12th Winter school on Mathematical Finance


Mike 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)
  1. Foundations
    • generic black-box approach
    • algorithmic differentiation
    • adjoints for higher-level linear algebra
    • automatic differentiation software
  2. 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
  3. 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
  4. 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
  5. SDEs and Monte Carlo methods: II
    • multiple payoffs
    • binning and correlation Greeks
    • non-smooth payoffs
Many references and background material can be found here.

Xunyu Zhou: Mathematical behavioural finance

This mini-course will cover the recent development of a rigorous mathematical treatment of behavioural finance, including the economic background, formulation of continuous-time behavioural portfolio choice models, Arrow-Debreu equilibrium and asset pricing, the quantile/distribution approach, and behavioural optimal stopping models. (slides 1, slides 2, slides 3,slides 4)

  1. Introduction
    • Expected Utility Theory
    • Expected Utility Theory Challenged
    • Alternative Theories for Risky Choice
    • Summary and Further Readings
  2. 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
  3. 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
  4. 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 lectures

Pierre 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)
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)

Karel in 't Hout: Alternating direction implicit schemes for multi-dimensional PDEs in finance

This talk deals with numerical methods for solving multi-dimensional time-dependent PDEs arising in financial option valuation theory. For their numerical solution, the well known method-of-lines approach is considered, whereby the PDE is first discretized in the spatial variables, yielding a large system of ordinary differential equations (ODEs) that is subsequently solved by applying a suitable time-discretization method. In general, the obtained systems of ODEs are very large and stiff and have a large bandwidth, so that standard implicit time-stepping methods such as Crank-Nicolson are usually not effective anymore. Accordingly, tailored numerical time-stepping methods are required. In the past decades, operator splitting schemes of the Alternating Direction Implicit (ADI) type have proven to be a successful tool for efficiently dealing with multi-dimensional PDEs in a variety of application areas. However, PDEs modeling option values often contain mixed-derivative terms, stemming from the correlations between the underlying Brownian motions, and ADI schemes were not originally developed to deal with such terms. In this talk we first present recent theoretical stability results for ADI schemes relevant to PDEs with mixed derivative terms. Subsequently, we study their performance in the application to the Heston-Hull-White model. This is a well-known three-dimensional time-dependent PDE. Besides mixed derivative terms it also possesses e.g. a degeneracy feature that will be explained. The pricing of both vanilla options and exotic (barrier) options shall be discussed. (slides)
Tinne Haentjens and Karel in 't Hout: Alternating direction implicit finite difference schemes for the Heston-Hull-White partial differential equation. J. Comp. Finan. 16, 83-110 (2012).

Ronnie Sircar: Portfolio optimization & stochastic volatility asymptotics

We discuss the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its time-scale of fluctuation. When volatility is fast mean-reverting, this is a singular perturbation problem for a nonlinear Hamilton-Jacobi-Bellman PDE, and when it slowly varying, it is a regular perturbation. These analyses can be combined for multiscale stochastic volatility models. The asymptotics shares remarkable similarities with the linear option pricing problem, using the properties of the Merton risk-tolerance function. We also discuss extensions that include transaction costs.
Jean-Pierre Fouque, Ronnie Sircar and Thaleia Zariphopoulou, Portfolio Optimization & Stochastic Volatility Asymptotics (preprint)

Agnès Sulem: A stochastic control approach to duality methods in finance cancelled

A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities:
(i) The optimal terminal wealth $X^*(T) : = X_{\phi^*}(T)$ of the classical problem to maximise the expected $U$-utility of the terminal wealth $X_{\phi}(T)$ generated by admissible portfolios $ \phi(t); 0\leq t \leq T$ in a market with the risky asset price process modelled as a semimartingale;
(ii) The optimal scenario $dQ^*/dP$ of the dual problem to minimise the expected $V$-value of $dQ/dP$ over a family of equivalent local martingale measures $Q$. Here $V$ is the convex dual function of the concave function $U$.
In this talk we consider markets modelled by Itô-Lévy processes, and we present a new approach to the above result in this setting, based on the maximum principle in stochastic control theory. An advantage with our approach is that it also gives an explicit relation between the optimal portfolio $\phi^*$ and the optimal scenario $Q^*$, in terms of backward stochastic differential equations. This is used to obtain a general formula for the optimal portfolio $\phi^*$ in terms of the Malliavin derivative. We illustrate the results with explicit examples. Joint work with Bernt Øksendal.

