MathFinance Conference, Frankfurt March 18th, 2008
Selected Applications of Optimization in Finance
March 18th, 2008 Dr. Jan H. Maruhn
Contents
1 Introduction and Motivation 2 Model Calibration based on Closed Form Solutions 3 Monte Carlo Calibration 4 Robust Static Hedging of Barrier Options 5 Gamma-Constrained Dynamic Hedging 6 Conclusions
March 18th, 2008 Dr. Jan H. Maruhn
2
Where is Optimization Used in Fin. Engineering?
After choosing a model reflecting the product‘s risk profile typically the following steps are carried out: 1. Calibrate the model to liquidly traded instruments 2. Use the model to price exotic options 3. Compute / identify optimal hedging strategies In all these areas optimization problems arise, which require numerical algorithms that are… Fast to handle quickly changing market data, a large number of underlyings and growing product complexity Robust to perform reliably in a highly automized framework and changing market conditions
March 18th, 2008 Dr. Jan H. Maruhn
3
Covered Optimization Methods
Problem classes naturally arising in the area of calibration / hedging: Nonlinear deterministic optimization: Model calibration via closed forms Semidefinite programming: Projections within feasible point algorithms Stochastic optimization: Calibration via Monte Carlo Adjoint techniques for the evaluation of gradients Robust/semi-infinite optimization: Static hedging of barrier options Optimal control: Dynamic hedging under Gamma constraints Efficient algorithms are required for the solution of all these problems See the talk of H. Mittelmann for available software
March 18th, 2008 Dr. Jan H. Maruhn
4
Contents
1 Introduction and Motivation 2 Model Calibration based on Closed Form Solutions 3 Monte Carlo Calibration 4 Robust Static Hedging of Barrier Options 5 Gamma-Constrained Dynamic Hedging 6 Conclusions
March 18th, 2008 Dr. Jan H. Maruhn
5
Calibration of the Time-Dependent Heston Model
Joint work with: F. Gerlich, A. Giese, E. Sachs Goal: Choose the model parameters x such that the model matches the market prices Cjobs of n given standard calls with strikes Kj and maturities Tj
s.t.
March 18th, 2008 Dr. Jan H. Maruhn
6
Formulation as a Standard Least Squares Problem
It is possible to derive a semi-closed form solution for the price of a standard Call option C by solving the partial differential equation
Caching of intermediate results (see F. Kilin) is applicable Hence the calibration problem can be rephrased as a (deterministic) nonlinear least squares problem
with residual function
March 18th, 2008 Dr. Jan H. Maruhn
7
Analysis of the Derivatives of the Objective
An analytic computation of the derivatives of yields
where JR(x) denotes the Jacobian of R(x). Since the residuals Rj(x) in the optimal point are usually quite small, we can approximate the Hessian of f by
We get a very good approximation of the second derivative by solely making use of first order information (Gauss Newton approximation)
March 18th, 2008 Dr. Jan H. Maruhn
8
Sketch of the Algorithm
Idea: Combine Gauss-Newton approximation of the Hessian with a feasible point trust region SQP algorithm developed by Wright and Tenny To compute a stationary point the SQP algorithm successively solves
x
Feasible set
where H is a Gauss-Newton-approximation of the Hessian of the Lagrangian
To preserve feasibility of the iterates we project the solution of (QP) onto the feasible set after each iteration.
March 18th, 2008 Dr. Jan H. Maruhn
9
The Feasible Set of the Heston Calibration Problem
Theorem: (Gerlich, Giese, M., Sachs, 2006) The set of Heston constraints is equivalent to
with an explicit solution of the associated projection problem. Furthermore, if additional lower and upper bounds are imposed on the parameters, we can solve the projection problem via the semidefinite program
March 18th, 2008 Dr. Jan H. Maruhn
10
Sample Iteration Run
Algorithm output for the calibration of the constant parameter Heston model to a dataset taken from Andersen and Brotherton (1997).
Calibration error at optimal solution
The calibration usually takes less than one second on a desktop PC A combination of nonlinear and semidefinite programming leads to a robust and rapidly converging algorithm
March 18th, 2008 Dr. Jan H. Maruhn
11
Other Available Algorithms
Other algorithms for the solution of nonlinear least squares problems: Quasi-Newton based nonlinear programming codes like SNOPT, IPOPT Derivative-free methods like simulated annealing In our experience Gauss-Newton methods are superior to Quasi Newton codes for the calibration of financial market models, because
Sketch of an ill-conditioned function
The residuals Rj(x) are usually small The Gauss-Newton approximation better captures the curvature of ill-conditioned objective functions But what about derivative-free algorithms?
