This is the second tutorial of two covering derivatives for the Level II CFA Exam.  The topics of options, swaps, and credit default swaps can be intimidating for some candidates, particularly given the volume of associated formulas.

One recommended prep approach is to start with the conceptual aspects of derivatives and then apply the formulas.  First, some exam questions are likely to be conceptual or at least require limited math.  By understanding the theory and concepts for options and swaps, candidates have positioned themselves for success on these questions.  Second, attempting to study the formulas before mastering the concepts may hinder the preparation progress if frustration settles in.

A total of two item sets (12 questions) on derivatives is a reasonable expectation.  Some percentage of those questions will come from options, swaps, and credit derivatives (i.e. CDS).  Application of the “Greeks”, from the Black-Scholes-Merton option pricing model is a likely suspect for one or two exam questions.

Derivatives are a middle tier subject in terms of importance.  If you want to pass the exam, derivatives does not need to be your strongest subject, but you cannot afford to leave too many points on the table either.  Take derivatives seriously and adapt a test tactic around your strengths and weaknesses that allows you to maximize points.

This tutorial aligns with Study Session 17 material in the Level II CFA Program Curriculum ©.

Material:

I.         Options

II.        Swaps

III.       Interest Rate Derivatives

IV.       Credit Derivatives

CFA Level 2: Derivatives Part 2 – Introduction

An option is a derivative instrument that gives its holder a right to buy or sell an asset, at a set price (the strike price) on a set date (contract expiration).

A call option gives its owner the right to buy; it is not a promise to buy. For example, a store holding an item for you for a fee is a call option.

Assume that you want to buy a stock after three months, currently trading at $100, but feel that the price of the stock will rise significantly. In this case, you can buy an option on the stock to buy the stock after three months at a price fixed today (say $120). After 3 months, if the spot stock price has actually risen above $120, you can exercise the option and get the stock at $120. If however the actual stock price doesn’t rise as expected (say it rises only upto $110), then the investor need not exercise the option and let it expire. He can instead buy the stock directly from the market at $110.

A put option gives its owner the right to sell; it is not a promise to sell. For example, a lifetime money back guarantee policy on items sold by a company is an embedded put option.

In the U.S., stock options are standardized and traded over exchanges (unlike a <a title="What are Forward Contracts?" href="https://financetrain.com/what-are-forward-contracts/">forward contract</a> or a swap, which are private over-the-counter instruments).

Just like a stock or a bond, an option is a binding contract with strict terms. The option buyer is required to pay a premium at the beginning to buy the option.

Options can be of two types: European and American.

European Options: European option contracts are the ones that can be exercised only at expiration.

American Options: American option contracts are the ones that can be sold at any time prior to expiration.

An option contract can be based on various underlying securities such as stocks, indexes, and bonds. Stock options are the most commonly traded option contracts. In general, one option contract is for 100 shares of stock.

Introduction to Options

The prices of put and call <a href="https://financetrain.com/introduction-to-options/" title="Introduction to Options">options</a> have an identity relationship through the concept of put-call parity.
<ul>
			<div class="formula1">c0 + X/(1+rF)T = p0 + S0
				<ul>
					<li>c0 = Current price of the European call</li>
					<li>p0 = Current price of the European put</li>
					<li>X = Strike price of the put and the call</li>
					<li>T = Time to expiration</li>
					<li>rF = Risk free rate</li>
					<li>S0 = Current spot price of the underlying asset</li>

				</ul>
			
			</div>

</ul>
The formula translation is: the price of a call with strike X plus the present value of strike price X equals the price of the put with strike X plus the current spot price.


<ul>
	<li>Synthetic Call Option: If an investor believes that a call option is over-priced, then he/she can sell the call on the market and replicate a synthetic call.</li>
		<ul>
			<li>Borrow the present value of the strike price at the risk free rate and purchase the underlying stock and a put.</li>
			<div class="formula1">c0 = p0 + S0 - X/(1+rF)T</div>
		</ul>
	<li>Synthetic Put Option: Similar to the synthetic call option. A synthetic put can be created by re-arranging the put-call parity relationship, if the trader believes the put is overvalued.</li>
	<li>Synthetic Stock: A synthetic stock can also be created by rearranging the put-call parity identity. In this case, the investor will buy the call, sell the put, and lend the present value of the strike at the current risk free rate.</li>
		<ul>
			<div class="formula1">S0 = c0 - p0 + X/(1+rF)T</div>
			<li>If the stock rises in value, then the long call will provide the upside; if the stock falls, then the short put will replicate the downside.</li>
		</ul>
	<li>Rationale for a "Synthetic"</li>
		<ul>
			<li>Rational investors would not arbitrarily enter into "synthetic" position; it is done to exploit a perceived mispricing. For example, if the investor believes that put call parity is showing that a stock's call is overvalued, then he/she may execute a synthetic call strategy.</li>
			<li>The key to a synthetic strategy is to buy the undervalued asset, sell the overvalued asset and invest or borrow the difference at the risk free rate.</li>
		</ul>

