“theory of storage and option pricing: determinants of implied skewness and kurtosis

“Theory of Storage and Option Pricing: Determinants of Implied Skewness and

Kurtosis

“Theory of Storage and Option Pricing: Determinants of Implied Skewness and

Kurtosis

Marin Bozic and T. Randall FortenberyUniversity of Wisconsin-Madison

Znanstveni utorak, Ekonomski Institut Zagreb15 lipanj, 2010

M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 2/49

Introduction

Life is risky if you are a grain farmer…

• Types of uncertainty:• production risk• price risk (input and output)• counterparty risk• cash flow risk

•What can be done about it:• “Take it easy approach” a.k.a. Do Nothing, sell for cash• Removing price risk but forfeiting opportunity as well: Hedging with futures• Removing downside risk, but preserving opportunity of favorable price change: Options!


Introduction

Toolbox for managing risks• Futures contract: A promise to sell (or buy) at a pre-specified quantity, price, location, and time in the future (“delivery month”)• removes (almost all) price risk• removes counterparty risk• removes the opportunity to make some extra profit if prices increase a lot

• Options contract: Gives holder a right, but not the obligation to sell (or buy) a futures contract with pre-specified quantity, at a pre-specified price (strike price), at any point before the delivery month. • unlike futures, does not remove the profit opportunity from favorable cash prices• unlike futures, it’s not “free”, you have to pay for the right upfront


Introduction

So what happens if I…


Introduction


We entered a Dec’ 10futures contract Promising to sellat a price of $3.70.Price is locked,no matter what.

We entered a Dec’ 10futures contract Promising to sellat a price of $3.70.Price is locked,no matter what.


Introduction


On 2/1/2010 we bought an option To sell at the price of $3.70.If futures price in Juneis better than that,we forfeit the option.If not, we exercise it.

On 2/1/2010 we bought an option To sell at the price of $3.70.If futures price in Juneis better than that,we forfeit the option.If not, we exercise it.


Introduction


Option is never thebest choice (ex post)But it’s rarely the worst.

Option is never thebest choice (ex post)But it’s rarely the worst.


Introduction

What determines the price of an option contract?

• Options are a sort of insurance. In good times you don’t exercise it, in bad times it may save your business. • If farmers are risk-averse, they will likely hedge to reduce the uncertainty.• Does that mean that price of options depend on risk-preferences of farmers? Not necessarily. In fact, some folks got the Nobel price for showing how price of options are arrived at, solely by arbitraging away riskless profits.


Introduction


• Merton (1973), Black & Scholes (1973), Black (1976) • Arbitrage Pricing Theory:• Start from reasonable if simplifying assumptions:• Prices can never be negative• Changes in prices are proportional to time passed• It is known how small change in price will affect the price of option

• key assumptions: normality, volatility assumed constant and independent of price level

, ~ 0,dF

dt dz dz N dtF

(1)

2 20

1ln ~ ln ,

2TF N F T


Introduction


• Black & Scholes do a nice trick to show that if an arbitrageur holds cleverly chosen proportions of options and underlying futures, then in the simple world described by Eq. (1) such portfolio may be made riskless. • Returns on riskless portfolio must match riskless rate of interest (holding treasury bills) and hence risk preferences are purged from the function determining price.

•

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rtT T T

V f r t F K

e Max F K f F F r T dF

(2)


Introduction

Inverting the option pricing function: getting • In reality, we do not observe volatility parameter directly. • However, we can invert the function by asking: which would produce the option prices we do observe, if Black’s model assumptions hold

Calculated option prices via Black-Scholes formulaT-t = 90/365 r = 5.32%σ = 36.1% Ft = $3.64

•Pretended we don’t know σ and estimated it via Newton-Raphson algorithm, for each strike price separately

• Key consequence of the theory: IV curve should be flat!

Calculated option prices via Black-Scholes formulaT-t = 90/365 r = 5.32%σ = 36.1% Ft = $3.64

•Pretended we don’t know σ and estimated it via Newton-Raphson algorithm, for each strike price separately

• Key consequence of the theory: IV curve should be flat!


