## Invited Session Mon.2.H 3027

#### Monday, 13:15 - 14:45 h, Room: H 3027

**Cluster 7: Finance & economics** [...]

### New developments in computational finance

**Chair: Thomas F. Coleman**

**Monday, 13:15 - 13:40 h, Room: H 3027, Talk 1**

**Thomas F. Coleman**

On the use of automatic differentiation to efficiently determine first and second derivatives in financial applications

**Coauthor: Xi Chen**

**Abstract:**

Many applications in finance require the efficient computation of the first derivatives (i.e., "Greeks'') and sometimes 2nd derivatives of pricing functions of financial instruments. This is particularly true in the context of portfolio optimization and hedging methodologies. Efficient and accurate derivative computations are required. If the target instruments are simple, e.g., vanilla instruments, then this task is simple: indeed, analytic formulae exist and can be readily used. However, explicit formulae for more complex models are unavailable and the accurate and efficient calculation of derivatives is not a trivial matter. Examples include models that require a Monte Carlo procedure, securities priced by the Libor market model, the Libor swap market model, and the copula model. A straightforward application of automatic differentiation (AD) is exorbitantly expensive; however, a structured application of AD can be very efficient (and highly accurate). In this talk we illustrate how these popular pricing models exhibit structure that can be exploited, to achieve significant efficiency gains, in the application of AD to compute 1st and 2nd derivatives of these models.

**Monday, 13:45 - 14:10 h, Room: H 3027, Talk 2**

**Raquel Joao Fonseca**

Robust value-at-risk with linear policies

**Coauthor: BerĂ§ Rustem**

**Abstract:**

We compute the robust value-at-risk in the context of a multistage international portfolio optimization problem. Decisions at each time period are modeled as linearly dependent on past returns. As both the currency and the local asset returns are accounted for, the original model is non-linear and non-convex. With the aid of robust optimization techniques, however, we develop a tractable semidefinite programming formulation of our model, where the uncertain returns are contained in an ellipsoidal uncertainty set. The worst case value-at-risk is minimized over all possible probability distributions with the same first two order moments. We additionally show the close relationship between the minimization of the worst case value-at-risk and robust optimization, and the conditions under which the two problems are equivalent. Numerical results with simulated and real market data demonstrate the potential gains from considering a dynamic multiperiod setting relative to a single stage approach.

**Monday, 14:15 - 14:40 h, Room: H 3027, Talk 3**

**Christoph Reisinger**

The effect of the payoff on the penalty approximation of American options

**Abstract:**

This talk combines various methods of analysis to draw a comprehensive picture of penalty approximations to the value, hedge ratio, and optimal exercise strategy of American options. While convergence of the penalised PDE solution for sufficiently smooth obstacles is well established in the literature, sharp rates of convergence and particularly the effect of gradient discontinuities (i.e. the omni-present `kinks' in option payoffs) on this rate have not been fully analysed so far. We use matched asymptotic expansions to characterise the boundary layers between exercise and hold regions, and to compute first order corrections for representative payoffs on a single asset following a diffusion or jump-diffusion model. Furthermore, we demonstrate how the viscosity theory framework can be applied to this setting to derive upper and lower bounds on the value. In a small extension, we derive weak convergence rates also for option sensitivities for convex payoffs under jump-diffusion models. Finally, we outline applications of the results, including accuracy improvements by extrapolation.