香港六合彩
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金月夜这句话是在说李哲羽吗?难道真如苏姬说的,只有香港六合彩这个傻瓜不知道,可是金月夜生病香港六合彩为什么要让香港六合彩去见香港六合彩圣诞节香港六合彩为什么愿意帮香港六合彩替香港六合彩补习,让香港六合彩去见金月夜在星星阁上,香港六合彩为什么不带香港六合彩走上那十一阶楼梯为什么?到底是为什么救命啊!救命啊!!香港六合彩的头真的好痛,香港六合彩不要再想了,不要再想了正当香港六合彩在摇晃着脑袋,想把李哲羽的影子从脑子里彻底除去,却发现校园公告栏前怎么围着那么多人?咦?要举办团体操比赛?好无聊哦!学校干吗老是出这种馊主意吗?呜原来是活动公告,好像是明德和崇阳要举行团体操比赛!正在香港六合彩纳闷的时候,学校的广播突然响了起来。苏佑慧同学请到校长办公室!咦?崔校长找香港六合彩?该不会是为了这次团体操比赛的事情吧香港六合彩拉了拉肩上的书包,朝校长办公室走了过去。佑慧,公告栏上的公告你看到了吗?崔校长笑着问。嗯,看到了。香港六合彩点点头回答。这次你把Sun邀请到了崇阳,做得非常出色!真不愧是明德之花啊!哦呵呵呵呵!崔校长您过奖了!香港六合彩有些得意地笑着说。呵呵,这次的团体操比赛对于香港六合彩崇阳来说是件头痛的事情啊。崔校长说着叹了口气。咦?这是为什么呢?因为崇阳的校风向来很自由,同学们都不太适应团体操这种需要一板一眼地按照节奏来完成的运动。香港六合彩不希望崇阳输掉这次比赛,所以把你这位明德之花找来,就是要和你商量一下怎么办。呜呼!这个崔校长说话老是绕来绕去的,绕得香港六合彩的头都晕了!还是白校长说话直接多了!崔校长,您的意思是希望香港六合彩来负责崇阳这一次团体操比赛是吗?嗯,香港六合彩的确是有这个意思。问题是没问题啦,只是香港六合彩担心,因为香港六合彩毕竟是从明德过来的,大家会不会听香港六合彩的安排崔校长意味深长地笑了笑。呵呵,这就是你这位明德之花需要经受考验的地方了!对了,香港六合彩和白凝校长通过电话,这次团体操明德的负责人是金月夜。香港六合彩就知道会是这个家伙!这个恶魔最近受尽明德女生的万千宠爱,最可恶的是香港六合彩也不知道香港六合彩哪句话是真的,哪句是假的,害香港六合彩误会了李哲羽!香港六合彩都不知道该拿什么脸去见香港六合彩了呜呜呜呜呜呼!金月夜,香港六合彩要你为此付出代价!崔校长,香港六合彩答应您,这次崇阳的团体操比赛香港六合彩来负责。崔校长眼睛一亮。那实在是太好了!苏佑慧同学,那香港六合彩先在这里预祝你成功!Five团体操团体操说起来,香港六合彩对团体操真是一点经验都没有,在加上现在香港六合彩又是客场作战,香港六合彩真的没有问题吗呸呸呸!苏佑慧,你乱七八糟地想些什么啊!你可是米兰公主耶!虽然考试经常输给金月夜那个臭猴子0.5分咳咳这句话是多余的总之,这种小场面对于你来说只是小儿科而已!好了,佑慧公主为了胜利努力吧!嚯呀呀呀呀!嗯?花?!哪里来的?!正当香港六合彩在为自己鼓劲的时候,一束蓝色的花突然出现在了香港六合彩的面前!香港六合彩转头一看!Sun?!呵呵!佑慧,是不是吓了一跳?香港六合彩今天来学校了!Sun说着把香港六合彩手上那束蓝色的玫瑰花塞到了香港六合彩的手上。哇!是Sun耶!香港六合彩送了一束蓝色妖姬给苏佑慧TRANSCRIPT
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Visual FAQ’s on Real OptionsVisual FAQ’s on Real Options Celebrating the Fifth Anniversary of the Website:Celebrating the Fifth Anniversary of the Website:
Real Options Approach to Petroleum InvestmentsReal Options Approach to Petroleum Investments
http://www.puc-rio.br/marco.ind/http://www.puc-rio.br/marco.ind/
By: Marco Antônio Guimarães DiasPetrobras and PUC-Rio, Brazil
Real Options 2000 ConferenceReal Options 2000 ConferenceCapitalizing on Uncertainty and Volatility in the New MillenniumCapitalizing on Uncertainty and Volatility in the New Millennium
September 25, 2000 September 25, 2000 Chicago Chicago
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Visual FAQ’s on Real OptionsVisual FAQ’s on Real Options Selection of frequently asked questions (FAQ’s) by
practitioners and academics Something comprehensive but I confess some bias in
petroleum questions Use of some facilities to visual answer
Real options models present two results: The value of the investment oportunity (option value)
How much to pay (or sell) for an asset with options? The decision rule (thresholds)
Invest now? Wait and See? Abandon? Expand the production? Switch use of an asset?
