Stochastic Oscillator for Financial Analysis
When using the Stochastic Indicator your basic aim is to buy (go long) when the stochastic becomes oversold (goes below 20%) and then the %K line crosses above the %D line at a market low, and to sell (go short) when the stochastic becomes overbought (goes above 80%) and then the %K line crosses below the %D line at a market high. Other indicators or candlestick patters can also be used to further confirm the trade. Below is a chart of a trade I recently took on GUD.AX using the Slow Stochastic in combination with support and resistance levels as well as a candlestick pattern.  When I conducted my stock search on the evening of 28th July 2015, GUD was one of the results from the search that I particularly liked. The Slow Stochastic had just made a crossover in the oversold area just as the price was bouncing off its recent lows at the support line. But what I really like about this opportunity was that there was also a bullish reversal candle (a Hammer) at this short term market bottom. The high for the day was $8.34, so I placed a stop buy order to buy the stock tomorrow if the price opened or moved above this high of today. So I would only buy if the stock hit or moved above $8.35 during the next days trading. So if the stock opened and stayed below the previous day's high I would not buy the stock and my order would be canceled at the end of the day. This is very important because it stops you from getting into trades that don't go in the direction you want. If this happens then you could check the chart again after market close to see if it is still worthwhile to place a new order for the next day. On the 29th July 2015 the stock did open higher at $8.42 (which is where my order got executed) and closed at $8.46. So I ended up buying slightly above my target price but it did move higher on the day, so a successful entry overall. I had placed my initial stop loss at $8.09 (just below the low of the previous day of $8.10). If the trade had gone against me in the following days the most I would have lost was 1% of my total account capital. My target for this trade was $9.86 (just below the resistance line near $10), which would represent a 5% gain against my total account capital (or approx. 16.7% gain on the trade). So my win to loss ratio was 5:1. As you can see from the chart, the next day gapped up at the open and prices continued moving up strongly during the day. The next day was a slightly negative day, and then a few days later, on the 5th August 2015 my target price of $9.86 was reached (with a high of $9.88 for that day), so my order was closed for a total profit of 16.7% on the face value of the trade. However, as I bought on margin (using CFDs) my actual profit on the initial margin I had invested was 167%. My total time in the trade was 7 days, and I spent about an hour in the evening doing my searches and placing my orders, then less than 10 minutes managing the trade each evening after market close. The stock did go up a bit further after my profit target was reached, but the day after that the price started to fall as the price hit the resistance line and the Slow Stochastic did a crossover in the overbought area. Overall, this was a very ideal trade and I was very happy with it. Not all my trades reach my profit targets and I may get stopped out at a smaller profit or I might make a small loss (as I move my stops up as the price moves up). The key to successful trading over the long term is to keep you losses small and let your profits run (with my longer term trend trading I don't have profit targets and let my profits run until I get stopped out, but with my shorter term trading I do use a profit target usually 5 or 6 times the size of my initial stop). If you have an average win to average loss ratio of 3:1 or higher and have a success rate of 50% winners you will make money over the long term. While trading in stochastic I've understood, one needs reference (SMA/EMA/Bolinger Band and even RSI) to verify trade prior entering it. Stochastic is nothing to do with price or volume it is about speed. Adjusting K% has ability to turn you from Day trader to -> swing trader to -> long term investor. So you adjust your k% according to chart time-frame. Stochastic setup for 1 min, 5 min ,15, 30, 60 min, daily, weekly, monthly, quarterly, half yearly and yearly are all different. If you try hopping from one time-frame to another just because it is below oversold or above overbought region with same K%, you may get confused. Worst you may not square-off your loss making trade. And rather not use excel; charts gives better visual for oscillators.
