Least Squares Monte Carlo and Energy Markets Matteo Tesser - - PowerPoint PPT Presentation

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Least Squares Monte Carlo and Energy Markets

Matteo Tesser matteo.tesser@fairmat.com

Fairmat SRL

10/05/2013

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Agenda

1

Introduction

2

Least-Squares Monte Carlo

3

Fairmat and Open Source

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Agenda

1

Introduction

2

Least-Squares Monte Carlo

3

Fairmat and Open Source

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Agenda

1

Introduction

2

Least-Squares Monte Carlo

3

Fairmat and Open Source

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Agenda

1

Introduction

2

Least-Squares Monte Carlo

3

Fairmat and Open Source

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Energy Markets and Quantitative Methods

Liberalization of energy markets has incremented the number of players and has introduced several distribution and procurement alternatives.

Producers have the possibility of offering bilateral contracts or of bidding offers to the markets Consumers, aside entering into bilateral contracts have a wider choice of suppliers including going directly to the market.

Financial products linked to energy are becoming more and more wide spread (see swing options) All the available alternatives need quantitative assessment in term of costs and risks

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Energy Markets and Quantitative Methods

Liberalization of energy markets has incremented the number of players and has introduced several distribution and procurement alternatives.

Producers have the possibility of offering bilateral contracts or of bidding offers to the markets Consumers, aside entering into bilateral contracts have a wider choice of suppliers including going directly to the market.

Financial products linked to energy are becoming more and more wide spread (see swing options) All the available alternatives need quantitative assessment in term of costs and risks

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Energy Markets and Quantitative Methods

Liberalization of energy markets has incremented the number of players and has introduced several distribution and procurement alternatives.

Producers have the possibility of offering bilateral contracts or of bidding offers to the markets Consumers, aside entering into bilateral contracts have a wider choice of suppliers including going directly to the market.

Financial products linked to energy are becoming more and more wide spread (see swing options) All the available alternatives need quantitative assessment in term of costs and risks

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Energy Markets and Quantitative Methods

Liberalization of energy markets has incremented the number of players and has introduced several distribution and procurement alternatives.

Producers have the possibility of offering bilateral contracts or of bidding offers to the markets Consumers, aside entering into bilateral contracts have a wider choice of suppliers including going directly to the market.

Financial products linked to energy are becoming more and more wide spread (see swing options) All the available alternatives need quantitative assessment in term of costs and risks

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Valuation of flexible contracts

Large consumers usually has several alternatives including the capability of self generating energy and the access to flexible contracts (containing contingent decisions). The value (and the risk) of flexibility must be assessed: to do that the decision maker must find the schedule of contingent decisions for current and future dates that maximizes the expected discounted rewards.

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Valuation of flexible contracts

Large consumers usually has several alternatives including the capability of self generating energy and the access to flexible contracts (containing contingent decisions). The value (and the risk) of flexibility must be assessed: to do that the decision maker must find the schedule of contingent decisions for current and future dates that maximizes the expected discounted rewards.

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Valuation of flexible contracts

Large consumers usually has several alternatives including the capability of self generating energy and the access to flexible contracts (containing contingent decisions). The value (and the risk) of flexibility must be assessed: to do that the decision maker must find the schedule of contingent decisions for current and future dates that maximizes the expected discounted rewards.

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Valuation of flexible contracts

Large consumers usually has several alternatives including the capability of self generating energy and the access to flexible contracts (containing contingent decisions). The value (and the risk) of flexibility must be assessed: to do that the decision maker must find the schedule of contingent decisions for current and future dates that maximizes the expected discounted rewards.

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Optimal Energy Procurement & Integrated risk management

Simulate in the same framework the cash-flows coming from the business activities and the (stochastic) cost from the energy procurement Price/Evaluate contracts (Risk Neutral) Evaluate cash-flows at risk (CFaR) under historical probabilities Optimize with respect to the available procurement alternatives Appropriate frameworks:

Multi-stage stochastic programming (Approximated) dynamic programming and Markov Decision Processes

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Optimal Energy Procurement & Integrated risk management

Simulate in the same framework the cash-flows coming from the business activities and the (stochastic) cost from the energy procurement Price/Evaluate contracts (Risk Neutral) Evaluate cash-flows at risk (CFaR) under historical probabilities Optimize with respect to the available procurement alternatives Appropriate frameworks:

Multi-stage stochastic programming (Approximated) dynamic programming and Markov Decision Processes

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Model specification

Let T = {tk, k = 1, ..., n}, with t0 = 0 and tn our planning horizon, be a date sequence. With r is indicated the discounting rate, and y represents an exogenous state variable. x = (y, o, s) is our augmented state where we add operating modes o, and other endogenous state variables s. With f (x, a) we indicate a function that provides a new state given the action a. at dates tk ∈ T , we have the stochastic cash-flows πk(x, a) function of our decision a ∈ A. Goal: find δ∗ such max

δ

V = E n

• k=0

e−rtkπi(xk, δk)

• ,

s.t.x′ = f (x, a)