Introduction

S&P 500: Typical Post-1987 Implied Volatility Curve

• USDA Crop reports “XXX”, published 15th of March, June, Sep, Dec• Intention to plant reports• Planted reports• harvest reports


Introduction

Smile, you’re on the Candid Camera!• Implied Volatility curve can also be convex and symmetric. • That’s why we usually call IV curves “Volatility smiles”

Source: Hull, J.C. (2006) – Options, Futures, and Other Derivatives, pg. 377


Introduction

How do we reconcile “smiles” with theory?

, ~ 0,dF

dt dz dz N dtF


Introduction

Option premium as an expectation of terminal payoff

• Theory implies terminal distribution of futures price is log-normal(Log-normal = log of terminal price is normally distributed)

• If the traders expect terminal distribution to be non-normal, option prices will reflect that. Implied volatility curve will not be flat.

2

*

ln ~ ln ,2

, max ,0

T t

r T tt t

F N F T t T t

V F t e E K F


Introduction

Enter higher moments…

For a standardized density: location

spread

asymmetry

tail behavior

For a standardized density: location

spread

asymmetry

tail behavior

2

3

4

0

1

E x

E x

E x

E x

Introduction

Simulating skewed leptokurtic distributions

Source: Rubinstein, M. (1998) Edgeworth Binomial Trees

Kurtosis



Introduction

Effects of non-normal kurtosis on volatility smiles

Calculated option prices via Edgeworth binomial treeT-t = 90/365 r = 5.32%σ = 36.1% Ft = $3.64

•Pretended we don’t know σ and estimated it via Newton-Raphson algorithm for each strike price separately, assuming Black’s model holds (S=0, K=3).

• Key consequences of non-normal kurtosis:

•Volatility Smiles!• Higher kurtosis

Deeper smile



• Key consequences of non-normal kurtosis:

•Volatility Smiles!• Higher kurtosis

Deeper smile


Introduction

Effects of negative skewness on volatility curve



• Key consequences of negative skewness:

•Downward sloping curve!• Stronger skewness

Deeper slope



• Key consequences of negative skewness:

•Downward sloping curve!• Stronger skewness

Deeper slope


Introduction

Volatility skew for agricultural grains: Corn

Date: Jun 26, 2006Contract: Corn, Dec ’06Futures Price: $2.49

Commodities tend to have positively sloped implied volatility curve. Our research will explain why.



Introduction

Effects of positive skewness on volatility curve



• Key consequences of positive skewness:

•Upward sloping curve!• Stronger skewness

Deeper slope



• Key consequences of positive skewness:

•Upward sloping curve!• Stronger skewness

Deeper slope


Introduction

Problems with Edgeworth expansion

•The higher the skewness of the terminal distribution is, the higher will be the difference between implied volatility for options low and high strikes.• But Edgeworth can only create up to 8-9 percentage point difference, while we observe 15% difference in IV, or even more.• If we try to push Edgeworth harder, approximation will fail, density may turn negative in tails• We needed a different method to approach non-normality of terminal log-prices…


Introduction

Introduction to Generalized Lambda Distribution (GLD)

We needed a distribution that would…• allow for high flexibility in first four moments four-parameter family • Would be easy to simulate closed form percentile function (inverse CDF)• would produce close-form solution to option pricing


Introduction


1. Flexibility: by tweaking parameters of the distribution, we can accommodate almost any desired shape. Alternatively, we can think of this as being able to approximate any other distribution up to first four moments


Introduction


2. Easy to simulate: closed form inverse CDF allows easy drawing from GLD with arbitrary shape.Problem: parameters correspond to moments in highly complicated way

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Introduction



431

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Introduction



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1 2 1 2yxB x y t t dt


Introduction



431

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Introduction



431

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Introduction


Black-Scholes formula for European call on non-dividend paying stock:

1 2

2

1 2 1

,

ln /2

,

rtC S t SN d Ke N d

S K r t

d d d tt

GLD formula for European call on futures contract

2 43

0, 0 1 2 2

11

1 02 3 4 3 4

1

1 111

, 1 1

rt rt

t

C F t F e G e KG G p K

p K p Kp K p KeG F p K


Introduction

Volatility skew for agricultural grains: Corn

Date: Jun 26, 2006Contract: Corn, Dec ’06Futures Price: $2.49




Introduction

Why do commodities have positive implied skewness?

