Why call option price increases with higher volatility
The mathematics make it easier to understand why this is the case. Using very bad shorthand, d1 and d2 are inputs into N(), and N() can be expressed as the probability of the expected value or the most probable value which in this case is the discounted expected stock price at expiration. d1 has two σs which is volatility in the numerator and one in the denominator.  Cancelling leaves one on top.  Calculating when it's infinity gives an N() of 1 for S and 0 for K, so the call is worth S and the put PV(K).  At 0 for σ, it's the opposite. More concise is that any mathematical moment be it variance which mostly influences volatility, mean which determines drift, or kurtosis which mostly influences skew are all uncertanties thus costs, so the higher they are, the higher the price of an option. Economically speaking, uncertainties are costs.  Since costs raise prices, and volatility is an uncertainty, volatility raises prices. It should be noted that BS assumes that prices are lognormally distributed.  They are not.  The closest distribution, currently, is the logVariance Gamma distribution. Understanding the BS equation is not needed. What is needed is an understanding of the bell curve.   You seem to understand volatility. 68% of the time an event will fall inside one standard deviation. 16% of the time it will be higher, 16%, lower.  Now, if my $100 stock has a STD of $10, there's a 16% chance it will trade above $110. But if the STD is $5, the chance is 2.3% per the chart below. The higher volatility makes the option more valuable as there's a highr chance of it being 'in the money.' My answer is an over simplification, per your request. I agree that high volatility just means the underlying stock price fluctuates more, and it does not imply if the stock is going up or down. But a high volatility in the price of an underlying also means that there is a higher chance that the underlying price could reach extreme prices (albeit in either direction). However, if you purchased a call option then if the underlying price reached an extremely high value, then you will be richly rewarded. But if the underlying price reached an extremely low value, you won't lose any more than the initial premium that you paid. There is no additional risk on your side, it's capped to the premium that you paid for the call option. It's this asymmetric outcome (Heads - I win, Tails - I don't lose) combined with high volatility that means that call options will increase in value when the underlying price becomes more volatile. If the optionality wasn't there then the price wouldn't be related to the volatility of the underlying. But that would be called a Future or a Forward :-) When volatility is higher, the option is more likely to end up in-the-money. Moreover, when it ends up in-the-money, it is likely to be over the strike price by a greater amount. Consider a call option. With high volatility, moves in the stock price are big - both up moves and down moves. If the stock moves up by a lot, the call option holder will benefit greatly. On the other hand, when the stock moves down, below a certain point the option holder does not care how big a down move the stock has. His downside is limited. Hence, the value of the option is increased by high volatility. I know everyone who searches this is looking for this answer. Bump so people are able to get this concept instead of looking all over the web for it. The entire premise of purchasing a call option is your expectation that the prices will rise. So even though there is a possibility of prices falling, you wouldn't mind paying higher premiums in a volatile market  for a call option because you're bullish and are expecting the volatility to eventually turn out in your favour i.e. prices to rise Let's say a stock trades at $100 right now, and you can buy a $100 call option. When you buy the call option (and the money you paid is gone), one of two things can happen: The share price goes up, or the share price goes down.  If the share price goes up, you profit. If the share price goes down, you don't lose! Because once the shares are below $100, you don't exercise the call option, and you don't lose any money.  So if you have a share that is rock solid at $100, you don't make money. If you have a share where the company owner took some ridiculous risk, and the shares could go to $200 or the company could go bankrupt, then you have a 50% chance to make $100 and a 50% chance to not lose anything. That's much more preferable.
"You may be thinking about this the wrong way. The yield (Return) on your investment is effectively the market price paid to the investor for the amount of risk assumed for participating. Looking at the last few years, many including myself would have given their left arm for a so-called ""meager return"" instead of the devastation visited on our portfolios.  In essence, higher return almost always (arguably always) comes at the cost of increased risk.  You just have to decide your risk profile and investment goals.  For example, which of the following scenarios would you prefer? Investment Option A  Treasuries, CD's Worst Case: 1% gain Best Case 5% gain   Investment option B  Equities/Commodities Worst Case: 25% loss Best Case: 40% gain" "As JoeTaxpayer says, there's a lot you can do with just the stock price. Exploring that a bit: Stock prices are a combination of market sentiment and company fundamentals. Options are just a layer on top of that. As such, options are mostly formulaic, which is why you have a hard time finding historical option data -- it's just not that ""interesting"", technically. ""Mostly"" because there are known issues with the assumptions the Black-Scholes formula makes. It's pretty good, and importantly, the market relies on it to determine fair option pricing. Option prices are determined by: Relationship of stock price to strike. Both distance and ""moneyness"". Time to expiration. Dividends. Since dividend payments reduce the intrinsic value of a company, the prospect of dividend payments during the life of a call option depresses the price of the option, as all else equal, without the payments, the stock would be more likely to end up in the money. Reverse the logic for puts. Volatility. Interest rates. But this effect is so tiny, it's safe to ignore. #4, Volatility, is the biggie. Everything else is known. That's why option trading is often considered ""volatility trading"". There are many ways to skin this cat, but the result is that by using quoted historical values for the stock price, and the dividend payments, and if you like, interest rates, you can very closely determine what the price of the option would have been. ""Very closely"" depending on your volatility assumption. You could calculate then-historical volatility for each time period, by figuring the average price swing (in either direction) for say the past year (year before the date in question, so you'd do this each day, walking forward). Read up on it, and try various volatility approaches, and see if your results are within a reasonable range. Re the Black-Scholes formula,  There's a free spreadsheet downloadable from http://optiontradingtips.com. You might find it useful to grab the concept for coding it up yourself. It's VBA, but you can certainly use that info to translate in your language of choice. Or, if you prefer to read Perl, CPAN has a good module, with full source, of course. I find this approach easier than reading a calculus formula, but I'm a better developer than math-geek :)" I understand that ITM have little time value, so they will have small time decay(theta), but why OTM has a lesser theta than ATM? The Time value represents uncertainty.  