Option value and thresholds are the focus of most visual FAQ’s
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Visual FAQ’s on Real Options: 1Visual FAQ’s on Real Options: 1 Are the real options premium important?
Real Option Premium = Real Option Value NPV
Answer with an analogy: Investments can be viewed as call options
You get an operating project V (like a stock) by paying the investment cost I (exercise price)
Sometimes this option has a time of expiration (petroleum, patents, etc.), sometimes is perpetual (real estate, etc.)
Suppose a 3 years to expiration petroleum undeveloped reserve. The immediate exercise of the option gets the NPV
NPV = V I
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Real Options PremiumReal Options Premium The options premium can be important or not, depending of the of the project moneyness
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Visual FAQ’s on Real Options: 2Visual FAQ’s on Real Options: 2 What are the effects of interest rate, volatility,
and other parameters in both option value and the decision rule?
Answer with “Timing Suite” Three spreadsheets that uses a simple model analogy
of real options problem with American call option Lets go to the Excel spreadsheets to see the effects
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Visual FAQ’s on Real Options: 3Visual FAQ’s on Real Options: 3 Where the real options value comes from?
Why real options value is different of the static net present value (NPV)?
Answer with example: option to expand Suppose a manager embed an option to expand into her
project, by a cost of US$ 1 million The static NPV = 5 million if the option is exercise today,
and in future is expected the same negative NPV Spending a million $ for an expected negative NPV: Is the
manager becoming crazy?
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Uncertainty Over the Expansion ValueUncertainty Over the Expansion Value Considering combined uncertainties: in product prices and demand, exercise price of the real option,
operational costs, etc., the future value (2 years ahead) of the expansion has an expected value of $ 5 million The traditional discount cash will not recommend to embed an option to expansion which is expected to be negative But the expansion is an option, not an obligation!
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Option to Expand the ProductionOption to Expand the Production Rational managers will not exercise the option to expand @ t = 2 years in case of bad news (negative value)
Option will be exercised only if the NPV > 0. So, the unfavorable scenarios will be pruned (for NPV < 0, value = 0) Options asymmetry leverage prospect valuation. Option = + 5
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Real Options Asymmetry and ValuationReal Options Asymmetry and Valuation
+
=Prospect Valuation
Traditional Value = 5
Options Value(T) = + 5
The visual equation for “Where the options value comes from?”
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E&P Process and OptionsE&P Process and Options Drill the wildcat? Wait? Extend? Revelation, option-game: waiting incentives
Oil/Gas SuccessProbability = p
Expected Volumeof Reserves = B
RevisedVolume = B’ Appraisal phase: delineation of reserves
Technical uncertainty: sequential options
Developed Reserves. Expand the production? Stop Temporally? Abandon?
Delineated but Undeveloped Reserves. Develop? “Wait and See” for better conditions? Extend the
option?