March 18th, 2008 Dr. Jan H. Maruhn
12
Comparison to Derivative-Free Algorithms
Many practitioners apply derivative-free global optimization algorithms because they believe that local minima pose severe problems in applications. But in contrast to local minima the global minima can move discontinuously with market data Benchmark: Direct search simul. annealing algorithm of Hedar and Fukushima
Statistics of optimal solutions for 100 randomly chosen start points of the algorithms
The calibration results of the FPSQP code are much more stable, although the DSSA algorithm took 40 times longer (Ø 3100 calls of f) to solve the problem
March 18th, 2008 Dr. Jan H. Maruhn
13
Time-Dependent Parameters
The time-dependent calibration problem is much more ill-conditioned and requires additional regularization.
Solution of unregularized problem Solution of regularized problem
The time-dependent parameters reflect the curvature of the vol. surface
March 18th, 2008 Dr. Jan H. Maruhn
14
Further Applications
The feasible point FPSQP algorithm can also be applied to Calibrate other models (local volatility, jump diffusions etc.) Minimize other residuals like differences of implied volatilities Sample results for the dataset taken from Andersen and Brotherton (1997):
Price fit of Heston model with constant parameters Price fit of time-dependent Heston model with lognormal jumps
March 18th, 2008 Dr. Jan H. Maruhn
15
Contents
1 Introduction and Motivation 2 Model Calibration based on Closed Form Solutions 3 Monte Carlo Calibration 4 Robust Static Hedging of Barrier Options 5 Gamma-Constrained Dynamic Hedging 6 Conclusions
March 18th, 2008 Dr. Jan H. Maruhn
16
Monte Carlo Calibration: Motivation
Joint work with: C. Käbe, E. Sachs Idea: If closed forms do not exist, calibrations via MC seem to be attractive Advantages: Easy to implement and flexible with respect to changes of the model dynamics Allows to calibrate models to exotic options Is also applicable in high dimensions (3 or more stochastic drivers) Drawbacks: MC is very slow, one pricing can take 10s. And a calibration usually requires O(#iter*[#params+1]) pricings (finite difference approx. of gradient in each iter) MC approximations are usually not differentiable (even for standard calls) Are MC calibrations feasible at all? To calibrate 40 parameters in 30 optimizer iterations we would need 30*(40+1) * 10s = 3h 25min !!!
March 18th, 2008 Dr. Jan H. Maruhn
17
The General Calibration Problem
Consider the problem of calibrating the model prices Ci(x) to a given set of call options Ciobs with strikes Ki and maturities Ti:
(P)
Example: A stoch. vol. model with lognormal distribution of the variance
March 18th, 2008 Dr. Jan H. Maruhn
18
Monte Carlo Discretization
An MC approximation with M samples and stepsize Δtn in step n leads to
(PM,Δt)
f(x) , fε (x)
Problem: Non-differentiabilities Maximum function z a max(z,0) Coefficients of the SDE, e.g. square root Smooth out the non-differentiabilities with appropriate smoothing polynomials
1 0.5 0
f(x)=max(0,x)
ε
-ε
-0.5 -1 -1
-0.5
0 x
0.5
1
March 18th, 2008 Dr. Jan H. Maruhn
19
Smoothed Optimization Problem and Convergence
Smoothing out the non-differentiabilities leads to the smooth problem
(PM,Δt,ε)
Now optimize for fixed (Mk,Δt ,ε ) to obtain a first order critical point xk. k k Question: Does (xk)k converge as Mk→1, Δtk → 0, εk → 0 ? Theorem: (Käbe, M., Sachs, 2007) If the smooth MC approximations and their derivatives converge uniformly, i.e.
then every limit point x* of (xk)k satisfies the first order optimality conditions of (P).