Short lectures

Lech Grzelak: An equity-interest rate hybrid model with stochastic volatility and the interest rate smile

In this presentation define an equity-interest rate hybrid model in which the equity part is driven by the Heston stochastic volatility Heston [1993], and the interest rate (IR) is generated by the displaced-diffusion stochastic volatility Libor Market Model Andersen and Andreasen [2002]. We assume a non-zero correlation between the main processes. A number of approximations leads to an approximating model which falls within the class of affine processes Duffie et al. [2000], for which we then provide the corresponding forward characteristic function. By using the appropriate change of measure and freezing of the Libor rates the dimension of the corresponding pricing PDE can be greatly reduced. We discuss in detail the accuracy of the approximations and the efficient calibration. Finally, by experiments, we show the effect of the correlations and interest rate smile/skew on typical equity-interest rate hybrid product prices. For a whole strip of strikes this approximate hybrid model can be evaluated for equity plain vanilla options in just milliseconds. (slides)
Lech A. Grzelak and Cornelis W. Oosterlee, On the Heston model with stochastic interest rates, SIAM J. Financial Math. Vol. 2, pp. 255-286
Lech A. Grzelak and Cornelis W. Oosterlee, An equity-interest rate hybrid model with stochastic volatility and the interest rate smile, The Journal of Computational Finance (1-33) Volume 15/Number 4

Alexander de Roode: Determinants of expected inflation in affine term structure models

Breakeven inflation is observable in financial markets with traded inflation-linked securities. However, the inflation risk premium and expected inflation cannot be directly obtained from the breakeven inflation without a structural model. Affine term structures models have been widely used in the literature on real interest rates to identify expected inflation. Most studies conjecture that expected inflation is largely determined by the Consumer price inflation due to the contractual agreements of inflation-linked securities. Other inflation measures may determine expected inflation as well. This paper answers the question whether these factors have an empirical relation to inflation in affine term structure models.
Our affine term structure model is estimated using a minimum Chi-squared methodology instead of a maximum likelihood estimation. Although this approach is asymptotically equivalent to MLE, it can reduce computational issues. (slides)

Marjon Ruijter: 2D-COS Method for Pricing Financial Options

The COS method for pricing European and Bermudan options with one underlying asset was developed in [F. Fang, C. W. Oosterlee, 2008] and [F. Fang, C. W. Oosterlee, 2009]. We extend the method to higher dimensions, with a multi-dimensional asset price process. The algorithm can be applied to, for example, pricing under the popular Heston stochastic volatility model, but also to pricing multi-color rainbow options. For the latter options the payoff depends on two or more different assets. It may involve, for example, an average or the maximum of several asset prices. For smooth density functions, the resulting method converges exponentially in the number of terms in the Fourier cosine series summations. The use of an FFT algorithm, for asset prices modelled by Levy processes, makes the algorithm highly efficient. We perform extensive numerical experiments. (slides)
M. J. Ruijter and C. W. Oosterlee, Two-Dimensional Fourier Cosine Series Expansion Method for Pricing Financial Options, SIAM J. Sci. Comput., 34(5), B642-B671, preprint

Kim Volders: Stability and convergence analysis of discretizations of the Black-Scholes PDE with the linear boundary condition

We consider the stability and convergence of numerical discretizations of the Black-Scholes partial differential equation (PDE) when complemented with the popular linear boundary condition. This condition states that the second derivative of the option value vanishes when the underlying asset price gets large and is often applied in the actual numerical solution of PDEs in finance. To our knowledge, the only theoretical stability result in the literature up to now pertinent to the linear boundary condition has been obtained by Windcliff, Forsyth and Vetzal (2004), who showed that for a common discretization a necessary eigenvalue condition for stability holds. In this presentation, we shall present sufficient conditions for stability and convergence when the linear boundary condition is employed. We deal with finite difference discretizations in the spatial (asset) variable and a subsequent implicit discretization in time. As a main result we prove that even though the maximum norm of $\exp(tM)$ ($t \geq 0$) can grow with the dimension of the semidiscrete matrix $M$, this generally does not impair the convergence behavior of the numerical discretizations. Our theoretical results are illustrated by numerical experiments.
Karel in 't Hout and Kim Volders, Stability and convergence analysis of discretizations of the Black-Scholes PDE with the linear boundary condition (arXiv preprint)

Poster presentations

Bastian 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

To the homepage of the Winter School on Mathematical Finance