R/stockmarket is probably the best.  When I have some free time I can test that strategy back as long as you would like in quantstrat, although the only problem with testing futures is joining the consolidated contracts, however I can test it with SPY as SPY returns and ES_F track very closely. "Check out Professor Damodaran's website: http://pages.stern.nyu.edu/~adamodar/ . Tons of good stuff there to get you started. If you want more depth, he's written what is widely considered the bible on the subject of valuation: ""Investment Valuation"". DCF is very well suited to stock analysis. One doesn't need to know, or forecast the future stock price to use it. In fact, it's the opposite. Business fundamentals are forecasted to estimate the sum total of future cash flows from the company, discounted back to the present. Divide that by shares outstanding, and you have the value of the stock. The key is to remember that DCF calculations are very sensitive to inputs. Be conservative in your estimates of future revenue growth, earnings margins, and capital investment. I usually develop three forecasts: pessimistic, neutral, optimistic. This delivers a range of value instead of a false-precision single number. This may seem odd: I find the DCF invaluable, but for the process, not so much the result. The input sensitivity requires careful work, and while a range of value is useful, the real benefit comes from being required to answer the questions to build the forecast. It provides a framework to analyze a business. You're just trying to properly fill in the boxes, estimate the unguessable. To do so, you pore through the financials. Skimming, reading with a purpose. In the end you come away with a fairly deep understanding of the business, how they make money, why they'll continue to make money, etc." Im creating a 5-year projection on Profit and loss, cash flow and balance sheet and i\m suppose to use the LIBOR (5 year forward curve) as interest rate on debt.  This is the information i am given and it in USD.  Thanks for the link.  I guess its the USD LIBOR today, in one year, in two years, three years, four years and five years "You make several good points.  I'll start with Black-Scholes; the arbitrage argument in Black-Scholes is between the option and a hypothetical and unobserved portfolio with identical cash flows (the replicating portfolio).  We then value the replicating portfolio via an equilibrium model.  When you use BS, there isn't necessarily *any* observed security whose is guaranteed not imply some arbitrage opportunity, because BS makes no reference to observed prices.  A no-arbitrage modification might look like this: ""I observe the prices (and implied volatilities) of some options, and use the implied volatilities to price another option (maybe with a different strike).  Doing so ensures that there is no arbitrage between my price and the market prices implied by my model.""  Realistically, the problem arbitrage-free models are addressing is that our models and assumptions are wrong, even though they're reasonable approximations a lot of the time.  A no-arbitrage model removes some set of obvious deficiencies, but at the cost of not being able to explain why things are priced as they are.  So, for instance, Vasicek won't reproduce the observed term structure, and Hull-White fixes this, but Hull-White doesn't explain where the term structure comes from (i.e., what the term structure *should* be)." If you're into math, do this thought experiment: Consider the outcome X of a random walk process (a stock doesn't behave this way, but for understanding the question you asked, this is useful): On the first day, X=some integer X1. On each subsequent day, X goes up or down by 1 with probability 1/2. Let's think of buying a call option on X. A European option with a strike price of S that expires on day N, if held until that day and then exercised if profitable, would yield a value Y = min(X[N]-S, 0). This has an expected value E[Y] that you could actually calculate. (should be related to the binomial distribution, but my probability & statistics hat isn't working too well today) The market value V[k] of that option on day #k, where 1 < k < N, should be V[k] = E[Y]|X[k], which you can also actually calculate. On day #N, V[N] = Y. (the value is known) An American option, if held until day #k and then exercised if profitable, would yield a value Y[k] = min(X[k]-S, 0).  For the moment, forget about selling the option on the market. (so, the choices are either exercise it on some day #k, or letting it expire) Let's say it's day k=N-1.  If X[N-1] >= S+1 (in the money), then you have two choices: exercise today, or exercise tomorrow if profitable. The expected value is the same. (Both are equal to X[N-1]-S). So you might as well exercise it and make use of your money elsewhere. If X[N-1] <= S-1 (out of the money), the expected value is 0, whether you exercise today, when you know it's worthless, or if you wait until tomorrow, when the best case is if X[N-1]=S-1 and X[N] goes up to S, so the option is still worthless. But if X[N-1] = S (at the money), here's where it gets interesting. If you exercise today, it's worth 0. If wait until tomorrow, there's a 1/2 chance it's worth 0 (X[N]=S-1), and a 1/2 chance it's worth 1 (X[N]=S+1). Aha! So the expected value is 1/2. Therefore you should wait until tomorrow. Now let's say it's day k=N-2.  Similar situation, but more choices: If X[N-2] >= S+2, you can either sell it today, in which case you know the value = X[N-2]-S, or you can wait until tomorrow, when the expected value is also X[N-2]-S. Again, you might as well exercise it now. If X[N-2] <= S-2, you know the option is worthless. If X[N-2] = S-1, it's worth 0 today, whereas if you wait until tomorrow, it's either worth an expected value of 1/2 if it goes up (X[N-1]=S), or 0 if it goes down, for a net expected value of 1/4, so you should wait. If X[N-2] = S, it's worth 0 today, whereas tomorrow it's either worth an expected value of 1 if it goes up, or 0 if it goes down -> net expected value of 1/2, so you should wait. If X[N-2] = S+1, it's worth 1 today, whereas tomorrow it's either worth an expected value of 2 if it goes up, or 1/2 if it goes down (X[N-1]=S) -> net expected value of 1.25, so you should wait. If it's day k=N-3, and X[N-3] >= S+3 then E[Y] = X[N-3]-S and you should exercise it now; or if X[N-3] <= S-3 then E[Y]=0.  