Deriving the put-call parity
Think of it this way: C + (-P) = forward contract. Work it out from there. Anyways, this stack is meant for professionals, not students, I think.
The crucial insight is that the alternative to early exercise of an American call is not necessarily to hold it to expiry, but to sell it. And selling it, at its value, is always better than exercising it.  Note that this holds only for options on assets that don't pay dividends. Here's the proof, using Put-Call-Parity. We know that at expiry T, we have (using a Call and a Put both struck at K): C(T) - P(T) = S(T) - K (if this is not clear to you, consider the case where S is less than, equal to, or greater than K at maturity, and go through each of them.) If the stock S doesn't pay any dividends (and there is no cost of carry etc.), we can replicate both sides now at time 0; we just buy one call, sell one put (that gives us the left hand side), buy the stock, and borrow money so that at time T we have to repay K (that gives us the right hand side). That means that now, we only need to borrow df * K, where df is the discount factor, and is less than one (assuming the good old pre-2009 world where interest rates are positive). Thus: C(0) - P(0) = S(0) - df * K.  Rearranging gives: C(0) = S(0) - df * K + P(0).  That's the value of the call, if we sell it (or hold it). However, if we exercise, we only get: C_ex = S(0) - K Now, we see that C(0) > C_ex, because we subtract less (df*K < K), and add P(0). "No, it means that is only the notional value of that underlying asset of that contract, generally. The contract specification itself is listed on the exchange's websites, and there are really no assumptions you can make about a particular contract. Where S&P futures have one set of specifications, such as what it actually represents, how many each contract holds, how to price profits and losses... a different contract, such as FTSE 100 stock futures have a completely different set of specifications. Anyway in this one example the s&p 500 futures contract has an ""initial margin"" of $19,250, meaning that is how much it would cost you to establish that contract. Futures generally require delivery of 1,000 units of the underlying asset. So you would take the underlying asset's price and multiple it by 1,000. (what price you use is also mentioned in the contract specification), The S&P 500 index  is $1588 you mentioned, so on Jun2013 you would have to delivery $1588 x 1000, or $1,588,000. GREAT NEWS, you only have to put up 1.2% in principal to control a 1.5 million dollar asset! Although, if even that amount is too great, you can look at the E-Mini S&P futures, which require about 1/10th the capital and delivery. This answer required that a lot of different subjects be mentioned, so feel free to ask a new question about the more specific topics." This isn't totally wrong- there are hedge funds that are long 150% of AUM and short 50%. However, Rentech has said that holding a position for 8 seconds is long for them, so that's not what they're doing. I'd assume the 4X leverage most just refers to option positions that have delta 4 on average. They also may be borrowing money, which they can probably do extremely cheaply since they have a 35 year track record showing they're essentially risk-free. 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. 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. The Sharpe ratio is, perhaps, the method you are looking for.  That said, not really sure beta is a meaningful metric, as there are plenty of safe bets to be made on volatile stocks (and, conversely, unsafe bets to be made on non-volatile ones). long deep ITM calls is equivalent to owning the equity.  You're going to pay alot and hence will start off in a hole already, and you aren't getting too much leverage there at all depending how deep ITM you go.  Covariance scales, but assuming B-S in order to get nice scaling and ignoring the risks you are actually taking with options (unlimited down-size ie you can lose your entire investment in the option, people forget this) will screw you unless you really know what you are dong.  Leverage means increasing your risk.  long dep ITM is not obtaining much leverage and therefore not risking too much.  but you aren't going ot get 3-4x leverage this way.  you get leverage by saying: oh, i have 100, i could invest in 1 share of stock OR I could buy 100 worth of some option.  If I pick a deep ITM (think strike = 0) it's identical to owing the stock.  If i pick ATM, i have a ~50/50 chance of wining, so i should be able to double my upside.  If I go OTM, i can increase my exposure to the upside while increasing hte chance that my options expire worthless.  So really, i have no idea why deep ITM do what you are trying to do.  and If you don't either, you probably shouldn't do it. "This chart concerns an option contract, not a stock. The method of analysis is to assume that the price of an option contract is normally distributed around some mean which is presumably the current price of the underlying asset. As the date of expiration of the contract gets closer the variation around the mean in the possible end price for the contract will decrease. Undoubtedly the publisher has measured typical deviations from the mean as a function of time until expiration from historical data. Based on this data, the program that computes the probability has the following inputs: (1) the mean (current asset price) (2) the time until expiration (3) the expected standard deviation based on (2) With this information the probability distribution that you see is generated (the green hump). This is a ""normal"" or Gaussian distribution. For a normal distribution the probability of a particular event is equal to the area under the curve to the right of the value line (in the example above the value chosen