Mean-Variance Hedging

Abstract

In Sect. 11.3 we discussed pricing and hedging of OTC derivatives from the option writer’s point of view. The approach was based on arbitrage arguments.

Appendices

Appendix 1: Problems

The exercises correspond to the section with the same number.

12.1 (Basis Risk)

Suppose that a claim is written on an asset $\widetilde S$ which cannot be traded. Instead, only a closely correlated security S₁ is liquid enough to be used for hedging purposes. Specifically, we consider discounted price processes

$$\displaystyle \begin{aligned} dS_1(t)&=S_1(t)(\mu dt+\sigma dW(t)),\\ d\widetilde S(t)&=\widetilde S(t)(\widetilde\mu dt+\widetilde\sigma d\widetilde W(t)), \end{aligned} $$

where $\mu ,\widetilde \mu ,\sigma ,\widetilde \sigma$ are constants and $(W,\widetilde W)$ is a two-dimensional Brownian motion with $E(W(1))=0=E(\widetilde W(1))$, $\mathrm {Var}(W(1))=1=\mathrm {Var}(\widetilde W(1))$, and $\mathrm {Cov}(W(1),\widetilde W(1))=\varrho$. Only S₁ is assumed to be tradable. For simplicity we assume S₁ to be a martingale, i.e. μ = 0.

The claim under consideration is of the form $H=f(\widetilde S(T))$ for some function $f:\mathbb R _+\to \mathbb R$. Derive expressions for the variance-optimal hedging strategy φ^⋆ of H and for the minimal quadratic risk ε²(φ^⋆). How do these quantities depend on ϱ?

12.2 (Reduction to the Martingale Case)

We consider the setup and the notation of Sect. 12.2. Suppose that $\mathscr {E}(-a\bullet S)$ is positive. We denote by

$$\displaystyle \begin{aligned}\widetilde S:=(\widetilde S_0,\dots,\widetilde S_d):={(S_0,\dots,S_d)\over\mathscr{E}(-a\bullet S)}\end{aligned}$$

the original price process, but discounted relative to the value process of the self-financing strategy φ with $\varphi _i:=a_i\mathscr {E}(-a\bullet S)_-$, i = 1, …, d and initial value 1. Accordingly, we write $\widetilde H:=H/\mathscr {E}(-a\bullet S)(T)$ for the payoff H, discounted relative to the same numeraire. Finally, we need the probability measure Q ∼ P with density

$$\displaystyle \begin{aligned}{dQ\over dP}:={\mathscr{E}(-a\bullet S)(T)^2\over E(\mathscr{E}(-a\bullet S)(T)^2) }.\end{aligned}$$

Show that

• 1. $\widetilde S$ is a Q-martingale, i.e. Q is an EMM for S relative to the tradable numeraire $\mathscr {E}(-a\bullet S)$;

• 2. the variance-optimal hedging strategy of $\widetilde H$ relative to $\widetilde S$ and measure Q coincides with the variance-optimal hedging strategy φ^⋆ of H relative to S and the original measure P;

• 3. the minimal quadratic risks are related via

  $$\displaystyle \begin{aligned}\widetilde\varepsilon^2(\varphi^\star)L(0)=\varepsilon^2(\varphi^\star)\end{aligned}$$

  where we write $\widetilde V_\varphi =\varphi ^\top \widetilde S$ for the discounted value of any self-financing strategy φ and $\widetilde \varepsilon ^2(\varphi ):=E_Q((\widetilde V_\varphi (T)-\widetilde H)^2)$ for its quadratic risk.

Note that the hedging problem for $\widetilde H$ relative to $\widetilde S$ and Q is of the simpler structure of Sect. 12.1.

Appendix 2: Notes and Comments

A hands-on introduction to regression in finance can be found in the textbook [54]. Quadratic hedging started with [118] in the martingale case and [79] for securities with drift. We refer to [233, 270] for an overview over the extensive literature in the field and to [55] for more recent publications.

As far as rigorous statements are concerned, Rule 12.1 is a special case of the general setup in Sect. 12.2. The operator formulation of the solution can be found in [37]. In particular, we refer to [37, Theorems 3 and 5] for rigorous versions of Rules 12.3, 12.4. Section 12.1.3 is based on the results of [178, 179, 231], but the idea to apply Laplace transforms to quadratic hedging goes back to [148]. The latter covers Example 12.9 as a special case. Section 12.2 closely follows the reasoning in [55], where rigorous statements and proofs can be found. Example 12.13 can be found in [148]. For extensions to the affine case see [56, 184]. Problem 12.1 is taken from [147]. The numeraire change approach of Problem 12.2 has been put forward by [128] in the continuous case and extended by [3] in the presence of jumps.