Option pricing: fundamentals

By roma, 16 August, 2020

Until 1970, there was no single method for pricing an option. The value of option was mainly based on market sentiment. In this situation, traders who predicted an increase in the share price preferred call options more than traders who had a negative outlook on the underlying asset. Some market players used their own rules based on observations. For example, the price of at-the-money options was calculated as a percentage of the price of the underlying asset.

The first step in understanding option models is to study the fundamental relationship between the option price before the expiration date and the price of the underlying asset. This article explores this relationship by testing the regularities applicable to European options that do not pay dividends.

 

Rule 1

The call option can never cost more than the underlying asset.

c ≤ S 

c - call option price
S - spot price of the underlying asset

If rule 1 is not followed, the trader would have an opportunity to earn risk-free profit. For example, a trader could buy a share for $100 and sell a call option on that share for $110. Thus, the trader owns the share and is fully protected regardless of whether this option is exercised or not. In this case, the trader will always earn profit regardless of the direction of the share price.

 

Rule 2

The minimum value of a European call option per share that does not pay dividends is either zero or the difference between the spot price of the underlying and the discounted strike value.

c ≥ max (0; S - Xe-rt)

Regarding the first part of the statement, the price of an option cannot drop below zero, even if the option is far out of the money. To check the second part of the law, let's consider in-the-money call with a strike of $130 and an expiration date in a year. The spot price of the underlying asset is $140 and the annual risk-free interest rate is 10%. Rule 2 states the following:
Minimum option price = $140 – $130/1.1 = $21.82

Let's say that the call option with a strike of $130 is traded on the exchange at $15. Then the trader will build an arbitrage strategy as follows:
- Short position on the stock for $140
- Buy a call option with a strike price of $130 at $15

The rest ($140 - $15 = $125) is deposited for one year at 10%.

In one year the deposit account will be $137.5. If after one year the stock is worth $150, the call option is exercised and the trader buys the stock for $130 and earns $137.5 – $130 = $7.5, as it is necessary to close a short position. If the share price drops to $120, the value of the option will go down to zero and the profit will be $137.5 – $120 = $17.5.

It turns out that when buying an option and selling a stock, the strategy always brings profit, which contradicts the laws of statistics. The catch is the option premium equal to $15. If the option is worth $21.82 or more, the payout of the strategy will bring either 0 or losses. Therefore, a call option must cost at least $20, which is the difference between the spot price of the underlying and the discounted strike value.

 

Rule 3

There is a link between European call and put options, which have the same strike price and expiration dates, which market participants and academics call put-call parity.

p + S = c + Xe-rt

p - put option price
c - call option price
S - spot price of the underlying asset
X - strike price

To check this formula, we will compare call and put options with a strike of $80 and maturity in one year. The stock's spot price is $90, the call option price is $25, and the annual risk-free interest rate is -10%. Discounted strike price is $72.73. What is the fair value of a put option?

Let's look at two portfolios:

Portfolio A:

Put with strike $80 + 1 stock

Portfolio B:

Call with strike 80 + deposit $72,73

Two portfolios have similar payouts at the end of the option one year later.

Scenario 1

For example, if the share price fell to $70 a year later, the value of portfolio A is $80:
$10 (put option price) + $70 (share price) = $80
While the value of portfolio B also equals $80:
$0 (call option price) + $80 (deposit at 10% p.a.) = $80

Scenario 2

If the share price rises to $110 in a year, then portfolio A is worth $110:
$0 (put option price) + $110 (one share price) = $110

The value of portfolio B is also $110: $30 (call option payout) + $80 (deposit at 10% p.a.).

The same value of two portfolios in all scenarios leads to the conclusion: the value of the portfolios should be equal at the moment of their formation.

The current value of portfolio B is $97.73: $25 (call option price) and $72.73 (deposit). Therefore, portfolio A should also be worth $97.73 today. Since the spot share price is $90, the put option price is $7.73. The put-call parity offers a similar result.

p + S = c + Xe-rt
p + $90 = $25 + $72,73
p = $7,73

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