###### Common Univariate Random Variables

After completing this reading, you should be able to: Distinguish the key properties... **Read More**

**After completing this reading, you should be able to:**

- Identify the six factors that affect an option’s price.
- 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, with dividends and without dividends and express it in terms of forward prices.
- Explain and assess potential rationales for using the early exercise features of American call and put options.
- Explain the relationship between options and forward prices.

There are six factors that impact the value of an option:

\(S\) = current stock price

\(K\) = 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

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

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).

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).

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.

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 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*.

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*.

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 there are higher chances of the option expiring in the money as volatility increases.

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

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; and

\({ S }_{ 0 }\) = value of the stock today.

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,

$$ \text{p}\le \text{K and P}\le \text{K} $$

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 a 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 price, that is.

$$ p\le K(1+r)^{-T} $$

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 than the present value of the exercise price. This is expressed as follows:

$$c+K(1+r)^{-T}\ge { S }_{ 0 }$$

Thus, the lower pricing bound of a European call option is given by:

$$c\ge max\left( { S }_{ 0 }-K(1+r)^{-T},0 \right)$$

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:

$$p+{ S }_{ 0 }\ge K(1+r)^{-T}$$

Thus, the lower pricing bound of a European put option is given by:

$$p \ge max\left( K(1+r)^{-T}-{ S }_{ 0 },0 \right) $$

American options can be exercised at any time on or before their maturity dates. The below illustration will tell us if it is prudent to exercise American options before their maturity dates.

A non-paying dividends American call option has a strike price of $50, with a three-month maturity period. The underlying stock has a current price of $80. Can an investor exercise this option before the three months maturity period?

Exercising the option now gives an investor a profit of $(80-50)=$30. However, this option should not be exercised before maturity if interest rates are positive.

The option holder is faced with two scenarios.

Further suppose that under scenario 1, at maturity, the price of the stock is (i) greater than $50 and (ii) less than $50:

For a stock price greater than $50, there will be no benefits in exercising the option before maturity as the strike price of $50 will still be paid.

If the stock price is less than $50, the investor incurs a loss of $50 – the current stock price. (Had the investor waited until maturity, then the option would not have been exercised.)

An investor loses insurance against losses by exercising an American option earlier than its maturity date. This insurance against losses is provided by the choices offered by an option (optionality).

Exercising the option now gives a profit equal to the option’s intrinsic value, i.e., the value of the option if it were to be exercised immediately; in this case, $(80-50) = $30.

If the investor sells the option, the profit will be the intrinsic value (in this case, $30) plus the time value (the premium that would have been paid if the option were at the money).

An investor has two portfolios:

- Portfolio X comprises a call option plus cash equal to the present value of the strike price.
- Portfolio Y comprises the stock.

The table below summarizes the possible outcomes when the stock price is (i) greater than and (ii) lesser than the strike price at maturity.

$$ \begin{array}{l|l|l} \text{} & \textbf{Portfolio X} & \textbf{Portfolio Y} \\ \hline \text{Comprises} & \begin{array}{l} \text{Call option plus cash} \\ \text{equal to the present} \\ \text{value of the strike} \\ \text{price.} \end{array} & \text{The stock.} \\ \hline \begin{array}{l} \text{Stock price larger} \\ \text{than the strike price} \end{array} & \begin{array}{l} \text{The option will be} \\ \text{exercised.} \\ \text{} \\ \text{The portfolio will} \\ \text{become worth the} \\ \text{stock price.} \end{array} & \begin{array}{l} \text{The portfolio is} \\ \text{worth the stock} \\ \text{prices.} \end{array} \\ \hline \begin{array}{l} \text{Stock price smaller} \\ \text{than the strike price} \end{array} & \begin{array}{l} \text{The option will not be} \\ \text{exercised.} \\ \text{} \\ \text{The portfolio will} \\ \text{become worth the} \\ \text{strike price.} \end{array} & \begin{array}{l} \text{The portfolio is} \\ \text{worth the stock} \\ \text{prices.} \end{array} \\ \hline \text{Value} & \begin{array}{l} \text{Call price + Present} \\ \text{value of the strike} \\ \text{price.} \end{array} & \text{Stock price.} \\ \end{array} $$

Therefore, at maturity, the value of portfolio X is either greater than or equal to the value of portfolio B, that is.

$$ C + K(1+r)^{-T} ≥ S_T $$

Rewriting gives:

$$ C ≥ S_T – K(1+r)^{-T} $$

But since option prices cannot be negative, the result can be expressed as:

$$ C ≥ max(S_T – K(1+r)^{-T}, 0) $$

Interest rates are positive and thus: \(K > K(1+r)^{-T} \)

The call price is, therefore, greater than the stock price minus the strike price.

Exercising an American call option earlier gives a profit equal to the stock price minus the strike price. The above proves that **American call options should not be exercised** earlier than their maturity dates.

Since a call option cannot be more worth than the stock price, the stock price is the upper bound of a call option.

The lower bound is obtained by subtracting the present value of the strike price from the stock price.

Employee stock options cannot be sold – they have to be exercised by the employee to which the stock option has been given.

As with all other call options, employee call options on stocks that pay no dividends should not be exercised before maturity.

Dividends reduce the price of a stock. For an American call dividend-paying option to be profitable, the option should be exercised just before the ex-dividend date. However, the option should not be exercised if the dividend is less than \(K-K^*\) where \(K^*\) is obtained by discounting \(K\) from the next ex-dividend date (option maturity) to the current ex-dividend date.

In the presence of dividends \((D)\), the lower bound for a European call option will be adjusted to incorporate dividends:

$$ c≥ S_0-K(1+r)^{-T}-D(1+r)^{-T}$$

In the presence of dividends, American call options can be exercised immediately.