</ul>

Synthetic Options and Rationale

<ul>
	<li>The following model can be used for <a href="https://financetrain.com/introduction-to-options/" title="Introduction to Options">options</a> on stocks, currencies, and commodities; points on interest rate option pricing will be made at the end of this section.</li>
	<li>A critical component for option pricing with the one period binomial model is the notion of constructing a hedged portfolio.</li>
	<div class="formula1">
	H = nS - c
	
	<ul>
		<li>H = The value of the hedged portfolio</li>
		<li>n = Hedge ratio</li>
		<li>S = Price of the underlying asset</li>
		<li>c = Call price</li>
	</ul>
	</div>
	
	<li>A hedged portfolio will be "riskless" in that there is a perfect balance between a long position in the underlying and a short position in the call, so the gain in one offsets the loss to the other.</li>
	
	<div class="formula1">
	n = (c+ - c-)/(S+ - S-)
	
	<ul>
		<li>c+ = The price of the call when the underlying rises to S+</li>
		<li>c- = The price of the call with the underlying falls to S-
		 
		Note that these values are "intrinsic" values (difference between the underlying's future price and the option's strike price). For example, if the strike price is $25/share and the upside is $30/share, then c+ is $5.
		</li>
		<li>S+ = The upside price of the underlying asset</li>
		<li>S- = The downside price of the underlying asset</li>
	</ul>
	</div>
	
	<li>Upside and downside for the underlying must be converted to 1 + the % price move</li>
		<ul>
			<li>u = (S+/S)</li>
			<li>d = (S-/S)</li>
			<li>S = Starting price for the underlying</li>
		</ul>
	<li>Because the hedged portfolio is a risk free portfolio, the rate of return on the hedged portfolio should equal the risk free rate.</li>
<li>The one period model can be expanded to multiple periods.</li>
</ul>

One Period Binomial Option Pricing Model

Call option price formula for the <a href="https://financetrain.com/one-period-binomial-option-pricing-model/" title="One Period Binomial Option Pricing Model">single period binomial option pricing model</a>:
<div class="formula1">
c = (&pi;c+ + (1-&pi;) c-) / (1 + r) 

<ul>
	<li>&pi; = (1+r-d) / (u-d)</li>
	<li>"&pi;" and "1-&pi;" can be called the risk neutral probabilities because these values represent the price of the underlying going up or down when investors are indifferent to risk.</li>
	<li>r = The risk free rate</li>
	<li>The same formula is applied for put options.</li>
	
</ul>
</div>
<ul>
	<li>Steps for solving the value of a call option with the single period binomial model:</li>

		<ul>
			<li>Calculate "u" and "d".</li>
			<li>Calculate "&pi;" (note: the risk free rate should be provided)</li>
			<li>Combine "&pi;" with c+ and c- to value the call.</li>
			<li>NOTE: This can be repeated for the put option.</li>
		</ul>

</ul>
<div class="special-area">
Test Tip: 
 
Whenever pricing options on an exam question, it is a good idea to give your answer the laugh test; in other words, does the answer you are calculating make sense given the data provided.
 
For example a call that is deep out of the money should be relatively inexpensive; whereas a call that is deep in the money should be close to its intrinsic value plus a small time premium.
</div>

Call Option Price Formula

<ul>
	<li>Similar to the applications of the interest rate tree discussed in fixed income, a variation of the <a href="https://financetrain.com/one-period-binomial-option-pricing-model/" title="One Period Binomial Option Pricing Model">binomial option pricing model</a> as presented can be used for options on bonds and interest rates.</li>
	<li>The analyst will need to:</li>
	
		<ol>
			<li>Create an interest rate tree of future spot rates.</li>
			<li>Calibrate the interest rate tree so current prices for recently issued bonds are correctly priced.</li>
			<li>Value the underlying bond with the calibrated interest rate tree.</li>
			<li>Apply the decision rule for the option on each tree node; the value of the option at each node will be its intrinsic value.</li>
			<li>Calculate the option's "fair" price by discounting the intrinsic option values through the tree.</li>
			
