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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}...


3

Let $\mathrm{d}r_t=\mu(t,r_t)\mathrm{d}t+\sigma(t,r_t)\mathrm{d}W_t$ be a model for the short rate under the risk-neutral measure $\mathbb{Q}$. Starting from the bond PDE \begin{align*} P_t + \mu(t,r) P_r + \frac{1}{2}\sigma(t,r)^2P_{rr} -rP=0, \end{align*} subject to $P(T,T)=1$ whose general solution is $P(t,T)=\mathbb{E}^\mathbb{Q}\left[e^{-\int_t^T r_u\...


3

Yes, LIBOR rates can be simulated using short rate models. Or rather, Libor rates can be obtained from simulated short rate values. Usually, you have formulas giving you the zero-coupon bond price as a function of the short rate. For affine models for example, this would be of the form: $$P(t, T) = e^{A(t, T) - r(t)B(t,T)}$$ (for example, for the one-factor ...


2

I solved by myself. The following is this solution. Let $T-t=s$, one reaches the following equation. \begin{eqnarray} B'(s) + \beta B(s) + \frac{1}{2} \sigma^2 B(s)^2 =1 \end{eqnarray} One finds out it is the Riccatti equation because of $A(s)=0$. Therefore, one reaches the following equation. \begin{eqnarray} B' = - \frac{1}{2} \sigma^2 B^2 - \...


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I solved from (2) to (4) by myself ! (2) My answer Use the result of (1) with keeping in mind that the following R.H.S is $\mathcal{F}_t $ measurable. \begin{eqnarray} V_t &=& E^{\mathbb{P}} \left[ \exp \left(- \int^T_t r_s ds \right) \cdot ( P(T,S) - K )^+ \middle| \mathcal{F}_t \right] \\ &=& P(t, T) E^{ \tilde{\mathbb{P}} } \left[ ( P(T,...


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 ...


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 ...


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I am not quite sure I get your question. You cannot solve the model in closed-form. What you get is that \begin{align*} r_t=r_0e^{-a t}+\frac{b}{a}(1- e^{-a t})+\sigma e^{-at}\int_0^te^{a u} \sqrt{r_u}\mathrm{d}W_u. \end{align*} Furthermore, you can get that $r_t$ follows a (non-central) chi-squared distribution and can compute the (conditional) moments of ...


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For Q1, Indeed the ratio of 2 zero coupon bonds associated with the forward is an exact lognormal process (Just apply Ito's lemma to the ratio, as you already know the dynamics of the 0 coupon bonds. You can disregard the drift term as the forward rate is a martingale in the bond forward measure.). The forward rate is then obtained by just adding a scalar, ...


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The average of simulated discount factors from the Hull-White model and market discount factor are the same in theory but very similar in the simulation due to numerical error. I draw one figure which compares two discount factors and shows their difference. red line : mean of simulated discount factors blue line : market discount factor green line : ...


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Note that \begin{align*} f(t, T) = f(0, T) + \int_0^t\alpha(u,T)du+\int_0^t\sigma e^{-a(T-u)}dW_u, \end{align*} where, based on this question, \begin{align*} f(0, T) = \int_0^T \theta(u) e^{-a(T-u)} du - \frac{\sigma^2}{2a^2}\big(e^{-a T} -1\big)^2 + e^{-a T} r_0. \end{align*} Note also that \begin{align*} \int_0^t\alpha(u,T)du &= \int_0^t\sigma(u,T)\...


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In practice, using sensitivity based methods (i.e. those methods @python_enthusiast mentioned in his comment) is still quite common, but it is going out of fashion. Given today's technical infrastructure (parallelisation, fast codes etc.) a risk simulatoin under full revaluation is - for most of the products - quite feasible in a risk controlling type Value ...


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quantlib does not seem able to handle negative rates The slides by Peter Caspers discuss 'QL_NEGATIVE_RATES" and related topics in a very accessible style.


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Analytically the Feller condition ($ 2 \kappa \theta > \sigma ^2)$ guarantees that the process doesn't become negative but this is not enough when you are simulating. Even if you choose parameters that satisfy the Feller condition, you still may have the problem of getting negative values inside the square root giving bad results. This is a consequence of ...


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If you don't want that your CIR process goes below zero, this condition should be satisfied: $$2k\theta>\sigma^2$$ Thus, you can't choose a very big value for volatility. It is limited by this condition.


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You can simply apply formula (3.4) in Brigo and Mercurio's book (page 56). There is a simple put-call parity for the prices of European-style options written on zero-coupon bonds, i.e. \begin{align*} \mathbf{ZBP}(t,T,S,X) = \mathbf{ZBC}(t,T,S,X) -P(t,S)+XP(t,T). \end{align*} The formula is kind of identical to the standard equity put call parity where you ...


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Treasury / OIS spread is simply the difference between a given Treasury bond's yield (typically the on-the-run Treasuries, like 2y, 5y, etc.) and the fixed rate on an OIS of a similar tenor. If you consider OIS to be a decent proxy for repo rates, the Treasury / OIS spread is a way of gauging how cheap / rich Treasuries are versus their funding. Typically, ...


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(Cumulative Integration Formula Replacing $du$ and $dB_s$) I have developed formulas to solve this by myself! \begin{eqnarray} \int^t_0 \int^u_0 dB_s \ du &=& \int^t_0 \int^u_s du \ dB_s \\ \int^T_t \int^u_0 dB_s \ du &=& \int^T_0 \int^u_s du \ dB_s - \int^t_0 \int^u_s du \ dB_s \end{eqnarray} Therefore, we can use the following formula ...


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(My answer) the Vasicek Bond Price and its Forward Price Recall the result of Exercise 5.2.(1) or Exercise 4.5.(10). \begin{eqnarray} P(t, T) &=& E \left[ \exp \left( - \int^T_t r_u du \right) \middle| \mathcal{F}_t \right] \\ &=& E \left[ \exp \left( - \int^T_t \left( e^{-bu} r_0 + \sigma \int^u_0 e^{-b(u-s)} dB_s \right) du\right) ...


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