March 18th, 2008 Dr. Jan H. Maruhn
20
Example: Constant Parameter Heston Model
Optimal solutions for different settings of Mk, Δ tk, εk:
The convergence of the solutions can also be confirmed in applications Calibrations based on M=100,000 paths already deliver good results! Speedups: Multi layer, var. reduction, store random numbers, parallelization
March 18th, 2008 Dr. Jan H. Maruhn
21
Adjoint-Based Monte Carlo Calibration
Even with the speedups mentioned before the calibration may still be too slow. The finite difference gradient approximation dominates the optimization Idea: Apply adjoint-techniques within the Monte Carlo calibration framework Remark: Giles/Glasserman recently used adjoints to compute Greeks Theorem: (Käbe, M., Sachs, 2007) The derivative of
can be computed via
where
is the solution of the adjoint equation
March 18th, 2008 Dr. Jan H. Maruhn
22
Benefits from the Adjoint Approach
A finite difference approximation of the gradient requires an MC simulation of
for each unit vector ei. For 40 parameters we need 40 forward solutions of the SDEs!! Instead the same gradient can be obtained with L backwards solutions of the adjoint equation
If L << num_parameters a significant speedup can be expected
March 18th, 2008 Dr. Jan H. Maruhn
23
Sample Iteration Run
Calibration of the lognormal variance model with 10 parameter time buckets:
Calibration with finite diffs. and one MC layer (M1,Δt1,ε1) = (100k, 9e-3, 3e-3) Adjoint-based calibration with two Monte Carlo layers (M1,Δt1,ε1) = ( 10k,3e-2,9e-3), (M2,Δt2,ε2) = (100k,9e-3,3e-3)
Observations: Computation time is reduced from > 3 hours to < 10 minutes Adjoints significantly reduce the computation time per iteration Multi layer methods reduce the number of expensive iterations MC calibration is feasible on a desktop PC!
March 18th, 2008 Dr. Jan H. Maruhn
24
Contents
1 Introduction and Motivation 2 Model Calibration based on Closed Form Solutions 3 Monte Carlo Calibration 4 Robust Static Hedging of Barrier Options 5 Gamma-Constrained Dynamic Hedging 6 Conclusions
March 18th, 2008 Dr. Jan H. Maruhn
25
Static Hedging: Motivation
Joint work with: A. Giese, E. Sachs Static hedging of an up-and-out call
Liquidate Calls!
Barrier D
Bond position remains
Constant portfolio
Strike K
t=0: Set up portfolio consisting of αi units of call Ci, i=1,…,n, and α0 units of bond B
March 18th, 2008 Dr. Jan H. Maruhn
26
Robust Static Super-Replication
Idea: Super-Replicate along the barrier and at time T
s.t.
where P½ IRk reflects a set of parametrized future volatility surface scenarios.
Source: Hans Buehler, Deutsche Bank AG
Interpretation as an "uncertain skew hedge" for reverse barriers Hedge is the solution of a semi-infinite optimization problem Existence, convergence, duality results can be derived (see M., Sachs, 2006)
March 18th, 2008 Dr. Jan H. Maruhn
27
How Can We Prevent Losses Beyond the Barrier?
Idea: Ask super-replication to hold in an interval [D,Smax] instead of at D only
Can be approximated by moving the barrier
Theoretical super-replication in jump-diffusion models requires Smax=1 (too conservative). In practice one may choose Smax=D+x%*D.
March 18th, 2008 Dr. Jan H. Maruhn
28
Empirical Hedge Performance
Joint work with: M. Nalholm, M. Fengler Investigate performance of robust static superhedge on a set of real world data.
Surprising insights: Lowest abs. dev. around the median Constant portfolio positions True super-replication on the barrier
March 18th, 2008 Dr. Jan H. Maruhn
29
Contents
1 Introduction and Motivation 2 Model Calibration based on Closed Form Solutions 3 Monte Carlo Calibration 4 Robust Static Hedging of Barrier Options 5 Gamma-Constrained Dynamic Hedging 6 Conclusions
March 18th, 2008 Dr. Jan H. Maruhn
30
Discontinuity Problem of Reverse Barriers
The discontinuous payoff leads to unbounded Greeks in the corner (D,T).
Payoff/value of a UOC in the BS model Static superhedge closes the discontinuity gap
Similar values along the barrier can be obtained by moving the barrier
March 18th, 2008 Dr. Jan H. Maruhn
31
What is the Optimal Regularization of the Payoff?
