# Tag Info

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 ...

4

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}^{\...

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 $\sigma=0.$

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 In the BS model there is the upper bound of the stock price, which can be proven by the fact the stock price bounds the call option pay-off. Here we are seeing a similar effect: the discounted rate corresponds to the stock price. 3 Typically, strategists run a regression of changes in implied vols against changes in rates. If rates are highly directional with implied vols (regression coefficient is positive and statistically significant), then it would imply a more lognormal relationship. If the two series are not correlated or very weakly correlated, then the relationship is ... 3 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 ... 2 These are relatively common, especially in convertible bonds. You are correct that the effective maturity of the bond becomes the call/put date. The reason for issuing them is fairly prosaic: a 10 year bond with a 3 year call/put date counts as a 10 year liability for accounting purposes, and of course a 3 year instrument for trading purposes. The latter ... 2 One of the most used interpolation techniques is the cubic spline interpolation. Here you can find an overview of that, while, on Mathworks.com, you can find the tutorial to implement that in Matlab directly simply by using the spline(x,Y,xx) command function. It is not difficult to implement and, moreover, it gives pretty reliable results. I never tried ... 2 American options pricing (swaption is just a kind of option) is a bit tricky due to the early exercise. Here is a page listing possible approaches, including some numeric methods, and some close form approximation formula. As I understand, lattice methods (tree, PDE discretization such as forward shooting) are fine to price American options. There're ... 2 Thanks to my research leader, I found what I missed.$V_{0,1}$is vol of swaption that matures at$T_0$which is not 0 (as I thought), rather it is maturity of the first libor. So$V_{0,1}$is the closest available point on market. And now this is all clear with table on page 323 in section 7.4.$V_{0,2}$is realy vol of swaption that matures at$T_0$=1y ... 2 well just take the Bachelier formula with$r=d=0S_0 = S_{\alpha,\beta}$and then multiply by the annuity. The annuity will be $$\sum \limits_i \tau_i P_{i+1}.$$ where$P_{i+1}$is the df for$t_{i+1}.$2 Pricing via characteristic functions arises naturally in models that involve Levy processes. Therefore I can see how Black's formula for swaptions can be generalized for Levy dynamics: As in Black's model take the annuity as numeraire, and define the relevant measure$Q$Black assumes that under this measure the swap rate is martingale GBM, that is to say$...

2

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....

1

Exploiting an arbitrage is straightforward. Constructing and noticing one is the hard part. In your case if you know that Swptn(K,T1,T2)+Swptn(K,T2,T3) >= Swptn(K,T1,T3), Simply sell Swptn(K,T1,T2)+Swptn(K,T2,T3) and buy Swptn(K,T1,T3). Sell the most expensive and buy the cheapest. L.

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Actually, I want to have the calibration model to calibrate parameter such as "a" and "sig" based on swaption volatilities and market price of swaption. For the trinomial model, I can manage to implement it.

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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 ...

1

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 ...

1

from a practitioner perspective, i can say there's no such thing as a 0 year swap (obviously). The shortest tenor that you could trade would be a contract on one month LIBOR or more likely 3 month LIBOR. Then the instrument you are asking about is a 5 year expiration caplet (payoff in 5 years = max (0, LIBOR- strike).)

1

One can write for the payoff of an swaption $$\sum_i\tau_i P_{i+1}(S_{\alpha,\beta}(T_\alpha)-K)^+$$ and therefore the pricing equation follows Joshi's explainations. To derive the above equation use that the swap rate is given by $$S_{\alpha,\beta} = \sum_i \frac{\tau_iP_{i+1}}{\sum_i\tau_iP_{i+1}}F^i,$$ where $F^i$ are the corresponding forward rates. ...

1

Given that you have swap rates and Cap prices (ATM, I assume), you can back out the IVs for the time periods using by bootstrapping. Strictly speaking, you would need Caplet prices for the given strikes. In such a case, You would look at the shortest dated cap and (assume) it is made up of only one caplet. You can then use black's formula and back out ...

1

I'm not sure that machine learning would lead to any practical solutions here. Do you really have enough data for that kind of techniques? I would suggest a different approach: assume that the exercise is optimal, but just based on a different cost function than the expected pay-off. If you can find a function that replicates well enough the past exercise ...

1

Your question is too broad, but I there is plenty of examples of uses of machine learning to mimic human behaviour. For instance deep learning has been used 25 years ago to read checks in banks, or support vector machines 15 years ago to implement artificial vision, or bayesian networks to mimic expert diagnosis. I guess it would not be that hard to use ...

1

UBS launched a series of these around 1997-98. One needs to see the Offering Memorandum to see the full details and the reference trigger for the put and call. In the case of the UBS bonds (they were the ibanker, not the issuer), they issues 3/10 and 3/30 put-table and callable bonds. What they really were (and what these probably are): the bond buyer ...

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