Whereas call option holders pay the strike price, put option holders receive the strike price. The decision to exercise an American put option, thus, is dependent on a trade-off between receiving the strike price early so as to reinvest it and benefitting from a very small probability that the stock price will be greater than the strike price at maturity.

Dividends make it undesirable for a put option to be exercised before maturity. In the presence of dividends:

$$ p ≥ max (K(1+r)^{-T} +D(1+r)^{-T}-S_0, 0) $$

Options with a longer maturity period are less likely to be exercised before their maturity date as there is enough time for the stock price to move above the strike price.

A put option holder is less likely to exercise the position earlier if the time to maturity increases, the stock price increases, the dividends to be received an increase, and if the interest rate decreases.

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.

In other words, put-call parity is the relationship between the price of a European put option and the price of a European call option, with the same strike price and time to maturity.

Consider the following portfolios:

- Portfolio \(A\): One call option plus an amount of cash equivalent to \(K(1+r)^{-T}\)
- 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,

$$ c+K(1+r)^{-T}=p+{ S }_{ 0 } $$

Where:

\(c\) = value of call option

\(K\) = strike price

\(p\) = value of put option

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

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 you also own a 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.

$$ \begin{array}{l|r|r|r|r} \textbf{Expiration Price} & \text{\$80} & \text{\$89} & \text{\$110} & \text{\$130} \\ \hline \textbf{Stock} & \text{\$80} & \text{\$89} & \text{\$110} & \text{\$130} \\ \hline \textbf{Put Option} & \text{\$10} & \text{\$1} & \text{\$0} & \text{\$0} \\ \hline \textbf{Portfolio} & \text{\$90} & \text{\$90} & \text{\$110} & \text{\$130} \\ \end{array} $$

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 you also own a zero-coupon, risk-free bond that matures for $90.

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

$$ \begin{array}{l|r|r|r|r} \textbf{Expiration Price} & \text{\$80} & \text{\$89} & \text{\$110} & \text{\$130} \\ \hline \textbf{Call Option} & \text{\$0} & \text{\$0} & \text{\$20} & \text{\$40} \\ \hline \textbf{Bond} & \text{\$90} & \text{\$90} & \text{\$90} & \text{\$90} \\ \hline \textbf{Portfolio} & \text{\$90} & \text{\$90} & \text{\$110} & \text{\$130} \\ \end{array} $$

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:

$$ c+K(1+r)^{-T}=p+{ S }_{ 0 } $$

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

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.

Applying the put-call parity relationship,

$$ c+K(1+r)^{-T}=p+{ S }_{ 0 } $$

Making \(P\) the subject,

$$ \begin{align*} p & =c+K(1+r)^{-T}-{ S }_{ 0 } \\ & =5+50(1.10)^{-0.25}-51 \\ & = 2.82 \end{align*} $$

If \(p\) is greater than or less than 2.82, there will be arbitrage opportunities.

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

- Buy call for \($5\)
- Short Put to realize \($3.50\)
- Short the stock to realize \($51\)
- Invest \($49.5\left( =51+3.50–5 \right) \) for 3 months, making \($50.69(=49.5(1.10)^{0.25})\)

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

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

- Receive $50.69 from the investment;
- Exercise the call to buy the stock for $50.
- Net profit = $0.69

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

- Receive $50.69 from investment,
- Put exercised by the holder: buy the stock for $50.
- Net profit = $0.69

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

$$ { S }_{ 0 }-K\le C-P\le { S }_{ 0 }- K(1+r)^{-T}$$

- 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.
- A dividend payment will reduce the lower pricing bound for a call and increase the lower pricing bound for a put.

$$

\begin{array}{c|c|c}

\textbf{Option} & \textbf{Minimum value} & \textbf{Maximum value} \\ \hline

\text{American call} & C\ge max\left( 0,{ S }_{ 0 }-K(1+r)^{-T}\right) & { S }_{ 0 } \\ \hline

\text{American put} & P\ge max\left( 0,{ K-S }_{ 0 } \right) & K \\ \end{array}

$$

For now, we have only dealt with calls and put options on stocks. Forward prices, for example, on commodities such as oil, can also be used to derive the price of call and put option prices on commodities or other assets that trade with forward contracts. Let’s define \(F\) as the forward price for a contract maturing at the same time as the options and \(F(1+r)^{-T}\) as the present value of F when discounted from the options’ maturity at the risk-free rate. Note that \(K\) is still the strike price of the option.

The put-call parity relationship is, therefore:

$$ c +K(1+r)^{-T}= p + F(1+r)^{-T}$$

Because the put price cannot be negative, a lower bound for a European call price can be deduced as:

$$ c ≥ F(1+r)^{-T}- K(1+r)^{-T}$$

Similarly, because the call price cannot be negative, the lower bound of the European put price is:

$$ p ≥ K(1+r)^{-T}- F(1+r)^{-T} $$

## Question

A one-year European put option on a non-dividend-paying stock with the strike at USD 50 currently trades at USD 5.55. The current stock price is USD 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.

A. USD 4.12

B. USD 2.50

C. USD 5.55

D. USD 2.93

The correct answer is

D.According to put-call parity,

$$ c+K(1+r)^{-T}=p+{ S }_{ 0 }$$

Making \(c\) the subject,

$$\begin{align*} c & =p+{ S }_{ 0 }-K(1+r)^{-T}\\ & =5.55+45-50(1.05)^{-1} \\ & =2.93\end{align*}$$

Where:

\(c\) = value of call option

\(K\) = strike price

\(p\) = value of put option

\(S_{0}\) = initial stock price