Hedging with small uncertainty aversion

We study the pricing and hedging of derivative securities with uncertainty about the volatility of the underlying asset. Rather than taking all models from a prespecified class equally seriously, we penalise less plausible ones based on their “distance” to a reference local volatility model. In the limit for small uncertainty aversion, this leads to explicit formulas for prices and hedging strategies in terms of the security’s cash gamma.

1. A related approach is Mykland’s “conservative delta hedging” [52, 53].

2. This terminology stems from the literature on robust control [33] and also from the robust representation of convex risk measures (see e.g. [25, Sect. 5.2]). Note that the penalty is not imposed on the agent, but on her fictitious adversary who minimises over \(P\). Hence, the penalty in (1.1) is added and not subtracted.

3. In view of the well-known deficiencies of local volatility models in capturing the dynamics of option prices (cf. e.g. [21, 30]) an extension to more general reference models is an important direction for future research; see Remark 2.7 for some further discussion.

4. A similar penalty has been used by [5] in the context of local volatility calibration with prior beliefs.

5. The term \(U'(Y_{t})\) in (1.2) renders the preferences invariant under affine transformations of the utility function; see also Remark 2.6.

6. Asymptotic analyses of option pricing and hedging problems with the worst-case approach have been carried out by [48, 2, 3, 26].

7. In Sect. 1, we only display the formulas for the special case of the penalty functional (1.2). Our analysis also applies to more general penalty functionals; cf. Sect. 2.3.

8. Our results can be formally linked to the UVM with a random, time-dependent volatility band depending on the option’s cash gamma and the agent’s uncertainty aversion; cf. Remark 3.7. Note that the cash gamma also plays a crucial role in the asymptotic analysis of other frictions such as discrete rebalancing [9], transaction costs [64], price impact [51], or jumps [13].

9. Specifically, the payoff is the Black–Scholes value of a standard put option with strike 100 and maturity 1 day.

10. As the option payoff is convex, this spread is simply the difference between the Black–Scholes values of the “smooth put” corresponding to the two endpoints of the volatility band.

11. Here and in the following, subscripts on functions denote the corresponding partial derivatives.

12. The strategy adjustment may become dependent on risk aversion for other penalty functionals; see the discussion following Theorem 3.4.

13. A rigorous verification of these results would proceed along the same lines as for the simpler benchmark case discussed here. In order not to drown the ideas in further technicalities resulting from even more state variables, regularity conditions, etc., we do not pursue this here.

14. Worst-case hedges for some exotics have been derived by [36, 12, 16, 15, 41, 40, 39, 62], for example.

15. This is the analogue of the Lagrangian uncertain volatility model [7]; also compare [54].

16. The same formula is obtained—mutatis mutandis—for exotics of Asian, lookback or barrier type, after adding the appropriate state variables to the reference cash gamma.

17. This error incurred from hedging with a misspecified volatility is well known in the literature and in practice. To the best of our knowledge, it appeared first in Lyons [48, Eq. (27)] (see also [47, Eq. (6.7)]).

18. Under each \(P\), the stochastic integrals of sufficiently integrable, progressively measurable integrands against Itô process integrators are constructed as \(\mathbb{F}\)-progressively measurable stochastic processes with \(P\)-a.s. continuous paths; see the discussion before Lemma 4.3.3 in Stroock and Varadhan [63].

19. Since \(\mathfrak{P}(\theta ,\sigma )\) may contain more than one measure, we also have to allow “nature” to choose a specific measure.

20. The penalty functional in the sense of the criterion (1.1) is \(\alpha (P) = E^{P} [ \mathfrak{a}(\sigma^{P}) ]\), where \(\sigma^{P}\) is the volatility of returns of \(S\) under \(P\).

21. Recall from footnote 2 that \(\mathfrak{a}\) penalises the fictitious adversary (“nature”) and not the agent.

22. Notably, [5] show that penalty functionals of this form can arise as the continuous-time limit of the relative entropy in a discrete-time approximation.

23. Using \(U'(y_{0})\) instead of \(U'(Y_{t})\) would yield the same expansion for \(v(\psi )\) as in Theorem 3.4 and, formally and at the leading order, the same almost optimal strategies and volatilities. This is because in the asymptotic limit for small uncertainty aversion, the P&L process converges to a constant.

24. In the context of robust portfolio choice, Maenhout [50] 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.

25. Here, we assume that all relevant partial derivatives exist; precise conditions are given in Assumption 3.2.

26. Hölder-continuity uniformly on \((0,T)\times (K ^{-1},K)\) suffices for this step.

27. Strictly speaking, \(\theta^{\psi }\) and \(\sigma^{\psi }\) have to be defined for every \(\omega \in \varOmega \), even those for which \(S\) or \(Y\) exceeds the bounds (3.1). Outside these bounds, however, the functions \(\bar{V}_{s}\), \(\widetilde{\theta }\), \(\bar{\sigma }\), \(\widetilde{\sigma }\) are not defined. As we only consider measures \(P\) such that (3.1) holds, we do not make explicit the corresponding straightforward adjustments, which would only hamper readability.

