Valuation
- Compare: Bond option: Valuation
The valuation of Swaptions is complicated in that the at-the-money level is the forward swap rate, being the forward rate that would apply between the maturity of the option - time m - and the tenor of the underlying swap such that the swap, at time m, would have an "NPV" of zero; see swap valuation. Moneyness, therefore, is determined based on whether the strike rate is higher, lower, or at the same level as the forward swap rate.
Addressing this, quantitative analysts value swaptions by constructing complex lattice-based term structure and short rate models that describe the movement of interest rates over time. However, a standard practice, particularly amongst traders, to whom speed of calculation is more important, is to value European swaptions using the Black model. For American- and Bermudan- styled options, where exercise is permitted prior to maturity, only the lattice based approach is applicable.
- To use the lattice based approach, the analyst constructs a "tree" of short rates consistent with today's yield curve and short rate (caplet) volatility, and where the final time step of the tree corresponds to the date of the underlying swap's maturity. Models commonly used here are Ho–Lee, Black-Derman-Toy and Hull-White. Using this tree, (1) the swap is valued at each node by "stepping backwards" through the tree, where at each node, its value is the discounted expected value of the up- and down-nodes in the later time step, added to which is the discounted value of payments made during the time step in question, and noting that floating payments are based on the short rate at each tree-node. Then (2), the option is valued similar to the approach for equity options: at nodes in the time-step corresponding to option maturity, value is based on moneyness; at earlier nodes, it is the discounted expected value of the option at the up- and down-nodes in the later time step, and, depending on option style, of the swap value at the node. For both steps, the discounting is at the short rate at the tree-node in question. (Note that the Hull-White Model returns a Trinomial Tree: the same logic is applied, although there are then three nodes in question at each point.)
- In valuing European swaptions using the Black model, the underlier is treated as a forward contract on a swap. Here, as mentioned, the forward price is the forward swap rate. The volatility is typically "read-off" a two dimensional grid of at-the-money volatilities as observed from prices in the Interbank swaption market. On this grid, one axis is the time to expiration and the other is the length of the underlying swap. Adjustments may then be made for moneyness; see Implied volatility surface under Volatility smile.
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