How does a delta signify the probability of expiring in the money
Just for clarification, delta and probability of expiring in the money are not the same thing. What the guy meant was that delta is usually a close enough approximation to the probability. One way to think about it is to look at the probabilities and deltas of In the Money, Out of the Money, and At the Money options. In these cases, the delta and probabilities are about the same. In fact if you look at an options chain with delta and probabilities, you can see that they are all about the same. In other words, there is a linear relationship between delta and probability. Here are a couple links to other answers around the web: Hope this answer helps!
When people voice or express racist discriminatory views for public consumption they are signifying to the world that they have lost all sense of the concept of socialization, and that they wish to be alone and isolated from civilized educated people.  Politics is knowing when to leave the world stage, and racism is not knowing you are just about to, Mr. Faber. Have fun counting your money you anal sadistic antisocial buffoon knuckledragger piece of sociopathic shit.  Bon Voyagee'.... You don't mention any specific numbers, so I'll answer in generalities. Say I buy a call option today, and I short the underlying stock with the delta. The value will be the value of the option you bought less the value of the stock you are short. (your premium is not included in the value since it's a sunk cost, but is reflected in your profit/loss) So, say I go out and adjust my portfolio, so I am still delta short in the underlying. It's still the value of your options, less the value of the underlying you are short. What is my PnL over this period? The end value of your portfolio less what you paid for that value, namely the money you received shorting the underlyings less the premium you paid for the option. If you haven't already, check out his earlier articles when he was at DealBreaker.com - GREAT stuff, my [favourite one](http://dealbreaker.com/2011/09/lets-just-go-ahead-and-assume-that-greek-letters-evil/) is when UBS had a delta one trader Kweku losing $2.1b from unauthorized/unreported trades in the midst of Dodd Frank/Volcker Rule was being discussed.   &gt; A thing that you probably do is go borrow one million shares of UBS and sell them short. This is a sensible thing to do. Why is it sensible? Well, for one thing, your desk is called “delta one,” and you’ve written a derivative, and you want to delta hedge that derivative, and so you pull out a Black-Scholes calculator and plug in a lot of variables and get stuck but then you look at your business card and you’re like, oh, shit, the answer is “one,” The short answer to your initial question is: yes. The option doesn't expire until the close of the market on the day of expiration. Because the option is expiring so soon, the time value of the option is quite small.  That is why the option, once it is 'in-the-money', will track so closely to the underlying stock price. If someone buys an in-the-money option on the day of expiration, they are likely still expecting the price to go up before they sell it or exercise it. Many brokers will exercise your in-the-money options sometime after 3pm on the day of expiration.  If this is not what you desire, you should communicate that with them prior to that day. The market can stay irrational longer than you can remain solvent -John Maynard Keynes The stocks could stagnate and trade in a thin range, or decline in value. You assume that your stocks will offer you ANY positive return for every month over 24 months. Just one month of negative returns puts you underwater. Thats whats wrong with it.  Even if you identified any stock that has been up every month for a consecutive 24 months in the past, there is nothing that says it will be so in the future, and a broad market selloff will effect both indexes as well as individual stocks.  Literally any adverse macroeconomic event in the next two years will put you underwater on your loan, no matter how much research you do on individual stocks. Makes sense.  I typed the previous reply on an ipad and was too lazy to go into the fact that as you point out the cash exits the balance sheet so in a DCF it doesn't get any weight in the terminal value calculation which makes up a significant chunk of an EV normally. "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" "Not cumulative volatility. It's cumulative probability density. Time value isn't linear because PDFs (probability distribution function) aren't linear. It's a type of distribution e.g. ""bell-curves"") These distributions are based on empirical data i.e. what we observe. BSM i.e. Black-Scholes-Merton includes the factors that influence an option price and include a PDF to represent the uncertainty/probability. Time value is based on historical volatility in the underlying asset price, in this case equity(stock). At the beginning, time value is high since there's time until expiration and the stock is expected to move within a certain range based on historical performance. As it nears expiration, uncertainty over the final value diminishes. This causes probability for a certain price range to become more likely.  We can relate that to how people think, which affects the variation in the stock market price. Most people who are hoping for a value increase are optimistic about their chances of winning and will hold out towards the end. They see in the past d days, the stock has moved [-2%,+5%] so as a call buyer, they're looking for that upside. With little time remaining though, their hopes quickly drop to 0 for any significant changes beyond the market price. (Likewise, people keep playing the lottery up until a certain age when they're older and suddenly determine they're never going to win.)  We see that reflected in the PDF used to represent options price movements. Thus your time value which is a function of probability decreases in a non-linear fashion.  Option price = intrinsic value + time value At expiration, your option price = intrinsic value = stock price - strike price, St >= K, and 0 for St < K."