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Implied Dynamics in the SV-LMM Framework

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Asymptotic Chaos Expansions in Finance

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Abstract

In this chapter we apply the generic term structure framework defined in Chap. 5 to the particular case of interest rates options, within a universal Stochastic Volatility Libor Market Model (SV-LMM). As in Chap. 6, our main focus is to solve the direct problem (generating the smile’s shape and dynamics from the model specification) up to the first layer (which includes the smile’s curvature and slope). We target some of the most liquid option types, namely caplets, swaptions and bond options. For technical reasons we exploit a model re-parametrisation via the rebased Zero Coupons, which allows us to recycle some of the SV-HJM results of Chap. 6. Likewise, in order to manage swaptions we use the basket results of Sect. 3.5. This enables us, in particular, to compute the systematic error of the usual frozen weights approximation.

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Notes

  1. 1.

    In practice we could restrict to a finite horizon, but this is a moot point.

  2. 2.

    With the proviso that the instantaneous forward rates \(f_t(s)\) for \(t\le s < \delta \) are also given.

  3. 3.

    However, some specific parametrisations might perform better in one framework than in the other.

  4. 4.

    Recall that these choices imply that we cannot represent all European options written on a running bond.

  5. 5.

    Proposition 5.2 [p. 297], Theorem 5.1 [p. 300] or Theorem 5.2 [p. 310], for instance.

  6. 6.

    These coefficients are \( \overrightarrow{\sigma }{}_t^{\delta ,L}(T), \,\overset{\Rightarrow }{a}_{2,t}^{\delta ,L}(T), \,\overset{\Rightarrow }{a}_{3,t}^{\delta ,L}(T) \ \text{ and } \ \overset{\Rrightarrow }{a}_{22,t}^{\delta ,L}(T). \)

  7. 7.

    To make the proxy dynamics (7.58) martingale, we would have to consider the following numeraire: \( N_t \,\overset{\vartriangle }{=}\,\prod _{i=1}^N\,\delta _i \,B_t(T_i) \). However, this process is not a recognised asset.

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    David Nicolay

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  1. David Nicolay
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Nicolay, D. (2014). Implied Dynamics in the SV-LMM Framework. In: Asymptotic Chaos Expansions in Finance. Springer Finance(). Springer, London. https://doi.org/10.1007/978-1-4471-6506-4_7

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