The payoff profile of the options can be used to create more complex positions. A long position in the underlying asset can be decomposed into a long call plus a short put with the same strike price and maturities.

Pay on purchase of a call option is as follows:

<img class="aligncenter size-full wp-image-4172" title="Call option Payoff" src="https://financetrain.com/wp-content/uploads/2012/02/Screen-Shot-2012-02-01-at-6.21.27-PM.png" alt="" width="432" height="223" />

Pay on sale of put option is as follows:

<img class="aligncenter size-full wp-image-4173" title="Put Option Payoff" src="https://financetrain.com/wp-content/uploads/2012/02/Screen-Shot-2012-02-01-at-6.21.41-PM.png" alt="" width="465" height="246" />

Assume that the strike price and maturities of both the options are the same, below is the payoff profile of combining both these options which is essentially the payoff of going long on the asset.

<img class="aligncenter size-full wp-image-4174" title="Screen Shot 2012-02-01 at 6.21.55 PM" src="https://financetrain.com/wp-content/uploads/2012/02/Screen-Shot-2012-02-01-at-6.21.55-PM.png" alt="" width="468" height="277" />

The long call provides the equivalent of the upside and the short put generates the downside risk of holding the asset. This is known as put-call parity.

Consider a portfolio of a long position in the call, a short position in the put and an investment that will ensure that they will be able to pay the exercise at maturity. The initial payoff of this position is as follows:
<ol>
	<li>Pay-off on the purchase of the call: <strong>-c</strong></li>
	<li>Pay-off on the sale of the put: <strong>+p</strong></li>
	<li>Payoff on the investment that will pay the strike at the maturity:  (-K present values to today)</li>
</ol>
The total payoff of this strategy is as follows:

<div class="formula1"><strong>(-c + p + (- K present values to today at the prevailing interest rate)</strong>

This payoff is equal to the payoff of going long on an asset = <strong>(-S)</strong></div>

There are two possible cases: the call is exercised or the put in exercised. In both the cases the final payoff is that of holding a long position in the asset.

<b>Scenario 1:</b> The Spot on the maturity date is less than the Strike.

In this situation the call is not exercised and the put is exercised. The final payoff is as follows:

<div class="formula1"><strong>Payoff of the call (0) + Payoff the put (- (K-Spot Rate At Maturity)) + Payoff from the investment (K)</strong>

The total payoff is the payoff of the Asset - <strong>S</strong></div>

<b>Scenario 2:</b> The Spot on the maturity date is greater than the Strike

In this situation the call is exercised and the put is not exercised. The final payoff is as follows:

<div class="formula1"><strong>Payoff of the call (Spot Rate at maturity - K) + Payoff the put ( 0) + Payoff from the investment (K)</strong>

The total payoff is the payoff of the Asset: <strong>S</strong></div>

With the Put-Call parity equation it is possible to find the value of a call given the put and the value of a put given the value of a call. This relationship holds only for European options and not for American options as American options can be exercised anytime between trade and maturity date.

This relationship can be used to determine the implied dividend yield from the market prices (the interest rate). This can also be used to determine the forward rate in dividend swaps, which are contracts where the payoff is indexed to the actual dividends paid over the horizon, minus the implied dividends.

Put Call Parity in Options

<h2><strong>Lower Bound</strong></h2>
We know that the value of an <a title="Introduction to Options" href="https://financetrain.com/introduction-to-options/">option</a> is equal to the sum of its intrinsic value and time value.

Since an option cannot sell below its intrinsic value, its value cannot be negative, Therefore, the lower bound for both American and European options is zero.
<h2><strong>Upper Bound</strong></h2>
<strong>Call Options</strong>

A call option provides the option buyer the right to buy the asset. For the option to have value, its price at any time must be lower than the underlying stock price at any time. This is because if the option price were higher than the stock price, it would be cheaper to just buy the asset directly in the spot market. Therefore, the maximum price for an option is equal to the stock price at that time. This applies to both American and European options.

<strong>Put Options</strong>

A put option provides the option buyer to sell the asset at the strike price. Since an American option can be exercised at any time, its maximum price can be equal to its strike price. However, for a European option, since it cannot be exercised before the expiry date, its maximum value will be equal to the present value of the strike price.

The following table summarizes the upper and lower bounds for these options.
<table border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td valign="top" width="142"></td>
<td valign="top" width="142"><strong>Minimum Value</strong></td>
<td valign="top" width="142"><strong>Maximum Value</strong></td>
</tr>
<tr>
<td valign="top" width="142">European Call</td>
<td valign="top" width="142">ct&gt;=0</td>
<td valign="top" width="142">ct&lt;=St</td>
</tr>
<tr>
<td valign="top" width="142">American Call</td>
<td valign="top" width="142">Ct&gt;=0</td>
<td valign="top" width="142">Ct&lt;=St</td>
</tr>
<tr>
<td valign="top" width="142">European Put</td>
<td valign="top" width="142">pt&gt;=0</td>
<td valign="top" width="142">pt&lt;=X/(1+rf)^(T-t)</td>
</tr>
<tr>
<td valign="top" width="142">American Put</td>
<td valign="top" width="142">Pt&gt;=0</td>
<td valign="top" width="142">Pt&lt;=X</td>
</tr>
</tbody>
</table>
A note about notations:
<ul>
	<li>c represents European call price (Capital C for American call)</li>
	<li>p represents European call price (Capital P for American put)</li>
	<li>S is the Spot price</li>
	<li>X is the Strike price</li>
	<li>T-t is the time remaining to maturity</li>
</ul>
These are the theoretical limits for the value of the options. However, for pricing purpose, we need to be more precise about the lower bound for the option’s price.

