Value of tokens bought at an older price
"You will make a profit in nominal dollars (or nominal units of whatever currency you used to buy the token).  Whether you'll make a profit in real dollars depends on inflation, and in practice whether it would be possible to sell your existing tokens to someone else for the new price. Suppose when the price was 50 U (50 ""units"", since you didn't specify a currency), you bought one token.  Today you can either spend 52 U for a token, and get a liter of milk, or you can spend your existing token (for which you paid 50 U) and get a liter of milk.  It looks like you are making a profit of 2 U by spending your token. However, whether that profit is real or illusory depends on what else you could do with the token.  For instance, suppose that, since the price of a token is now 52 U, you will have no trouble finding someone who wants to buy your token from you for 52 U.  If you sell your token for 52 U, you'll still only be able to buy 1 L of milk.  So if you measure your wealth in milk, you have made no profit: in the past you had a token representing 1 L of milk, and today you still have a token representing 1 L of milk. Suppose now that in the past, when a token cost 50 U, a hamburger also cost 50 U.  Suppose further that a hamburger now costs 52 U.  So you can sell your token for 52 U, but that 52 U will still only buy you one hamburger.  So, again, if you measure your wealth in hamburgers, your have made no profit.  In the past, you could have sold your token and bought a hamburger; today, you can still sell your token and buy a hamburger, and you'll have nothing left over, so you have gained nothing. If, on the other hand, the price of a hamburger today is still 50 U, then you call sell your token for 52 U, buy a hamburger for 50 U, and still have 2 U left over.  You have made a profit. What this all goes to show is that, in practice, the idea of ""profit"" depends on the overall economy, and whether you could exchange the currency units you have in your possession for a greater quantity of goods than you could in the past.  Whether this is possible depends on the relative changes in price of various goods.  In other words, if you get your money by selling Product A, and later you buy Product B, you may or may not make a profit depending on how the prices of the two products moved relative to one another. Also, in your hypothetical setup, the ""currency"" (the token) is directly linked to the value of a single good, so you can always at least get 1 L of milk for your token.  Most real currency is not bound to specific goods like your milk token, so it is possible for your currency to lose value in an absolute sense.  For instance, suppose you sell a book for $5.  The $5 is not a ""book token"" and you cannot rely on being able to exchange it for a book in the future; in the future, all books may cost $10, and the prices of all goods may rise similarly, so your currency will actually be worth less no matter how you try to use it.  This could happen with the milk token if the milkman announces that henceforth 1 L of milk will cost 2 tokens; your existing token suddenly loses half its value. In sum, it is easy to calculate whether you made a profit in currency units.  What is harder is to calculate whether you made a profit in ""real terms"" (often referred to as ""real dollars"" or ""inflation-adjusted dollars"", or the equivalent in your favorite currency).  The reason this is hard is because the idea of ""real dollars"" is fundamentally linked to the possibility of exchanging currency for goods (and services), and so it depends what goods you're buying.  Inflation statistics published by governments and the like use a ""basket"" of goods to approximate the overall price movements in the economy as a whole."
"Yes, long calls, and that's a good point. Let's see... if I bought one contract at the Bid price above...          $97.13 at expiry of $96.43 option = out of the money =- option price(x100)          = $113 loss.     $97.13 at expiry of $97.00 option = out of the money =- option price(x100)          = $77 loss.     $97.13 at expiry of $97.14 option = in the money by 1-cent=$1/contract profit - option price(x100) = $1-$58          = $57 loss   The higher strike prices have much lower losses if they expire with the  underlying stock at- or near-the-money. So, they carry ""gentler"" downside potential, and are priced much higher to reflect that ""controlled"" risk potential.  That makes sense. Thanks." "The textbook answer would be ""assets-liabilities+present discounted value of all future profit"".  A&L is usually simple (if a company has an extra $1m in cash, it's worth $1m more; if it has an extra $1m in debt, it's worth $1m less). If a company with ~0 assets and $50k in profit has a $1m valuation, then that implies that whoever makes that valuation (wants to buy at that price) really believes one of two things - either the future profit will be significantly larger than $50k (say, it's rapidly growing);  or the true worth of assets is much more - say, there's some IP/code/patents/people that have low book value but some other company would pay $1m just to get that. The point is that valuation is subjective since the key numbers in the calculations are not perfectly known by anyone who doesn't have a time machine, you can make estimates but the knowledge to make the estimates varies (some buyers/sellers have extra information), and they can be influenced by those buyers/sellers; e.g. for strategic acquisitions the value of company is significantly changed simply because someone claims they want to acquire it. And, $1m valuation for a company with $500m in profits isn't appropriate - it's appropriate only if the