Topic 3: Quantifying uncertainty from data
Shauna Phillips
School of Economics
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AREC3005 Agricultural Finance & Risk
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Dr Shauna Phillips (Unit Coordinator) Phone: 93517892
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Uncertainty estimation from data
› There are two ways of thinking about this uncertainty – that it is: – Static:
– Time doesn’t matter
– Our expectation for the next time period is always the mean – …irrespective of what we have observed during this period
– Dynamic:
– Time matters
– Our expectation for the next time period depends on what we have observed during this period and what we know about the ability of the system to drift
– …possibly, towards the mean, if there is one – i.e. mean reversion
Static characterisation of uncertainty
› Random variables are driven by events – time does not matter
– This interpretation is completely appropriate for the casino, where the cards and
dice are not changing in a systematic way
– However, if we have data which is evolving through time
– Data gathered now may not look anything like data collected in the past
– For example, due to climate change, economic growth, and non-stationary processes more generally
Static characterisation of uncertainty
› We can summarise uncertainties from data using:
– Estimates of the moments of observed distributions, assuming the distribution is
stationary
– Arranging the data into density and cumulative density functions or histograms, and interpreting
– Possibly, by fitting some standard distribution function, such as Normal, Gamma, or Inverse Gaussian, to the data
Normal (Gaussian) distribution
Gamma distribution
Inverse gamma distribution
Inverse Gaussian distribution
Rain data BOM
April rain (mm) sydney 1901-2015
0-30 31-60 61-90 91-120 121-150 151-180 181-210 211-240 241-270 271-300 301-330 331-360 361-390 391-420 421-450
25 20 15 10
Orroroo wheat T/ha
0-0.5 >0.5-1 >1-1.5 >1.5-2 >2-2.5 >2.5-3 >3-3.5 >3.5-4 >4-4.5 >4.5-5
Orroroo wheat T/ha
40 35 30 25 20 15 10
Clare wheat T/ha
0-0.5 >0.5-1 >1-1.5 >1.5-2 >2-2.5 >2.5-3 >3-3.5 >3.5-4 >4-4.5 >4.5-5
Clare wheat T/ha
Static CDF for Clare wheat
40 35 30 25 20 15 10
Orroroo DSE/ha
Orroroo DSE/ha
25 20 15 10
Dynamic characterisation of uncertainty
› Kolmogorov defined dynamic probabilities in 1931
› Stochastic process: a collection, over time, of realisations of a random
variable that characterises a system
– In discrete time, a stochastic process is given by a sequence of random variables, and the time series associated with these random variables
– Deterministic counterpart: a deterministic process which can evolve only one way through time
Static vs. Dynamic
› Random variables (static) are driven by events – Time does not matter
› Stochastic processes (dynamic) are driven by information
– Information includes all the events that have happened, all the events that
haven’t happened, and when the events did or didn’t happen
– As information accumulates, uncertainty about the future is resolved
– Time matters
Example: Static
› Suppose there are two urns
– One urn containing red and black balls – The other urn contains two dice
› If neither urn is shaking, expectations and probabilities are driven by the event of a hand reaching in and drawing out a ball or a die
– The urns can remain for a millisecond or a millennium, and the expectations and probabilities will be the same, regardless of when the hand reaches in
– In this case, time does not matter
Example: Dynamic
› Next, suppose the urns are shaking
– The red and black balls will bounce around, but they won’t have changed
their colours
– Expectations and probabilities will be unchanged, regardless of when the hand reaches in
– The dice, however, will have changed their spots and/or orientations
– Their spots/orientations will change very few times in a millisecond
– Their spots/orientations will change a near-infinite number of times in over a millennium
– Expectations and probabilities depend upon the elapsed time – In this case, time matters
Stochastic processes
› Stochastic processes describe systems that are shocked by external forces and set in motion
– If someone describes a process in terms of (1) what might happen and (2) when in might happen, they are describing a stochastic process
– For example:
– What’s the price of barley?
– Now, or at harvest time? – Should I sell my stocks?
– Now or later in the year? – What’s the chance of rain?
– Today, next week or after climate change? – Will the glaciers melt?
– I don’t know: if so, I probably won’t be around to drown
› There are a number of stochastic processes which may be used as models of the data
› series. In real options analyses, a stochastic process known as Geometric Brownian
› Motion (GBM) is almost always used, due to the existence of the well- known analytical
› solution for option pricing called the Black-Scholes Formula. Another well known
› stochastic process is the Ornstein-Uhlenbeck (OU) process, although an analytical
› solution for option pricing is not known. These two processes have very different
› properties and their application needs to be carefully considered.
