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Q: Why didn't my option move as much as the underlying stock?
A: Options will not move as much as their underlying stock unless they are in-the-money and/or very close to expiration. There are valid mathematical reasons for this. The amount an option can be expected to move (all other conditions being equal) given a 1-point move in the underlying stock is called delta. Delta is derived from the Black-Scholes formula for pricing options and represents roughly how much the option behaves like the underlying stock. A delta of .50, for example, means that an option can be expected (all other things being equal) to move about fifty cents for every $1 move in the underlying stock. Delta will change with time to expiration as the option moves more in- or out-of-the-money, and will also be affected by the volatility of the underlying stock.
Q: When an underling security (e.g. QQQ) is up (e.g. +1.36), why do some call options actually fall in value for the day?
A: There are 6 inputs that determine an option's value: stock price, strike price, time to expiration, interest rate, dividend yield and volatility (over the life of the option). Normally, if the stock price goes up and the other factors remain the same then a call option also goes higher. So if the call option has gone down, then one of the other factors must have changed.
The passage of time can certainly push an option's value lower, although sometimes it can have the opposite effect. A dividend payment will also have an impact. But the real wild card is volatility. Sometimes, the market will bid up the volatility in anticipation of a market-moving event such as earnings release or a major speech by a Very Important Person. After the event, the volatility often drops sharply, especially if the event failed to have the expected impact.
Q: I'm curious why when interest rates rise, option prices also rise? This seems odd since stock prices drop in this situation.
A: I believe youa re asking why a CALL option price rises when interest rates rise. That is because options are priced on a risk-neutral basis, i.e. on a delta-neutral or fully hedged basis. So a long call would be paired with a short-stock, and a short-stock position generates interest revenue. That makes the call option worth more. If interest rates go up, the interest revenue from the short stock position increases, which makes the call worth still more. Note that for put options it works the opposite way. Dividends also work in the opposite direction.
A stock's value is equal (theoretically) to the present value of all its future dividends, so an increase in the interest rate used to discount the future dividends will reduce the value of the stock. When someone says higher interest rates make call options worth more, there is an implicit assumption of ALL OTHER THINGS BEING EQUAL. As a practical matter, all other things are rarely equal, and the decline in a stock's price due to an interest rate increase will often overwhelm the effect of the higher interest rate on the option itself.
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