📝 Abstract

We discuss utility based pricing and hedging of jump diffusion processes with emphasis on the practical applicability of the framework. We point out two difficulties that seem to limit this applicability, namely drift dependence and essential risk aversion independence. We suggest to solve these by a re-interpretation of the framework. This leads to the notion of an implied drift. We also present a heuristic derivation of the marginal indifference price and the marginal optimal hedge that might be useful in numerical computations.

💡 Analysis

We discuss utility based pricing and hedging of jump diffusion processes with emphasis on the practical applicability of the framework. We point out two difficulties that seem to limit this applicability, namely drift dependence and essential risk aversion independence. We suggest to solve these by a re-interpretation of the framework. This leads to the notion of an implied drift. We also present a heuristic derivation of the marginal indifference price and the marginal optimal hedge that might be useful in numerical computations.

📄 Content

Utility based pricing and hedging of jump diffusion processes with a view to applications Jochen Zahn Courant Research Centre “Higher Order Structures” University of G¨ottingen Bunsenstraße 3-5, D-37073 G¨ottingen, Germany jzahn@uni-math.gwdg.de October 30, 2018 Abstract We discuss utility based pricing and hedging of jump diffusion pro- cesses with emphasis on the practical applicability of the framework. We point out two difficulties that seem to limit this applicability, namely drift dependence and essential risk aversion independence. We suggest to solve these by a re-interpretation of the framework. This leads to the notion of an implied drift. We also present a heuristic derivation of the marginal indifference price and the marginal optimal hedge that might be useful in numerical computations. 1 Introduction The applicability of the Black–Scholes framework for the pricing and hedging of derivative claims crucially depends on the assumption of market completeness, i.e., the possibility to replicate claims and thus eliminate risk. This assumption is not fulfilled if the asset process is driven by more than one source of risk or when market imperfections such as transaction costs are not negligible. One then speaks of an incomplete market in which investors may attribute different prices to derivatives, according to their risk preferences. As an example, let us consider a jump diffusion process, i.e., the asset S evolves according to dSt = µSt−dt + σSt−dWt + (eJt −1)St−dNt. (1) Here Wt is a Wiener and Nt a Poisson process with frequency λ. The random variable Jt determines the relative size eJt −1 of the jump. The oldest and probably most popular approach for the pricing and hedging of a claim on such an asset is Merton’s [11]. There, the investor sets up a portfolio Π consisting 1 arXiv:1106.1395v2 [q-fin.CP] 4 Dec 2012 of the claim with value V and a quantity −∆of assets, so that the evolution of the portfolio is given by dΠt = dVt −∆tdSt

 ∂tVt(St) + µSt∂SVt(St) + σ2 2 S2 t ∂2 SVt(St) −∆tµSt  dt (2)

• σSt {∂SVt(St) −∆t} dWt +  Vt(eJtSt) −Vt(St) −∆t(eJt −1)St

dNt It is in general not possible to eliminate jump and diffusion risk at the same time, so some “optimal” choice is necessary. Merton’s proposal is to hedge only the diffusion risk and to diversify the jump risk, i.e., to set ∆t = ∂SVt. The above then yields dΠt =  ∂tVt(St) + σ2 2 S2 t ∂2 SVt(St)  dt +  Vt(eJtSt) −Vt(St) −(eJt −1)St∂SVt(St)

dNt. If jump risk is diversified, the investor does not need any risk premium for taking this risk, i.e., the expected value of dΠt should vanish. Thus, we obtain the partial integro-differential equation (PIDE) 0 = ∂tVt(S) + σ2 2 S2∂2 SVt(S) − Z (ez −1)dν(z)  S∂SVt(S) + Z {Vt(ezS) −Vt(S)} dν(z). (3) Here ν is the cumulative jump frequency distribution, i.e., for an interval I with characteristic function χI, ν(χI) gives the frequency of jumps of size in I. In particular ν(R) = λ. Two remarks are in order here:

1. The diversification of jump risk is problematic not only in our model (as there is only one asset), but also in practice: In a typical market crash, jumps occur in the whole market, so diversification may well turn out to be accumulation of risk.
2. Merton’s proposal coincides with a naive interpretation of the Black– Scholes framework which states that for risk-neutral pricing one simply has to adjust the drift term such that the expected drift vanishes (in dis- counted units), and that the appropriate hedging strategy is given by the derivative of the price. In particular, the real-world drift does not enter the price, which is a benefit, as it is notoriously hard to estimate. Note that the assumption that diversification is possible is crucial here, since otherwise one could not invoke no-arbitrage arguments to set the expected re- turn of the portfolio to zero. If one drops this assumption, then the investor should (i) try to find an optimal balance between diffusion and jump risk and (ii) value the remaining risk in order to obtain a risk premium. A popular 2 framework that achieves (i) is minimal variance pricing and hedging, cf. [16, 5] and references therein. There, the investor tries to minimize the variance of the expected returns. It has the advantage that no new concepts have to be intro- duced. However, the choice of a quadratic criterion is somewhat arbitrary and penalizes profits as well as losses. Furthermore, in the case of a jump diffusion, the framework in general yields a signed risk-neutral measure, i.e., there would be positive claims which have a negative value in the framework. Finally, the framework only tackles (i), but does not yield a price for the remaining risk. A framework that achieves (i) and (ii) at one stroke is utility based pricing and hedging. There, the investor is equipped with a concave von Neumann utility function U(XT ) that assigns an economic value to the wealth XT at the investment horizon T. Risk aversion is encoded in the concavity of U which entails that th

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