profits are expected to drop to zero within a couple years; a stagnant but stable company with $500m profits would be worth at least $5m and potentially much more." "Below I will try to explain two most common Binomial Option Pricing Models (BOPM) used. First of all, BOPM splits time to expiry into N equal sub-periods and assumes that in each period the underlying security price may rise or fall by a known proportion, so the value of an option in any sub-period is a function of its possible values in the following sub period. Therefore the current value of an option is found by working backwards from expiry date through sub-periods to current time. There is not enough information in the question from your textbook so we may assume that what you are asked to do is to find a value of a call option using just a Single Period BOPM. Here are two ways of doing this: First of all let's summarize your information: Current Share Price (Vs) = $70 Strike or exercise price (X) = $60 Risk-free rate (r) = 5.5% or 0.055 Time to maturity (t) = 12 months Downward movement in share price for the period (d) = $65 / $70 = 0.928571429 Upward movement in share price for the period (u) = 1/d = 1/0.928571429 = 1.076923077 ""u"" can be translated to $ multiplying by Vs => 1.076923077 * $70 = $75.38 which is the maximum probable share price in 12 months time. If you need more clarification here - the minimum and maximum future share prices are calculated from stocks past volatility which is a measure of risk. But because your textbook question does not seem to be asking this - you probably don't have to bother too much about it yet. Intrinsic Value: Just in case someone reading this is unclear - the Value of an option on maturity is the difference between the exercise (strike) price and the value of a share at the time of the option maturity. This is also called an intrinsic value.  Note that American Option can be exercised prior to it's maturity in this case the intrinsic value it simply the diference between strike price and the underlying share price at the time of an exercise. But the Value of an option at period 0 (also called option price) is a price you would normally pay in order to buy it. So, say, with a strike of $60 and Share Price of $70 the intrinsic value is $10, whereas if Share Price was $50 the intrinsic value would be $0. The option price or the value of a call option in both cases would be fixed.  So we also need to find intrinsic option values when price falls to the lowest probable and rises to the maximum probable (Vcd and Vcu respectively) (Vcd) = $65-$60 = $5 (remember if Strike was $70 then Vcd would be $0 because nobody would exercise an option that is out of the money) (Vcu) = $75.38-$60 = $15.38 1. Setting up a hedge ratio:  h = Vs*(u-d)/(Vcu-Vcd) h = 70*(1.076923077-0.928571429)/(15.38-5) = 1 That means we have to write (sell) 1 option for each share purchased in order to hedge the risks. You can make a simple calculation to check this, but I'm not going to go into too much detail here as the equestion is not about hedging. Because this position is risk-free in equilibrium it should pay a risk-free rate (5.5%). Then, the formula to price an option (Vc) using the hedging approach is: (Vs-hVc)(e^(rt))=(Vsu-hVcu) Where (Vc) is the value of the call option, (h) is the hedge ratio, (Vs) - Current Share Price, (Vsu) - highest probable share price, (r) - risk-free rate, (t) - time in years, (Vcu) - value of a call option on maturity at the highest probable share price. Therefore solving for (Vc): (70-1*Vc)(e^(0.055*(12/12))) = (75.38-1*15.38) => (70-Vc)*1.056540615 = 60 => 70-Vc = 60/1.056540615 => Vc = 70 - (60/1.056540615)  Which is similar to the formula given in your textbook, so I must assume that using 1+r would be simply a very close approximation of the formula above. Then it is easy to find that Vc = 13.2108911402 ~ $13.21 2. Risk-neutral valuation: Another way to calculate (Vc) is using a risk-neutral approach. We first introduce a variable (p) which is a risk-neutral probability of an increase in share price. p = (e^(r*t)-d)/(u-d) so in your case: p = (1.056540615-0.928571429)/(1.076923077-0.928571429) = 0.862607107 Therefore using (p) the (Vc) would be equal: Vc = [pVcu+(1-p)Vcd]/(e^(rt)) => Vc = [(0.862607107*15.38)+(0.137392893*5)]/1.056540615 => Vc = 13.2071229185 ~ $13.21 As you can see it is very close to the hedging approach. I hope this answers your questions. Also bear in mind that there is much more to the option pricing than this. The most important topics to cover are: Multi-period BOPM Accounting for Dividends Black-Scholes-Merton Option Pricing Model" In order to see whether you can buy or sell some given quantity of a stock at the current bid price, you need a counterparty (a buyer) who is willing to buy the number of stocks you are wishing to offload. To see whether such a counterparty exists, you can look at the stock's order book, or level two feed. The order book shows all the people who have placed buy or sell orders, the price they are willing to pay, and the quantity they demand at that price. Here is the order book from earlier this morning for the British pharmaceutical company, GlaxoSmithKline PLC.  