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3

You don't need all the discount factors. You just need the currency basis swap market, which exists precisely for this purpose. For example if the 5 yr eur/usd currency basis is -25, it means that you can exchange a euribor-25 liability for a usd libor flat liability. These swaps also have an exchange of principal amounts at the start and end to convert ...


0

As You Know, CIR model is the square root process given by the following stochastic differential equation $$d{{r}_{t}}=\kappa (\theta -{{r}_{t}})dt+\sigma \sqrt{{{r}_{t}}}d{{W}_{t}}$$ Let $\Theta=(\kappa,\theta,\sigma)$. It is well-known that conditional on a realized value of $r_t$, the random variable $2c_t\,r_{t+\Delta t}$ follows a non-central ...


1

Anything that is used for discounting is by definition an "interest rate". But then the question arises what is the appropriate choice of interest rate to use for discounting pension liabilities. There are many possibilities (many interest rates). Some want to use the expected return on the stock market as the interest rate. That is a very bad choice ...


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You earn coupons on a corporate bond portfolio and in this sense corporate bond yield is an interest rate. But it is important (especially in liability driven investment) to recognise that corporate bond yield has two quite different components: credit spread and riskfree interest rate. To quote from Wikipedia Corporate bond: "High Grade corporate bonds ...


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Thank you for your answer @MarkJoshi. I followed you advice and achieved in deriving the approximation formula. However, I can not fully understand why the fact that Black's formula is linear in $\sigma$ for ATM strikes causes the Rebonato approximation only to be accurate for ATM strikes and not OTM and ITM strikes. I would be grateful if somebody can ...


2

You almost get there. However, you ca not conclude that $\rho^2$ is a constant based on $(10)$. Note that, from your $(7)$ and $(8)$, \begin{align*} \frac{\rho(z_t)^2}{\beta} e^{\beta \tau} (e^{\beta \tau} - 1) = -h'(\tau)+e^{\beta \tau}h'(0). \end{align*} Taking derivative with respect to $\tau$ on both sides, we obtain that \begin{align*} ...


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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 an $n$-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 ...


1

I think you are partially correct, but here's the way we look at these and we make markets in many of these products. There's a concept called "arbitrage-free pricing". Essentially there should no way to trade both sides of this and to make risk-free money. So let's take the front month gold contract, the June (GCM6). The first delivery date for it is ...


1

I'm going to try to answer my own question here and hopefully someone can come by and confirm I'm right before I accept my own answer. The key to a forward contract is there is no immediate exchange of goods or money. Just an agreement to do so at a later date. To incentivize the seller of the contract it's important to remember if the transaction ...


3

Here's my 2 cents: a) Conditional expectations can always be seen as martingales (this is a direct consequence of the tower property). Thus, we here have that $$ M_t := E^*[e^{-\lambda {r_{T}}}|r_t] $$ is a martingale. Applying Itô's lemma to $M_t = f_{\lambda}(t,r_t)$ as you did is a good starting point. But doing this, leaves you with an SDE, not a ...


3

Let $$ f_{\lambda}(t,r)=E^{(t,r)}\left[e^{-\lambda r_{T}}\right] $$ where $E^{(t,r)}$ denotes the expectation conditional on $r_{t}=r$. We assume $f$ is smooth for the remainder. Let $\theta=T\wedge\inf\left\{ s>t\colon\left|r_{s}-r\right|>1\right\} $. By the Markov property of $\{r_{t}\}$, $$ ...


0

"Calibrating the model to them via regression"... Vasicek has just three parameters: $\sigma$ (vol), $r_L$ (long rate) and $a$ (mean reversion) and a time-dependent short rate $r_0$. Can you specify what you are thinking of in terms of a regression model to estimate these? Obviously $r_0$ will not be a stable parameter. If you want to estimate $\sigma$, ...


2

From Equation (6), $B(t,T)=-t+c(T)$ for some function $c(T)$. $1=P(t,t)=e^{-A(t,t)-(c(t)-t)r_t}$ or $A(t,t)+(c(t)-t)r_t=0,\,\forall (r_t,t)$. So $c(t)=t, A(t,t)=0,\forall t$. For Equation (8) you have missed the square on $\sigma$ and a factor of $\frac13$. Then you just need to substitute in the function for $b(s)$ and integrate the following to get the ...


1

In order to define option price we should follow Black Scholes construction to construct riskless portfolio at t then to state that instantaneous rate of return of this portfolio equal risk free rate r ( t ) where r is a random on [ t , t + dt ] interval. We actually then arrive at the problem which could not be embedded in BS pricing world.


3

it certainly works best at the money. Why? I think it comes from the fact that Black's formula is approximately linear at the money. The approximation $$ \frac{1}{\sqrt{2\pi}} \operatorname{SR} \sigma \sqrt{T} A, $$ with $A$ the annuity is remarkably good. One way of deducing these formulas is to do an asymptotic/Taylor expansion about $\sigma=0.$


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 ...


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, ...


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 ...


0

I think (1) is the issue. You need to compare market normal vols to normal vols implied by the sabr model. (2) is not the issue - these vols look reasonable. By the way we express normal vols in bp per annum, not percent!



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