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Erschienen in: Finance and Stochastics 4/2017

07.09.2017

Model uncertainty, recalibration, and the emergence of delta–vega hedging

verfasst von: Sebastian Herrmann, Johannes Muhle-Karbe

Erschienen in: Finance and Stochastics | Ausgabe 4/2017

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Abstract

We study option pricing and hedging with uncertainty about a Black–Scholes reference model which is dynamically recalibrated to the market price of a liquidly traded vanilla option. For dynamic trading in the underlying asset and this vanilla option, delta–vega hedging is asymptotically optimal in the limit for small uncertainty aversion. The corresponding indifference price corrections are determined by the disparity between the vegas, gammas, vannas and volgas of the non-traded and the liquidly traded options.

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Fußnoten
1
Vega is the sensitivity of the Black–Scholes price with respect to changes in the volatility parameter.
 
2
Davis [19, Sect. 2. (b)], Musiela and Rutkowski [44, Sect. 7.1.8] and Wilmott [58, Sect. 7.10.5] raise the same concern.
 
3
For simplicity, we restrict ourselves to vanilla options in this introduction. Our main result, Theorem 4.5, is also applicable to a wide range of exotic options like barrier options, lookback options, Asian options, forward-start options, and options on the realised variance of the stock.
 
4
As is customary in asymptotic analysis, the powers of the processes \(\sigma^{P}\), \(\nu^{P}\), \(\eta^{P}\) and \(\xi^{P}\) in the dynamics of \((S,\Sigma )\) are chosen so that all of them have a nontrivial effect on the leading-order term in the asymptotic expansions below. Using the uncorrelated volatility \(\sqrt{\xi^{P}}\) of implied volatility instead of the uncorrelated squared volatility \(\xi^{P}\) would only generate a higher-order effect in these expansions; cf. Remark 3.2 for the details.
 
5
Here, the partial derivatives \(\mathcal{C}_{\Sigma}\), \(\mathcal{C}_{SS}\), \(\mathcal{C}_{S\Sigma}\) and \(\mathcal {C}_{\Sigma \Sigma}\) of \(\mathcal{C}\) are evaluated in \((t,S_{t},\Sigma_{t})\).
 
6
The local martingale property of the liquidly traded assets is sufficient to exclude arbitrage opportunities. It also ensures that the agent has no incentive to invest in the market but only uses it as a hedging instrument for the non-traded option; cf. Remark 2.1.
 
7
In contrast, most of the literature on hedging under model uncertainty studies variants of the uncertain volatility model introduced by Avellaneda et al. [5] and Lyons [40]. These and many more recent studies (e.g. [24, 21, 46, 49, 9, 47]) look for hedging strategies that dominate the payoff of the non-traded option almost surely for every model of a prespecified class. This worst-case approach corresponds to preferences with infinite risk and uncertainty aversion.
 
8
We refer to [28, Sect. 1] for more details on these preferences and their relation to the standard expected utility framework as well as the worst-case approach.
 
9
Our analysis also applies to somewhat more general penalty terms; cf. (2.13)–(2.15). The inclusion of the term \(U'(Y^{\boldsymbol{\varpi}}_{t})\) is not crucial but has some appealing properties. For instance, it renders the preferences invariant under affine transformations of the utility function; cf. Remark 2.6 for more details.
 
10
Asymptotic analyses of the uncertain volatility model have been carried out by [40, 2, 3, 23].
 
11
A second-order expansion and a next-to-leading order optimal strategy are obtained in [28, Theorem 3.4], where only the stock but no additional vanilla option is used for dynamic hedging.
 
12
Delta is the sensitivity of a Black–Scholes option value with respect to changes in the price of the underlying.
 
13
The vega of the underlying is obviously zero.
 
14
Gamma, vanna and volga are the second-order partial derivatives \(\partial^{2}/\partial S^{2}\), \(\partial^{2}/(\partial S\partial\Sigma)\) and \(\partial^{2}/\partial\Sigma^{2}\) of the Black–Scholes value of an option.
 
15
In contrast, if there is no liquidly traded call available as a hedging instrument, then the option’s cash gamma is the only greek that appears in the probabilistic representation of the cash equivalent [28].
 
16
According to formula (1.6), a short net volga position is only exposed to the part of the volatility of implied volatility that is uncorrelated with the underlying. However, it can be seen from the proof that the correlated volatility of implied volatility has the same effect, albeit only at the order \(O(\psi^{2})\).
 
17
See e.g. [52, 19, 20] for precise conditions.
 
18
Semi-static hedging problems have also been analysed numerically in the context of the Lagrangian uncertain volatility model [6, 4].
 
19
General superhedging duality results in the semi-static context have been obtained, among others, by [1, 7, 22, 25, 10]; see also the references therein.
 
20
Other early articles on risk-neutral dynamics for stochastic implied volatility models include [41, 11, 39]. For more recent developments on arbitrage-free market models for (parts of or the whole) option price surface, we refer the reader to [55, 54, 13, 35, 14, 15, 36] and the references therein.
 