		</ol>


</ul>

Binomial Interest Rate Options Pricing

<ul>
 <li>Commonly called "Black-Scholes" outside the CFA exam world.</li>
 <li>BSM is a model for deriving the price of an option.</li>
 <li>Assumptions</li>
	<ul>
		<li>Stock returns are lognormally distributed.</li>
		<li>The risk free rate is known and stays constant during the option term.</li>
		<li>The stock's volatility is known and stays constant during the option term.</li>
		<li>Transaction costs are omitted from the model.</li>
		<li>Dividends are excluded.</li>
		<li>The options cannot be exercised until expiration (i.e. the option is European).</li>
	</ul>
 <li>Weaknesses in BSM Assumptions</li>
	<ul>
		<li>It is good for candidates to understand that BSM is not without its flaws; it is after all a theoretical model.</li>
		<li>The risk free rate is not always "known" and may not stay constant for the option's life; further an investor is unlikely to be able to borrow at the risk free rate.</li>
		<li>A stock's volatility is not likely to remain constant over the life of the option.</li>
		<li>In reality, investors face transaction costs.</li>
		<li>Many stocks pay dividends.</li>
		<li>The investor may be trading American options and not European style options.</li>
	</ul> 
 <li>BSM Formula: 

Using the Black-Scholes model, the price of a call option is calculated using the following formula:

<img class="size-full wp-image-1437 aligncenter" title="call-price" src="https://financetrain.com/wp-content/uploads/2010/12/call-price.gif" alt="" width="229" height="44" />
Where:
C is the price of the call option
S is the price of the underlying stock
X is the option exercise price
r is the risk-free interest rate
T is the current time until expiration
N() is the area under the normal curve
d1 =  [ ln(S/X) + (r + σ2/2) T ] / σ T1/2
d2 =  d1 - σ T1/2
Similarly, the price of the put option is:
<img class="size-full wp-image-1438 aligncenter" title="put-price" src="https://financetrain.com/wp-content/uploads/2010/12/put-price.gif" alt="" width="257" height="41" />

The CFAI may include the BSM formula as part of the candidate body of knowledge, so technically, it could be required on the exam. However, given the time constraints of the exam, calculation of the BSM formula is unlikely.</li> 

		<li>Candidates are encouraged to focus on understanding the BSM and its applications (especially the "Greeks"), rather than the formula itself.</li>
 
</ul>
<h2>Underlying Asset Cash Flows and Option Prices</h2>
<ul>
 <li>While BSM assumes that the underlying asset does not have any cash flows, in reality many stocks pay dividends.</li>
 <li>Holding other variables constant, dividend payments reduce the value of calls and increase the value of puts. This is because stock prices are adjusted down once the stock goes ex-dividend.</li>
 <li>BSM can be adjusted for a dividend payment by reducing the spot price (S0) of the stock by the present value of the dividend payment expected to take place during the option's life.</li>
 <li>For stock's a continuously compounded dividend yield can represent the cash flow; for currencies, a continuously compounded interest rate can be applied.</li>
 
</ul>

Black-Scholes-Merton (BSM) Option Pricing Model

The Black-Scholes-Merton model has six inputs (or five, if gamma is considered a sub-part of delta); five are known as the Greeks.
<ol>
	<li>Delta: The change in the option price per one dollar change in the underlying stock's price; alternatively, the change in the option price equals the change in the underlying multiplied by the option's delta.</li>
	<li>Gamma: The sensitivity of delta to the change in a price of the underlying asset.</li>
	<li>Theta: An option's price is affected by the amount of time to expiration; the longer the time, the more valuable the option. Theta is a measure of the rate of the time value decay of the option.</li>
	<li>Rho: This is an option's price sensitivity to a change in the risk free rate. Rho is commonly small and a European option's price is not heavily sensitive to a change in the interest rate.</li>
	<li>Vega: This is volatility sensitivity; specifically vega is option price sensitivity to the standard deviation of the asset's return. As volatility increases, options become more valuable.</li>
	<li>Strike (or Exercise) Price: This is the price at which the underlying asset can be bought (long call) or sold (long put) by the option holder. Excluding other factors, as strike prices go up calls are worth less and puts are worth more.</li>
</ol>

Summary of Input Impacts on Option Prices:

<table class="mytable">
<tr>
<th scope="colspan">INCREASE TO</th>
<th scope="colspan">CALL PRICE</th>
<th scope="colspan">PUT PRICE</th>
</tr>
<tr>
 <th scope="row" class="spec">Dividend</th>
 <td>Down</td>
 <td>Up</td>
</tr>
<tr>
 <th scope="row" class="specalt">Interest Rates</th>
 <td class="alt">Up</td>
 <td class="alt">Down</td>
</tr>
<tr>
 <th scope="row" class="spec">Volatility</th>
 <td>Up</td>
 <td>Up</td>

</tr>
<tr>
 <th scope="row" class="specalt">Underlying Asset Price</th>
 <td class="alt">Up</td>
 <td class="alt">Down</td>
</tr>
<tr>
 <th scope="row" class="spec">Time to Expiration</th>
 <td>Up</td>
 <td>Up</td>

