# Call price and put relationship

### Arbitrage Strategies and Price Relationships

We examine the asymptotic behaviour of the call price surface and the associated Black-Scholes implied volatility surface in the small time to expiry limit under. shown that both American call and put options have values larger than their This paper develops pricing relationships for European and American call and put. Prior to learning the relationships between call and put values, we'll review a The call and put would have the same strike price and the same expiration.

Because you have sold the option, which has now decreased in value your short option position has benefited from an upward move in the underlying asset.

Due to the association of position delta with movement in the underlying, it is common lingo amongst traders to simply refer to their directional bias in terms of deltas.

Example, instead of saying you have bought put options, you would instead say you are short the stock. Because a downward movement in the stock will benefit your purchased put options.

**Put-call parity - Finance & Capital Markets - Khan Academy**

Hedge Ratio Option contracts are a derivative. This means that their value is based on, an underlying instrument, which can be a stock, index or futures contract. Call and put options therefore become a sort of proxy for long or short position in the underlying.

Buying a call benefits when the stock price goes up and buying a put benefits when the stock price goes down. However, we know now that the price movement of the options doesn't often align point for point with the stock; the difference in the future movement being the delta. The delta therefore tells the trader what the equivalent position in the underlying should be. For example, if you are long call options showing a delta of 0.

To make the comparison complete, however, you need to consider the option contract's "multiplier" or contract size. To read more on using the delta for hedging please read: This page explains in more detail the process of delta neutral hedging your portfolio and is the most common of the option strategies used by the institutional market.

You have a synthetic long stock position in the August options and a synthetic short stock position in the December options.

You are just spreading one month versus the other. If you are dealing with futures options, where these option months would be based on different futures month, then you are essentially just trading one month's value versus another. But when you are looking at a stock, this opens up some interesting issues. Think about what this position actually is. Once you reach the August expiration, you will be left with a long position in the stock. If the stock price moves up, the call option will be in-the-money and you would exercise it.

If the stock price falls, the put will be in-the-money and you will be assigned. Either way, you will be long the stock. The December position is just the opposite—you would be short the stock after expiration. Once you reach this date, the short position would, of course, cancel out the long position. In a way you could say the jelly roll is a position where you arrange to be long the underlying at some point in the future and then hold it for a short period only. Now we need to think about how the position should be priced.

It all has to do with carrying costs and dividends.

## Long and Short of Option Delta

Carrying costs refers to the cost of holding the stock for the four-month period from August to December. With the current risk-free interest rate at about 1. If it is less there would be an arbitrage opportunity.

You can see how a change in just one of the component options would provide the smart traders out there with an opportunity. As they began to take advantage of it, they would drive prices back in line. So any deviation from "fair value" would not last very long. Keep in mind that the reason we are looking at this strategy is not for you to turn around and trade it.

### Interpreting Options Price and Open Interest Relationship

It is to simply understand how the option prices are related. It is easy to see the effect transaction costs can have. You simply need to make sure that the profit from any of these strategies is greater than your transaction costs. Since is worthwhile to do the trade only if you can at least cover your transaction costs, options prices can thus be out of line and stay that way even in the long term by the amount of those transaction costs.

Dividends are a little more complicated, but still pretty easy to factor in. Let's look at an example of a reversal on a stock that pays dividends. You would need to reinvest the proceeds from the short sale of stock at a sufficiently high interest rate to cover any dividends, and still have more than enough to pay the strike price on the expiration date of your synthetic long position.

Thus you need to pay a bit less for your calls and sell your puts for a little higher price. For the jelly roll, if the stock pays a dividend during the holding period August to December in the above example it would change the relative value of the position in each expiration month. The impact would be to lower the fair value of the jelly roll by the amount of the dividend.

Increasing dividends will boost the value of the front month options and reduce the value of the back month options, reducing the cost of the roll. I would also like to mention again the importance of correct dividend information when modeling these arbitrage strategies on a stock that pays dividends.

After researching and inputting the correct upcoming dividends, those apparent opportunities will disappear. Conclusion I have tried to demonstrate all the relationships between the various securities — the underlying stock, the call options and the put options, as well as the different expiration months — using various arbitrage strategies to explain the fundamental relationships that underlie their pricing.

If you are not on the trading floor, the probability of finding one of these arbitrage trades is very small. Off-floor traders can look for undervalued opportunities, but with electronic trading there are thousands of other traders, often using specialized computer software applications, watching market quotes for bargains. That has reduced the life of a mispriced quote to just seconds, often less than one second.

The short life of these opportunities means that if you find one you need to jump on it immediately. That can be dangerous because it doesn't leave you time to do the thorough research you should to see what is causing g those "mispriced" options.

Chances are you will find many apparent opportunities are simply due to a stale or incorrect quote. Keep in mind the danger of incorrect data such as dividends and how incomplete information can make a fairly valued situation look like a risk-free opportunity. Simply leave the unknown variable as 0 and it will automatically be calculated by the program.

- American call options
- Put price when maturity tends to infinity
- Interpreting Options Price and Open Interest Relationship

Do note that only one unknown variable is supported at one time. Binomial Option Pricing For many years, financial analysts have difficulty in developing a rigorous method for valuing options. This model is famously known as the Black Scholes model. The model has a name "Binomial" because of its assumptions of having two possible states. Basically, the Binomial Option Pricing and Black Scholes models use the simple idea of setting up a replicating portfolio which replicates the payoff of the call or put option.

The value of the portfolio is then observed to be the value or cost of the options. Binomial Option Pricing - Call Option This worksheet sets up a replicating portfolio by borrowing money at the risk free rate and purchasing an amount of the actual stock to replicate the payoff of the Call Option. It then calculates the value price of the Call Option through observing the value of the portfolio.