Abstract
In this chapter we start by considering a generalised framework, encompassing in particular the Caplets and Swaptions markets, but potentially applicable to other products. This is made possible because these payoffs, as well as the martingale method used to price them, are very similar. Hence the main requirement is to find the correct numeraire and pricing measure. The difference with the single underlying setting of Part I is that we are now dealing with a collection of underlyings, for instance the forward Libor or forward par swap rates. Each of these underlying has its own numeraire, is martingale under the associated measure, and defines a specific strike-continuum of vanilla options. Hence we end up with associated collections of numeraires, measures and options. All these families are parametrised by their own list of maturities, which we will naturally extend to a common maturity continuum. We end up naturally with a term structure (TS) framework, and in solving the direct and indirect problems we will point to the structural difference simpler single-underlying environment of Part I.
With regard to the association between different maturities, we start by considering a fairly general case, before restricting ourselves to the practical situation presented by Caplet and Swaptions.
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Notes
- 1.
The continuous nature of any European option family is understood as the permanent availability of a common bid/ask tradeable price, for every maturity and for every strike.
- 2.
Meaning \(\overrightarrow{a}_{1,t}(T)\), \(\overset{\Rightarrow }{a}_{2,t}(T)\), \(\overset{\Rightarrow }{a}_{3,t}(T)\), \(\overset{\Rrightarrow }{a}_{22,t}(T)\) and all subsequent offspring.
- 3.
With the tensorial family \(\overset{\Rrightarrow }{a}_{22,t}(T)\).
- 4.
Which is coherent with the low liquidity of the IR volatility market. At the time of writing, “vol bonds”, “vol swaps” or “vol caps” are certainly not vanilla products and even less advertised as such.
- 5.
Determined via a trial-and-error method or a more complex optimisation routine. These optimisation procedures are typically non-trivial because the “market error” value function is a priori non-convex w.r.t. the model parameters.
- 6.
a.k.a. “moment” or “couple”.
- 7.
\( \frac{V_t(X_t(T),T,K)}{N_t(T)}\,= \,C^{BS}\,\left( \,X_t(T),K,\varSigma _t(X_t(T),T,K)\cdot \sqrt{T-t} \, \right) \) (see p. 289).
- 8.
These are dynamic features, not to be confused with maturity-wise shapes such as “twist” or “flattening”. The heave is a uniform vertical movement of the smile, analogous to a parallel movement of the yield curve in an HJM model. The roll is a uniform motion around the ATM axis, and flapping refers to the dynamics of the curvature.
- 9.
Assuming enough liquidity.
- 10.
This inference obviously has to be conducted maturity-by-maturity. The respective moneyness of the options should be roughly 90–95, 100 and 105–110 % to expect a decent precision from the finite difference approximations.
- 11.
The ECB for instance uses standing facilities (marginal lending and deposit) which refer to the overnight rate, as well as open market operations (main refinancing, longer-term refinancing, fine-tuning and structural) which are mainly associated to maturities ranging from 1 week to 6 months.
- 12.
Such as \(\Vert \overrightarrow{\sigma }_t(t)\Vert ^{-3} \overrightarrow{\sigma }_t(t)^\bot \left[ \overset{\Rightarrow }{a}_{2,t}(t) + \overset{\Rightarrow }{a}_{2,t}(t)^\bot \right] \overset{\Rightarrow }{a}_{2,t}(t)\overrightarrow{\sigma }_t(t)\), for instance.
- 13.
In \((t,0,0)\).
- 14.
Equivalent to the sliding underlying map \(\theta \rightarrow \widetilde{X}_t(\theta )\).
- 15.
At least in our framework, see the discussion on mid-curves.
- 16.
Constant Maturity Swap.
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Nicolay, D. (2014). Volatility Dynamics in a Term Structure. In: Asymptotic Chaos Expansions in Finance. Springer Finance(). Springer, London. https://doi.org/10.1007/978-1-4471-6506-4_5
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