Abstracts

## MinicoursesMike Giles: Adjoint methods for option pricing, Greeks and calibration
using PDEs and SDEs
In computational finance it is very important to be able to compute the sensitivity of option prices to various input parameters. As well as being used to compute the so-called Greeks for risk hedging, they are also used for calibrating models to market prices. Adjoint methods are a well-established mathematical approach for efficiently computing sensitivities when there are multiple input parameters, but only one output quantity. In this case, the computational cost is similar to the original pricing calculation, whereas the standard linear sensitivity approach would have a cost proportional to the number of inputs. In this series of lectures, I will discuss the mathematical foundations for adjoints methods, concentrating on the discrete level, not the differential level (i.e. finite difference and recurrence equations, rather than PDEs) and the use of automatic differentiation software to generate the adjoint code. I will then discuss its application to both finite difference methods for PDEs, and Monte Carlo methods for SDEs. (slides 1, slides 2, slides 3) - Foundations
- generic black-box approach
- algorithmic differentiation
- adjoints for higher-level linear algebra
- automatic differentiation software
- PDEs and finite difference methods: I
- formulation of adjoint PDEs and finite difference methods
- financial application
- possible advantages for pricing calculation
- FDE sensitivities for linear explicit discretisations
- PDEs and finite difference methods: II
- nonlinear implicit equations
- what can go wrong?
- calibration using Fokker-Planck discretisation
- Greeks using Black-Scholes discretisation
- local volatility example
- SDEs and Monte Carlo methods: I
- Monte Carlo simulation and augmented state
- LRM and pathwise sensitivity approaches
- adjoint pathwise approach
- use of automatic differentiation software
- storage / re-computation tradeoff
- local volatility example, revisited
- SDEs and Monte Carlo methods: II
- multiple payoffs
- binning and correlation Greeks
- non-smooth payoffs
References
Many references and background material can be found here.
- Introduction
- Expected Utility Theory
- Expected Utility Theory Challenged
- Alternative Theories for Risky Choice
- Summary and Further Readings
- Portfolio Choice under RDUT - Quantile Formulation
- Formulation of RDUT Portfolio Choice Model
- Quantile Formulation
- Solutions
- Quantile Formulation as a General Approach
- Summary and Further Readings
- Market Equilibrium and Asset Pricing under RDUT
- An Arrow-Debreu Economy
- Individual Optimality
- Representative RDUT Agent
- Asset Pricing
- CCAPM and Interest Rate
- Equity Premium and Risk-Free Rate Puzzles
- Summary and Further Readings
- Portfolio Choice under CPT
- Formulation of CPT Portfolio Choice Model
- Ill-posedness
- Divide and Conquer
- Solutions to GPP and LPP
- Grand Solution
- Continuous Time and Time Inconsistency
- Summary and Further Readings
## Special invited lecturesPierre Collin-Dufresne:
Insider trading, stochastic liquidity and equilibrium prices
We extend Kyle's (1985) model of insider trading to the case where liquidity provided by noise traders follows a general stochastic process. Even though the level of noise trading volatility is observable, in equilibrium, measured price impact is stochastic. If noise trading volatility is mean-reverting, then the equilibrium price follows a multivariate `stochastic bridge' process, which displays stochastic volatility. This is because insiders choose to optimally wait to trade more aggressively when noise trading activity is higher. In equilibrium, market makers anticipate this, and adjust prices accordingly. More private information is revealed when volatility is higher. In time series, insiders trade more aggressively, when measured price impact is lower. Therefore, execution costs to uninformed traders can be higher when price impact is lower. (slides) References
Pierre Collin-Dufresne and Vyacheslav Fos, Do prices reveal the presence of informed trading? (preprint)
Pierre Collin-Dufresne and Vyacheslav Fos, Insider Trading, Stochastic Liquidity and Equilibrium Prices (preprint)
## Short lectures
## Poster presentationsBastian Gross (University of Trier): Adjoint Monte-Carlo technique for calibration of financial market models
Tinne Haentjes (University of Antwerp): ADI schemes for the numerical solution of the Heston-Hull-White PDE
Iker Perez Lopez (University of Nottingham): Selling a stock over random horizons
Amirhossein Sadoghi (Frankfurt school of finance and management): Multiple exercise options approach for optimum strategy in market order execution by Monte Carlo techniques
Illia Simonov (University of Leoben): A numerical
approximation of the solution to parabolic SPDEs forced by a Lévy Noise |

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