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Model specification

Let T = {tk, k = 1, ..., n}, with t0 = 0 and tn our planning horizon, be a date sequence. With r is indicated the discounting rate, and y represents an exogenous state variable. x = (y, o, s) is our augmented state where we add operating modes o, and other endogenous state variables s. With f (x, a) we indicate a function that provides a new state given the action a. at dates tk ∈ T , we have the stochastic cash-flows πk(x, a) function of our decision a ∈ A. Goal: find δ∗ such max

δ

V = E n

• k=0

e−rtkπi(xk, δk)

• ,

s.t.x′ = f (x, a)

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Agenda

1

Introduction

2

Least-Squares Monte Carlo

3

Fairmat and Open Source

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Least-Squares Monte Carlo

The optimal procurement problem stated in the slide before can be modeled as a Markov Decision Process and hence solved by using (approximated) dynamic programming. The Least-Squares Monte Carlo (LSMC) approach, an algorithm which efficiently implements dynamic programming withing Monte Carlo simulation, has become a popular method for pricing American options (with Longstaff & Schwartz (2001)). In Gamba (2003); Gamba & Tesser (2009) we extended the LSMC algorithm for the solution of general Markov Decision Processes (MDPs).

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Least-Squares Monte Carlo

The optimal procurement problem stated in the slide before can be modeled as a Markov Decision Process and hence solved by using (approximated) dynamic programming. The Least-Squares Monte Carlo (LSMC) approach, an algorithm which efficiently implements dynamic programming withing Monte Carlo simulation, has become a popular method for pricing American options (with Longstaff & Schwartz (2001)). In Gamba (2003); Gamba & Tesser (2009) we extended the LSMC algorithm for the solution of general Markov Decision Processes (MDPs).

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Least-Squares Monte Carlo

The optimal procurement problem stated in the slide before can be modeled as a Markov Decision Process and hence solved by using (approximated) dynamic programming. The Least-Squares Monte Carlo (LSMC) approach, an algorithm which efficiently implements dynamic programming withing Monte Carlo simulation, has become a popular method for pricing American options (with Longstaff & Schwartz (2001)). In Gamba (2003); Gamba & Tesser (2009) we extended the LSMC algorithm for the solution of general Markov Decision Processes (MDPs).

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Least Squares Monte Carlo and MDP - I

The value of the project V is the fixed point of the Bellman

• perator

Vti(x) = maxa∈A{π(x, a) + eti+1−tiEx[V (f (x, a))]} and the related optimal decision schedule is δti(x) = arg max

a∈A {π(x, a) + eti+1−tiEx[V (f (x, a))]}

where Ex[.] is the expectation with respect to p(.|x, a).

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Least Squares Monte Carlo and MDP - II

The contract continuation value V at the generic date ts ≥ 0, for generic values of y,o and s is: Vti(xti) = E∗

i

• n
• k=i

e−r(tk−ti)πti(xti, δ(xti))

• ,

(1) where with E∗

i we indicate the expectation with respect to f

conditioned at the information at date i. Given that x follows a Markovian process, the needed information for calculating expectations of V evolution is present in the current values of x.

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Least Squares Monte Carlo and MDP - III

The approach is based on the idea that the conditional expectations present in (1) can be approximated by a linear regression defined on a given basis of functions of variables xti. With V H

i

we indicate the H-components approximation of the continuation value Vi using the base Φ = {φ1, ..., φH} of functions:

• V H

s (ω) = H

• h=0

ah

s φh(xi(ω)),

(2) where as = (a1

s, . . . , aH s ) is the vector given by the projection

• f V on the basis Φ.

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

ADP vs multi-stage stochastic programming

Plus:

Reduction of the curse-of-dimensionality which occur when building a proper multi-stage event-tree. Ability to model several decision stages with tenth of thousands of scenarios, allowing more robust estimates

• f risk measures (CVaR, CFaR)

Minus:

Limitation to MDPs: stochastic programming can solve more complex models

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

ADP vs multi-stage stochastic programming

Plus:

Reduction of the curse-of-dimensionality which occur when building a proper multi-stage event-tree. Ability to model several decision stages with tenth of thousands of scenarios, allowing more robust estimates

• f risk measures (CVaR, CFaR)

Minus:

Limitation to MDPs: stochastic programming can solve more complex models

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

ADP vs multi-stage stochastic programming

Plus:

Reduction of the curse-of-dimensionality which occur when building a proper multi-stage event-tree. Ability to model several decision stages with tenth of thousands of scenarios, allowing more robust estimates

• f risk measures (CVaR, CFaR)

Minus:

Limitation to MDPs: stochastic programming can solve more complex models

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

From Monte Carlo simulation to stochastic programming

When the problem cannot be modeled using the MDP paradigm we can use Monte Carlo simulation and multi-state stochastic programming jointly by:

1

Simulating underlying or payoff using the MDP with Monte Carlo simulation framework

2

Discretizing the price realizations using a scenario reduction technique (See Dupa˘ cov´ a et al. (2003))

3

From the event tree setting up a multi-stage stochastic programming problem

4

Find optimal policies of procurement and generation

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Additional issues

Aside proposing a coherent decision model, one of the most crucial points in solving those types of problem is the model calibration and validation Stochastic models must be calibrated to available market data. Risk neutral models are calibrated by matching the

• bserved price of traded financial instruments like caps,

swaps, options, futures, forward curves. Real world models are calibrated by fitting historical series Sometimes it is necessary to cross validate the decisions results by using different approaches or different theoretical models. Scenario generation: take into account no-arbitrage constraints!

