Commodity Modeling What worked well worked poorly or

Commodity Modeling What worked well, worked poorly or didn’t work at all Alexander Eydeland Morgan Stanley IPAM, May 2015

Evolution of Models • Black-Scholes: – constant volatility – option value reflects the cost of delta hedging (only) – But we do vega hedging, thus, contradicting the assumption of constant volatility – Moreover, what is the cost of vega hedging and why is it not reflected in the option value?

Evolution of Models • Stochastic Volatility Models (SV) – Take care of non-constant volatility and vega hedging – But we now face a new set of problems: • We have introduced a number of additional model parameters (eg. , vol-of-vol, correlation between price and volatility processes, strength of volatility mean reversion, etc. ) • Calibrating these parameters is a difficult but mostly manageable task. The problem is that these parameters change from day to day, sometimes, significantly. Hence, they require hedging. Again, we run into contradiction since, while calculating option values, we have assumed that these parameters are constant. • What is the cost of this hedging and how to incorporate it into the option valuation?

Evolution of Models • Chasing the “hedging” tail – Similar problems exist for other modeling approaches (jump-diffusion, regime switching, stochastic time change models, etc. ) – We choose these methods to model observed market behavior – fat tails, volatility smile, jumps, spikes, etc. – However, each model brings with it a number of model parameters presumed constant in option valuation. These parameters need to be calibrated on market prices. – Needless to say, these parameters change from day to day (thus, contradicting the assumptions) and, hence, need to be hedged. – Perpetual question: what is the cost of this hedging and do we need a model of evolution of these parameters to answer this question?

Evolution of Models • More serious problem – In commodity markets we find ourselves in the situation of “incompleteness” even for models with relatively few parameters to hedge, i. e. , there are no market instruments to hedge all we want to hedge – In this case the usage of risk-neutral methodology is questionable. We may need to use physical measure – The non-hedgeable model parameters calibrated on market data may be substantially different from the realized values – For example, if in a SV model (e. g. , Heston) the vol-of-vol parameter calibrated on the implied option volatility smile is significantly different from the realized vol-of-vol, we risk to seriously overestimate the structures that are convex in volatility

Vol-of-Vol: realized vs implied

Evolution of Models • Modeling challenges: – Issue of parameter non-uniqueness – In many commonly used models (SV, jump) the shapes of generated volatility smiles diverge from typically observed shapes in the OTM region

Model vs Market

Evolution of Models • Local Volatility Model (LV) – Pros: very economical as parameters are concerned – “Cons”: Volatility evolution is induced by the current implied volatility structure – What is correct volatility dynamics in commodity markets (sticky strike, sticky delta, etc. ) ? – Empirical evidence or even a hint – none – Moreover, LV is designed for spot processes. In many commodity markets spot evolution is meaningless (no liquid spot prices). Futures prices have very few early exercise option prices available.

Stable Empirical Properties • Do commodity vols exhibit in a stable way such properties as “reverse leverage”, i. e. , negative correlation between price and vol movs? Probably, no.

Stable Empirical Properties • Samuelson effect • Example: #2 Heating Oil and #6 Fuel Oil At-the-Money Implied Volatility Curves on 21/10/2003

Samuelson effect • The historical graphs of the implied volatility ratios of WTI futures contracts as a function of time to expiry T-t 1

Samuelson Effect • Samuelson effect is typically represented as • Resulting deterministic volatility dynamics works reasonably well for pricing a wide range of derivatives capturing “first order” effects

Correlation Structure: single commodity • ρ is either instantaneous correlation at time s or cumulative over the interval [t, s]; T 1 and T 2 - expiries of futures contracts • Requirements (crude, products, NG, power): – For fixed t and T 2 – T 1 – For fixed t, s, T 1 – For fixed t and T 2 – T 1 – For fixed t and s the matrix is positive definite in T 1 and T 2 • Finding analytical representation – a challenge

Correlation Structure: two commodities power/NG correlation structure • Capturing the structure needed for correctly pricing tolling deals, power plants, etc. Still a challenge. The best approach – hybrid (supply/demand + market) model Not for redistribution

Criteria for Choosing and Validating a Model • Ability to match the market data – most models used in practice do reasonably well • The best test: Historical hedge performance – Time consuming but very useful test. Recommended for complex models – The test yields the distribution of the residuals. The properties of this distribution are important in determining the quality of the model • Example: SV vs log-normal with re-calibrated implied vol. In our tests: no difference

Post 2008 world • What discount factor to use? What is a risk-free rate? • Adjustments: – – – CVA Cost of funding Liquidity adjustment Concentration adjustment FVA: estimated exit price correction due to credit quality of potential counterparty • A thoroughly commodity problem: physical assets are not marked-to-market while hedges are

Commodity Derivatives: Financial Products • Payout is well defined, financially settled • Examples: – Listed and OTC options: APOs (average price options), CSOs (calendar spread options), swaptions – Baskets, Indices, Options on baskets (related to component performance: best-of, Himalayan) – Modeling is not much different from other markets

Commodity Derivatives: Real Options

Commodity Derivatives: Physical Products • Payout at expiration is not well defined; a result of complex process involving optimization, logistics, transportation, dispatch, procurement, etc. • The majority of energy derivatives are spread options • Spread Options – Power plant -- spark spread option (option on the spread between power prices and fuel(s)+emission price – Storage -- calendar spread option – Transmission -- geographical spread option – Refinery -- crack spread option

Modeling Real Options: New Challenges • Non-market/non-price parameters – Examples: • Load growth • Changes in the market structure • ISO activity – Standard models cannot capture these phenomena • Regulatory actions – Examples: • • New emission markets CER approval Price caps etc. • Geology, meteorology, technology

Successful methodologies: Hybrid Power Price Model Power is a function of principal drivers 1. Demand 2. Fuel Prices 3. Outages

Hybrid Power Price Model uses fundamental and market data • sgen - function determined by technical characteristics of all power plants (efficiency, operational constraints, etc. ) • D - demand • U - fuel(s) used • Ω - outages

Hybrid Model generates realistic paths Actual prices vs. Modeled prices

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