10

From an equities perspective, there are two concepts that should not be confused in my opinion and context should make the distinction self-explicit: Forward variance swap volatility (A) Forward implied volatility smile (B) I really recommend reading Bergomi's "Stochastic Volatility Modeling" which is an excellent book for equity practitioners. The topics ...


8

Practically, few things in real life have convenient closed-form calculations. Instead, you price some exotic, then you bump the various inputs, one or several at a time, up and down, by various small amounts, and re-price. There are seldom any short-cuts. (Autodiff can sometimes be a shortcut.) This Wikipedia article actually has a good list of commonly ...


7

Currently the USD 10Y swaprate is $2.93 \%$ and the ATMF 1Yx10Y implied volatility (relative) is $22.5 \%$ which corresponds to the Black model (absolute) volatility of about $4.15$ bp/day. The 1Y swaprate is $2.60 \%$ and the ATMF 10Yx1Y implied volatility is $25.0 \%$ which corresponds to the Black model volatility of about $4.10$ bp/day. With the ...


7

It is possible, yes, but it requires assumptions. But, philosophically speaking, this is the case as with all pricing, of any instrument. For example, given only the price of a 6Y and 7Y IRS can you correctly price the 6.5Y IRS rate? Well, yes you can, but it depends upon your assumptions about interpolation which is a subjective choice. Lets look ...


7

The CMS represents the value of a swap rate for any point in time, i.e. we are interested in extrapolating the density of the swap rate in a similar way as the IBOR rate. Let us start with the fair value of a swaption under the annuity measure $\mathcal{A}$ with tenor at time $\tau$: $$\mathcal{A}(t)\mathbb{E}^\mathcal{A}_t[(\mathcal{S}(\tau)-k)^+]$$ Instead ...


7

If they were a bank, or insurer, utility etc, then some regulator would likely encourage them do everything that others do, whether they like it or not, or whether it makes any sense. But if no regulator tells them to... and if if they don't have exposure to interest rates beyond 1 year... well, let's look at USD 1 year swap rate, for example, https://fred....


6

CME publishes its volatility surface daily on their FTP: CME Vol Surface. Unfortunately I know of no open APIs that would get you historical data. I'd recommend looking at Barclays Live or Morgan Markets if you don't have access to Bloomberg (volatilty data quality is higher on these dealer sites anyways).


6

I think you did something wrong in translating the input to numerics. As pointed out by dm63 normal vols are quoted in basis points. Using equation A.67a) from the Hagan paper you linked we see (setting $\beta = 0$) $$\sigma_N(K) = \alpha\frac{\xi}{x(\xi)}\left[1+\frac{2-3\rho^2}{24}\nu^2\tau_{exp}\right]$$ where $\tau_{exp} = 0.25$ in your example and ...


6

As the swap rate is not tradable, the delta hedge ratio with respect to the spot swap rate is not really useful. However, note that \begin{align*} V_0 &= \sum_{i=\alpha+1}^{\beta}\tau_i P(0, T_i)\big[S_{\alpha, \beta}(0)N(d_1) - k N(d_2) \big]\\ &= \sum_{i=\alpha+1}^{\beta}\tau_i P(0, T_i) S_{\alpha, \beta}(0)N(d_1) - N(d_2) k \sum_{i=\alpha+1}^{\...


6

There is nothing in simple cubic spline fitting routines that would prevent arbitrage. Even with conscientious use of knot points and smoothing techniques you may end up with simple spread and local volatility arbitrage conditions. Stochastic volatility models on the other hand can explicitly constrain your solutions to prevent call/ put spread arbitrage at ...


6

At most banks, swaption traders have models that allow non atm volatilities to be controlled by two parameters. Specifically , a parameter to control the smile (richness of out of the money options) and the skew (whether implied vol is upward or downward sloping as a function of strike ). Look up papers on the SABR model. In practice, one would ...


6

The advantage of cash-settled swaptions is that the payoff only depends on one variable: the corresponding swap rate which is directly observable in the market: $$ \mathrm{Payoff}(T) = f(S_T) = A^{\mathrm{Cash}}(S_T)\max(S_T - K,0) $$ The payoff of a physical swaption on the other hand depends on the physical annuity which is not directly observable. You ...


5

You are asking about the term structure of lognormal implied volatilities for European swaptions, which is a two dimensional function (expiration and tenor). First expiration: typically (but not always), implied volatilities are increasing in the 0 to 6 month sector, because the immediate future is often more predictable than the medium term. At some ...


5

In swaptions, there is the expiration of the swaption into an underlying swap. When the dealers provide the vol surface, in the first column, they typically put the expiry of the swaption from earliest to farthest. Along the top row, they put maturity of the underlying swap from shortest to farthest. So when the dealers describe the upper left having high ...


