{"id":6746,"date":"2020-11-15T10:37:07","date_gmt":"2020-11-15T10:37:07","guid":{"rendered":"http:\/\/onlineclassesguru.com\/?p=6746"},"modified":"2020-11-15T10:37:07","modified_gmt":"2020-11-15T10:37:07","slug":"monte-carlo-simulations","status":"publish","type":"post","link":"https:\/\/onlineclassesguru.com\/index.php\/2020\/11\/15\/monte-carlo-simulations\/","title":{"rendered":"Monte-Carlo simulations"},"content":{"rendered":"<style type=\"text\/css\"><\/style><p>Real Options<\/p>\n<p>Real Options: Types<\/p>\n<p>Expand<br \/>\nDelay<br \/>\nAbandon<br \/>\nUsing a Binomial Model<\/p>\n<p>Sometimes using a closed-form solution like the Black-Scholes formula is impossible.<br \/>\nMany options (especially in energy) don\u0092t have the properties necessary for the Black-Scholes formula to apply.<br \/>\nIn many cases, you can use a binomial model with Monte-Carlo simulations.<br \/>\nThat\u0092s where we\u0092re heading next.<br \/>\nBinomial Model<\/p>\n<p>To make the multi-period problem more manageable, allow the binomial tree to recombine (we call it recombining binomial lattice)<br \/>\nt=0<\/p>\n<p>t=1<\/p>\n<p>t=2<\/p>\n<p>t=3<\/p>\n<p>=&gt;<\/p>\n<p>t=0<\/p>\n<p>t=1<\/p>\n<p>t=2<\/p>\n<p>t=3<\/p>\n<p>*<\/p>\n<p>Binomial Model<\/p>\n<p>A single node at a random point in time t:<br \/>\nt<\/p>\n<p>t+1<\/p>\n<p>S<\/p>\n<p>Su=u\u00b7S<\/p>\n<p>Sd=d\u00b7S<\/p>\n<p>t<\/p>\n<p>t+1<\/p>\n<p>c<\/p>\n<p>cu<\/p>\n<p>cd<\/p>\n<p>*<\/p>\n<p>Binomial Model<br \/>\nFormula to find c<br \/>\nCox, Ross, and Rubinstein (1979)<br \/>\n*<\/p>\n<p>Risk-Neutral Probabilities<\/p>\n<p>The terms in brackets in the formula on the previous slide are called \u0093risk-neutral probabilities\u0094 of stock price going \u0093up\u0094 and \u0093down\u0094.<br \/>\nExample<br \/>\nSince the probabilities are \u0093risk-neutral\u0094 we can discount at the risk-free rate!<br \/>\nDetermining u and d<\/p>\n<p>What if we have multiple intervals, say 90 dates over 3 months?<br \/>\nLet n=number of steps per year. So if we had 90 dates over 3 months, we will have 90*4 = 360 steps over a full year, then n=360.<br \/>\nSuppose we can estimate volatility of the underlying (annualized!) Say volatility is 30%.<br \/>\nTurns out the formulas for u and d are as follows:<br \/>\nu = e?\/?n<br \/>\nd=1\/u<br \/>\nExample: u = exp(.3\/sqrt(360)) = 1.015937<br \/>\nAnd d = 1\/1.015937 = 0.984313<br \/>\nAnother Way to Find u and d<\/p>\n<p>Alternatively, T = time to expiration in years, and k = number of steps before expiration. If we had 90 dates over 3 months, then T = 3\/12 = 0.25 years, and k=90. Say annualized volatility is 30%. We can calculate<br \/>\nu = e?*?(T\/k)<br \/>\nd=1\/u<br \/>\nSo u = EXP(.3*sqrt(0.25\/90)) = 1.015937<br \/>\nAnd d = 1\/1.015937 = 0.984313<br \/>\nImportant<\/p>\n<p>Note that in the binomial formula, volatility is annualized (per year); and risk-free rate is per period (step)!<br \/>\nValuing a Gold Mine<\/p>\n<p>Woe Is Me gold mine: founded in 1878<br \/>\nPlayed out by 1908 but is reopened occasionally depending on the price of gold<br \/>\nIt owns land but worth very little<br \/>\nCash and liquid securities $3 million<br \/>\nNot mining now<br \/>\nHowever, has market capitalization of over $100 million. Why?<br \/>\nGold Mine (Continued)<\/p>\n<p>Costs $0.5 million to reopen the mine, $0.25 million to shut it down (fixed cost)<br \/>\nAssume that for technical reasons the mine cannot be closed \/ reopened more frequently than twice a year<br \/>\nProduction is 5,000 ounces per year (2,500 ounces per half year)<br \/>\nExtraction costs are $350 per ounce (variable cost)<br \/>\nAccording to terms of lease, cannot stockpile gold: must sell all gold produced.