Properties of Stock Options

After completing this reading, you should be able to:

  • Identify the six factors that affect an option’s price and describe how these six factors affect the price for both European and American options.
  • Identify and compute upper and lower bounds for option prices on non-dividend and dividend-paying stocks.
  • Explain put-call parity and apply it to the valuation of European and American stock options.
  • Explain the early exercise features of American call and put options.

Six Factors Affecting Option Prices

There are six factors impact the value of an option:

\(S\) = current stock price

\(X\) = strike price of the option

\(T\) = time to expiration of the option

\(r\) = short-term risk-free interest rate \(over\) \(T\)

\(D\) = present value of the dividend of the underlying stock

\(\alpha\) = expected volatility of stock prices over \(T\).

Current stock price

The value of all call options increases (decreases) as \(S\) increases (decreases). For put options, the value of the put decreases (increases) as \(S\) increases (decreases).

The strike price of the option

For call options, the value decreases (increases) as the strike price increases (decreases). For put options, the value increases (decreases) as the strike price increases (decreases).

Time expiration

With American style options as the time to expiration increases, the value of the option increases. With more time, there are higher chances of the option moving in-the-money.

  • As the time to expiration increases, the value of a call option increases.
  • As the time to expiration increases, the value of a put option also increases.

However, the same does not apply to European-style options, precisely when the underlying has scheduled dividends. For example, assume we have a two-month call option and a four-month call with the same exercise price \(K\), and the same underlying stock. Assume further that a sizeable dividend is expected in three months. The ex-dividend stock price and call price will decrease. As such, the two-month call could actually be more valuable than the four-month call.

Risk-free Rate over the Lifespan of the Option

Here, the simplest way to think about this is as a rate of return on a stock. Let’s say you have the choice between buying a bond worth $1000 or one share of stock priced at $1000. If you know the risk-free rate of interest is a known 5%, you would expect the stock price to increase by more than 5% on average. Otherwise, why would you buy a share of stock instead of investing in a risk-free bond? Therefore,

  • As the time the risk-free rate increases, the value of a call option increases.
  • However, as the risk-free rate increases, the value of a put option decreases.

Dividends

Payments from an underlying may include dividends. As we’ve seen previously, immediately after payment of a dividend the stock price falls by the amount of the dividend. However, the benefits of these cash flows to the holders of the underlying security do not pass to the holder of a call option. Therefore,

  • As dividends increase, the value of a call option decreases.
  • However, as dividends increase, the value of a put option decreases.

Expected volatility of stock price over time

Volatility is considered the most significant factor in the valuation of options. As volatility increases, the value of all options increases. Since the maximum loss for the buyer of a call or put option is limited to the premium paid we can conclude that as volatility increases, there are higher chances of the option expiring in-the-money.

  • As volatility increases, the value of a call option increases.
  • As volatility increases, the value of a put option increases.

Option pricing Bounds

Let:

c = value of a European call option.

C = value of an American call option.

p = value of a European put option.

P = value of an American put option.

\({ S }_{ T }\) = value of the stock at expiration.

A call option gives the holder the right to buy the stock at a specified price. The value of the call is always less than the value of the underlying stock. Thus,

$$ c\le { S }_{ 0 }\quad and\quad C\le { S }_{ 0 } $$

If the value of a call were to be higher than the value of the underlying stock, arbitrageurs would sell the call and buy the stock, earning an instant risk-free profit in the process.

A put option gives the holder the right to sell the underlying stock at a specified price. The value of a put is always less than the strike price. Thus,

$$ p\le K\quad and\quad P\le K $$

If the value of a put were to be higher than the strike price, everyone would move swiftly to sell the option and then invest the proceeds at the risk-free rate throughout the life of the option.

European options can only be exercised at expiration. As such, the value of a European put is always less than the present value of the strike.

$$ P\le K{ e }^{ -tT } $$

Lower pricing bounds for European calls on nondividend-paying stocks

Call options can never be worth less than zero as the call option holder cannot be forced to exercise the option. The lowest value of a call option has a price which is the maximum of zero and the underlying price less the present value of the exercise price. This is written as follows:

$$ c+{ e }^{ -tT }\ge { S }_{ 0 } $$

$$ c\ge max\left( { S }_{ 0 }-K{ e }^{ -tT },0 \right) $$

Lower pricing bounds for European puts on nondividend-paying stocks

A put option has an analogous result. A put option can never be worth less than zero as the option owner cannot be forced to exercise the option. The lowest value of a put option is the maximum of zero and the present value of the exercise price less the value of the underlying. This is expressed as follows:

$$ p+{ S }_{ 0 }\ge K{ e }^{ -rT } $$

$$ p\ge max\left( K{ e }^{ -rT }-{ S }_{ 0 },0 \right) $$

Put-call Parity In European Options

Put call parity states that the price of a call option implicitly informs a certain price for the corresponding put option with the same strike and expiration, and vice versa.