28. Mutatis mutandis, the threshold 1 in the definition of the stopping time \(\tau \) can be replaced by any other constant. The same modification also appears in the asymptotic analysis of models with transaction costs [44].

29. A sufficient condition for \(\widetilde{w}_{sy} = 0\) is that \(f''(t,s,y;\varsigma )\) does not depend on \(y\).

30. A second-order expansion for the ask price can also be obtained, but does not offer much additional insight.

31. This symmetry generally breaks down for the second-order term \(\widehat{w}\); cf. the corresponding source term (3.5). Hence, for a second-order expansion of the indifference bid price, we have to use the \(\widehat{w}\) corresponding to the negative of the option.

32. This specification is a particular case of the general “random \(G\)-expectation” [56].

33. [48] imposes bounds on the instantaneous variance of prices instead of the volatility of returns. Hence, the PDE for \(\widetilde{V}\) there looks slightly different. The PDE presented here is a slight generalisation of the one derived in [26].

34. For example, if \(N\) is integrable under \(P^{\theta ,\sigma }\) for each \((\theta ,\sigma ) \in \mathcal{A} \times \mathcal{V}\), then \(v\) is a well-defined number in the extended real line \([-\infty ,+\infty ]\).

35. We obtain the same results if we interchange the order of the infimum and the supremum. In the language of two-player, zero-sum stochastic differential games, this indicates that the game “has a value”.

36. For the heuristic derivation in this section, we tacitly assume that for each \((\theta ,\sigma )\), \(P^{\theta ,\sigma }\) attains the infimum in (2.13), so that the additional infimum over measures in (2.13) disappears.

37. This covers most of the specific choices that are dealt with in the following subsections, except for additional state variables needed for some exotic options in Sect. 4.2. To explain the general procedure, we first focus here on the easiest case with just two state variables, the stock price \(S\) and the P&L process \(Y\).

38. Note that this hedge \(\bar{\varDelta }\) reflects the option’s sensitivity to price moves in the underlying both directly through \(S\) and indirectly through the additional state variable \(A\).

39. Portfolios including exotic options can be treated along the same lines; we do not pursue this here to ease notation.

40. That is, the local volatility model is calibrated to the observed market prices of the liquid options at time 0.

41. Portfolios of barrier options as in [4] or other exotics can be treated along the same lines, but require a more extensive notation.

42. For instance, if \(\theta \) is of finite variation, then the stochastic integral can be defined pathwise via the integration by parts formula.

43. Unlike [14], we disregard bid-ask spreads for the liquidly traded options.

44. A Mathematica file containing these calculations is available from the authors upon request.

The authors thank Ibrahim Ekren, Paolo Guasoni, Martin Herdegen, David Hobson, Jan Kallsen, Ariel Neufeld, Marcel Nutz, Oleg Reichmann, Martin Schweizer and H. Mete Soner for fruitful discussions, and two referees and an Associate Editor for helpful comments.

The first author gratefully acknowledges financial support by the Swiss Finance Institute.

Authors and Affiliations

1. Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, 48109, USA

   Sebastian Herrmann & Johannes Muhle-Karbe

2. Department IV – Mathematics, Universität Trier, Universitätsring 19, 54296, Trier, Germany

   Frank Thomas Seifried

Corresponding author

Correspondence to Sebastian Herrmann.

Appendix A: Calculus

The following result is an extension of the implicit function theorem that allows the defining function to depend on a parameter. In particular, it provides parameter-independent bounds for the first and second derivatives of the implicitly defined functions.

Lemma A.1

Let \(\varLambda \neq \emptyset \) be a set, and \(U\), \(V\) open subsets of ℝ. For each \(\lambda \in \varLambda \), let \(F_{\lambda }: U \times V \to \mathbb{R}\) and \(y_{\lambda }: U \to V\) be twice continuously differentiable functions with

If there are constants \(M_{0} > 1\) and \(M_{1} \geq 0\) such that for each \(\lambda \in \varLambda \) and all \((x,y) \in U \times V\),

then there is a constant \(\widetilde{M} > 0\) such that for all \(\lambda \in \varLambda \) and \(x\in U\),

Moreover, if \(\frac{\partial^{2} F_{\lambda }}{\partial y^{2}} \equiv 0\), then for all \(\lambda \in \varLambda \) and \(x\in U\),

Proof

Taking the derivative of (A.1) with respect to \(x\) yields

Solving this for \(y_{\lambda }'(x)\) and using the bounds (A.2) then gives

Taking the derivative in (A.3), we obtain for all \(x\in U\) that

Again, solving for \(y_{\lambda }''(x)\) and using the bounds (A.2) and (A.4) gives

for some sufficiently large constant \(\overline{M}\). Finally, if \(\frac{\partial^{2} F_{\lambda }}{\partial y^{2}} \equiv 0\), then it is easily seen that the quadratic term in (A.5) vanishes. □

