1. pricing an option

Pricing an Option

Whether or not you choose to believe this, option pricing models are founded on this premise. The result is that at the money put and call prices for the same stock with the same expiration should trade at the same price.

A call holder profits if the underlying stock price rises, without the risk of downside but pays a premium. A stock holder, on the other hand, profits if the underlying stock price rises but loses when the stock price falls. The stock holder can protect against losses by purchasing a put. The holder of the stock and long put profits when the stock rises without the risk of downside but pays a premium. Sounds just like holding a call? These positions are synthetically equivalent.

	STOCK = CALL - PUT	(a long stock position equals a long call and short put)
	CALL = STOCK + PUT	(a long call position equals a long stock and long put)
	PUT = CALL - STOCK	(a long put equals a short stock position and long call)
	-PUT = STOCK - CALL	(a covered call is equivalent to short put)
	-CALL = -STOCK - PUT	(a covered put is equivalent to short call)

A trader wishing to hold a stock and insure against loss with a long put need not buy the stock and put at all, just buy a call; the risk profile is the same, the commissions are usually cheaper, and much less capital is required. Buy the call, invest the remaining capital.

If the above relationships do not hold true, then a profit can be made by taking a long position on one side of the equation and shorting the other. For example, AAPL is trading at $50 and the three month at the money call is trading at $4. The three month at the money put should trade at $4 as well, but what if were trading at $5 (you will be hard pressed to find such a discrepancy in the market). One could guarantee a profit if they were to:

• Sell the put for $5
• Buy the call for $4
• Short the stock.

The proceeds, referred to as the net credit, from the option transactions are $1. The short stock transaction covers the potential loss of the short put, and the net credit from the sale of the put more than covers the cost of the call. The position will not change in value as the stock price moves, and the $1 profit is locked in. This arbitrage rarely occurs with enough margin to be profitable for the typical public trader, since prices are electronically monitored and professional traders will trade these positions back to parity in return for very small, but risk free, profits.

Puts and calls on the same stock are priced similarly, and at the money puts and calls will trade at the same price. This is known as put-call parity.

There are differences in holding a stock versus its synthetic equivalent. Call holders do not collect dividends, but the call will be discounted somewhat to account for the pending decline in the stock price due to a dividend payout. Similarly, put holders do not pay dividends and puts have additional premium over a call when a dividend is pending. Also, option holders have no voting rights.

Synthetic equivalents can be used to lock in the price of a stock over time, or capitalize on dividends when it is advantageous to collect dividends over capital gains. Depending on your tax situation, taxes on dividends may be lower than taxes on capital gains. A trader may take advantage of this by buying stock that pays a large dividend, buying the put to protect against loss, and selling an in the money call to pay for the put. The position will lose value when the dividend is paid, which may be treated as a short term capital loss and used to offset other capital gains. The dividend will compensate for the loss and be taxable at a different, possibly lower, rate. Stocks that go ex-dividend near the end of the year may offer the added advantage of paying the dividend in the next calendar year, which can defer the tax liability. A trader realizing a capital loss typically does not wish to receive dividends in lieu of gains, since dividends will always be taxable and capital loss deductions are limited. Check with your tax advisor before entering such a trade.