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1

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 $\sigma=0.$


1

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 $$u(r(t),t)=\mathbf E[u(r(s),s)\big|r(t)],\, \forall t<s. \tag{1}$$ 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, ...


1

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!


0

Let $\sigma(F,K)$ be the SABR implied vol. In the shifted model, the formula essentially becomes $\sigma(F+x,K+x)$ (you have to shift the strike as well). So to answer your question in the ATM vol calibration you take $K=F$ in order to have $F+x=K+x$. There is no need to "reconcile" anything as it is just a model. Once you have your model, you have to ...


1

He meant to compute the swaption price given by (2.4a), that is, \begin{align*} V_{opt} = L_0 E\left((R_s(\tau)-R_{fix})^+ \mid \mathcal{F}_0 \right). \end{align*} Under the swap measure (i.e., with $L_t$ as the numeraire), the swap rate process $\{R_s(t), t \ge 0\}$ is a martingale, and is assumed to be of the form \begin{align*} R_s(\tau) = R_s^0 e^{\sigma ...


0

The best option would be to bootstrap a curve. But lacking shorter interest rates, this doesn't appear to be possible. Linear interpolation is the next best alternative. Since the BUBOR rates are all annualized, and so long as you're using the portion of the year for the given leg's day count convention, using the given rate (in this case the bootstrapped ...


1

Let's recall the definition of a Martingale first: it is a stochastic process $X(t)$ that has the following property: let $0 \leq t < T$ two real numbers. Let $\mathcal{F}_t$ be a filtration for the process $X$ at time $t$. We have then: $$ \mathbb{E}[X(T)|\mathcal{F}_t] = X(t) $$ Now, if you use Black's model, you describe your asset price using a ...


0

The result depends on where you see 3y and 4y forward rates. 1 year forward rate is $7.01\% = (1+6\%)^2/(1+5\%)-1$ 2 year forward rate is $9.03\% = (1+7\%)^3/(1+6\%)^2-1$ If you assume that forwards are flat after 2nd year: 3yFwd=4yFwd=2yFwd = $9.03\%$ then your 5y spot becomes $((1+5\%)*(1+7.01\%)*(1+9.03\%)^3)^{1/5} - 1 = 7.81\%$. If you assume that ...


2

Who knows what the 5 year zero coupon rate is in that case, there could be an event 4.5 years out that will have serious interest rate implications that we don't know about. The only thing you can do with these three numbers is extrapolate and say the rate should 9%. You should be aware of what assumptions you're making when you do something like that, but ...


1

I edit this answer to give more details. The process for $r$ above is the geometric Brownian motion (GBM) used to model stock prices in the Black-Scholes framework. Thus the question is about (the expectation of the) exponetial of the integral of GBM. The intergral of GBM is closely connected to Asian options. Thus one can study the literature about this ...


0

If you model the spot price of the stock, then it is just the spot price (what else could be more accurate?). If you model the forward price of a stock, then you most probably want to apply cost-of-carry (in order to avoid arbitrge). If there are no dividends in your spot, then the forward price for time $T$ is $$ F_T = S_0 rT $$ where $r$ is a rate that ...


1

The risk free rate is used to get the present value of future payoff, so you should use the rate of a risk-free instrument (e.g. a Treasury note) that has roughly the same maturity of the option you are valuing. If you option expires in a time that does not have an exact Treasury instrument, you can get a rough approximation by interpolating between two ...



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