But if X[N-3] = S+2 then there's an expected value E[Y] of (3+1.25)/2 = 2.125 if you wait until tomorrow, vs. exercising it now with a value of 2; if X[N-3] = S+1 then E[Y] = (2+0.5)/2 = 1.25, vs. exercise value of 1; if X[N-3] = S then E[Y] = (1+0.5)/2 = 0.75 vs. exercise value of 0; if X[N-3] = S-1 then E[Y] = (0.5 + 0)/2 = 0.25, vs. exercise value of 0; if X[N-3] = S-2 then E[Y] = (0.25 + 0)/2 = 0.125, vs. exercise value of 0. (In all 5 cases, wait until tomorrow.) You can keep this up; the recursion formula is E[Y]|X[k]=S+d = {(E[Y]|X[k+1]=S+d+1)/2 + (E[Y]|X[k+1]=S+d-1) for N-k > d > -(N-k), when you should wait and see} or {0 for d <= -(N-k), when it doesn't matter and the option is worthless} or {d for d >= N-k, when you should exercise the option now}. The market value of the option on day #k should be the same as the expected value to someone who can either exercise it or wait. It should be possible to show that the expected value of an American option on X is greater than the expected value of a European option on X. The intuitive reason is that if the option is in the money by a large enough amount that it is not possible to be out of the money, the option should be exercised early  (or sold), something a European option doesn't allow, whereas if it is nearly at the money, the option should be held, whereas if it is out of the money by a large enough amount that it is not possible to be in the money, the option is definitely worthless. As far as real securities go, they're not random walks (or at least, the probabilities are time-varying and more complex), but there should be analogous situations. And if there's ever a high probability a stock will go down, it's time to exercise/sell an in-the-money American option, whereas you can't do that with a European option. edit: ...what do you know: the computation I gave above for the random walk isn't too different conceptually from the Binomial options pricing model. This would be a nice Raspberry Pi project for Mathematica, which comes bundled free on the Raspbian OS. You can program it up and leave it running.  It's not expensive and doesn't use much power. A program to monitor stock prices or volume could be written as simply as :-  This checks the volume of trades of Oct 2014 US crude oil futures every 30 seconds and sends an email if the volume jumps by more than 100. The financial data in this example is curated from Yahoo.  If specific data is not available or not updated frequently enough, if you can find an alternative online data source it's usually possible read the data in.  For example, this is apparently real-time data :- {Crude Oil, 92.79, -0.67, -0.71%} After leaving the above program running while writing this the volume of trades has risen like so :-  Edit I just set this running on a Raspberry Pi.  I had to use gmail for the email setup as described in this post: Configuring Mathematica to send email from a notebook.  Anyway, it's working.  Hope I don't get inundated with emails. ;-) Your question is a bit odd in that you are mixing long-term fundamental analysis signals which are generally meant to work on longer time frames with medium term trading where these fundamental signals are mostly irrelevant.  Generally you would buy-and-hold on a fundamental signal and ride the short-term fluctuations if you believe you have done good analysis. If you would like to trade on the 2-6 month time scale you would need a signal that works on that sort of time scale.  Some people believe that technical analysis can give you those kind of signals, but there are many, many, many different technical signals and how you would trade using them is highly dependent on which one you believe works. Some people do mix fundamental and technical signals, but that can be very complicated.  Learning a good amount about technical analysis could get you started.  I will note, though, that studies of non-professionals continuously show that the more frequently people trade the more on they underperform on average in the long term when compared with people that buy-and-hold. An aside on technical analysis: michael's comment is generally correct though not well explained.  Say Bob found a technical signal that works and he believes that a stock that costs $10 dollars should be $11.  He buys it and makes money two months later when the rest of the market figures out the right price is $11 and he sells at that price.  This works a bunch of times and he now publishes how the signal works on Stack Exchange to show everyone how awesome he is.  Next time, Bob's signal finds a different stock at $10 that should be $11, but Anna just wrote a computer program that checks that signal Bob published faster than he ever could.  The computer program buys as much as it can in milliseconds until the price is $11.  