21
The parametrisation in terms of the squared volatility of implied volatility is explained in Remark 3.2.
 
22
For example, [33] find in a Lévy model that the (drift-dependent) variance-optimal hedge is virtually identical to the (drift-independent) Black–Scholes delta hedge.
 
23
See Assumption 4.2 for the precise details.
 
24
Recall that \(M\) is the running maximum of \(S\) and \(A\) is a general state variable with dynamics of the form (2.8) which can track exotic features of the option like the average stock price or the stock price at an intermediate time; cf. [28, Sect. 4.2] for examples.
 
25
For locally bounded, progressively measurable integrands, the stochastic integrals in (2.11) are well defined under each measure in \(\mathfrak{P}^{0}\). The delta–vega hedge considered in our main result, Theorem 4.5, is even continuous.
 
26
In [28], only the underlying but no liquid call is available for dynamic hedging, and the spot volatility is the only control variable of the fictitious adversary.
 
27
Note that the penalty is imposed on the fictitious adversary who chooses the model \(P\) after the agent has chosen her trading strategy \({\boldsymbol{\varpi}}\). Alternatively, it can be interpreted as a fictitious bonus for the agent.
 
28
More general functions \(f\) are considered in [28], where it becomes apparent that only the locally quadratic structure at the minimum matters for the leading-order asymptotics.
 
29
Formally, this corresponds to directly imposing the penalty in monetary terms, i.e., inside the utility function in (2.12).
 
30
Using \(U'(Y_{0})\) instead of \(U'(Y^{{\boldsymbol{\varpi}},P}_{t})\) would yield the same expansion for \(v(\psi)\) as in Theorem 4.5. Formally, the delta–vega hedge and the candidate optimal controls for the fictitious adversary would still be leading-order optimal. This is because the P&L process converges to a constant in the limit of small uncertainty aversion. Consequently, one could also remove \(U'(Y^{{\boldsymbol{\varpi}},P}_{t})\) from the penalty term by replacing the matrix \(\Psi\) by \(\Psi/U'(Y_{0})\). Then \(U'(Y_{0})\) would reappear in the candidate feedback control for the fictitious adversary and hence also in the cash equivalent \(\widetilde{w}_{0}\). Keeping \(U'(Y^{{\boldsymbol{\varpi}},P}_{t})\) in the penalty term avoids that the candidate optimal controls depend on the current P&L of the agent. This avoids some mathematical subtleties in the formulation of the hedging problem; cf. [28], where the P&L process \(Y\) lives on the canonical space so that (progressively measurable) controls may depend on \(Y\).
 
31
In the context of robust portfolio choice, Maenhout [43] also observes that some modification of the standard (non-wealth-dependent) entropic penalty is reasonable to avoid that the agent’s uncertainty aversion wears off as her wealth rises, and tackles this effect by directly modifying the HJBI equation.
 
32
In view of [28], it is expected that \(\psi\) (and not e.g. \(\psi^{1/2}\) or \(\psi^{2}\)) is the correct power for the expansion of the value function. Alternatively, one could write \(\psi ^{\alpha}\) instead of \(\psi\) in (3.5) and then find \(\alpha= 1\) by matching the powers of the penalty term and the drift term of the P&L process in the expansion of the HJBI equation in such a way that the optimisation over \(\widetilde{\boldsymbol{\zeta}}\) becomes nontrivial.
 
33
Here and in the following, we assume that all relevant partial derivatives of \(\mathcal{C}\) and \(\mathcal{V}\) exist; precise conditions are given in Assumption 4.2 below.
 
34
This holds e.g. for a “smooth put”, whose payoff is the Black–Scholes put value with some arbitrarily short maturity.
 
35
See also [28, Remark 3.2] for a discussion of such regularity assumptions in a similar setting.
 
36
Recall from Remark 4.3 (a) that the delta–vega hedge \({\boldsymbol{\varpi}}^{\star}\) can always be included into the set of trading strategies \(\mathfrak {Y}\) by making the constant \(K_{\mathfrak{Y}}\) from Assumption 4.2 (a) larger if necessary. The existence of a candidate asymptotic model family is discussed in Sect. 4.3.
 
37
With a slight abuse of notation, \({\boldsymbol {\varpi}}^{\star}_{t}\) always denotes the time-\(t\) value of the process \({\boldsymbol{\varpi}}^{\star}\) and not the partial derivative of the function \({\boldsymbol{\varpi}}^{\star}\) with respect to the first variable.
 
38
Note that the definition of \(H^{\psi}\) already contains the candidate first-order expansion \(w^{\psi}\) of the value function and thus does not feature a general solution function and its derivatives as arguments.
 
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Metadaten
Titel
Model uncertainty, recalibration, and the emergence of delta–vega hedging
verfasst von
Sebastian Herrmann
Johannes Muhle-Karbe
Publikationsdatum
07.09.2017
Verlag
Springer Berlin Heidelberg
Erschienen in
Finance and Stochastics / Ausgabe 4/2017
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI
https://doi.org/10.1007/s00780-017-0342-6

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