Pricing Natural Gas Storage Using Dynamic Programming

Transcription

1 Pricing Natural Gas Storage Using Dynamic Programming Sergey Kolos 1 1 The presentation is by Markets Quantitative Analysis, part of Citigroup Global Markets' sales and trading operations. 10/21/2011 Sergey Kolos Storage by DP

2 Outline 1 2 Finding Markovian Approximate Dynamic

3 Outline 1 2 Finding Markovian Approximate Dynamic

4 Why Storage Is Valuable Characteristics: volume and rates of injection/withdrawal Storage value: ability to buy/sell gas in dierent times NG prices are seasonal, which makes NG storage particularly valuable (signicant intrinsic value): Extrinsic value: no commitment at t 0 to inj/withdrawal schedule - optionality

5 Why Storage Is Valuable Characteristics: volume and rates of injection/withdrawal Storage value: ability to buy/sell gas in dierent times NG prices are seasonal, which makes NG storage particularly valuable (signicant intrinsic value): Extrinsic value: no commitment at t 0 to inj/withdrawal schedule - optionality

6 Why Storage Is Valuable Characteristics: volume and rates of injection/withdrawal Storage value: ability to buy/sell gas in dierent times NG prices are seasonal, which makes NG storage particularly valuable (signicant intrinsic value): Extrinsic value: no commitment at t 0 to inj/withdrawal schedule - optionality

7 Why Storage Is Valuable Characteristics: volume and rates of injection/withdrawal Storage value: ability to buy/sell gas in dierent times NG prices are seasonal, which makes NG storage particularly valuable (signicant intrinsic value): Extrinsic value: no commitment at t 0 to inj/withdrawal schedule - optionality

8 Outline 1 2 Finding Markovian Approximate Dynamic

9 Optimization Problem for Pricing a Storage Contract Policy p - using all available information available at t decide how much to inject/withdraw over next time step Price vector z - all information necessary to describe future price movements (can be just a set of futures prices) State vector x - total information available at t consists of storage level and z Cashow c (t,p,x) - cashow at time t given policy value and prices (example: F t maxinj t costs t ) The goal is to nd policy function p (t,x) such that the expected value of storage is maximal: [ ] v = max p(t,x) E T t=0 df t c (t,x t,p (t,x t ))

10 Optimization Problem for Pricing a Storage Contract Policy p - using all available information available at t decide how much to inject/withdraw over next time step Price vector z - all information necessary to describe future price movements (can be just a set of futures prices) State vector x - total information available at t consists of storage level and z Cashow c (t,p,x) - cashow at time t given policy value and prices (example: F t maxinj t costs t ) The goal is to nd policy function p (t,x) such that the expected value of storage is maximal: [ ] v = max p(t,x) E T t=0 df t c (t,x t,p (t,x t ))

11 Optimization Problem for Pricing a Storage Contract Policy p - using all available information available at t decide how much to inject/withdraw over next time step Price vector z - all information necessary to describe future price movements (can be just a set of futures prices) State vector x - total information available at t consists of storage level and z Cashow c (t,p,x) - cashow at time t given policy value and prices (example: F t maxinj t costs t ) The goal is to nd policy function p (t,x) such that the expected value of storage is maximal: [ ] v = max p(t,x) E T t=0 df t c (t,x t,p (t,x t ))

12 Optimization Problem for Pricing a Storage Contract Policy p - using all available information available at t decide how much to inject/withdraw over next time step Price vector z - all information necessary to describe future price movements (can be just a set of futures prices) State vector x - total information available at t consists of storage level and z Cashow c (t,p,x) - cashow at time t given policy value and prices (example: F t maxinj t costs t ) The goal is to nd policy function p (t,x) such that the expected value of storage is maximal: [ ] v = max p(t,x) E T t=0 df t c (t,x t,p (t,x t ))

13 Optimization Problem for Pricing a Storage Contract Policy p - using all available information available at t decide how much to inject/withdraw over next time step Price vector z - all information necessary to describe future price movements (can be just a set of futures prices) State vector x - total information available at t consists of storage level and z Cashow c (t,p,x) - cashow at time t given policy value and prices (example: F t maxinj t costs t ) The goal is to nd policy function p (t,x) such that the expected value of storage is maximal: [ ] v = max p(t,x) E T t=0 df t c (t,x t,p (t,x t ))

14 Dynamic Programming This optimization problem is usually solved using dynamic programming: v t (x) = max{c (t,x,p) + df t E [v t+1 (χ t (x,p,ε))]} p where χ t (x,p,ε) describes how state x changes from t to t + 1, given policy p and a random shock ε