Specified stochastic processes
› Three well-known stochastic processes are the: – Ornstein-Uhlenbeck process
– Geometric Brownian Motion
– Arithmetic Brownian Motion
› Ergodic systems can be attracted toward a dynamic equilibrium: – e.g. Orstein-Uhlenbeck process
› Non-ergodic systems have no dynamic attractor, and will either explode or implode:
– e.g. Arithmetic Brownian Motion and Geometric Brownian Motion
Ornstein-Uhlenbeck
› The most common process for application to ‘mean-reverting’ natural/agricultural systems
dx =(−x)dt+dW ttt
xt is the random variable (rainfall, temperature, yield)
μ is the dynamic attractor (mean/average)
is the rate of return to the attractor
σ is the variability of the process (akin to standard deviation) dW is a Wiener increment (Brownian motion/White noise)
Ornstein-Uhlenbeck and the Normal distribution
Geometric Brownian Motion
› Geometric Brownian Motion is widely used in finance
– Black-Scholes formula for pricing European options, the Merton optimal consumption and portfolio rules, and the Dixit and Pindyck method for real options
› Geometric Brownian Motion is used, not because it is realistic, but because it is better understood than other stochastic processes, and because many terms will cancel in a theoretical model
Geometric Brownian Motion
dx =xdt+xdW tttt
Estimating stochastic processes
Example: Clare wheat yield (t/ha) 1900-2007
1960 1980 2000 2020
G(x) H(x) y
Example: Clare wheat yields phase diagram
Example: Clare wheat yields – OU estimation
dx =(−x)dt+dW ttt
dx =0.92(3.56−x)dt+0.68dW ttt
0.915711 3.556011 0.678892
Example: Clare wheat yield phase diagram with estimated OU process
g(x) h(x) dx/dt
dx =0.92(3.56−x)dt+0.68dW ttt
› Whether the GBM or OU process provides a more appropriate characterisation of the
› system can be seen immediately in Figures 9 and 10. The OU process fitted to the
› wheat gross margins data is more appropriate. We observe that agricultural systems
› tend to revert to equilibrium and do not grow exponentially
Probabilities from the stochastic process
› There are two ways in which we can represent probabilities: – Probability of a set of states for a given time
– Probability of a set of times for a given state
› These concepts are orthogonal and, in principle, if you know one, you should be able to estimate the other
– In reality, this is much harder
State-probabilities from stochastic processes
State-probabilities from OU process
State-probabilities from GBM process
Example: State-probabilities from Clare wheat yields
Initial Transition Invariant Threshold
f(s,x,t,y)
2.0 3.0 4.0
Example: Cumulative state-probabilities from Clare wheat yields
1.0 0.9 0.8 0.7 0.6
F(s,x,t,y) 0.5 0.4 0.3 0.2 0.1 0.0
Initial Transition Invariant Threshold
2.0 3.0 4.0 5.0 6.0
Static vs. Dynamic CDF for Clare wheat
Time-probabilities
› We can think of time as a random variable simply because we cannot be certain how long we will need to wait before the system delivers us to a given state
– This ‘waiting time’ is a function of the current state, the location of the state and the system dynamics
– In the mathematics literature, these functions/times are known as first passage or first hitting times
– First passage times are represented by a distribution of time, for a given current state, from which we may take expectations
Time-probabilities from GBM processes
Time-probabilities from OU processes
› Unfortunately, there is no analytical solution for OU processes, but we can approximate the time-probability by employing Monte-Carlo simulation methods
– First, we simulate 10,000 OU processes for given parameters
– Second, we count the time elapsed before the system delivers us to our
nominated state, for each of the 10,000 simulations
– Third, we plot time counts and estimate the distribution
Time-probabilities from OU processes
Example: Clare time-probability wheat yields
For x = 1.18, y = 4.925, t = 0 and s = 100 years
Example: Clare cumulative time-probability wheat yields
For x = 1.18, y = 4.925, t = 0 and s = 100 years
Example: Price of hard red winter wheat, Kansas
› Prices are good examples of unbounded, non-attracted systems – i.e. Non-ergodic systems
› Example: data from 1970-2011 for hard red winter wheat prices in Kansas in $/t
Example: Price of hard red winter wheat, Kansas
1975 1980 1985
1990 1995 2000
2005 2010 2015
G(x) H(x) y
Example: Phase diagram of wheat price, Kansas
g(x) h(x) dx/dt
dx =0.07xdt+0.27xdW tttt
100 80 60 40 20 0 -20 -40 -60 -80 -100
0 50 100 150 200 250 300 350
› Figures 9 and 10 show the GBM and OU processes estimated using the gross margin
› data for wheat at Clare. Standard error bars of the estimation appear as red dotted
› lines around the fitted process which appears as a solid red line.
Example: State-probabilities of price wheat Kansas
f(s,x,t,y)
0.6 0.4 0.2 0.0
Initial Transition
1.0 1.5 2.0
*All prices have been divided by 100
Example: Cumulative state-probabilities
1.0 0.9 0.8 0.7 0.6
F(s,x,t,y) 0.5 0.4 0.3 0.2 0.1 0.0
Initial Transition
1.0 1.5 2.0
*All prices have been divided by 100
Example: Time-probabilities
For x = $100, y = $500, t = 0 and s = 100 years
Example: Time-probabilities
For x = $100, y = $500, t = 0 and s = 100 years
Hardaker, Huirne and Anderson (2004) Coping with Risk in Agriculture, CAB International.
– Chapter 4
Sanderson et al. (2015) A real options analysis of Australian wheat production under climate change. Australian Journal of Agricultural and Resource Economics, 60(1).
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