Utility-Based Valuation and Hedging of Derivatives

Abstract

Valuation based on quadratic hedging is easy to understand and mathematically tractable compared to other approaches. Its economic justification is less obvious.

Appendices

Appendix 1: Problems

The exercises correspond to the section with the same number.

13.1

Determine the approximate price and hedge of Sect. 13.1.3 for a European call in the Heston model of Sect. 8.2.4. For simplicity assume that the discounted stock is a martingale or, more specifically, S₀ = 1 and μ = 0, δ = −1∕2 in (8.11).

Hint: Apply the results of Sect. 12.1.3.

13.2

Determine the approximate price and hedge in the sense of Sect. 13.2 for a European call in the Heston model of Sect. 8.2.4. As in the previous exercise assume that the discounted stock is a martingale. Compare the results of Problems 13.1 and 13.2.

13.3 (Worst and Tail Conditional Expectation)

Suppose that the loss H has a continuous law, i.e. a continuous cumulative distribution function. Show that in this case AVaR_α(H) = WCE_α(H) = TCE_α(H), where the worst conditional expectation at level α is defined as

$$\displaystyle \begin{aligned}\mathrm{WCE}_\alpha(H):= \sup\big\{E(H|F):F\mbox{ event with }P(F)\geq1-\alpha\big\}\end{aligned}$$

and the tail conditional expectation at level α as

$$\displaystyle \begin{aligned}\mathrm{TCE}_\alpha(H):=E(H|H\geq\mathrm{VaR}_\alpha(H)).\end{aligned}$$

13.4 (Basis Risk)

Repeat the comparative study of Example 13.29 for the basis risk example of Problem 12.1 and Examples 13.3, 13.11. Specifically, consider a market composed of a riskless bond S₀(t) = e^rt with constant interest rate r and two assets with 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 S₀, S₁ are supposed to be tradable. One may think of an index $\widetilde S$ and an exchange-traded fund (ETF) S₁ on this index. For simplicity, we assume $\mu =\widetilde \mu =0$, i.e. $S_1, \widetilde S$ are martingales. The parameters σ = 0.104, $\widetilde \sigma =0.103$, ϱ = 0.993 have been estimated from S&P 500 data and a corresponding ETF in the period January 1, 2016 till December 31, 2017. The initial values are S₁(0) = 266.86 and $\widetilde S(0)=2673.61$.

We consider two contingent claims, namely an at-the-money put with discounted payoff $\widehat H_1:=(\widetilde S(0)-\widetilde S(T))^+$ and an at-the-money call with discounted payoff $\widehat H_2:=(\widetilde S(T)-\widetilde S(0))^+$. The goal is to determine the various option prices and hedging strategies parallel to Example 13.29. For the naive Black–Scholes approximation simply suppose that $S_1/S_1(0)\approx \widetilde S/\widetilde S(0)$, i.e. $\sigma \approx \widetilde \sigma =0.103, \varrho \approx 1$. The calibration approach should be left out because it is not obvious how to obtain, say, the correlation parameter ϱ from liquid option prices. How do the resulting figures change if correlation amounts to ϱ = 0.70 instead of the value from above?

Appendix 2: Notes and Comments

Utility-based derivative pricing is discussed in [45, 117, 242]. For applied aspects of risk measures see also [216]. The utility indifference approach in finance goes back at least to [145]. Rigorous statements are to be found in [19, 25, 74, 253] and others. The dual characterisation (13.6) in Rule 13.1 is stated in [74, equation (5.6)]. For Rule 13.2 see [19, (3.8)]. Example 13.3 is due to [141]. For versions of Rule 13.5 we refer to [19, 74, 253]. The other limiting case of Rule 13.6 is treated in [20, 180, 212] in various levels of generality and in [198, 199] for general utility functions. The approach of running P&L is pursued in [164]. For pricing small numbers of claims see also [166] which, however, takes an equilibrium point of view which differs from the approach taken here.

The theory of risk measures goes back to [4] in the coherent case and to [116, 121, 139] in the convex extension. For Sect. 13.3 we suggest [117] as a prime reference where rigorous statements on risk measures can be found. For the convenience of the reader we point at corresponding statements: for Definition 13.14 see [117, Section 4.1], for Example 13.15 see [117, Definition 4.45], for Example 13.16 see [117, Definition 4.48], for Example 13.17 see [117, Example 4.13], for Definition 13.18 see [117, Section 4.1], for Example 13.19 see [117, Example 4.10], for Rule 13.20 see [117, Propositions 4.6, 4.7 and Exercise 4.1.4], for Example 13.21 see [117, Definition 4.112], for Rule 13.22 see [117, Proposition 4.15, Theorems 4.16, 4.22, 4.33, and Remark 4.17], for Example 13.23 see [117, Theorem 4.52], for Example 13.24 see [117, Example 4.34], for Definition 13.25 see [117, Section 4.8], and for Problem 13.3 see [117, Example 4.41 and Corollaries 4.54, 4.68]. Risk measures in the context of hedging is discussed in detail in [11, 12, 286]. Specifically, we refer to [12, Corollary 3.7] and [286, Theorem 3.2] for Rule 13.27, and to [12, Proposition 5.3] for Example 13.28.