Consider the following portfolios:

  • Portfolio \(A\): One call option plus an amount of cash equivalent to \({ e }^{ rT } K\)
  • Portfolio \(B\): One put option plus one share

Since the options are European, they cannot be exercised prior to maturity. Thus, put-call parity demands that the value of the two portfolios today is the same. Expressed mathematically,

$$ C+K{ e }^{ -rT }=P+{ S }_{ 0 } $$

Where:

\(C\) = value of call option

\(K\) = strike price

\(P\) = value of put option

\({ S }_{ 0 }\) = initial stock price

Example

On the expiration date, the put-call parity is now

 $$ C+K=P+{ S }_{ T } $$

because we do not have to use the present value of the bond.

Let’s say you own a stock trading at $100 and own the put option with an expiration price of $90.

Let’s now look at what happens to this two-asset portfolio if the prices at expiration are $80, $89, $110 or $130.

Exp Price

$80

$89

$110

$130

Stock

$80

$89

$110

$130

Put Option

$10

$1

$0

$0

Portfolio

$90

$90

$110

$130

As you can see from the table above, when you own a put and a stock, you have what is called a protective put. The price can never get below the price floor ($90 in our example), but you still have unlimited profit on the upside.

Now, let’s say you own a call with an expiration price of $90 and own a zero-coupon, risk-free bond that matures for $90. 

With the same expiration prices as the previous table, we now have:

Exp date Price

$80

$89

$110

$130

Call Option

$0

$0

$20

$40

Bond

$90

$90

$90

$90

Portfolio

$90

$90

$110

$130

As we can see from the two tables, the portfolio value at expiration for the same expiration prices is the same whether we own the stock plus the put or the call plus the risk-free bond.

However, before the expiration date, we have to discount the present value of the bond so the put-call parity is:

$$ C+K{ e }^{ -rT }=P+{ S }_{ 0 } $$

Applying put-call parity

Let’s now use an example to illustrate the put-call parity and see how we could exploit arbitrage opportunities in the options market

Example

A stock currently sells for \($51\). A 3-month call option on the stock, with a strike price of \($50\), has a price of \($5\). Assuming a \(10\%\) continuously compounded risk-free rate, determine the price of the associated put option.

$$ C+X{ e }^{ -rT }=P+{ S }_{ 0 } $$

Making \(P\) the subject,

$$ P=C+X{ e }^{ -rT }-{ S }_{ 0 } $$

$$ =5+50{ e }^{ -0.1\times 0.25 }-51 $$

$$ = 2.77 $$

If \(P\) is greater than or less than 2.77, there will be arbitrage opportunities.

For example, assume \(P = 3.50\). The following arbitrage opportunities would present themselves:

  1. Buy call for \($5\)
  2. Short Put to realize \($3.50\)
  3. Short the stock to realize \($51\)
  4. Invest \($49.5\left( =51+3.50–5 \right) \) for 3 months, making \($50.75(=49.5{ e }^{ 0.1\times 0.25 })\)

Let \({ S }_{ T }\) be the price of the stock at expiry.

If \({ S }_{ T }>50\),

  • Receive $50.75 from the investment;
  • Exercise the call to buy the stock for $50.
  • Net profit = $0.75

If \({ S }_{ T }<50\),

  • Receive $50.75 from investment,
  • Put exercised by the holder: buy the stock for $50.
  • Net profit = $0.75

Put-call Parity In American Options

Put call parity is only valid for European options. For American options with the possibility of early exercise, the relationship turns into an equality:

$$ { S }_{ 0 }-X\le C-P\le { S }_{ 0 }-X{ e }^{ -rT } $$

Effect of Dividends

  • When a stock pays a dividend, its value must decrease by the amount of the dividend. This increases the value of a put option and decreases the value of a call option.
  • The payment of a dividend will reduce the lower pricing bound for a call and increase the lower pricing bound for a put.

Lower Bounds of American Options

$$
\begin{array}{|c|c|c|}
\hline
Option & Minimum \quad value & Maximum \quad value \\ \hline
American \quad call & C\ge max\left( 0,{ S }_{ 0 }-X{ e }^{ -rT } \right) & { S }_{ 0 } \\ \hline
American \quad put & P\ge max\left( 0,{ X-S }_{ 0 } \right) & X \\ \hline
\end{array}
$$

Question

A one-year European put option on a non-dividend-paying stock with strike at \(USD\quad 50\) currently trades at \(USD \quad 5.55\). The current stock price is \(USD \quad 45\). The stock exhibits an annual volatility of \(30\%\). The annual risk-free interest rate is \(5\%\), compounded continuously.

Determine the price of a European call option on the same stock with the same parameters as those of this put option.

  1. USD 4.12
  2. USD 2.50
  3. USD 5.55
  4. USD 2.99

The correct answer is D.

According to put call parity,

$$ C+X{ e }^{ -rT }=P+{ S }_{ 0 } $$

Making \(C\) the subject,

$$ C=P+{ S }_{ 0 }-X{ e }^{ -rt } $$

$$ =5.55+45-50{ e }^{ -0.05×1 } $$

$$ =2.99 $$


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