15 Outline 1 2 Finding Markovian Approximate Dynamic

16 Modelling Spot Most common approach Pro: simple direct modelling of dynamics, suitable for dynamic programming Cons: Need signicant number of parameters and factors to match observed term structures of volatility and correlation Hard to model seasonality Hard to calibrate to current market conditions (futures and volatilities cannot be matched exactly) Tricky and slow to compute sensitivities to futures prices and volatilities

17 Modelling Spot Most common approach Pro: simple direct modelling of dynamics, suitable for dynamic programming Cons: Need signicant number of parameters and factors to match observed term structures of volatility and correlation Hard to model seasonality Hard to calibrate to current market conditions (futures and volatilities cannot be matched exactly) Tricky and slow to compute sensitivities to futures prices and volatilities

18 Modelling Spot Most common approach Pro: simple direct modelling of dynamics, suitable for dynamic programming Cons: Need signicant number of parameters and factors to match observed term structures of volatility and correlation Hard to model seasonality Hard to calibrate to current market conditions (futures and volatilities cannot be matched exactly) Tricky and slow to compute sensitivities to futures prices and volatilities

19 Modelling Spot Most common approach Pro: simple direct modelling of dynamics, suitable for dynamic programming Cons: Need signicant number of parameters and factors to match observed term structures of volatility and correlation Hard to model seasonality Hard to calibrate to current market conditions (futures and volatilities cannot be matched exactly) Tricky and slow to compute sensitivities to futures prices and volatilities

20 Modelling Spot Most common approach Pro: simple direct modelling of dynamics, suitable for dynamic programming Cons: Need signicant number of parameters and factors to match observed term structures of volatility and correlation Hard to model seasonality Hard to calibrate to current market conditions (futures and volatilities cannot be matched exactly) Tricky and slow to compute sensitivities to futures prices and volatilities

21 Modeling Forward Curve HJM type model df t,t F t,t = σ q (t,t )dw q,t q (S t = F t,t) Pros: Direct calibration using market data (futures and volatilities are calibrated exactly) Can explain term structures of volatilities and correlations Con: Non-markovian (i.e. in general, to explain future prices of N contracts, one needs N state variables)

22 Modeling Forward Curve HJM type model df t,t F t,t = σ q (t,t )dw q,t q (S t = F t,t) Pros: Direct calibration using market data (futures and volatilities are calibrated exactly) Can explain term structures of volatilities and correlations Con: Non-markovian (i.e. in general, to explain future prices of N contracts, one needs N state variables)

23 Modeling Forward Curve HJM type model df t,t F t,t = σ q (t,t )dw q,t q (S t = F t,t) Pros: Direct calibration using market data (futures and volatilities are calibrated exactly) Can explain term structures of volatilities and correlations Con: Non-markovian (i.e. in general, to explain future prices of N contracts, one needs N state variables)

24 Modeling Forward Curve HJM type model df t,t F t,t = σ q (t,t )dw q,t q (S t = F t,t) Pros: Direct calibration using market data (futures and volatilities are calibrated exactly) Can explain term structures of volatilities and correlations Con: Non-markovian (i.e. in general, to explain future prices of N contracts, one needs N state variables)

25 Common Solution To Last Con Markovian Form of Exact Dynamics To make HJM type model Markovian can choose factors to be sum of exponentials - number of state variables (dimension of z) equals the number of dierent powers in exponents. Problems: For two factor model with all months having same exponents - dimension of z is 3 (too high) In reality exponents are slightly dierent for dierent months, so loose markovian property

26 Common Solution To Last Con Markovian Form of Exact Dynamics To make HJM type model Markovian can choose factors to be sum of exponentials - number of state variables (dimension of z) equals the number of dierent powers in exponents. Problems: For two factor model with all months having same exponents - dimension of z is 3 (too high) In reality exponents are slightly dierent for dierent months, so loose markovian property

27 Common Solution To Last Con Markovian Form of Exact Dynamics To make HJM type model Markovian can choose factors to be sum of exponentials - number of state variables (dimension of z) equals the number of dierent powers in exponents. Problems: For two factor model with all months having same exponents - dimension of z is 3 (too high) In reality exponents are slightly dierent for dierent months, so loose markovian property

28 My Solution Markovian Form of Approximate Dynamics Main idea: Instead of trying to get exact markovian approximation, decide on dimension of state variable, and nd markovian dynamics that is as close to exact dynamics as possible.

