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“Theory of Storage and Option Pricing: Determinants of Implied Skewness and Kurtosis. Marin Bozic and T. Randall Fortenbery University of Wisconsin-Madison Znanstveni utorak , Ekonomski Institut Zagreb 15 lipanj , 2010. Types of uncertainty: production risk - PowerPoint PPT Presentation
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“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!
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 3/49
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 4/49
Introduction
So what happens if I…
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 5/49
Introduction
So what happens if I…
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.
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 6/49
Introduction
So what happens if I…
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.
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 7/49
Introduction
So what happens if I…
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.
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 8/49
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.
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 9/49
Introduction
What determines the price of an option contract?
• 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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 10/49
Introduction
What determines the price of an option contract?
• 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.
•
00
, , , ,
,0 ; , , ,
t
rtT T T
V f r t F K
e Max F K f F F r T dF
(2)
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 11/49
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!
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 12/49
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 13/49
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 14/49
Introduction
How do we reconcile “smiles” with theory?
, ~ 0,dF
dt dz dz N dtF
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 15/49
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 16/49
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 17/49
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 18/49
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
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 19/49
Introduction
Effects of negative skewness on volatility curve
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 negative skewness:
•Downward sloping curve!• Stronger skewness
Deeper slope
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 negative skewness:
•Downward sloping curve!• Stronger skewness
Deeper slope
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 20/49
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.
Commodities tend to have positively sloped implied volatility curve. Our research will explain why.
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 21/49
Introduction
Effects of positive skewness on volatility curve
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 positive skewness:
•Upward sloping curve!• Stronger skewness
Deeper slope
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 positive skewness:
•Upward sloping curve!• Stronger skewness
Deeper slope
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 22/49
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…
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 23/49
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 24/49
Introduction
Introduction to Generalized Lambda Distribution (GLD)
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 25/49
Introduction
Introduction to Generalized Lambda Distribution (GLD)
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
431
12
1p pF p
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 26/49
Introduction
Introduction to Generalized Lambda Distribution (GLD)
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
431
12
1p pF p
1 2/A
3 4
1 1
1 1A
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 27/49
Introduction
Introduction to Generalized Lambda Distribution (GLD)
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
431
12
1p pF p
2 2 22/B A
1
1
3 43 4 0
1 12 1 ,1 2 , 1
1 2 1 2yxB x y t t dt
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 28/49
Introduction
Introduction to Generalized Lambda Distribution (GLD)
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
431
12
1p pF p
33
3 3 2 32
3 2C AB A
3 4 3 43 3
1 13 1 2 ,1 3 1 ,1 2
1 3 1 3C
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 29/49
Introduction
Introduction to Generalized Lambda Distribution (GLD)
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
431
12
1p pF p
2 44
4 4 42
4 6 3D AC A B A
3 4 3 4 3 43 3
1 14 1 3 ,1 4 1 ,1 3 6 1 2 ,1 2
1 4 1 4D
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 30/49
Introduction
Introduction to Generalized Lambda Distribution (GLD)
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 31/49
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.
Commodities tend to have positively sloped implied volatility curve. Our research will explain why.
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 32/49
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!
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 33/49
Source: Williams & Wright (1991) Storage and Commodity Markets
Tighter market • Higher expected price• Larger variance• Higher (more positive) skewness
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 34/49
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.
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 35/49
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 36/49
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 37/49
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 38/49
Volatility surface for December contracts: 2004
Introduction
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 39/49
Volatility surface for December contracts: 2005
Introduction
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 40/49
Volatility surface for December contracts: 2006
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 41/49
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 42/49
Introduction
What is the influence of uncertainty in demand?
CornDec ’09
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 43/49
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
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 44/49
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.
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 47/49
Introduction
Skewness vs. market “tightness” 1995-2009
0.27
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 45/49
Introduction
Testing main hypothesis
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 46/49
Introduction
Testing main hypothesis
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 47/49
Introduction
Testing main hypothesis
M. Bozic and T. R. Fortenbery Theory of Storage and Option Pricing 48/49
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]