That uncertainty decreases the farther away from ATM you get (in either direction).  At-the-money, there is roughly a 50% chance that the option expires worthless. As you get deeper in-the-money, the change that is expires worthless decreases, so there is less uncertainty (there is more certainty that the option will pay off).  As you go deeper OTM, the probability that the option expires worthless increases, so there is also less uncertainty. At the TTM decreases, the uncertainty (theta) decreases as well, since there is less time for the option to cross the strike from either direction.  Similarly, as volatility decreases, theta decreases, since low-volatility stocks have a less change of crossing the strike. If we were to observe some call price (e.g., 15), and then derived implied volatilities from the BS formula depending on different strike prices but fixed maturity (i.e, maturity = 1, and strike goes from 80 to 140??), would we then see a smile? Yes. Market prices for various strikes and a given maturity often have higher implied volatilities from the Black-Scholes model away from at-the-money.   It is not accounted for in the Black-Scholes model in the fact that volatility is not a function of strike, so volatility is assumed to be constant across strikes, but the market does not price options that way. I don't know that a quantitative theory has ever been proven; I've always just assumed that people are willing to pay slightly more for options deep in or out of the money based on their strategy, but I have no evidence to base that theory on. Not really. This just shows you might be able to fill huge orders a little less cheaply now, but then somebody else gets to fill it at a better price and the HFT/Market Maker took on some risk for some profit. That's pretty much the definition of providing liquidity. How is this raising prices? It's not. Raising prices would be if HFT bought like 2mil worth of stock and just held it for some time, then tried to sell it to you later at a better price, but that's just investing. Options pricing is based on the gap between strike and the current market, and volatility. That's why the VIX, a commonly accepted volatility index, is actually just a weighted blend of S&P 500 future options prices.  A general rise in the price of options indicates people don't know whether it will go up or down next, and are therefore less willing to take that risk.  But your question is why everything underwater in the puts chain went higher, and that's simple: now that Apple's down, the probability of falling a few more points is higher. Especially since Apple has gone through some recent rough times, and stocks in general are seen as risky these days. On expiry, with the underlying share price at $46, we have : You ask :  How come they substract 600-100. Why ? Because you have sold the $45 call to open you position, you must now buy it back to close your position.  This will cost you $100, so you are debited for $100 and this debit is being represented as a negative (subtracted); i.e., -$100 Because you have purchased the $40 call to open your position, you must now sell it to close your position.  Upon selling this option you will receive $600, so you are credited with $600 and this credit is represented as a positive (added) ; i.e., +$600. Therefore, upon settlement, closing your position will get you $600-$100 = $500.  This is the first point you are questioning.   (However, you should also note that this is the value of the spread at settlement and it does not include the costs of opening the spread position, which are given as $200, so you net profit is $500-$200 = $300.) You then comment :  I know I am selling 45 Call that means : As a writer: I want stock price to go down or stay at strike. As a buyer: I want stock price to go up. Here, note that for every penny that the underlying share price rises above $45, the money you will pay to buy back your short $45 call option will be offset by the money you will receive by selling the long $40 call option.  Your $40 call option is covering the losses on your short $45 call option. No matter how high the underlying price settles above $45, you will receive the same $500 net credit on settlement.  For example, if the underlying price settles at $50, then you will receive a credit of $1000 for selling your $40 call, but you will incur a debit of $500 against for buying back your short $45 call.  The net being $500 = $1000-$500.  This point is made in response to your comments posted under Dr. Jones answer. "SELL -10 VERTICAL $IYR 100 AUG 09 32/34 CALL @.80 LMT 1) we are talking about options, these are a derivative product whose price is based on 6 variables. 2) options allow you to create risk out of thin air, and those risks come with shapes, and the only limit is your imagination (and how much your margin/borrowing costs are). Whereas a simple asset like the shares for $IYR only has a linear risk profile. stock goes up, you make money, stock goes down, you lose money, and that risk graph looks linear. a ""vertical"" has a nonlinear risk profile 3) a vertical is a type of ""spread"" that requires holding options that expire at the same time, but at different strike prices. 3b) This particular KIND of vertical is called a bear call spread (BCS). Since you are bearish (this makes money if the stock goes down, or stays in a very specific range) but are using calls which are a bullish options product. 4) -10 means you are selling the vertical. +10 means you are buying the vertical. A ""long"" vertical is initiated by buying an option closer to the money, and selling an option at a higher strike price. This would be +X A ""short"" vertical is initiated by selling an option closer to the money and buying an option at a higher strike price. The quantity would be -X 5) 32/34 stands for the strike prices. so you would be selling 10 call options at the 32 strike price, and buying 10 call options at the 34 strike price, both options expire in August 6) LMT stands for limit order, and $.80 is the limit order price that is desired. OPENING a vertical spread requires knowledge of options as well as how to send orders. MANAGING a vertical requires even more finesse, as you can ""leg-in"" and ""leg-out"" of spreads, without sending the entire order to the exchange floor at once. There is much to learn."