is 122.49). This area can be computed with the formula:  This formula is called the probability density for x, where x is the value (122.49 in the example above). Tau (T) is the reciprocal of the variance (which can be computed from the standard deviation). Mu (μ) is the mean. The main assumption such a calculation makes is that the price of the asset will not change between now and the time of expiration. Obviously that is not true in most cases because the prices of stocks and bonds constantly fluctuate. A secondary assumption is that the distribution of the option price around the mean will a normal (or Gaussian) distribution. This is obviously a crude assumption and common sense would suggest it is not the most accurate distribution. In fact, various studies have shown that the Burr Distribution is actually a more accurate model for the distribution of option contract prices." There is no margin call. Inverse ETFs use derivatives that would lose value in the case you describe though this doesn't force a margin call as you may be misunderstanding how these funds are constructed. The quant aspect '''''. This is the kind of math I was wondering if it existed, but now it sounds like it is much more  complex in reality then optimizing by evaluating different cost of capital. Thank you for sharing "JoeTaxpayer nailed it.  Here's another way to look at it: Generally, we invest in something, then might leave it there for a few years, then take it out, but don't touch it in between.  In that case, to get the final amount X(N), we need to take the initial amount, then multiply by growth in the first year, then multiply by growth in the second year, etc. So, for three years, we have: X(3) = X(0) * G(1) * G(2) * G(3) = X(0) * ""average annual growth"" ^ 3 So, here, we see that we want the average annual growth to the power three equal to the product of the annual growth rates, thus, geometric mean: geometric mean = (G(1) * G(2) * G(3)) ^ (1/3) On the other hand, consider a situation where I have three investments X,Y,Z over one year. Now I have, after one year: X(1)+Y(1)+Z(1) = X(0)*G(1,X) + Y(0)*G(1,Y) + Z(0)*G(1,Z) = ( X(0)+Y(0)+Z(0) ) * ""average annual growth"" Now, in this case, if we assume X(0) = Y(0) = Z(0) = 1, i.e. I put equal amounts in each, we see that the average annual growth rate we want in this case is the arithmetic mean: arithmetic mean = (G(1,X) + G(1,Y) + G(1,Z)) / 3 (if we had unequal amounts at the beginning, it would be a weighted average). TL;DR:" "According to the book of Hull, american and european calls on non-dividend paying stocks should have the same value. American puts, however, should be equals to, or more valuable than, european puts. The reason for this is the time value of money. In a put, you get the option to sell a stock at a given strike price. If you exercise this option at t=0, you receive the strike price at t=0 and can invest it at the risk-free rate. Lets imagine the rf rate is 10% and the strike price is 10$. this means at t=1, you would get 11.0517$. If, on the other hand, you did'nt exercise the option early, at t=1 you would simply receive the strike price (10$). Basically, the strike price, which is your payoff for a put option, doesn't earn interest. Another way to look at this is that an option is composed of two elements: The ""insurance"" element and the time value of the option. The insurance element is what you pay in order to have the option to buy a stock at a certain price. For put options, it is equals to the payout= max(K-S, 0) where K=Strike Price and St= Stock price.  The time value of the option can be thought of as a risk-premium. It's difference between the value of the option and the insurance element. If the benefits of exercising a put option early (i.e- earning the risk free rate on the proceeds) outweighs the time value of the put option, it should be exercised early. Yet another way to look at this is by looking at the upper bounds of put options. For a european put, today's value of the option can never be worth more than the present value of the strike price discounted at the risk-free rate. If this rule isn't respected, there would be an arbitrage opportunity by simply investing at the risk-free rate. For an american put, since it can be exercised at any time, the maximum value it can take today is simply equals to the strike price. Therefore, since the PV of the strike price is smaller than the strike price, the american put can have a bigger value. Bear in mind this is for a non-dividend paying stock. As previously mentioned, if a stock pays a dividend it might also be optimal to exercise just before these are paid." Thanks. Just to clarify I am looking for a more value-neutral answer in terms of things like Sharpe ratios. I think it's an oversimplification to say that on average you lose money because of put options - even if they expire uselessly 90% of the time, they still have some expected payoff that kicks in 10% of the time, and if the price is less than the expected payoff you will earn money in the long term by investing in put options (I am sure you know this as a PhD student I just wanted to get it out there.)I guess more formally my question would be are there studies on whether options prices correspond well to the diversification benefits they offer from an MPT point of view. The initial position is worth 40000. You post 50% margin, so you deposited 20000 and borrowed 20000. 6% of 20000 is 1200. Can you give more detail on the problem? If you can model it with a one step binomial tree, then the price is favourable as long as the chance to multiply is P(S^1 = 10 S^0 ) &gt; 0.1.  If you don't know the probabilities, then the usual go-to is to determine what probability space is that would lead to an expected profit (plus an error, and a cushion for risk aversion if the bet is sufficiently large).