Appendix B: Stochastic differential equations

Fix an abstract filtered probability space \((\varOmega , \mathcal{F}, \mathbb{F}= (\mathcal{F}_{t})_{t\geq 0}, P)\) carrying an \(r\)-dimensional Brownian motion \(W = (W^{1}_{t},\ldots , W^{r} _{t})_{t\geq 0}\). The goal of this section is to prove the existence of solutions for a class of stochastic differential equations (SDEs) whose coefficients change at a stopping time. More precisely, we consider SDEs of the form

where \(\xi \) is an \(\mathbb{R}^{d}\)-valued \(\mathcal{F}_{0}\)-measurable random vector, \(X = (X^{1}_{t},\ldots ,X^{d}_{t})_{t\geq 0}\) is a continuous semimartingale in \(\mathbb{R}^{d}\), and

Here, \(\tau \) is a stopping time for the filtration induced by the canonical process on \(C(\mathbb{R}_{+};\mathbb{R}^{d})\), \(\sigma^{(1)}\), \(b ^{(1)}\) are functions on \(\mathbb{R}_{+}\times C(\mathbb{R}_{+}; \mathbb{R}^{d})\) that are progressive for the same filtration, and \(\sigma^{(2)},b^{(2)}\) are measurable functions on \(\mathbb{R}_{+} \times \mathbb{R}^{d}\); all codomains are understood to be of suitable dimension.

First of all, note that we cannot apply general existence results directly to the coefficients \(\sigma \) and \(b\) since the stopping time \(\tau \) is typically not a continuous function on \(C(\mathbb{R}_{+}; \mathbb{R}^{d})\). However, existence for this type of SDE is of course expected, provided that solutions exist for both sets of coefficients separately. The obvious idea is to solve the SDE for the first set of coefficients, stop the solution at \(\tau \), and solve from there the SDE with the second set of coefficients. This can be made precise as follows.

Theorem B.1

Suppose that the process \(X^{(1)}\) on \((\varOmega ,\mathcal{F}, \mathbb{F},P)\) satisfies

Moreover, assume that there is a constant \(K > 0\) such that for all \(t, t' \geq 0\) and \(x, x' \in \mathbb{R}^{d}\),

where \(\vert \cdot \vert \) denotes the Euclidean norm in the suitable dimension. Then there is a continuous, \(\mathbb{F}\)-adapted, \(\mathbb{R}^{d}\)-valued process \(X\) satisfying (B.1).

Proof

The solution prior to time \(\hat{\tau }:= \tau (X^{(1)})\) is already given. To construct the part of the solution after time \(\hat{\tau }\), consider the time-shifted filtration \(\widehat{\mathbb{F}}= ( \widehat{\mathcal{F}}_{t})_{t \geq 0}\) defined by \(\widehat{\mathcal{F}}_{t} = \mathcal{F}_{\hat{\tau }+ t}\) and the time-shifted \((\widehat{\mathbb{F}},P)\)-Brownian motion \(\widehat{W} = (\widehat{W}_{t})_{t \geq 0}\) defined by \(\widehat{W}_{t} := W_{ \hat{\tau }+ t} - W_{\hat{\tau }}\). By our assumptions (B.4) and (B.5), the coefficients of the SDE

fulfil the standard Lipschitz and linear growth assumptions that guarantee the existence of a \(P\)-a.s. unique strong solution for any \(\widehat{\mathcal{F}}_{0}\)-measurable random vector \(\zeta \) in \(\mathbb{R}^{d+1}\). In particular, there exists an \(\widehat{\mathbb{F}}\)-progressive process \(\widehat{Y} = (\widehat{X}, \widehat{A})\) for the initial condition \(\zeta := (X^{(1)}_{ \hat{\tau }},\hat{\tau })\). Clearly, \(\widehat{A}_{t} = \hat{\tau }+ t\) plays the role of the shifted time variable. A simple time change now yields that \(X^{(2)}_{t} := \widehat{X}_{t-\hat{\tau }} \mathbf{1} _{\lbrace t \geq \hat{\tau }\rbrace }\) is \(\mathbb{F}\)-progressive and satisfies

Finally, we verify that the process \(X_{t} := X^{(1)}_{t} \mathbf{1} _{\lbrace t < \hat{\tau }\rbrace } + X^{(2)}_{t}\mathbf{1}_{\lbrace t \geq \hat{\tau }\rbrace }\) is a solution to the original SDE (B.1). As \(X^{(2)}_{\hat{\tau }} = X^{(1)}_{\hat{\tau }}\),

Plugging in (B.3) and (B.6) gives

As \(X^{(1)}_{s} = X_{s}\) on \(\lbrace s \leq \hat{\tau }\rbrace \), Galmarino’s test implies that \(\hat{\tau }= \tau (X^{(1)}) = \tau (X)\). Moreover, since \(\sigma^{(1)}\) is progressive, \(\sigma^{(1)}(s,X^{(1)})\) only depends on the path of \(X^{(1)}\) up to time \(s\). Using also the definition of \(\sigma \) in (B.2), we obtain

Using this and the analogous statement for the drift coefficients, we see from (B.7) that \(X\) is a solution to (B.1). □

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