• Essential role of storage• Imagine that a corn yield is normally distributed. I.e. it’s equally likely that harvest will be better than usual and that it will be worse than usual.• If harvest is very good, we don’t have to eat it all right away, we can store it• If harvest is bad, we have only so much of last crop’s stocks we can use in addition to new crop.• Normal shocks to yield will lead to skewed shocks to price!


Source: Williams & Wright (1991) Storage and Commodity Markets

Tighter market • Higher expected price• Larger variance• Higher (more positive) skewness


Introduction

Main hypothesis - Contribution of this paper

• Literature has worked out that rational expectations competitive model is able to replicate time-series properties of spot commodity price: mean-reversion, sudden peaks with gradual decline, skewness• We were not able to test so far if market expectations were in alignment with that model

• We propose here a way to test exactly this:• if model works, then in periods where inventories are low, and buffers to bad harvest are less effective, market should expect higher skewness in terminal price distribution.


Introduction

Data

• Tick data for futures and options for corn, wheat and soybeans for 1995-2009, donated by Chicago Mercantile Exchange: 90+ million data points

•Options matched with last observed futures trade

• Frequency reduced to 15 minutes interval•11,139 files.

• For each option, implied volatility calculated using Black’s model modified to account for early exercise (Cox Ross Rubinstein binomial trees, 500 steps)

• Price of equivalent European option calculated


Introduction

What influences expected price distribution

• Major role USDA reports play: • Quarterly stock reports

“Grain Stocks”, Jan 11, Mar 31, Jun 30, Sep 30

• USDA surveys on intended acres planted“Prospective plantings”, March 31

• Report on actual acres planted“Acreage”, June 30

•Report on how crops are progressing“Crop progress”, weekly (Monday, 4pm), Apr-Dec


Introduction

Design specifications and current prices of corn futures and options

• Months traded: March, May, July, September, December• Contract size: 5000 bushels• Options expire: The last Friday preceding the first day of the delivery month by at least two business days.

•June, 14 2010, closing prices:• December 2010 Futures: 3.72 $/bu• Options:• $3.60 Call: $0.355 ($1,775 per contract)• $3.90 Put: $0.397 ($1,985 per contract)

Introduction


Volatility surface for December contracts: 2004

Introduction




Introduction

Skewness vs. market “tightness” 1997-2005

Lower ending stocks relative to demand are associated with higher difference in implied volatility between high and low strike prices

0.81


Introduction

What is the influence of uncertainty in demand?

CornDec ’09


Introduction

Estimation of GLD option pricing model

• Used least squares to estimate lambda parameters that best matchObserved option prices for a certain contract on a given day

* Software used: excel solver, Fortran .dll libraries for computationally intensive elements* Results may be sensitive to starting values* calculated implied skewness and kurtosis for Corn December contracts, as priced from June, 10 to June, 20 of each year (i.e. before Acreage report)* averaged implied moments over that 9 trading days


Introduction

Testing main hypothesis

• Implied skewness regressed on intended increase in planted acreage and ending stocks-to-use• if hypothesis has any merit, we should see statistically significant, negative parameters for both explanatory variables.


Introduction

Skewness vs. market “tightness” 1995-2009

0.27


Introduction



Introduction

Conclusions

•Deviations from classical Black’s model can be explained by theory of storage – asymmetric response of price to symmetric shocks to crop yield under rational expectations induces expected terminal distribution to be positively skewed and leptokurtic to the extent that exceeds skewness and kurtosis of lognormal distribution

• We find the way to extend the test of rational expectations competitive model from time-series properties of spot price to market expectations of future spot price

• Next challenge: try to separate option premium due to market expectations from “liquidity premium” priced in options that are very much out-of-money, and are thinly traded.

Thank You!

______________________________________________________________

Theory of Storage and Option Pricing: Determinants of Implied Skewness and Kurtosis

Marin Bozic and T. Randall FortenberyUniversity of Wisconsin-Madison

Contact Info:

Marin BozicResearch Assistant

Department of Agricultural and Applied EconomicsUniversity of Wisconsin-Madison

Email: [email protected]

“theory of storage and option pricing: determinants of implied skewness and kurtosis

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