Bob goes to buy, but now it is too late the price is already $11 and he can't make any money. Eventually, people learn to anticipate/adjust for this signal and even Anna's algorithms don't even work anymore and the hunt for new signals starts again. "While derivative pricing models are better modeling reality as academia invests more into the subject, none sufficiently do. If, for example, one assumes that stock returns are lognormal for the purposes of pricing options like Black Scholes does, the only true dependent variable becomes log-standard deviation otherwise known as ""volatility"", producing the infamous ""volatility smile"" which disappears in the cases of models with more factors accounting for other mathematical moments such as mean, skew, and kurtosis, etc.  Still, these more advanced models are flawed, and suffer the same extreme time mispricing as Black Scholes.  In other words, one can model anything however one wants, but the worse the model, the stranger the results since volatility for a given expiration should be constant across all strikes and is with better models. In the case of pricing dividends, these can be adjusted for the many complexities of taxation, but the model becomes ever more complex and extremely computationally expensive for each eventuality.  Furthermore, with more complexity in any model, the likelihood of discovering a closed form in the short run is less. For equities in a low interest rate, not high dividend yield, not low volatility, low dividend tax environment, the standard swap pricing models will not provide results much different from one where a single low tax rate on dividends is assumed. If one is pricing a swap on equity outside of the bounds above, the dividend tax rate could have more of an effect, but for computational efficiency, applying a single assumed dividend tax rate would be optimal with D*(1-x), instead of D in a formula, where D is the dividend paid and x is the tax rate. In short, a closed form model is only as good as its assumptions, so if anomalies appear between the actual prices of swaps in the market and a swap model then that model is less correct than the one with smaller anomalies of the same type.  In other words, if pricing equity swaps without a dividend tax rate factored more closely matches the actual prices than pricing with dividend taxes factored then it could be assumed that pricing without a dividend tax factored is superior.  This all depends upon the data, and there doesn't seem to be much in academia to assist with a conclusion. If equity swaps do truly provide a tax advantage and both parties to a swap transaction are aware of this fact then it seems unlikely swap sellers wouldn't demand some of the tax advantage back in the form of a higher price.  A model is no defense since volatility curves persist despite what Black Scholes says they should be." James Roth provides a partial solution good for stock picking but let's speed up process a bit, already calculated historical standard deviations: Ibbotson, very good collection of research papers here, examples below Books As you point out, the moving average is just MA(k)t = (Pt-1 + … + Pt-k )/k and is applied in technical analysis (TA) to smooth out volatile (noise) price action. If it has any logic to it, you might want to think in terms of return series (Pt - Pt-1 / Pt-1) and you could hypothesize that prices are in fact predictable and will oscillate below and above a running moving average.  Below is a link to a study on MA trading rules, published in the Journal of Finance, with the conclusion of predictive power and abnormal returns from such strategies. As with any decision made upon historical arguments, one should be aware of structural changes and or data mining.  Simple  technical trading rules and the stochastic properties of stock  returns Brock, W., J. Lakonishok and B. Le Baron, 1992, Simple technical trading rules and the stochastic properties of stock returns, Journal of Finance, 47, 1731-64. MA rules betterthan chance in US stock market, 1897-1986 I don't know whether you are new to TA or not, but a great commercial site, with plenty of computer-generated signals is FinViz. "I'm just trying to visualize the costs of trading. Say I set up an account to trade something (forex, stock, even bitcoin) and I was going to let a random generator determine when I should buy or sell it. If I do this, I would assume I have an equal probability to make a profit or a loss. Your question is what a mathematician would call an ""ill-posed problem.""  It makes it a challenge to answer.  The short answer is ""no.""  We will have to consider three broad cases for types of assets and two time intervals. Let us start with a very short time interval.  The bid-ask spread covers the anticipated cost to the market maker of holding an asset bought in the market equal to the opportunity costs over the half-life of the holding period.  A consequence of this is that you are nearly guaranteed to lose money if your time interval between trades is less than the half-life of the actual portfolio of the market maker.  To use a dice analogy, imagine having to pay a fee per roll before you can gamble.  You can win, but it will be biased toward losing. Now let us go to the extreme opposite time period, which is that you will buy now and sell one minute before you die.  For stocks, you would have received the dividends plus any stocks you sold from mergers.  Conversely, you would have had to pay the dividends on your short sales and received a gain on every short stock that went bankrupt.  