Let's start by looking at the left-hand blue part of the book, beneath the yellow strip. This is called the Buy side. The book is sorted with the highest price at the top, because this is the best price that a seller can presently obtain. If several buyers bid at the same price, then the oldest entry on the book takes precedence. You can see we have five buyers each willing to pay 1543.0 p (that's 1543 British pence, or £15.43) per share. Therefore the current bid price for this instrument is 1543.0. The first buyer wants 175 shares, the next, 300, and so on. The total volume that is demanded at 1543.0p is 2435 shares. This information is summarized on the yellow strip: 5 buyers, total volume of 2435, at 1543.0. These are all buyers who want to buy right now and the exchange will make the trade happen immediately if you put in a sell order for 1543.0 p or less. If you want to sell 2435 shares or fewer, you are good to go. The important thing to note is that once you sell these bidders a total of 2435 shares, then their orders are fulfilled and they will be removed from the order book. At this point, the next bidder is promoted up the book; but his price is 1542.5, 0.5 p lower than before. Absent any further changes to the order book, the bid price will decrease to 1542.5 p. This makes sense because you are selling a lot of shares so you'd expect the market price to be depressed. This information will be disseminated to the level one feed and the level one graph of the stock price will be updated. Thus if you have more than 2435 shares to sell, you cannot expect to execute your order at the bid price in one go. Of course, the more shares you are trying to get rid of, the further down the buy side you will have to go. In reality for a highly liquid stock as this, the order book receives many amendments per second and it is unlikely that your trade would make much difference. On the right hand side of the display you can see the recent trades: these are the times the trades were done (or notified to the exchange), the price of the trade, the volume and the trade type (AT means automatic trade).  GlaxoSmithKline is a highly liquid stock with many willing buyers and sellers. But some stocks are less liquid. In order to enable traders to find a counterparty at short notice, exchanges often require less liquid stocks to have market makers. A market maker places buy and sell orders simultaneously, with a spread between the two prices so that they can profit from each transaction. For instance Diurnal Group PLC has had no trades today and no quotes. It has a more complicated order book, enabling both ordinary buyers and sellers to list if they wish, but market makers are separated out at the top. Here you can see that three market makers are providing liquidity on this stock, Peel Hunt (PEEL), Numis (NUMS) and Winterflood (WINS). They have a very unpalatable spread of over 5% between their bid and offer prices. Further in each case the sum total that they are willing to trade is 3000 shares. If you have more than three thousand Dirunal Group shares to sell, you would have to wait for the market makers to come back with a new quote after you'd sold the first 3000. You may refer to project http://jstock.sourceforge.net. It is open source and released under GPL. It is fetching data from Yahoo! Finance, include delayed current price and historical price. According to Active Equity Management by Zhou and Jain: When a stock pays dividend, the adjusted price in Yahoo makes the following adjustment: Let T be the ex-dividend date (the first date that the buyers of a stock will not receive the dividend) and T-1 be the last trading day before T. All prices before T are adjusted by a multiplier (C_{T-1} - d_T)/C_{T-1}, where C_{T-1} is the close price at T-1 and d_T is the dividend per share. This, of course means that the price before T decreases. Yes, because most were holding it from 50-300 dollars cost pre-split, giving them a very healthy dividend yield today.  The question is upside if you buy today, not upside from 5 years ago.  I bought it at 300 too, but today other stocks are making me money... The value of the asset doesn't change just because of the exchange rate change. If a thing (valued in USD) costs USD $1 and USD $1 = CAN $1 (so the thing is also valued CAN $1) today and tomorrow CAN $1 worth USD $0.5 - the thing will continue being worth USD $1. If the thing is valued in CAN $, after the exchange rate change, the thing will be worth USD $2, but will still be valued CAN $1. What you're talking about is price quotes, not value. Price quotes will very quickly reach the value, since any deviation will be used by the traders to make profits on arbitrage. And algo-traders will make it happen much quicker than you can even notice the arbitrage existence. You can't. If there was a reliable way to identify an undervalued stock, then people would immediately buy it, its price would rise and it wouldn't be undervalued any more. "To a certain degree ""the only sure thing I know is the price I paid for the stock is the fair price at the time I buy it"" is absolutely right, by definition, and by the law of the free and efficient market and forces of supply and demand, freedom of public information about share price sensitive information, etc, etc, etc, and you've made a good point that eludes many investors I'd say. However, in practise, the market has many participants, and they will all be arriving at a different idea of what the ""fair price"" is by way of a slightly different analysis and slightly different information. In theory they all have the same information, but unfortunately in practise there is always some disparity. When one participant feels a stock is undervalued though the last thing they want to do is say so, instead they will start buying stock. They might feel it is undervalued by 20%, but that doesn't mean they'll keep buying and buying until it gets to 20%, they might push the price up just a little, then let the price drift down again, buy some more, relax, buy some more, etc. Over time the price will rise of course because the supply will become weaker, but even if the participant is correct about the 20% the price might have only risen 7% by the time they acquire all the stock they want given their risk models, market exposure and margin guidelines, etc, and it might be more than a year later before the price has actually risen to 20%, presumably because more and more other market participants have come to the same conclusion. The opposite can obviously also happen, a participant might dump stock it feels is over valued long before it hits the values it believes in. So right away you can see that pricing might not really reflect value, or ""fair price""." &gt; “If the [Black-Scholes] formula is applied to extended time periods….it can produce absurd results.”  &gt; -Warren Buffett, 2008 Letter to Berkshire Hathaway Shareholders  I will give your question more thought, and come back with a quantitative solution. It may be most fruitful to apply a backward-induction options pricing model with detailed scenario-based discounted cash flow valuation models supported by pro forma financial statement and investment analysis.    Nonetheless, my initial reaction is inline with Warren Buffett's belief that in the long-run an assumption of the Black-Scholes options pricing model is invalid (see [here](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.613.1657&amp;rep=rep1&amp;type=pdf) and [here](http://www.elon.edu/docs/e-web/academics/business/economics/faculty/kurt/OptionValuationBuffetCritique_123113.pdf)). The perceived invalid assumption is that the distribution of expected future stock prices is *not* log-normal in the long-run. This non-log-normal view is especially true for a single company stock.   This invalid assumption results in over-valuation of options prices from Black-Scholes - which makes it much better to sell long-term options than to buy them if market participants are using Black-Scholes pricing models.  Again, without having done the math yet, my gut tells me if the option seller is using Black-Sholes pricing for long-dated options, you would be best to avoid buying them as the prices will be inflated compared to an economic reality fair price. The problem you're talking about can be handled the same way now until better alternatives appear. I'm simply talking about improving the existing infrastructure to carry out these trades. Instead of holding a number on charles schwab account, you hold a token which is the same as a stock certificate of ownership. "I don't have a formula for anything like this, but it is important to note that the ""current value"" of any asset is really theoretical until you actually sell it. For example, let's consider a house.  You can get an appraisal done on your house, where your home is inspected, and the sales of similar houses in your area are compared.  However, this value is only theoretical.  If you found yourself in a situation where you absolutely had to sell your house in one week, you would most likely have to settle for much less than the appraised value. The same hold true for collectibles.  If I have something rare that I need cash for immediately, I can take it to a pawn shop and get cash.  However, if I take my time and locate a genuinely interested collector, I can get more for it. This is comparable to someone who holds a significant percentage of shares in a publicly held corporation.  If the current market value of your shares is $10 million, but you absolutely need to sell your entire stake today, you aren't going to get $10 million.  But if you take your time selling a little at a time, you are more likely to get much closer to this $10 million number. A ""motivated seller"" means that the price will drop." Many of the Bitcoin exchanges mimic stock exchanges, though they're much more rudimentary offering only simple buy/sell/cancel orders.  It's fairly normal for retail stock brokerage accounts to allow other sorts of more complex orders, where once a certain criteria is met, (the price falls below some $ threshold, or has a movement greater than some %) then your order is executed.  The space between the current buy order and the current sell order is the bid/ask spread, it's not really about timing.  Person X will buy at $100, person Y will sell at $102.  If both had a price set at $101, they would just transact.  Both parties think they can do a little bit better than the current offer.  The width of the bid/ask spread is not universal by any means.  The current highest buy order and the current lowest sell order, are both the current price.  The current quoted market price is generally the price of the last transaction, whether it's buy or sell. "The answer is yes. And the reason is if today's interest rates are lower than the interest rate (coupon) at which the bond was issued. The bond's ""lifetime value"" is 100 cents on the dollar. That's the principal repayment that the investor will get on the maturity date. But suppose the bond's coupon rate is 4% while today's interest rate is 3%. Then, people who bought the bond at 100 would get 4% on their money, while everyone else was getting 3%. To compensate, a three year bond would have to rise to almost 103 so that the so-called yield to maturity"" would be 3%. Then there would be a ""capital loss"" (from almost 103 to 100) that would exactly offset the extra interest, that is 1% ""more"" for three years. If today's interest rates are negative (as they were from time to time in the 1930s, and in the present decade), the ""negative"" interest rates will prevent the buyer from getting the ""lifetime value"" (as defined by the OP) of principal plus interest over the original life of the bond. This happens in a ""flight to quality"" situation, where people are willing to take a (small) capital loss on Treasuries in order to prevent a large possible loss from bank failures like those that took place in 2008."