# Valuation of options

In option pricing theory, there are basically two approaches to determine the fair option price:

- With the help of estimates without making assumptions about possible future share prices and their probabilities ( distribution -free no-arbitrage relations, content of this article)
- Due to possible stock prices and risk-neutral probabilities. These include the binomial model and the Black- Scholes model

In the distribution-free no-arbitrage relations it comes to using no-arbitrage arguments, to find bounds for call and put value. For the determination of no-arbitrage relations (without assumption of a particular distribution ), it is assumed that the available instruments stocks, zero coupon bonds and call and put options are different series.

- The options are not dividend- protected.
- It is frictionless markets, that is, there are no room charges; it is abstracted from a buy-sell spread.
- There is no tax effect arising from differences in taxation of corporate level and the investor level.
- Short sales are possible borrowing rate equals interest earned.
- There is no risk of default.

## Upper and lower bounds

### Calls

The uncertain value of a European call at the exercise date can not be greater than the value of the stock; because the call includes the right to buy the stock at a predetermined price. This relation in the time of exercise must also apply at the initial instant. So the current share price is greater than the call price.

A European call is at least as large as the share price (before dividend payments ) minus the discounted strike price and the discounted dividend. The call can never have a negative value; it is a right without duties ( limited liability ).

An American option is at least as much value as a European call and the difference between the current stock price and the strike price, he could already be applied today.

### Puts

A European put is not worth more than the discounted strike price, an American put no more than the base price.

A European put is at least as large as the discounted strike price minus the stock price, plus the discounted dividend. The Putwert is at least zero.

An American put is at least as much value as a European put and the difference between the strike price and the current share price.

## Estimates depending on the strike price

Monotony in the basic price

A ( European ) call option for a stock (call) with a lower strike price is more expensive than an otherwise completely identical option with a higher strike price. A call is the right to buy a stock at the previously agreed strike price. This right is all the more valuable, the " cheaper " option holder may purchase the share ( higher intrinsic value, that is, the difference between the current share price and the strike price. This is also true for put options for a stock (Put ), with a higher strike price a higher value implies.

Option value difference

In addition, a statement may be on the value of acceptable options based on the difference between the exercise prices (higher minus lower ) make. This is larger than the difference of the call with the lower strike price and the call with the higher strike price in the case of call options. In the case of puts the difference between the exercise price is less than the difference of the puts ( lower with higher minus exercise price).

Convexity in the exercise price

A combination of two calls ( or puts) with different strike prices is more expensive than an option with the average base price of the two weighted options. Option strategy that can be formed in this context, the butterfly spread.

## Estimates as a function of the option period

We must distinguish between American and European options.

An American option with a longer maturity is at least as valuable as a corresponding call with a shorter term. The right one shares at any time to buy a submitted exercise price, the higher the value, the longer this right can be exercised. The reverse is true for puts.

For European options then we have to distinguish whether and when a dividend is paid. Here volatility effects and interest rate effects are observed:

- A European call with a longer maturity is worth more than a call with a shorter term, if the record date outside the interval between the two exercise dates is.
- If the dividend date, however, between the two exercise dates, no definitive statement can be made. The amount of the dividend determines the dominating effect.

- In the case of puts, it is even possible that the longer-term put is worth less than the short-term. This is dependent on the current price of the stock. If this is greater than the strike price as a longer running put is rewarding.
- If, however, the current share price is much less than the strike price, the put is so "deep in the money" so the relation is possible due to stronger discounting: In extreme cases, the stock price is zero. If the put exercised at an earlier time, it must not be so heavily discounted. The proceeds shall not be increased. So true in this case that the put with a longer maturity is worth less.
- This is however only an estimate based on today known data. The opposite does not automatically mean that an exercise would be optimal.

## Relationships between call and Putwerten

There are European put and call options considered with the same underlying, strike price and maturity. Substituting calls and puts to hedge one a stock position ( short call, long put, equity long) can be in the case of European options derive the put-call parity. This is based on the law of one price. This relationship was described by Hans Stoll (1969, Journal of Finance ) for the first time.

The statement is the following: A European put has the value of a portfolio of European call minus the current stock price, plus the T periods discounted base price.

Summary

## Arbitrage strategies

In trading is the buying price (Ask) equal to the selling price (bid ). Thus one of the above assumptions is resolved. Arbitrage opportunities arise in the following situations:

- Option trading
- Capital Market Theory