# Option premium boundaries. Minimum and Maximum Value of European/American Options

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Options Arbitrage As derivative securities, options differ from futures in a very important respect.

## Minimum and Maximum Value of European/American Options - Finance Train

They represent rights rather than obligations — calls gives you the right to buy and option premium boundaries gives you the right to sell. Consequently, a key feature of options is that the losses on an option position are limited to what you paid for the option, if you are a buyer.

Since there is usually an underlying asset that is traded, you can, as with futures, construct positions that essentially are riskfree by combining options with the underlying asset. Exercise Arbitrage The easiest arbitrage opportunities in the option market exist when options violate simple pricing bounds.

No option, for instance, should sell for less than its exercise value. In other words, you invest nothing today and are guaranteed a positive payoff in the future. You could construct a similar example with puts.

This lesson is part 2 of 6 in the course Option Valuation Lower Bound We know that the value of an option is equal to the sum of its intrinsic value and time value.

The arbitrage bounds work best for non-dividend paying stocks and for options that can be exercised only at expiration European options. Most options in the real world can be exercised only at expiration American options and are on stocks that pay dividends.

### CALL OPTION BOUNDARIES

Even with these options, though, you should not see short term options trading violating these bounds by option premium boundaries margins, partly because exercise is so rare even with listed American options and dividends tend to be small.

As options become long term and dividends become larger and more uncertain, you may very well find options that violate these pricing bounds, but you may not be able to profit off them. Replicating Portfolio One of the key insights that Fischer Black and Myron Scholes had about options in the s that revolutionized option pricing was that a option premium boundaries composed of the underlying asset and the riskless asset could be constructed to have exactly the same cash flows as a call or put option.

This portfolio is called the replicating portfolio. In fact, Black and Scholes used the arbitrage argument to derive their option pricing model by noting that since the replicating portfolio and the traded option had the same cash flows, they would have to sell at the same price.

Updated May 21, What are Boundary Conditions? Boundary conditions are the maximum and minimum values used to indicate where the price of an option must lie. Boundary conditions are used to estimate what an option may be priced at, but the actual price of the option may be higher or lower than what is set as the boundary condition. For all options contracts, the minimum boundary value is always zero, since options cannot be priced at negative money. Meanwhile, maximum boundary values will differ depending on the whether the option is a call or put, and if it is an American or European style option.

To understand how replication works, let us consider a very simple model for stock prices where prices can jump to one of two points in each time period. This model, which is called a binomial model, allows us to model the replicating portfolio fairly easily.

Since we know the cashflows on the option with certainty at expiration, it is best to start with the last period and work back through the binomial tree. Step 1: Start with the end nodes and work backwards. The value of the call therefore has to be the same as the cost of creating this position.

Since the cashflows on the two positions are identical, you would be exposed to no risk and make a certain profit. Again, you would not have been exposed to any risk.

You could construct a similar example using puts. The replicating portfolio in that case would be created by selling short on the underlying stock and lending the money at the riskless rate.

### Boundary Conditions

Again, if puts are priced at a value different from the replicating portfolio, you could capture the difference and be exposed to no risk. What are the assumptions that underlie this arbitrage? The first is that both the traded asset and the option are traded and that you can trade simultaneously in both markets, thus locking in your profits.

The second is that there are no or at least very low transactions costs. If transactions costs are large, prices will have to move outside the band created by these costs for arbitrage to be feasible.

## CALL OPTION BOUNDARIES Homework Help - Finance Assignment Help

The third is that you can borrow at the riskless rate and sell short, if necessary. If you cannot, arbitrage may no longer be feasible.

Arbitrage across options When you have multiple options listed on the same asset, you may be able to take advantage of relative mispricing — how one option is priced relative to another - and lock in riskless profits.

We will look first at the pricing of calls relative to puts and then consider how options with different exercise prices and maturities should be option premium boundaries, relative to each other. Put-Call Parity When you have a put and a call option with the same exercise price and the same maturity, you can create a riskless position by selling the call, buying the put and buying the underlying asset at the same time.

To see why, consider selling a call and buying a put with exercise price K and expiration date t, and simultaneously buying the underlying asset at the current price S. The payoff from this position is riskless and always yields K at expiration t.