Because you have to pay interest on short sales and dividends passed, you will lose money on a net basis to the market maker.  Maybe you are seeing a pattern here.  The phrase ""market maker"" will come up a lot. Now let us look at currencies.  In the long run, if the current fiat money policy regime holds, you will lose a lot of money.  Deflation is not a big deal under a commodity money regime, but it is a problem under fiat money, so central banks avoid it.  So your long currency holdings will depreciate.  Your short would appreciate, except you have to pay interest on them at a rate greater than the rate of inflation to the market maker. Finally, for commodities, no one will allow perpetual holding of short positions in commodities because people want them delivered.  Because insider knowledge is presumed under the commodities trading laws, a random investor would be at a giant disadvantage similar to what a chess player who played randomly would face against a grand master chess player.  There is a very strong information asymmetry in commodity contracts.  There are people who actually do know how much cotton there is in the world, how much is planted in the ground, and what the demand will be and that knowledge is not shared with the world at large.  You would be fleeced. Can I also assume that probabilistically speaking, a trader cannot do   worst than random? Say, if I had to guess the roll of a dice, my   chance of being correct can't be less than 16.667%. A physicist, a con man, a magician and a statistician would tell you that dice rolls and coin tosses are not random.  While we teach ""fair"" coins and ""fair"" dice in introductory college classes to simplify many complex ideas, they also do not exist.  If you want to see a funny version of the dice roll game, watch the 1962 Japanese movie Zatoichi.  It is an action movie, but it begins with a dice game.   Consider adopting a Bayesian perspective on probability as it would be a healthier perspective based on how you are thinking about this problem.  A ""frequency"" approach always assumes the null model is true, which is what you are doing.  Had you tried this will real money, your model would have been falsified, but you still wouldn't know the true model. Yes, you can do much worse than 1/6th of the time.  Even if you are trying to be ""fair,"" you have not accounted for the variance. Extending that logic, then for an inexperienced trader, is it right to   say then that it's equally difficult to purposely make a loss then it   is to purposely make a profit? Because if I can purposely make a loss,   I would purposely just do the opposite of what I'm doing to make a   profit. So in the dice example, if I can somehow lower my chances of   winning below 16.6667%, it means I would simply need to bet on the   other 5 numbers to give myself a better than 83% chance of winning. If the game were ""fair,"" but for things like forex the rules of the game are purposefully changed by the market maker to maximize long-run profitability.  Under US law, forex is not regulated by anything other than common law.  As a result, the market maker can state any price, including prices far from the market, with the intent to make a system used by actors losing systems, such as to trigger margin calls.  The prices quoted by forex dealers in the US move loosely with the global rates, but vary enough that only the dealer should make money systematically.  A fixed strategy would promote loss. You are assuming that only you know the odds and they would let you profit from your 83.33 percentage chance of winning. So then, is the costs of trading from a purely probabilistic point of   view simply the transaction costs? No matter what, my chances cannot   be worse than random and if my trading system has an edge that is   greater than the percentage of the transaction that is transaction   cost, then I am probabilistically likely to make a profit? No, the cost of trading is the opportunity cost of the money.  The transaction costs are explicit costs, but you have ignored the implicit costs of foregone interest and foregone happiness using the money for other things. You will want to be careful here in understanding probability because the distribution of returns for all of these assets lack a first moment and so there cannot be a ""mean return.""  A modal return would be an intellectually more consistent perspective, implying you should use an ""all-or-nothing"" cost function to evaluate your methodology." "You're talking about modern portfolio theory. The wiki article goes into the math. Here's the gist: Modern portfolio theory (MPT) is a theory of finance that attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets. At the most basic level, you either a) pick a level of risk (standard deviation of your whole portfolio) that you're ok with and find the maximum return you can achieve while not exceeding your risk level, or b) pick a level of expected return that you want and minimize risk (again, the standard deviation of your portfolio). You don't maximize both moments at once. The techniques behind actually solving them in all but the most trivial cases (portfolios of two or three assets are trivial cases) are basically quadratic programming because to be realistic, you might have a portfolio that