex function increases with volatility. Note that this result is model independent. 4 The claim that interest rates don't follow long term trends is not consistent with observed data. The idea of mean reversion is that interest rates do not rise or fall without bound, but are limited by economic and political factors. But there is no indication that this oscillation of short rates should happen around a constant mean. Allowing the mean ... 4 We consider the first one, that is, X_t = X_s + \mu (t-s) + \sigma (W_t-W_s), for t>s. Then, \begin{align*} P(X_t \le y \mid X_s) &= P(X_t-\mu(t-s)-X_s \le y-\mu(t-s)-X_s \mid X_s)\\ &=P(\sigma(W_t-W_s) \le y-\mu(t-s)-X_s\mid X_s)\\ &=\Phi\left(\frac{y-\mu (t-s) -X_s}{\sigma\sqrt{t-s}}\right). \end{align*} That is, \begin{align*} P(X_t \... 4 Calibrate to many observed curves, over all kinds of shapes: flat, normal, inverted, and humped, and measure and compare the model fitting errors. If you can't find all the shapes in history, make them up as possible scenarios (stresses). Classical Vasicek and CIR are parsimonious (have few parameters), so not very good at properly matching today's full ... 3 The paper continues "The quantity p(Y) provides the loan default probability under the given scenario." But the default probability is 0.001, not 0.999 as in the IRB version. So G(0.999) = -G(1 - 0.999) and that is where the minus comes in. 3 Yes you can! Any SDE that has an analytic solution can be simulated exactly. The vasicek model has dynamics dr=a(b-r)dt+\sigma dW_t. By Ito's lemma,d\left(e^{at}r\right)=e^{at}\left(a(b-r)dt+\sigma dW_t\right) +a e^{at} r dt$$Simplifying,$$d\left(e^{at}r\right)=e^{at} ab +e^{at}\sigma dW_t$$Integrating,$$e^{aT} r_T=r_0+b(e^{aT}-1)+\sigma \int_0 ... 3 The derivation in Appendix A of the paper Valuation of Equity-Indexed Annuities under Stochastic Interest Rates that you mentioned is Wrong: the Girsanov transformation is applied to ann$-dimensional Brownian motion, where the components are independent. However, for the case here with$n=2$, the Brownian motions are dependent, we can not naively combine ... 3 Vasnicek by itself does not specify what form the change of measure should be and how you should parameterise the market price of risk. A very natural parameterisation is affine in the factor, i.e., $$dW^* = dW + (\lambda_0+\lambda_1 r) dt$$ where$W$is the Wiener process under$\mathbb{Q}$and$W^*$for$\mathbb{P}$. Effectively, under$\mathbb{P}$... 3 You know the bond price formula takes this form:$P \left( t, T \right)= A \left( t, T \right) e^{ -r_{t} B \left(t, T \right) }$Now apply Ito's lemma, so you will get after some manipulation:$\frac{dP}{P}= \left(\frac{1}{A} \frac {\partial A}{\partial t} -r \frac {\partial B}{\partial t} - \kappa \theta B + \kappa r B+ \frac{1}{2} B^2 {\sigma}^2\right)...

3

You can first compute the average PD - few choices would be: Simple average of the individual PDs Exposure weighted average of the PDs If the PDs range is too large, then you might want to bucket them and apply the Vasicek formula to each bucket - this is how Basel approaches it. Once you have the PD and LGD, then you can solve for the correlation. This is ...

2

Vasicek model has parameters, which allow it to be calibrated to market prices (this means it becomes risk-neutral) or, if you'd like to, to history (and it becomes real-world model). Example of calibration to history see here: http://www.sitmo.com/article/calibrating-the-ornstein-uhlenbeck-model/ Me thoughts on calibration to market see here: http://...

2

Your SDE has no closed-form solution, so you'll have to apply the Euler method to obtain an approximate terminal distribution. Once you have the terminal distributions, any time series you want to validate has a highly multivariate probability density (due to the fact that each day's data comes from a slightly different distribution). You can transform ...

2

For starters, the short rate model you mention in equation (1) is Cox-Ingersoll-Ross while the bond price in equations (2)-(4) correspond to the Vacisek model. So there is a problem somewhere, I would go for a typo in (1). Second, what you wrote seems fine to me, so there must definitely be yet another typo in your solution manual. Note that if there is no ... 2 In practice, you can calibrate to either 1 month libor or 3 month libor, but not both. That's because there's a basis swap between 1 month libor and 3 month libor that can't be explained by your model. 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 that symbol means "the min of". So for example, if:s<t$, then$s$^ t = s. If you look in any book for the Covariance of a BM, you will see that same symbol and how to work with it. Cheers. 2 Here is a derivation for the Ornstein-Uhlenbeck process. Solution to the SDE $$dX_t = \theta(\mu-X_t) dt + \sigma dW_t$$ subject to the initial condition$X_0=x$has the form $$X_t= \mu + (x - \mu)e^{-\theta t} + \sigma\int_0^t e^{-\theta (t-s)}dW_s.\qquad$$ We need to calculate density function$p(t,x,y)$of the conditional distribution$(X_t|X_0=x)$.$...

2

$\theta(t) - a(t) r(t)$ is the risk neutral drift. The Hull & White models posits the dynamics $dr(t) = (\theta(t) - a(t) r(t)) dt + \sigma dW(t)$ under the risk neutral measure $P$ and then calibrates $\theta(t)$ so that the risk neutral condition $$E^P\left[e^{-\int_0^T r(u) du} \right]=\text{discount}(T)$$ is satisfied and $P$ is indeed the risk ...

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

there are many ways to solve Vasicek system, for me personally I markov short rate approach. Without going into the details of proofs: Note that eurodollar future is calculated under risk neutral Q measure of libor rate at each settlement $t_{fix}$ (on three months interval each) libor rate $l(t_{fix}) = \frac{1}{tenor} e^{A_{diff} - B_{diff} * r(t_{fix})}... 2 Although it's been a long time this question has been asked, I'd like to propose an answer in case someone was looking for the same thing. First, I think there's a confusion between$P(t,T)$and$DF(t,T)$. The former is the$t-$price of a contract paying$1$unit of currency at date$T$while the later is the (stochastic) discount factor at$t$for flows ... 2 Let$r(s)$be the process of a short rate. Then, by risk neutral pricing, $$P(t,T) = \mathbb{E}^\mathbb{Q}\left[ \exp\left( -\int_t^T r(s)\mathrm{d}s\right) \Bigg| \mathcal{F}_t\right].$$ Thus, the zero-coupon bond is determined completely by the short rate process. Here,$P(t,T)$denotes the time$t$price of a zero-coupon bond maturing at time$T\$. You ...

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