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Option to Expand the ProductionOption to Expand the Production Analyzing a large ultra-deepwater project in Campos Basin,
Brazil, we faced two problems: Remaining technical uncertainty of reservoirs is still important. In
this specific case, the better way to solve the uncertainty is by looking the production profile instead drilling additional appraisal wells
In the preliminary development plan, some wells presented both reservoir risk and small NPV. Some wells with small positive NPV (not “deep-in-the-money”) and
others even with negative NPVDepending of the initial production information, some wells can be
not necessary Solution: leave these wells as optional wells
Small investment to permit a fast and low cost future integration of these wells, depending of both market (oil prices, costs) and the production profile response
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Modeling the Option to ExpandModeling the Option to Expand Define the quantity of wells “deep-in-the-money” to start the
basic investment in development Define the maximum number of optional wells Define the timing (or the accumulated production) that the
reservoir information will be revealed Define the scenarios (or distributions) of marginal production of
each optional well as function of time. Consider the depletion if we wait after learn about reservoir
Add market uncertainty (reversion + jumps for oil prices) Combine uncertainties using Monte Carlo simulation (risk-
neutral simulation if possible, next FAQ) Use optimization method to consider the earlier exercise of the
option to drill the wells, and calculate option value Monte Carlo for American options is a frontier research area Petrobras-PUC project: Monte Carlo for American options
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Visual FAQ’s on Real Options: 4Visual FAQ’s on Real Options: 4 Does risk-neutral valuation mean that investors are
risk-neutral? What is the difference between real simulation and
risk-neutral simulation?
Answers Risk-neutral valuation (RNV) does not assume investors or
firms with risk-neutral preferences RNV does not use real probabilities. It uses risk neutral
probabilities (“martingale measure”) Real simulation: real probabilities, uses real drift Risk-neutral simulation: the sample paths are risk-adjusted.
It uses a risk-neutral drift: ’ = r
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Geometric Brownian Motion Simulation Geometric Brownian Motion Simulation The real simulation of a GBM uses the real drift . The price at future time t is
given by:
Pt = P0 exp{ () t + t By sampling the standard Normal distribution N(0, 1) you get the values forPt With real drift use a risk-adjusted (to P) discount rate
The risk-neutral simulation of a GBM uses the risk-neutral drift ’ = r . The price at t is:
Pt = P0 exp{ (r ) t + t With risk-neutral drift, the correct discount rate is the risk-free interest rate.
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Risk-Neutral Simulation x Real Simulation Risk-Neutral Simulation x Real Simulation For the underlying asset, you get the same value:
Simulating with real drift and discounting with risk-adjusted discount rate ( where ) Or simulating with risk-neutral drift (r ) but discounting with the risk-free interest rate (r)
For an option/derivative, the same is not true: Risk-neutral simulation gives the correct option result (discounting with r) but the real simulation does not gives the correct value (discounting with ) Why? Because the risk-adjusted discount rate is “adjusted” to the underlying asset, not to the option
Risk-neutral valuation is based on the absence of arbitrage, portfolio replication (complete market) Incomplete markets: see next FAQ
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Visual FAQ’s on Real Options: 5Visual FAQ’s on Real Options: 5 Is possible to use real options for incomplete markets? What change? What are the possible ways?
Answer: Yes, is possible to use. For incomplete markets the risk-neutral probability (martingale measure) is not unique So, risk-neutral valuation is not rigorously correct because there is a lack of market values Academics and practitioners use some ways to estimate the real option value, see next slide
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Incomplete Markets and Real OptionsIncomplete Markets and Real Options In case of incomplete market, the alternatives to real options
valuation are: Assume that the market is approximately complete (your
estimative of market value is reliable) and use risk-neutral valuation (with risk-neutral probability)
Assume firms are risk-neutral and discount with risk-free interest rate (with real probability)
Specify preferences (the utility function) of single-agent or the equilibrium at detailed level (Duffie)
Used by finance academics. In practice is difficult to specify the utility of a corporation (managers, stockholders)
Use the dynamic programming framework with an exogenous discount rate
Used by academics economists: Dixit & Pindyck, Lucas, etc.Corporate discount rate express the corporate preferences?
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Visual FAQ’s on Real Options: 6Visual FAQ’s on Real Options: 6 Is true that mean-reversion always reduces the
options premium? What is the effect of jumps in the options premium?