Idea: Search optimal dynamic superhedge subject to Gamma constraints (compare Schmock, Shreve, Wystup (2000)) Based on the maximum principle this leads to an optimal control problem:
Numerical solution via deterministic nonlinear optimization methods
March 18th, 2008 Dr. Jan H. Maruhn
32
Example: UOC with K=100%, D=120%, T=1, σ=20%
Comparison of optimized u(¢) with the u resulting from moving the barrier: Optimal control u(¢)
Intrinsic value
Moving the barrier
Intrinsic value
The optimized u(¢) first decays slowly, but then quickly drops to zero
March 18th, 2008 Dr. Jan H. Maruhn
33
Contents
1 Introduction and Motivation 2 Model Calibration based on Closed Form Solutions 3 Monte Carlo Calibration 4 Robust Static Hedging of Barrier Options 5 Gamma-Constrained Dynamic Hedging 6 Conclusions
March 18th, 2008 Dr. Jan H. Maruhn
34
Conclusions
Feasible point SQP methods combined with SDP projections can efficiently solve calibration problems Based on adjoint techniques Monte Carlo calibrations are feasible on a usual desktop PC Robust static super-replication leads to a linear semi-infinite optimization problem Dynamic superhedging problems often can be transformed to deterministic control problems Optimization problems appear in many areas of financial mathematics
March 18th, 2008 Dr. Jan H. Maruhn
35
Contact
Dr. Jan H. Maruhn Financial Engineering Equities and Hybrids Structured Products Development – MMP11 UniCredit Markets & Investment Banking Bayerische Hypo- und Vereinsbank AG Arabellastrasse 12 D-81925 Munich Tel. +49 89 378-13123 Fax +49 89 378-3313123 jan.maruhn@hvb.de
March 18th, 2008 Dr. Jan H. Maruhn
36
References
Andersen, L., Brotherton-Ratcliffe, R. The equity option volatility smile: an implicit finite difference approach. The Journal of Computational Finance, Vol. 1, No. 2, pp. 5-32, Winter 1997/98. Gerlich, F., Giese, A. M., Maruhn, J. H. and Sachs, E. W. Parameter Identification in Stochastic Volatility Models with Time-Dependent Model Parameters (submitted). Technical Report, University of Trier, 2006. Giles, M. and Glasserman, P. Smoking adjoints: Fast Monte Carlo greeks. Risk magazine, January 2006. Hedar, A. and Fukushima, M. Hybrid simulated annealing and direct search method for nonlinear unconstrained global optimization. Optimization Methods and Software, 17, pp.891-912, 2002. Käbe, C., Maruhn, J. H. and Sachs, E. W. Adjoint Based Monte Carlo Calibration of Financial Market Models (submitted). Technical Report, University of Trier, 2008. Maruhn, J. H. and Sachs, E. W. Robust Static Hedging of Barrier Options in Stochastic Volatility Models (submitted). Technical Report No. 06-3, University of Trier, 2006. Maruhn, J. H. Robust Static Super-Replication of Barrier Options. PhD thesis, University of Trier, Germany, 2007. Schmock, U., Shreve, S. E. and Wystup, U. Valuation of exotic options under shortselling constraints. Finance and Stochastics VI, 2, 2002. Wright, S. J. and Tenny, M. J.: A feasible Trust-Region Sequential Quadratic Programming Algorithm. SIAM Journal of Optimization, Vol.14, No.4, pp.1074-1105 , 2004.
March 18th, 2008 Dr. Jan H. Maruhn
37
Disclaimer
The information in this publication is based on carefully selected sources believed to be reliable but we do not make any representation as to its accuracy or completeness. Any opinions herein reflect our judgement at the date hereof and are subject to change without notice. Any investments discussed or recommended in this report may be unsuitable for investors depending on their specific investment objectives and financial position. Any reports provided herein are provided for general information purposes only and cannot substitute the obtaining of independent financial advice. Private investors should obtain the advice of their banker/broker about any investments concerned prior to making them. Nothing in this publication is intended to create contractual obligations on any of the entities composing UniCredit Markets & Investment Banking Division which is composed of (the respective divisions of) Bayerische Hypo- und Vereinsbank AG, Munich, Bank Austria Creditanstalt AG, Vienna, UniCredit Banca Mobiliare S.p.A., Milan and certain of their subsidiaries. Bayerische Hypo- und Vereinsbank AG is regulated by the German Financial Supervisory Authority (BaFin), Bank Austria Creditanstalt AG is regulated by the Austrian Financial Market Authority (FMA) and UniCredit Banca Mobiliare S.p.A. is regulated by Commissione Nazionale per le Società e la Borsa (Consob). Note to UK Residents: In the United Kingdom, this publication is being communicated on a confidential basis only to clients of UniCredit Markets & Investment Banking Division (acting through Bayerische Hypo- und Vereinsbank, London Branch ("HVB London") and/or CA IB International Markets Limited and/or CAIB Corporate Finance Limited) who (i) have professional experience in matters relating to investments being investment professionals as defined in Article 19(5) of the Financial Services and Markets Act 2000 (Financial Promotion) Order 2005 ("FPO"); and/or (ii) are falling within Article 49(2) (a) – (d) ("high net worth companies, unincorporated associations etc.") of the FPO (or, to the extent that this publication relates to an unregulated collective scheme, to professional investors as defined in Article 14(5) of the Financial Services and Markets Act 2000 (Promotion of Collective Investment Schemes) (Exemptions) Order 2001 and/or (iii) to whom it may be lawful to communicate it, other than private investors (all such persons being referred to as "Relevant Persons"). This publication is only directed at Relevant Persons and any investment or investment activity to which this publication relates is only available to Relevant Persons or will be engaged in only with Relevant Persons. Solicitations resulting from this publication