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Additional issues

Aside proposing a coherent decision model, one of the most crucial points in solving those types of problem is the model calibration and validation Stochastic models must be calibrated to available market data. Risk neutral models are calibrated by matching the

• bserved price of traded financial instruments like caps,

swaps, options, futures, forward curves. Real world models are calibrated by fitting historical series Sometimes it is necessary to cross validate the decisions results by using different approaches or different theoretical models. Scenario generation: take into account no-arbitrage constraints!

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Additional issues

Aside proposing a coherent decision model, one of the most crucial points in solving those types of problem is the model calibration and validation Stochastic models must be calibrated to available market data. Risk neutral models are calibrated by matching the

• bserved price of traded financial instruments like caps,

swaps, options, futures, forward curves. Real world models are calibrated by fitting historical series Sometimes it is necessary to cross validate the decisions results by using different approaches or different theoretical models. Scenario generation: take into account no-arbitrage constraints!

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Additional issues

Aside proposing a coherent decision model, one of the most crucial points in solving those types of problem is the model calibration and validation Stochastic models must be calibrated to available market data. Risk neutral models are calibrated by matching the

• bserved price of traded financial instruments like caps,

swaps, options, futures, forward curves. Real world models are calibrated by fitting historical series Sometimes it is necessary to cross validate the decisions results by using different approaches or different theoretical models. Scenario generation: take into account no-arbitrage constraints!

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Agenda

1

Introduction

2

Least-Squares Monte Carlo

3

Fairmat and Open Source

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

The above principles are implemented in Fairmat

Fairmat is a tool for modeling and assessing particular types of decision under uncertainty problems arising in finance and management sciences. Applications include derivative contracts pricing, capital budgeting problems and commodity contracts valuations. Key features are: Flexibility (it’s a modeling tool) Independent from numerical methods (it’s a description language) Lower barrier to entry (MSC students, MBA students, Analysts) Extensibility (third parties can add new capabilities through a plug-in system) Free and multi-platform, with commercial versions available

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

The above principles are implemented in Fairmat

Fairmat is a tool for modeling and assessing particular types of decision under uncertainty problems arising in finance and management sciences. Applications include derivative contracts pricing, capital budgeting problems and commodity contracts valuations. Key features are: Flexibility (it’s a modeling tool) Independent from numerical methods (it’s a description language) Lower barrier to entry (MSC students, MBA students, Analysts) Extensibility (third parties can add new capabilities through a plug-in system) Free and multi-platform, with commercial versions available

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Available theoretical models

One of the peculiar feature of Farmat is that it possible to valuate the same contract/problem using different theoretical models: Interest Rates: Black, CIR, Hull-White, Pelsser Squared Gaussian, Libor Market Model, Equity: GBM, Variance-Gamma, Heston, Dupire Commodity: Lucia-Swchartz, Swchartz-Smith, Ornstein-Uhlenbeck Inflation: Jarrow-Yildirim, Dai-Singleton

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Fairmat and open source development

A part of our solution is open source, see: https://github.com/fairmat In a similar way to the Google Summer of Code, we sponsor the development of open source model and software by offering bounties for completed projects. See http://www.fairmat.com/developers/ projects-ideas/. Furthermore we provide revenue sharing plans and we

• ffer internships.

if you are interested to contribute contact us.

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

Thank you for your attention!

For any question please do not hesitate to contact me at matteo.tesser@fairmat.com

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden

Introduction Least-Squares Monte Carlo Fairmat and Open Source Riferimenti bibliografici

References

Dupa˘ cov´ a, J., Gr¨

• we-Kuska, N., & R¨
• mish, W. (2003).

Scenario reduction in stochastic programming: An approach using probability metrics. Mathematical Programming, 95, 493-511. Gamba, A. (2003). Real options valuation: A monte carlo

• approach. faculty of management, university of calgary wp
• no. 2002/3. EFA 2002 Berlin Meetings Presented Paper.

Available at SSRN: http://ssrn.com/abstract=302613 or http://dx.doi.org/10.2139/ssrn.302613. Gamba, A., & Tesser, M. (2009). Structural estimation of real

• ptions models. Journal of Economic Dynamics and

Control, 33. Longstaff, F., & Schwartz, E. (2001). Valuing American

• ptions by simulation: a simple least-squares approach.

Review of Financial Studies, 14(1), 113–147.

The contents of this presentation are property of Fairmat SRL, editing and distribution are forbidden