Answers: First, we’ll see some different processes to model the
uncertainty over the oil prices (for example) Second, we’ll compare the option premium for an
oilfield using different stochastic processesAll cases are at-the-money real options (current NPV = 0)The equilibrium price is 20 $/bbl for all reversion cases
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Geometric Brownian Motion (GBM)Geometric Brownian Motion (GBM) This is the most popular stochastic process, underlying the famous Black-Scholes-Merton
options equation GBM: expected curve is a exponential growth (or decrease); prices have a log-normal distribution in
every future time; and the variance grows linearly with the time
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In this process, the price tends to revert toward a long-run average price (or an equilibrium level) P. Model analogy: spring (reversion force is proportional to the distance between current position and the
equilibrium level). In this case, variance initially grows and stabilize afterwards Charts: the variance of distributions stabilizes after ti
Mean-Reverting ProcessMean-Reverting Process
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Nominal Prices for Brent and Similar Oils (1970-1999)Nominal Prices for Brent and Similar Oils (1970-1999) We see oil prices jumps in both directions, depending of the kind of abnormal news:
jumps-up in 1973/4, 1978/9, 1990, 1999; and jumps-down in 1986, 1991, 1997
Jumps-up Jumps-down
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Mean-Reversion + Jumps for Oil PricesMean-Reversion + Jumps for Oil Prices Adopted in the Marlim Project Finance (equity
modeling) a mean-reverting process with jumps:
The jump size/direction are random: ~ 2N
In case of jump-up, prices are expected to double OBS: E()up = ln2 = 0.6931
In case of jump-down, prices are expected to halve OBS: ln(½) = ln2 = 0.6931
where:(the probability of jumps)
(jump size)
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Equation for Mean-Reversion + JumpsEquation for Mean-Reversion + Jumps The interpretation of the jump-reversion equation is:
mean-reversion drift:positive drift if P < Pnegative drift if P > P
{uncertainty overthe continuous
process (reversion){variation of thestochastic variablefor time interval dt
uncertainty overthe discreteprocess (jumps)
continuous (diffusion) process
discreteprocess(jumps)
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Mean-Reversion x GBM: Option PremiumMean-Reversion x GBM: Option Premium The chart compares mean-reversion with GBM for an at-the-money project at current 25 $/bbl
NPV is expected to revert from zero to a negative value
Reversion in all cases: to 20 $/bbl
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Mean-Reversion with Jumps x GBMMean-Reversion with Jumps x GBM Chart comparing mean-reversion with jumps versus GBM for an at-the-money project at current 25 $/bbl
NPV still is expected to revert from zero to a negative value
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Mean-Reversion x GBMMean-Reversion x GBM Chart comparing mean-reversion with GBM for an at-the-money project at current 15 $/bbl (suppose)
NPV is expected to revert from zero to a positive value
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Mean-Reversion with Jumps x GBMMean-Reversion with Jumps x GBM Chart comparing mean-reversion with jumps versus GBM for an at-the-money project at current 15 $/bbl (suppose)
Again NPV is expected to revert from zero to a positive value
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Visual FAQ’s on Real Options: 7Visual FAQ’s on Real Options: 7 How to model the effect of the competitor entry
in my investment decisions?
Answer : option-games, the combination of the real options with game-theory
First example: Duopoly under Uncertainty (Dixit & Pindyck, 1994; Smets, 1993)Demand for a product follows a GBMOnly two players in the market for that product (duopoly)
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Duopoly Entry under UncertaintyDuopoly Entry under Uncertainty The leader entry threshold: both players are indifferent about to be the leader or the follower.
Entry: NPV > 0 but earlier than monopolistic case
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Other Example: Oil Drilling GameOther Example: Oil Drilling Game Oil exploration: the waiting game of drilling Two companies X and Y with neighbor tracts and correlated oil prospects: drilling reveal information
If Y drills and the oilfield is discovered, the success probability for X’s prospect increases dramatically. If Y drilling gets a dry hole, this information is also valuable for X. Here the effect of the competitor presence is the opposite: to increase the value of waiting to invest
Company X tractCompany X tract Company Y tractCompany Y tract
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Visual FAQ’s on Real Options: 8Visual FAQ’s on Real Options: 8 Does Real Options Theory (ROT) speed up the
firms investments or slow down investments?