<br \/>\nLease is for the next 100 years<br \/>\nGold Mine (Continued)<\/p>\n<p>We can view the mine as a package of call options on the price of gold.<br \/>\nWhat is (are) the exercise price(s)?<br \/>\nIs there a \u0093maturity date\u0094?<br \/>\nGold Mine (Continued)<\/p>\n<p>This is a real option problem.<br \/>\nLet\u0092s use common sense first. Should the mine be reopened if the price of gold is $350.15?<br \/>\nWhat if the price is $360? Remember volatility!<br \/>\nSuppose the mine is now open, and price drops to $345? Does it make sense to immediately shut down?<br \/>\nNeed to find two threshold prices, popen and pclose, at which the \u0093open mine\u0094 option is sufficiently in and out of the money, respectively. Volatility is key!<br \/>\nGold Mine (Continued)<\/p>\n<p>Common sense tells us that the threshold prices depend on volatility. How?<br \/>\nWe also know that the threshold prices depend on the fixed costs of opening and closing the mine. How?<br \/>\nNext, we value the call option. Refer to the following diagram.<br \/>\nGold Mine (Continued)<\/p>\n<p>Mine is Open<\/p>\n<p>Choices:<br \/>\nMine is Closed<\/p>\n<p>Keep Open<\/p>\n<p>Close at $.25 million<\/p>\n<p>Reopen at $.5 million<\/p>\n<p>Keep Closed<\/p>\n<p>Gold Mine (Continued)<\/p>\n<p>Assume risk-free interest rate of 3.4% per six months<br \/>\nAssume volatility for the price of gold 20% per year<br \/>\nSuppose current price of gold is $350<br \/>\nConsider a 6-month-step binomial tree, 100 years long (200 total steps).<br \/>\nu = e0.20\/?2 = 1.1519<br \/>\nd=1\/u = 0.8681<br \/>\nGold Mine (Continued)<\/p>\n<p>Compute risk-neutral probabilities. Answer: about 58% of a rise in gold price. (Do yourself!)<br \/>\nNext, we need a computer simulation. Simulate, say 5000 paths for gold price<br \/>\nNext, pick some guesses for popen and pclose<br \/>\nFor each chosen pair, and for each of the 5000 paths, calculate PV(cash flows) discounting at 3.4%. Average. Maximize with respect to popen and pclose.<br \/>\nThe maximum value of PV(Cash Flows) = Value of Real Option<br \/>\nGold Mine (Continued)<\/p>\n<p>This example is not easy to implement<br \/>\nBut it is widely used in practice to value real options<br \/>\nBetter to use than Black-Scholes due to the complicated structure<br \/>\nValue PUD as a Call Option<\/p>\n<p>S = DCF from the resource once developed<br \/>\nK = cost of developing the resource<br \/>\nr = risk-free interest rate, as usual<br \/>\nT = time to lease expiration<br \/>\ny = cost of delay, if any (analogous to dividend yield)<br \/>\n? = volatility of commodity price (???)<br \/>\nCall Option with Dividends<\/p>\n<p>Development Lag<\/p>\n<p>What if you \u0093exercise the option\u0094 by developing the resource, but the cash flows don\u0092t start until N years after development?<br \/>\nDevelopment lag = N<br \/>\nIn that case, discount S back N years at the rate y before plugging in the formula on the previous slide.<br \/>\nIt\u0092s like you incur additional \u0093cost of delay\u0094 for N more years<br \/>\nExample<br \/>\nSource: Aswath Damodaran, \u0093Real Options\u0094<\/p>\n<p>Oil property has an estimated reserve of 50 million bbls<br \/>\nThe cost of developing the reserve is expected to be $600 million<br \/>\nThe development lag is two years<br \/>\nLease expires in 20 years<br \/>\nOil margin (i.e., price \u0096 production cost) = $12\/bbl<br \/>\nOnce developed, the net production revenue each year will be 5% of the value of the reserves.<br \/>\nThe riskless rate is 8%<br \/>\nThe variance in ln(oil prices) is 0.03<br \/>\nValue the real option to develop this reserve.