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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

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

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

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

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.


1

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.


1

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).)



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