Answer: depends of the kind of investment ROT speeds up today strategic investments that create
options to invest in the future. Examples: investment in capabilities, training, R&D, exploration, new markets...
ROT slows down large irreversible investment of projects with positive NPV but not “deep in the money”
Large projects but with high profitability (“deep in the money”) must be done by both ROT and static NPV.
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Visual FAQ’s on Real Options: 9Visual FAQ’s on Real Options: 9 Is possible real options theory to recommend
investment in a negative NPV project?
Answer: yes, mainly sequential options with investment revealing new informations Example: exploratory oil prospect (Dias 1997)
Suppose a “now or never” option to drill a wildcatStatic NPV is negative and traditional theory recommends to give
up the rights on the tractReal options will recommend to start the sequential investment,
and depending of the information revealed, go ahead (exercise more options) or stop
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Sequential Options (Dias, 1997)Sequential Options (Dias, 1997)
Traditional method, looking only expected values, undervaluate the prospect (EMV = 5 MM US$): There are sequential options, not sequential obligations; There are uncertainties, not a single scenario.
( Wildcat Investment )
( Developed Reserves Value )
( Appraisal Investment: 3 wells )
( Development Investment )
Note: in million US$“Compact Decision-Tree”
EMV = 15 + [20% x (400 50 300)] EMV = 5 MM$
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Sequential Options and UncertaintySequential Options and Uncertainty Suppose that each
appraisal well reveal 2 scenarios (good and bad news)
development option will not be exercised by rational managers
option to continue the appraisal phase will not be exercised by rational managers
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Option to Abandon the ProjectOption to Abandon the Project Assume it is a “now or never” option If we get continuous bad news, is better
to stop investment Sequential options turns the EMV to a
positive value The EMV gain is 3.25 5) = $ 8.25 being:
(Values in millions)
$ 2.25 stopping development
$ 6 stopping appraisal
$ 8.25 total EMV gain
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Visual FAQ’s on Real Options: 10Visual FAQ’s on Real Options: 10 Is the options decision rule (invest at or above the
threshold curve) the policy to get the maximum option value?
How much value I lose if I invest a little above or little below the optimum threshold?
Answer: yes, investing at or above the threshold line you maximize the option value.
But sometimes you don’t lose much investing near of the optimum (instead at the optimum) Example: oilfield development as American call option.
Suppose oil prices follow a GBM to simplify.
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Thresholds: Optimum and Sub-OptimaThresholds: Optimum and Sub-Optima The theoretical optimum (red) of an American call option (real option to
develop an oilfield) and the sub-optima thresholds (~10% above and below)
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Optima RegionOptima Region Using a risk-neutral simulation, I find out here that the optimum is over a “plateau” (optima region) not a “hill” So, investing ~ 10% above or below the theoretical optimum gets rough the same value
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Real Options PremiumReal Options Premium Now a relation optimum with option premium is clear: near of the point A (theoretical threshold) the option premium
can be very small.
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Visual FAQ’s on Real Options: 11Visual FAQ’s on Real Options: 11 How Real Options Sees the Choice of Mutually
Exclusive Alternatives to Develop a Project?
Answer: very interesting and important application Petrobras-PUC is starting a project to compare
alternatives of development, alternatives of investment in information, alternatives with option to expand, etc.
One simple model is presented by Dixit (1993). Let see directly in the website this model
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ConclusionsConclusions The Visual FAQ’s on Real Options illustrated:
Option premium; visual equation for option value; uncertainty modeling; decision rule (thresholds); risk-neutral x real simulation/valuation; Timing Suite; effect of competition; optimum problem, etc.
The idea was to develop the intuition to understand several results in the real options literature
The use of real options changes real assets valuation and decision making when compared with static NPV
There are several other important questions The Visual FAQ’s on Real Options is a webpage with a growth
option! Don’t miss the new updates with the new FAQ’s at:
http://www.puc-rio.br/marco.ind/faqs.html