Option price theory

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

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

The distribution-free no-arbitrage relationships it comes through no-arbitrage to find -Argumenten bounds for call and put value. For the determination of no-arbitrage relationships (without the requirement of a specific distribution) it is assumed that the available instruments are stocks, zero bonds as well as calls and puts of different series.

• The options are not protected against dividends .
• These are frictionless markets, i. i.e. there are no fees; it is abstracted from a buy-sell spread.
• There is no tax effect resulting from different taxation at company level and investor level.
• Short sales are possible, debit interest equals credit interest.
• There is no risk of failure.

Upper and lower bounds

Calls

Upper bounds

The uncertain value of a European call at the time of exercise cannot be greater than the value of the share; because the call includes the right to buy the stock at a predetermined price. This relationship at the time of exercise must also apply at the start time. So the current share price is greater than the call price.

Lower bounds

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

An American call is worth at least as much as a European call and the difference between the current share price and the strike price, it could be exercised today.

Puts

Upper bounds

A European put is no more than the discounted strike price, and an American put is no more than the strike price.

Lower bounds

A European put is at least as large as the discounted strike price minus the share price plus the discounted dividend. The put value is at least zero.

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

Estimates depending on the base price

Monotony in the base price

A (European) call option for a share ( 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 share at the pre-determined strike price. This right is worth the more, the “cheaper” the option holder can acquire the share (higher intrinsic value, i.e. the difference between the current share price and the base price). This also applies to put options for a share ( put ), where a higher strike price implies a higher value.

Option value difference

In addition, a statement can be made about value limits for options based on the difference in exercise prices (higher minus lower). In the case of call options, this is greater than the difference between the call with the lower exercise price and the call with the higher strike price. In the case of puts, the difference between the exercise prices is smaller than the difference between the puts (with a higher minus lower exercise price).

Convexity in the strike price

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

Estimates depending on the option period

A distinction must be made here between American and European options.

An American call with a longer term is worth at least as much as a corresponding call with a shorter term. The right to buy a share at any time at a given exercise price is worth the more, the longer this right can be exercised. The reverse is true for puts.

In the case of European options , a distinction must be made between whether and when a dividend is paid. Here, volatility effects and interest rate effects must be taken into account:

• A European call with a longer term is worth more than a call with a shorter term if the dividend date is outside the interval between the two exercise dates.
• However, if the dividend date is between the two exercise dates, no definitive statement can be made. The amount of the dividend determines the dominant effect.

• In the case of puts, it is even possible that the longer-running put is worth less than the short-term one. This depends on the current price of the share.
  • If this is greater than the base price, a longer put is more worthwhile.
  • If, on the other hand, the current share price is much lower than the base price, i.e. if the put is “deep in the money”, the relationship is possible due to stronger discounting: In the extreme case, the share price is zero. If the put is exercised at an earlier point in time, it does not have to be discounted as much. The proceeds cannot be increased. So in this case, the longer term put is worth less.
  • However, this is only an estimate based on data known today. The opposite does not automatically mean that exercising would be optimal.

Relationships between call and put values

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

The statement is as follows: A European put has the value of a portfolio of European calls minus the current share price plus the base price discounted T periods.

Summary

${\displaystyle C_{0}\geq S_{0}-E\ }$

${\displaystyle C_{0}(E_{1})\geq C_{0}(E_{2})\quad {\text{mit}}\quad E_{1}<E_{2}}$

${\displaystyle E_{2}-E_{1}\geq C_{0}(E_{1})-C_{0}(E_{2})\ }$

${\displaystyle C_{0}(E_{1})+C_{0}(E_{3})\geq C_{0}(E_{2})\quad {\text{mit}}\quad E_{2}=E_{1}+E_{3}}$

${\displaystyle C(T_{1})\leq C(T_{2})\quad {\text{mit}}\quad T_{2}>T_{1}}$

Arbitrage strategies

In trading, the buy price (ask) is not the same as the sell price (bid). This resolves one of the above assumptions. Arbitrage opportunities arise in the following situations:

${\displaystyle C_{\mathrm {ask} }<S_{\mathrm {bid} }-E}$

${\displaystyle C_{\mathrm {ask} }(E_{1})<C_{\mathrm {bid} }(E_{2})}$