5

If the question is how one defines Greeks for interest rate options, then it is a relatively straightforward extension of the concept from the basic idea for say equity options. They are defined as sensitivities to the inputs that go into pricing an option. Any half-decent interest rate derivatives book (search for interest rate modelling on Amazon, say) ...


5

In a past life, I was an equity strategist at a sell-side bulge bracket firm. In 2008 (obviously) the bank decided to take a long hard look at the funding costs of its derivatives books. So they appointed an MD in IBD/corporate finance who would obviously lack “skin in that game”, and I was his chosen “research guy” (deliberately not then from a fixed income ...


5

There are no no-arbitrage conditions on ATM vols of swaptions with different expiries/tenors, because the underlying swaps forward rates are different instruments. There are conditions however for these vols to be compatible with specific IR models. For instance when calibrating a Hull & White model on a set of coterminal swaptions, it sometimes happens ...


4

Market practitioners do the following: Correlation is calibrated most often by looking at historical correlations between liquid par swap rate pairs. One could look at implied correlations within options on the yield curve (eg 10 yr minus 2yr) also. Swaption calibration should be done by comparing straddle prices in the market to prices produced by the ...


4

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 $\sigma=0.$


4

The market standard formula approximation for cash settled swaptions applies Black/shifted Black/Bachelier around the forward swap rate so that with this formula parity between payer and receiver swaptions occurs around the forward swap rate, and in particular the zero wide collar struck at the forward swap rate is worth zero (a zero wide collar is the ...


4

You can only infer forward vol by pairing a mid-curve option with a spot option. It's easier to go through an example (I'll use 5y x 5y vol since I have the sketch below handy...) One decomposition of the 5y5y spot vol is as follows: 1y forward 4y x 5y vol: this is the implied vol of an option starting in 1 year, expiring 4 years thereafter, and eventually ...


4

Using Taylor polynomials of 2nd order:$$V(r+h)\approx V(r) + \frac{\partial{V}}{\partial{r}}h +\frac{1}{2}\frac{\partial^2{V}}{\partial{r}^2}h^2$$ $$V(r-h)\approx V(r) - \frac{\partial{V}}{\partial{r}}h +\frac{1}{2}\frac{\partial^2{V}}{\partial{r}^2}h^2$$ The sum of the previous 2 equation will give us gamma as: $$Gamma = \frac{\partial^2{V}}{\partial{r}^2} ...


4

Forget for a moment that your option is delivering the immediate entrance in a swap (if the swaption is physically settled) or the cash amount of the swap (if the swaption is cash-settled), as your question doesn't depend on this fact, and take a "general" 1Y option. Your today's (date $t_0$) cube loses the "swap tenor dimension" and becomes a today's ...


4

I want to propose a different answer here. I think mathematical expectation (under any measure) is not used in valuing an interest swap. Years ago I used to explain swaps to beginners by speaking in terms of expectation (perhaps because that is how I learned it myself, although I am not sure). "We see in this example that the market expects future Libor ...


4

I’m guessing you are finding that your model overvalues Bermudan receiver options and probably undervalues Bermudan payer options. The rationale for this has more to do with supply and demand than theory. That’s because every time a callable bond is issued and swapped, dealers buy Bermudan receiver options, so there’s a huge supply. For Bermudan payers ...


4

This question has partially already been answered here. Let's do a simple example to illustrate the idea though. Take a 5y Bermudan callable S/A USD bond. How would you reconstruct the multi-call feature using vanilla swaptions? Well, the issuer can call every 6 months so they're long a 6m4.5y swaption on the first call date. For the second call date they're ...


3

The Black 76 swaption formula works for all these cases. The expiration time T= 1mo, 2mo or 3mo but the forward rate of the swap is the same in each case. The market will place different implied volatilities on these 3 options, according to the expectations of realized volatility in these 3 time periods.


3

Consider a payer swaption with maturity $T_0$ and strike $K$. Here the strike $K$ is the fixed rate paid on the fixed leg of the underlying fixed-for-floating swap with reset dates $T_0, \ldots, T_{n-1}$ and payment dates $T_1, \ldots, T_n$, where $0<T_0 < \cdots < T_n$. We assume that the swap exchanges the payments $L(T_{i-1}; T_{i-1}, T_i)\Delta ...


3

I'm giving no assurance that this model is rigorous/functional. It also appears that time steps are severely limited. In general, though, the only way to ensure that something is created well is to create it yourself. I have been burned by canned functions/models in the past, so I avoid them whenever able or if I'm doing anything that is actually important....


3

I have traded swaptions for many years. The answer is that it is not possible to calculate exactly the implied volatility for a European option on an amortizing swap from the matrix of non-amortizing swaption volatilities. This is because there is a dependence on the correlation structure in addition to the volatility structure. Depending on the nature ...


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