will only be responded to if the person concerned is a Relevant Person. Other persons should not rely or act upon this publication or any of its contents. The information provided herein (including any report set out herein) does not constitute a solicitation to buy or an offer to sell any securities. The information in this publication is based on carefully selected sources believed to be reliable but we do not make any representation as to its accuracy or completeness. Any opinions herein reflect our judgement at the date hereof and are subject to change without notice. We and/or any other entity of the UniCredit Markets & Investment Banking Division may from time to time with respect to securities mentioned in this publication (i) take a long or short position and buy or sell such securities; (ii) act as investment bankers and/or commercial bankers for issuers of such securities; (iii) be represented on the board of any issuers of such securities; (iv) engage in "market making" of such securities; (v) have a consulting relationship with any issuer. Any investments discussed or recommended in any report provided herein may be unsuitable for investors depending on their specific investment objectives and financial position. Any information provided herein is provided for general information purposes only and cannot substitute the obtaining of independent financial advice. HVB London is regulated by the Financial Services Authority for the conduct of investment business in the UK. CA IB International Markets Limited, London, and CA IB Corporate Finance Limited, London, two subsidiaries of Bank Austria Creditanstalt AG, are authorised and regulated by the Financial Services Authority. Notwithstanding the above, if this publication relates to securities subject to the Prospectus Directive (2005) it is sent to you on the basis that you are a Qualified Investor for the purposes of the directive or any relevant implementing legislation of a European Economic Area ("EEA") Member State which has implemented the Prospectus Directive and it must not be given to any person who is not a Qualified Investor. By being in receipt of this publication you undertake that you will only offer or sell the securities described in this publication in circumstances which do not require the production of a prospectus under Article 3 of the Prospectus Directive or any relevant implementing legislation of an EEA Member State which has implemented the Prospectus Directive. Note to US Residents: The information provided herein or contained in any report provided herein is intended solely for institutional clients of UniCredit Markets & Investment Banking Division acting through Bayerische Hypo- und Vereinsbank AG, New York Branch and HVB Capital Markets, Inc. (together "HVB") in the United States, and may not be used or relied upon by any other person for any purpose. It does not constitute a solicitation to buy or an offer to sell any securities under the Securities Act of 1933, as amended, or under any other US federal or state securities laws, rules or regulations. Investments in securities discussed herein may be unsuitable for investors, depending on their specific investment objectives, risk tolerance and financial position. In jurisdictions where HVB is not registered or licensed to trade in securities, commodities or other financial products, any transaction may be effected only in accordance with applicable laws and legislation, which may vary from jurisdiction to jurisdiction and may require that a transaction be made in accordance with applicable exemptions from registration or licensing requirements. All information contained herein is based on carefully selected sources believed to be reliable, but HVB makes no representations as to its accuracy or completeness. Any opinions contained herein reflect HVB's judgement as of the original date of publication, without regard to the date on which you may receive such information, and are subject to change without notice. HVB may have issued other reports that are inconsistent with, and reach different conclusions from, the information presented in any report provided herein. Those reports reflect the different assumptions, views and analytical methods of the analysts who prepared them. Past performance should not be taken as an indication or guarantee of further performance, and no representation or warranty, express or implied, is made regarding future performance. HVB and/or any other entity of UniCredit Markets & Investment Banking Division may from time to time, with respect to any securities discussed herein: (i) take a long or short position and buy or sell such securities; (ii) act as investment and/or commercial bankers for issuers of such securities; (iii) be represented on the board of such issuers; (iv) engage in "market-making" of such securities; and (v) act as a paid consultant or adviser to any issuer. The information contained in any report provided herein may include forward-looking statements within the meaning of US federal securities laws that are subject to risks and uncertainties. Factors that could cause a company's actual results and financial condition to differ from its expectations include, without limitation: Political uncertainty, changes in economic conditions that adversely affect the level of demand for the company‘s products or services, changes in foreign exchange markets, changes in international and domestic financial markets, competitive environments and other factors relating to the foregoing. All forward-looking statements contained in this report are qualified in their entirety by this cautionary statement. UniCredit Markets & Investment Banking Division Bayerische Hypo- und Vereinsbank AG, Munich; Bank Austria Creditanstalt AG, Vienna as of 10 April 2008
March 18th, 2008 Dr. Jan H. Maruhn
38