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Discrete approximation of risk-based prices under volatility uncertainty

Jonas Blessing, Michael Kupper, Alessandro Sgarabottolo

Abstract

We discuss the asymptotic behaviour of risk-based indifference prices of European contingent claims in discrete-time financial markets under volatility uncertainty as the number of intermediate trading periods tends to infinity. The asymptotic risk-based prices form a strongly continuous convex monotone semigroup which is uniquely determined by its infinitesimal generator and therefore only depends on the covariance of the random factors but not on the particular choice of the model. We further compare the risk-based prices with the worst-case prices given by the $G$-expectation and investigate their asymptotic behaviour as the risk aversion of the agent tends to infinity. The theoretical results are illustrated with several examples and numerical simulations showing, in particular, that the risk-based prices lead to a significant reduction of the bid-ask spread compared to the worst-case prices.

Discrete approximation of risk-based prices under volatility uncertainty

Abstract

We discuss the asymptotic behaviour of risk-based indifference prices of European contingent claims in discrete-time financial markets under volatility uncertainty as the number of intermediate trading periods tends to infinity. The asymptotic risk-based prices form a strongly continuous convex monotone semigroup which is uniquely determined by its infinitesimal generator and therefore only depends on the covariance of the random factors but not on the particular choice of the model. We further compare the risk-based prices with the worst-case prices given by the -expectation and investigate their asymptotic behaviour as the risk aversion of the agent tends to infinity. The theoretical results are illustrated with several examples and numerical simulations showing, in particular, that the risk-based prices lead to a significant reduction of the bid-ask spread compared to the worst-case prices.

Paper Structure

This paper contains 20 sections, 12 theorems, 173 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Market model and pricing operators
  3. Asset distribution and sublinear expectations
  4. Continuous time limits and Chernoff-type approximations
  5. Dynamically consistent pricing operators
  6. Agent's preferences and indifference pricing
  7. Main results
  8. Examples and numerical illustrations
  9. Implementation
  10. Binomial model
  11. Several linear models
  12. Uncertain binomial model
  13. Bid-ask spread
  14. Proof of the main results
  15. Proof of Theorem \ref{['thm:main']}
  16. ...and 5 more sections

Key Result

Lemma 2.4

Under the assumption that conditions ($\hat{\rm p}$ 1) and ($\hat{\rm p}$ 2) are satisfied, condition ($\hat{\rm p}$ 3) is equivalent to In particular, $p_s(p_t f)=p_{s+t}f$ for all $s,t\in\mathcal{T}(h)$ and $f\in{\rm C}_{\rm b}$.

Figures (5)

  • Figure 1: Convergence of the binomial model to the Bachelier model for a butterfly option\ref{['fn:butterfly']}. Maturity: $T=0.5$; number of time steps: $n=200$; parameters of the process: $\sigma=20\%$, $\mu=5\%$; risk aversion: $\alpha=1$.
  • Figure 2: Convergence of several linear models to the Bachelier model for a butterfly option\ref{['fn:butterfly']}; Parameters: $n=100$, $\sigma=20\%$, $\mu=5\%$, $\alpha=1$. Figure (a) displays the pricing functional for the maturity $T=0.5$; figure (b) shows the corresponding Bachelier implied volatilities.
  • Figure 3: Impact of the level of uncertainty on the risk-based ask price for a butterfly option\ref{['fn:butterfly']}. Parameters: $T=0.5$, $n=100$, $\mu=5\%$, $\sigma_0=20\%$, $\alpha=1$.
  • Figure 4: Impact of the risk aversion parameter on the risk-based ask price for a butterfly option\ref{['fn:butterfly']} and comparison with the worst-case bound. Parameters: $T=0.5$, $n=100$, $\mu=5\%$, $\sigma_0=20\%$, $u=3\%$. Figure (a) also displays the payoff function while figure (b) shows more levels of risk aversion.
  • Figure 5: (a) Impact of the risk aversion on the risk-based bid price for a butterfly option\ref{['fn:butterfly']}. (b) Comparison of the risk-based bid-ask bounds with the worst-case bid-ask bounds. Parameters: $T=0.5$, $n=100$, $\mu=5\%$, $\sigma_0=20\%$, $u=3\%$.

Theorems & Definitions (28)

  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.4
  • proof
  • ...and 18 more