a) doesn't allow short sales for all instruments, and/or b) has some securities that can't be held in fractional amounts (like ETF's or bonds). Then there isn't a closed form solution and you need computational techniques like mixed integer quadratic programming Plenty of firms and people use these techniques, even in their most basic form. Also your terms are a bit strange: It has correlation table p11, p12, ... pij, pnn for i and j running from 1 to n This is usually called the covariance matrix. I want to maximize 2 variables. Namely the expected return and the additive inverse of the standard deviation of the mixed investments. Like I said above you don't maximize two moments (return and inverse of risk). I realize that you're trying to minimize risk by maximizing ""negative risk"" so to speak but since risk and return are inherently a tradeoff you can't achieve the best of both worlds. Maybe I should point out that although the above sounds nice, and, theoretically, it's sound, as one of the comments points out, it's harder to apply in practice. For example it's easy to calculate a covariance matrix between the returns of two or more assets, but in the simplest case of modern portfolio theory, the assumption is that those covariances don't change over your time horizon. Also coming up with a realistic measure of your level of risk can be tricky. For example you may be ok with a standard deviation of 20% in the positive direction but only be ok with a standard deviation of 5% in the negative direction. Basically in your head, the distribution of returns you want probably has negative skewness:  because on the whole you want more positive returns than negative returns. Like I said this can get complicated because then you start minimizing other forms of risk like value at risk, for example, and then modern portfolio theory doesn't necessarily give you closed form solutions anymore. Any actively managed fund that applies this in practice (since obviously a completely passive fund will just replicate the index and not try to minimize risk or anything like that) will probably be using something like the above, or at least something that's more complicated than the basic undergrad portfolio optimization that I talked about above. We'll quickly get beyond what I know at this rate, so maybe I should stop there." Not to be flippant, but I hope you're asking for llfinance to cover stochastic calculus on his blog, and not for someone here to give you a primer on it.  If you like, I can give you an explanation for why stochastic calculus has to be different from regular calculus. Algorithmic trading essentially banks on the fact that a price will fluctuate in tiny amounts over short periods of time, meaning the volatility is high in that given time frame. As the time frame increases the efficiency of algorithmic trading decreases and proper investment strategies such as due diligence, stock screening, and technical analysis become the more efficient methods.  Algorithms become less effective as the time frame increases due to the smoothing effect of volatility over time. Writing an algorithm that could predict future long-term prices would be an impossible feat because as the time frame is scaled up there are far less price fluctuations and trends (volatility smooths out) and so there is little to no benchmark for the formulas.  An algorithm simply wouldn't make sense for a long-term position. A computer can't predict, say, the next quarter, an ousted CEO, a buyout, or anything else that could effect the price of the security, never mind the psychology behind it all. Vice versa, researching a company's fundamentals just to bank on a 0.25% daily swing would not be efficient. Tax advantages or not, it is the most efficient methods that are preferred for a given time-scale of trading. Here's another attempt at explanation: it's basically because parabolas are flat at the bottom. Let me explain.  As you might know, the variance of the log stock price in Black Scholes is vol^2 * T, in other words, variance of the log stock price is linear in time to expiry. Now, that means that the standard deviation of your log stock price is square root in time. This is consequential.  For normally distributed random variables, in 68% of cases we end up within one standard deviation. So, basically, we expect our log stock price to be within something something times square root of T. So, if your stock has a vol of 16%, it'll be plus/minus 32% in 4 years, plus/minus 16% for one year, plus/minus 8% for 3m, plus/minus 4% for 3-ish weeks, and plus/minus 1% for a business day. As you see, the decay is slow at first, but much more rapid as we get closer.  How does the square root function look? It's a sideways parabola. As we come closer to zero, the slope of the square root function goes to infinity. (That is related to the fact that Brownian motion is almost surely no-where differentiable - it just shoots off with infinite slope, returning immediately, of course :-) Another way of looking at it is the old traders rule of thumb that an at-the-money option is worth approximately S * 0.4 * vol * sqrt(T). (Just do a Taylor expansion of Black Scholes). Again, you have the square root of time to expiry in there, and as outlined above, as we get closer to zero, the square root drops slowly at first, and then precipitously.