###### Linear Regression

After completing this reading, you should be able to: Describe the models that... **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. A key question, however, is whether an investor who owns an American option should take up the offer to exercise early even when the option is deep in the money. Let’s look at an example, assuming the stock does not pay dividends.

A non-paying dividends American call option has a strike price of $50 and expires in three months. The underlying stock has a current price of $80. Assuming no dividends, should the investor exercise this option before expiry?

Looking at the situation, it appears the decision should be straightforward for the investor: exercise the option now, sell the stock immediately, and generate a profit of $30. Exercising the option would appear even more attractive when we consider that each option represents 100 shares of the stock, which means the investor would make a profit of $3,000 from just one option. Even with this profit, the option should not be exercised before maturity if interest rates are **positive**. In order to understand why this is the case, note that the option owner is in either one of two situations.

**Situation 1**: The investor wants a long position in the stock

**Situation 2**: The investor does not want a long position in the stock.

If the investor is in Situation 1, they would exercise the option but opt not to sell the stock. It would be best for them to hold the option until it expires. To better understand this, consider two outcomes for Situation 1:

**Outcome 1**: The stock price is in the money (greater than $50) on the expiry date.

**Outcome 2**: The stock price is out of the money (less than $50) on the expiry date.

Under Outcome 1, it would be **suboptimal** to exercise the option early, as waiting until the option expires would allow the investor to earn **additional interest** by investing the strike price for two months at the risk-free rate.

How about Outcome 2? If the option is OTM at expiry yet it has already been exercised, the investor would incur a loss of $50 – S, where s is the stock price at the expiry date. If the investor foregoes early exercise and waits until maturity, they would not exercise the option. Instead, they would let it expire worthless and in so doing avoid this loss.

The key point here is that keeping the option unexercised until maturity gives the option holder insurance against

the value of the stock falling below $50. As soon as the option is exercised, this insurance is lost.

Shifting our focus to situation 2 where the investor has no motivation to hold the stock, exercising early and selling the stock would appear to be the best decision, but it isn’t. The optimal decision would be to sell the option. But why? When the option is exercised immediately, it yields a profit equal to the intrinsic value (which in this case is $30). Selling the option, however, earns a profit of the intrinsic value plus what’s known as the time value. The time value is the insurance against the value of the stock falling below $50, as mentioned above.

Let’s consider two portfolios an investor can hold until maturity of an American call option

**Portfolio 1**: A call option plus cash equal to the present value of the strike price.

**Portfolio 2**: The stock

Under Portfolio 1, the investor can invest the cash amount today at the risk-free rate. By the time the option matures, the cash amount would be equal to the strike price. If the option is in the money at maturity, the investor can use the proceeds from the cash investment to exercise the option at the strike price. This would automatically increase the value of Portfolio 1 to the strike price, but the investor would still have the choice to sell the stock, further increasing the value of the portfolio. If the option is out of the money at maturity, it expires worthless, but the value of the portfolio would still have grown to the strike price thanks to the cash investment.

If follows that in all circumstances, Portfolio 1 would be worth **at least as much** as the stock. On the other hand, Portfolio 2 would be worth the stock price prevailing at maturity of the option. In summary, we can infer that at maturity,

Value of Portfolio 1 ≥ Value of Portfolio 2

This must be true today, otherwise an arbitrage opportunity would arise. If Portfolio 2 is worth more than Portfolio

1 today, an arbitrageur can buy Portfolio 1 and short Portfolio 2. This would create a position that would never lead to a loss and would sometimes lead to a profit.

Today, Portfolio 1 is worth:

Call Price + PV(*K*)

where PV represents “present value” and *K* is the strike price.

On the other had, the value of Portfolio 2 today is the stock price, *S*. We can therefore infer that:

Alternatively,

Call Price ≥ *S* – PV(*K*) ………Equation y

We know that the option price can never be negative. Thus,

Call Price > max(*S* – PV(*K*), 0)

working under the assumption that interest rates are positive,

K > PV(*K*)

Equation y above therefore implies that:

Call Price > S – K

Exercising the option today would mean that the call price equals S – K. It follows that the call option should **never be exercised early**.

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