29 Finding Markovian Approximate Dynamic How to Identify a Good? Consider forward curve realized at time t that has just 2 contracts and simulate possible curves using MC Consider markovian approximation with state dimension dim (z) = 1 Markovian approximation space on a cloud of realized exact curves of MC paths: F F 1

30 Finding Markovian Approximate Dynamic How to Identify a Good? Consider forward curve realized at time t that has just 2 contracts and simulate possible curves using MC Consider markovian approximation with state dimension dim (z) = 1 Markovian approximation space on a cloud of realized exact curves of MC paths: F F 1

31 Finding Markovian Approximate Dynamic How to Identify a Good? Consider forward curve realized at time t that has just 2 contracts and simulate possible curves using MC Consider markovian approximation with state dimension dim (z) = 1 Markovian approximation space on a cloud of realized exact curves of MC paths: F F 1

32 Outline Finding Markovian Approximate Dynamic 1 2 Finding Markovian Approximate Dynamic

33 Finding Markovian Approximate Dynamic How to Find Good? We look for makovian approximation in the following form: f t = F0 exp(g t + U t z t ) where F0- current forward prices, g t and U t are a vector and a matrix parametrizing the approximation, z t - state vector. Assuming z t is distributed close to standard normal, U t can be computed through PCA of the original covariance matrix and g t can be selected to ensure E [f T ] = F 0,T Markovian approximation of a curve F t is computed using a projection (derived through log-least-squares): ( ) ( ) 1 z t = U T U t t U T t ln F t g t F0

34 Finding Markovian Approximate Dynamic How to Find Good? We look for makovian approximation in the following form: f t = F0 exp(g t + U t z t ) where F0- current forward prices, g t and U t are a vector and a matrix parametrizing the approximation, z t - state vector. Assuming z t is distributed close to standard normal, U t can be computed through PCA of the original covariance matrix and g t can be selected to ensure E [f T ] = F 0,T Markovian approximation of a curve F t is computed using a projection (derived through log-least-squares): ( ) ( ) 1 z t = U T U t t U T t ln F t g t F0

35 Finding Markovian Approximate Dynamic How to Find Good? We look for makovian approximation in the following form: f t = F0 exp(g t + U t z t ) where F0- current forward prices, g t and U t are a vector and a matrix parametrizing the approximation, z t - state vector. Assuming z t is distributed close to standard normal, U t can be computed through PCA of the original covariance matrix and g t can be selected to ensure E [f T ] = F 0,T Markovian approximation of a curve F t is computed using a projection (derived through log-least-squares): ( ) ( ) 1 z t = U T U t t U T t ln F t g t F0

36 Outline Finding Markovian Approximate Dynamic 1 2 Finding Markovian Approximate Dynamic

37 Finding Markovian Approximate Dynamic What is a Dynamics of Markovian Variables? and what is its connection to exact dynamics Suppose at time t 1 the state of the world is described by vector z t 1. This vector can be used to produce approximate forward curve f t 1. Use exact dynamics to produce forward curve F t. Project this curve to markov approximation to obtain z t. The resulting dynamics has the following simple linear form: z t = b t + A t ε + B t z t 1

38 Finding Markovian Approximate Dynamic What is a Dynamics of Markovian Variables? and what is its connection to exact dynamics Suppose at time t 1 the state of the world is described by vector z t 1. This vector can be used to produce approximate forward curve f t 1. Use exact dynamics to produce forward curve F t. Project this curve to markov approximation to obtain z t. The resulting dynamics has the following simple linear form: z t = b t + A t ε + B t z t 1

39 Finding Markovian Approximate Dynamic What is a Dynamics of Markovian Variables? and what is its connection to exact dynamics Suppose at time t 1 the state of the world is described by vector z t 1. This vector can be used to produce approximate forward curve f t 1. Use exact dynamics to produce forward curve F t. Project this curve to markov approximation to obtain z t. The resulting dynamics has the following simple linear form: z t = b t + A t ε + B t z t 1

40 Finding Markovian Approximate Dynamic What is a Dynamics of Markovian Variables? and what is its connection to exact dynamics Suppose at time t 1 the state of the world is described by vector z t 1. This vector can be used to produce approximate forward curve f t 1. Use exact dynamics to produce forward curve F t. Project this curve to markov approximation to obtain z t. The resulting dynamics has the following simple linear form: z t = b t + A t ε + B t z t 1

41 The approach is very good at capturing extrinsic value (close or exedes rolling intrinsic) It is very fast (fraction of a second for 5 year deal) Greeks are very stable Posibility to extend to more complex deal types

42 The approach is very good at capturing extrinsic value (close or exedes rolling intrinsic) It is very fast (fraction of a second for 5 year deal) Greeks are very stable Posibility to extend to more complex deal types

43 The approach is very good at capturing extrinsic value (close or exedes rolling intrinsic) It is very fast (fraction of a second for 5 year deal) Greeks are very stable Posibility to extend to more complex deal types

44 The approach is very good at capturing extrinsic value (close or exedes rolling intrinsic) It is very fast (fraction of a second for 5 year deal) Greeks are very stable Posibility to extend to more complex deal types

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