<br \/>\nCost of Delay<\/p>\n<p>In this example, delaying exercise is costly because the lease clock is ticking. In the worst case scenario, if the company doesn\u0092t develop the reserve for 20 years, the rights to the property will be relinquished and the reserve will be worthless to the firm.<br \/>\nOnce developed, each year the firm expects to net out 5% of the total value of reserve. Hence, y=5%. Every year of delaying development costs the company 5% of reserve life.<br \/>\nWhen Does y=0?<\/p>\n<p>If the company were assumed to have indefinite rights to the property once it\u0092s developed, then the company would waste no money by delaying development.<br \/>\nSuppose the company has 5 years to start drilling, or else it will lose the lease. However, once drilled, the property can be kept until the reserve is fully depleted, which is estimated to be 20 years after the production commences.<br \/>\nIn this case, waiting to develop all the way until year 5 doesn\u0092t cost the company anything: it will still enjoy the same reserve life of 20 years whether it develops the reserve in year 1 or in year 5. Then y = 0.<br \/>\nReserve Value Once Developed, S<\/p>\n<p>Value of the reserve if developed today is $12*50 = $600 million<br \/>\nAlternatively, we could use the information about future cash flows from the reserve and find the value using DCF. Here, we are making a simplification.<br \/>\nDon\u0092t forget the development lag. Instead of $600 million, we have to discount = 600*e(-0.05*2) = $542.90<br \/>\nIf we assume that all cash flows occur at year-end (instead of continuously), then we have to use 600\/(1.05)2 = $544.22. Let\u0092s use that number for the Black-Scholes formula.<br \/>\nNotice the cost of developing today $600 is greater than the value of reserve, $544.22. The real option is out-of-the-money. But it still has value!!<br \/>\nReal Option<\/p>\n<p>Plugging all inputs into the call option with dividend formula, we have (please verify yourself)<br \/>\nEven though this reserve costs more to develop than it\u0092s currently worth under current oil prices, it\u0092s not worthless to the company. It has the value of the real option because the company has the right to develop it once it becomes profitable. This right has value.<br \/>\nd1 1.035919<br \/>\nd2 0.261323<br \/>\nC 97.09589<\/p>\n<p><center><a href=\"http:\/\/onlineclassesguru.com\/orders\/ordernow\"><img decoding=\"async\" src=\"https:\/\/encrypted-tbn0.gstatic.com\/images?q=tbn:ANd9GcTyj99p60XCLyLk1htB7-1neRt8-2QdnenNlQ&usqp=CAU\"target=\"_http:\/\/onlineclassesguru.com\/orders\/ordernow\"\/><\/center><p>","protected":false},"excerpt":{"rendered":"<p>Real Options Real Options: Types Expand Delay Abandon Using a Binomial Model Sometimes using a closed-form solution like the Black-Scholes formula is impossible. Many options (especially in energy) don\u0092t have the properties necessary for the Black-Scholes formula to apply. In many cases, you can use a binomial model with Monte-Carlo simulations. That\u0092s where we\u0092re heading&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-6746","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Monte-Carlo simulations - onlineclassesguru<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/onlineclassesguru.com\/index.php\/2020\/11\/15\/monte-carlo-simulations\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Monte-Carlo simulations - onlineclassesguru\" \/>\n<meta property=\"og:description\" content=\"Real Options Real Options: Types Expand Delay Abandon Using a Binomial Model Sometimes using a closed-form solution like the Black-Scholes formula is impossible. 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