For an at-the-money option or an out-of-the-money option, the lower bound is zero. For example, for an out-of-the-money call option strike price (X) is higher than the spot price (X). So, if someone purchased this option, he will still be better off by buying the option from the spot market.

For an in-the-money option, the lower bound will be the difference between the strike price (X), and the spot price (S). For in-the-money call options, X is lower than S, and if the price was lower than S-X, the a person can buy the call option and immediately exercise it to make a profit. Apart from this we also need to consider the present value of the exercise price.

We are not getting into the derivation of the minimum and maximum values, however it is important to know what they are. The following table summarizes the lower and upper price bounds for American and European options.
<table border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td valign="top" width="105"></td>
<td valign="top" width="179"><strong>Minimum Value</strong></td>
<td valign="top" width="142"><strong>Maximum Value</strong></td>
</tr>
<tr>
<td valign="top" width="105">European Call</td>
<td valign="top" width="179">ct&gt;= Max(0,S-X/(1+rf)^T-t)</td>
<td valign="top" width="142">ct&lt;=St</td>
</tr>
<tr>
<td valign="top" width="105">American Call</td>
<td valign="top" width="179">Ct&gt;= Max(0,S-X/(1+rf)^T-t)</td>
<td valign="top" width="142">Ct&lt;=St</td>
</tr>
<tr>
<td valign="top" width="105">European Put</td>
<td valign="top" width="179">pt&gt;= Max(0, X/(1+rf)^T-t - S)</td>
<td valign="top" width="142">pt&lt;=X/(1+rf)^(T-t)</td>
</tr>
<tr>
<td valign="top" width="105">American Put</td>
<td valign="top" width="179">Pt &gt;= Max(0,X-S)</td>
<td valign="top" width="142">Pt&lt;=X</td>
</tr>
</tbody>
</table>
Let’s take a few examples to understand these formulas.

<strong>Example 1</strong>

Calculate the minimum and maximum price of 6-month American and European calls with a strike price of 90 and currently trading at 96. Assume a risk-free rate of 6%.

<strong>American Call</strong>

Minimum: Max(0,S-X/(1+rf)^T-t) = Max (0,96-90/(1.06)^0.5) = 8.58

Maximum:  Ct&lt;=St = 96

<strong>European Call</strong>

Minimum: Max(0,S-X/(1+rf)^T-t) = Max (0,96-90/(1.06)^0.5) = 8.58

Maximum:  Ct&lt;=St = 96

<strong>Example 2</strong>

Calculate the minimum and maximum price of 6-month American and European puts with a strike price of 95 and currently trading at 90. Assume a risk-free rate of 6%.

<strong>American Put</strong>

Minimum: Max(0,X-S) = Max (0,95-90) = 5

Maximum: X = 95

<strong>European Put</strong>

Minimum: Max(0, X/(1+rf)^T-t - S) = Max (0,95/(1.06)^0.5 - 90) = 2.27

Maximum: X/(1+rf)^(T-t) = 95/(1.06)^0.5 = 92.27

Minimum and Maximum Value of European/American Options

Let’s look at the impact of exercise price and time to expiration on option prices.

<strong>Exercise Price</strong>
<ul>
	<li>The price of a put option increases with the increase in exercise price. This is because the put option holder can sell the stock at a higher price.</li>
	<li>The price of a call option increases with the decrease in exercise price. This is because the call option holder can sell the underlying at a lower price.</li>
</ul>
To remember it easily, we can say that put option prices are directly related to exercise price and call option prices are inversely related to exercise price.

<strong>Time to Expiry</strong>

For all options (except European Puts), the longer time to expiry will increase the option’s value. This is because of increased volatility.

In case of European options, this condition doesn’t always hold true.

Remember that for European puts: pt&gt;= Max(0, X/(1+rf)^T-t - S)

So, on one side the longer time to expiry increases volatility, which increases the option’s value, on the other side it decreases the present value of the payoff, which decreases the option’s value.

We cannot be certain whether the value will increase or decrease. To be certain about the impact on the option value, we will need to look at both volatility and discount rate. If volatility is high and discount rate is low, then the option value will increase. If volatility is low and discount rate is high, then the option value will decrease.

Impact of Exercise Price and Time to Expiry on Option Prices

In this article, we will look at how we can seek arbitrage opportunities by using the put-call parity equation. As we know, the put-call parity equation is represented as follows:

c + PV(K) = p + s

If the prices of put and call options available in the market do not follow the above relationship then we have an arbitrage opportunity that can be used to make a risk-free profit. In the above equation the left side of the equation represents a fiduciary call and the right side of the equation is called a protective put. Depending on the asymmetry we can take our positions to earn a risk-free profit. We buy the underpriced side and sell the overpriced side. Let’s take an example to understand this.

