從對數正態波動率報價中獲取掉期期權價格

I am working with the following dataset from quandl: https://www.quandl.com/databases/CSWO (I’m using the sample dataset only). My question is how to obtain the swaption prices from the quotes given. The dataset gives me the following information for each contract:

1. Currency (in the sample data set only Australian dollars).
2. Option tenor. I will denote it with [Math Processing Error] $ T_{\text{option}} $ .
3. Swap tenor. I will denote it with [Math Processing Error] $ T_{\text{swap}} $ .
4. How much the option is in/out of the money given in basis points, e.g. P100 means that the strike is 100 basis points above the ATM strike (Is the ATM strike equal to the current forward rate [Math Processing Error] $ F(t; T_{\text{option}}, T_{\text{option}}+T_{\text{swap}})) $ for the time interval [Math Processing Error] $ [T_{\text{option}}, T_{\text{option}}+T_{\text{swap}}] $ ?)

我獲得掉期期權價格的方法如下：對數正態波動率報價意味著黑色掉期期權公式用於計算隱含波動率。公式是（例如參見https://courses.maths.ox.ac.uk/node/view_material/3748中的第 19 頁）：

[Math Processing Error]$$ V^{\text{payer swaption}}(t) = A(t)\left\lbrack R^(t)N(d_1)-RN(d_2)\right\rbrack $$ 在哪裡 [Math Processing Error]$$ d_1 = \frac{\log\left(\frac{R^(t)}{R}\right)+\frac{1}{2}\sigma^2(T_0-t)}{\sigma (T_0-t)}, \quad d_2 = d_1 -\sigma\sqrt{T_0-t} $$ 和 $$ A(t) = \sum_{i=1}^n\delta_kP(t,T_k) $$ 與付款日期[Math Processing Error] $ T_k $ 和 $ \delta_k = T_k-T_{k-1} $ （數據集文件中沒有提及付款頻率。我怎麼知道使用的頻率？）。[Math Processing Error] $ T_0 $ 是可以行使交換權的時間。 $ R $ 是交換的罷工和 $ R^*(t) $ 是該時間段的遠期匯率 $ [T_0,T] $ 在哪裡 $ T=T_n $ 是互換到期的時間。現在的問題是如何在上述公式中插入給定的數據。我會這樣做：

1. 為了 $ R^*(t) $ 選擇從掉期開始的掉期利率[Math Processing Error] $ T_{\text{option}} $ 並成熟於[Math Processing Error] $ T_{\text{option}}+T_{\text{swap}} $ .
2. 放 $ R=R^*(t)\pm \text{basis points offset of ATM strike} $ .
3. 為了[Math Processing Error] $ \sigma $ 選擇引用的隱含波動率。
4. 計算 $ A(t) $ 首先必須引導一條零曲線並獲得 $ A(t) $ 從那個零曲線。我用什麼工具來引導零曲線，我該怎麼做？

您隱含的 vols 絕對是對數正常的嗎？是否應用了對數正態移位？如果不是，您將很難計算掉期期權的對數 (0.3%/-0.7%)，它是歐元和日元等貨幣的 P-100，在某種程度上是英鎊和美元。

如果沒有折扣元素，掉期價格通常非常有用。

例如考慮兩個價格；

5y30y: with 50bps normal vol or, say, 40 logvol might be priced at 35bps. 
Factoring the PV01 of a 1mm 5y30y swap of, say, 2300 gives a cash value of 80,500 (2300 x 35).
This is a cash price of 805bps of notional.

對比

5y5y: with 50bps normal vol or, say, 40 logvol might be priced at 35bps.
Factoring the PV01 of 1mm notional, say, 450 gives a cash value of 15,750.
This is a cash price of 157.5bps of notional.

因此，監控上述情況實際上更難標準化和評估以現金價格而不是利率價格（在這種情況下相同為 35 個基點）的掉期期權。

It depends on how precise you want to be with calculating your discount curve, but a very rough measure will be to get swaption prices for 1Y, 2Y 3Y etc and then use the formula from wiki IRS pricing, to determine discount factor points which you could interpolate, either log-linearly or log-cubically:

[Math Processing Error]$$ R = \frac{x_0-x_{n_2}}{\sum_{i=1}^{n_1} d_i x_i} $$

引用自：https://quant.stackexchange.com/questions/39123