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[Nazariy]

Not sure whether this qualifies as maths but I will try,

If you have a call and a put with the same strike, ceteris paribus, which one is more expensive?

 

[HallsofIvy]

Not sure whether this qualifies as maths but I will try,

If you have a call and a put with the same strike, ceteris paribus, which one is more expensive?

For those who are not familiar with the stock market, a "put" is an option to sell a specified stock at a given price before a certain date. A "call" is the right to buy a specified stock at a given price before a certain date. If you have reason to believe a stock is going to rise sharply in price in, say, the next week, you could purchase some of that stock- but if you were wrong, and the price of the stock fell, you might lose a large amount of money. You could, instead, buy a "call" allowing you to purchase that stock, at the current price, some time in the future. Then if the price of the stock rises, you "exercise the call", buying at the old, lower price and immediately selling at the new, higher price. On the other hand, if the price falls or simply does no rise high enough, you simply do not exercise the call, losing only the cost of the option.

On the other hand, if you believe a stock is going to fall in price, you can buy a "put" which would allow you to sell at today's price in the future. If the stock does fall, you exercise the put, buying at the new, lower, price and immediately selling at the old, higher price, making a profit even though the price has dropped. If the price does not fall, do not exercise the option, again losing only the cost of the option.

Of course to buy a "call" there has to be someone willing to sell it to you. That is, someone willing to sell you at today's price in the future. Of course, that person also wants to make a profit so he must believe the stock price will fall, or at least not rise more than would cover the cost of the call. The more likely the stock price is to rise, the more a call will cost.

Here, the question is "Suppose a stock is, in everyone's estimate, equally likely to rise or fall (essentially what "certeris paribus" means- all other things being equal), should it cost more to buy a put or a call?" Suppose the price of a stock is currently $100. If I buy a "call" for x dollars and the stock goes up to $102, I have made $2, minus the amount of the call, 2- x, for each share. That is a profit of \(\displaystyle \frac{2- x}{100}\) percent. On the other hand, if I buy a put for y dollars and the stock drops to 98$, I will still have made a profit of $2 minus the cost of the put, 2- y, on the sale, for a profit of \(\displaystyle \frac{2- y}{98}\) percent.

Notice the difference in denominators. That is the amount I actually had to pay for the stock when I exercised the call or put. To make the same profit either way, we must have \(\displaystyle \frac{2- x}{100}= \frac{2- y}{98}\). That is equivalent to 98(2- x)= 100(2- y), 196- 98x= 200- 100y,
100y= 98x+4, so y= .98x+ .04. y will be larger than x if and only if .98x+ .04> x or .04> .02x, x< 2. Of course, that "2" is the amount the stock rose or fell. You certainly won't buy a "call" for more than you think the stock will go up so generally, y, the cost of a put, will be more than x, the cost of a call.

 

[Nazariy]

For those who are not familiar with the stock market, a "put" is an option to sell a specified stock at a given price before a certain date. A "call" is the right to buy a specified stock at a given price before a certain date. If you have reason to believe a stock is going to rise sharply in price in, say, the next week, you could purchase some of that stock- but if you were wrong, and the price of the stock fell, you might lose a large amount of money. You could, instead, buy a "call" allowing you to purchase that stock, at the current price, some time in the future. Then if the price of the stock rises, you "exercise the call", buying at the old, lower price and immediately selling at the new, higher price. On the other hand, if the price falls or simply does no rise high enough, you simply do not exercise the call, losing only the cost of the option.

On the other hand, if you believe a stock is going to fall in price, you can buy a "put" which would allow you to sell at today's price in the future. If the stock does fall, you exercise the put, buying at the new, lower, price and immediately selling at the old, higher price, making a profit even though the price has dropped. If the price does not fall, do not exercise the option, again losing only the cost of the option.

Of course to buy a "call" there has to be someone willing to sell it to you. That is, someone willing to sell you at today's price in the future. Of course, that person also wants to make a profit so he must believe the stock price will fall, or at least not rise more than would cover the cost of the call. The more likely the stock price is to rise, the more a call will cost.

Here, the question is "Suppose a stock is, in everyone's estimate, equally likely to rise or fall (essentially what "certeris paribus" means- all other things being equal), should it cost more to buy a put or a call?" Suppose the price of a stock is currently $100. If I buy a "call" for x dollars and the stock goes up to $102, I have made $2, minus the amount of the call, 2- x, for each share. That is a profit of \(\displaystyle \frac{2- x}{100}\) percent. On the other hand, if I buy a put for y dollars and the stock drops to 98$, I will still have made a profit of $2 minus the cost of the put, 2- y, on the sale, for a profit of \(\displaystyle \frac{2- y}{98}\) percent.

Notice the difference in denominators. That is the amount I actually had to pay for the stock when I exercised the call or put. To make the same profit either way, we must have \(\displaystyle \frac{2- x}{100}= \frac{2- y}{98}\). That is equivalent to 98(2- x)= 100(2- y), 196- 98x= 200- 100y,
100y= 98x+4, so y= .98x+ .04. y will be larger than x if and only if .98x+ .04> x or .04> .02x, x< 2. Of course, that "2" is the amount the stock rose or fell. You certainly won't buy a "call" for more than you think the stock will go up so generally, y, the cost of a put, will be more than x, the cost of a call.

An extensive read! What do you think about forward curves? Say the forward curve is in backwardation, I would assume that in that case put would be priced higher. If however forward is in contango, reverse is true and call is more expensive (both with same strikes). I think this is a valid point. However, I am interested whether there are other reasons.

I was also thinking from the persepctive that if you are long put, your upside is capped, whereas if you are long call it is not. However, the reverse is true if you are short the vanillas, if you are short call your downside is unlimited, if you are short put your downside is capped. So it is unclear from such reasoning whether put or call with the same strike is more expensive.