Let’s say that we have we have the following information for a call and a put option on XYZ stock.

Exercise price: $100

Call option price: $7

Put option price: $5

Risk-free rate: 8%

Current market price of XYZ: $98

Time to maturity: 0.5 years

Let’s plug these values in the put-call parity equation:

7 + 100/(1.08)^0.5 = 5 + 99

103.225 = 104

As we can see, the right hand side is greater than the left hand side by (104 – 103.225) = 0.775

To make use of this arbitrage opportunity, we will buy the fiduciary call and sell the protective put.
<ol>
	<li><strong>Sell the protective put:</strong> We sell a put option and receive the $5 premium. We also short sell the ABC stock and receive $99. The total cash inflow is $104.</li>
	<li><strong>Buy fiduciary call</strong><b>: </b>We payout a total of $103.225 for the fiduciary call option. That is we pay $7 as premium for the call option and invest 96.225 in a bond for 6 months at 8%.</li>
	<li><strong>Net cash inflow:</strong> Our net cash inflow is (104 – 103.225) = $0.775</li>
</ol>
When the time to exercise these options comes, there are two scenarios, the actual spot price will either be above $100 or below $100, let’s see how our arbitrage will work out in each scenario:
<table border="1" cellspacing="0" cellpadding="0">
<tbody>
<tr>
<td valign="top" width="308"><strong>Underlying price is above $100</strong></td>
<td valign="top" width="308"><strong>Underlying value is below $100</strong></td>
</tr>
<tr>
<td valign="top" width="308">
<ol>
	<li>Receive $100 from the bond</li>
	<li>Exercise call option, pay $100 and receive ABC stock</li>
	<li>Deliver the stock to cover the short sale</li>
	<li>The put option expires without being exercised</li>
	<li>No net income or loss</li>
</ol>
</td>
<td valign="top" width="308">
<ol>
	<li>Receive $100 from the bond</li>
	<li>Put is in-the-money and is exercised. You pay $100 and take delivery of XYZ stock</li>
	<li>Deliver the stock to cover the short sale</li>
	<li>The call option expires without being exercised</li>
	<li>No net income or loss</li>
</ol>
</td>
</tr>
</tbody>
</table>
As you can see, in both the scenarios, there is no net inflow or outflow at maturity. Our risk-less profit is $0.775 that we made in the beginning. This example used European options. If we had to form a similar strategy with American options, it would be much more complicated.

Put Call Parity and Arbitrage Opportunity

As we know, unlike a European option, the holder of an American option can exercise the option before the expiry date. Because of this additional benefit of being able to exercise the option early, an American option is always more expensive than a European option. However, is this benefit of any real use, that is, is there a situation where the option holder will get a better payoff by exercising the option early? The answer is NO. You should never early exercise an American option, especially if it’s a non-dividend paying stock. Let’s look at the reasoning behind this.

The option has intrinsic value and time value. The intrinsic value of the option is always greater than 0. Along with that the cash has time value, so you would rather delay paying the strike price by exercising it as late as possible. You could use that money to earn interest.

So, a positive intrinsic value plus time value implies that you are better off selling the option rather than exercising it early. This is true for a non-dividend paying stock.

However, for a dividend paying stock, the only time it may pay to exercise a call option is the day before the stock goes ex-dividend, and only if the dividend minus the cost of carry is less than the corresponding Put. By exercising, the option holder may forego the time value but will make up from the dividend received. We have used the word 'may' because the dividend may not be high enough to justify the early exercise.

Should American Options be Exercised Early?

Both interest rates and underlying stock’s volatility have an influence on the option prices.

<strong>Impact of Interest Rates</strong>

When interest rates increase, the call option prices increase while the put option prices decrease.

Let’s look at the logic behind this. Let’s say you are interested in buying a stock which sells at $10 per share. You buy 1,000 shares at $10 each with a total investment of $10,000. Instead of directly buying the stock, you could also have purchases a call option selling for only $1, making a total investment of $1 x 1,000 = $1,000. If you choose to buy the call option instead of the underlying stock directly, you could have used the remaining $9,000 to earn some interest. The higher the interest rates, the higher your interest income would be. This makes the call option more attractive and more expensive.

For put options, the opposite holds true, that is, the higher the interest rates the lower the put option price. This is because if interest rates are high you will have to hold the asset for a longer time to deliver it under the put option. Simply selling the asset and using the proceeds to invest at a higher rate would be a better option. This makes the put option less attractive and hence less costly when interest rates are high.

This said, the impact of interest rates on option prices is minimal.

<strong>Impact of Volatility</strong>

Unlike interest rates, volatility significantly affects the option prices. The higher the volatility of the underlying asset, the higher is the price for both call options and put options. This happens because higher volatility increases both the up potential and down potential. The upside helps calls and downside helps put options.

How Interest Rates and Volatility Affect Option Prices?

Option Valuation

This frontmatter description will appear below the title

COURSE

Option Valuation

Lessons

R Programming Bundle: 25% OFF