Option Calculators and Stock Screeners
 Symbol Lookup

# chapter4/1 pricing an option

Chapter 4

#### Pricing an Option

A stock's price is generally considered to be the price of the last trade.

For options, the price of an option contract can change even if the option itself does not trade. Option prices will change for various reasons including a change in the underlying stock price, the passage of time, or even changes in the expected future value of the underlying stock.

The theoretical price is the price computed by plugging the strike, expiration, volatility, interest rate and cost to carry into a model such as Black Scholes. It is the price, in theory, which the option should trade based on the historical movement of the underlying stock.

The parameter that is most subjective when pricing an option is the volatility. There are many methods for deriving the volatility. The simplest method uses the standard historical volatility at a predetermined time interval (e.g. 1 year, 90, 60, or 30 day volatility). Another approach is to imply the volatility from other similar, actively trading option contracts with the same underlying.

Is it important to understand the concept of put/call parity. Calls and puts should have equal time premium regardles of whether the stock is rising or falling.

The probability of a stock's price moving up is equal to it moving down.

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.