</tr>
<tr>
 <th scope="row" class="specalt">Exercise/Strike Price</th>
 <td class="alt">Down</td>
 <td class="alt">Up</td>
</tr>
</table>

Black-Scholes-Merton Model and the Greeks

<ul>
	<li>Traders and securities dealers can use an option's delta to create hedges for the price risk exposure that they have in other <a href="https://financetrain.com/introduction-to-options/" title="Introduction to Options">option</a> or underlying asset positions.</li>
		<div class="formula1">
			Example, assume that a securities dealer has sold (short position) 100 call options on Ford Motors (NYSE: F) and that each option represents 100 shares. Thus, the dealer's net stock exposure is 10,000 shares. If the options on F have a delta of 0.8, then the dealer can counter his short call position by buying 8,000 shares of F. Thus if the stock increases by $1, the increase in the value of the long stock position should offset the loss on the short call option position.
		</div> 
	<li>Delta is dynamic, though, and will change as the underlying asset price changes. Therefore delta hedging requires constant readjustment.</li>
		<ul>
			<li>Because of this, delta hedging is also called dynamic hedging.</li>	
		</ul>		
	<li>Delta hedging loses its effectiveness when an underlying asset has large price moves, either up or down.</li>
	<li>As previously discussed, gamma represents the sensitivity of delta to a change in the underlying asset's price.</li>
		<ul>
			<li>When an option is either deep in or deep out of the money, gamma will be close to zero as it will take a large price move to impact the value of delta; an option's gamma is comparable to a bond's convexity.</li>
			<li>When an option is at the money and expiration is close, then gamma tends to be high.</li>
			<li>When gamma is high, delta becomes a less accurate measure of option price sensitivity to the underlying asset and delta hedging loses accuracy as a hedging technique.</li>
			
		</ul>

</ul>

Dynamic Delta Hedging & Gamma Related Issues

<ul>
 <li>A key challenge in using either <a href="https://financetrain.com/black-scholes-merton-bsm-option-pricing-model/" title="Black-Scholes-Merton (BSM) Option Pricing Model">Black-Scholes-Merton</a> or a <a href="https://financetrain.com/one-period-binomial-option-pricing-model/" title="One Period Binomial Option Pricing Model">binomial option pricing model</a> is accurately estimating the stock's return volatility.</li>
 <li>Two common approaches for estimating volatility:</li>
	<ol>
		<li>Historical Approach: This assumes that past volatility is representative of future volatility.</li>
			<ul>
				<li>For BSM, the annualized standard deviation of price returns is applied.</li>
				<li>&sigma;annual = &sigma;periodic * &radic;periods per year</li>
				<li>250 trading days per year is a convention when the periodic standard deviation is daily; 12 is applied for monthly data; 52 is applied for weekly data.</li>
			</ul>
		<li>Implied Volatility Approach: This takes the current market price of the option and back solves for the implied value of volatility via the option pricing model.</li>
			<ul>
				<li>The output of the implied volatility approach is an estimate for volatility that equates the BSM price of the option to the market price of the option.</li>
				<li>Analysts and traders can use this approach to form opinions as to whether an option price is too high or too low based on their own expectations for volatility relative to the implied volatility priced into the option.</li>
			</ul>			
	</ol>

</ul>

Estimating Volatility for Option Pricing

<ul>
	<li>An option can be created for a <a href="https://financetrain.com/what-are-forward-contracts/" title="What are Forward Contracts?">forward contract</a>.</li>
		<ul>
			<li>Recall that at contract initiation a forward will have a value of zero.</li>
		</ul>
	<li>European options on forwards will have the same price as European options on <a href="https://financetrain.com/introduction-to-futures-contracts/" title="Introduction to Futures Contracts">futures contracts</a>; but American options on forwards will have a different price from American options on futures.</li>
	<li>The payoff values for options on forwards are:</li>
		<div class="formula1">
		Call payoff = Max[0,ST-X]
		 Put payoff = Max [0,X-ST]
		 ST = Underlying's price at expiration
		 X = Strike price of option

		
		</div>
	<li>Put-Call Parity for Options on Forwards:</li>
		<div class="formula1">
		p0 = c0 + ((X - F(0,T))/(1+rF)T)

			<ul>
			<li>p0 = Today's price for a European put on a futures contract</li>
			<li>c0 = Today's price for a European call on a futures contract</li>
			<li>X = The strike price</li>
			<li>F(0,T) = Price at time "0" for a forward expiring at time "T"</li>
			<li>X - F(0,T) = The face value in long position of a zero coupon bond</li>
			</ul>
			
			 Excluding storage costs, F(0,T) can be substituted with the formula for pricing a forward: S0(1+rF)T
			
			 With the put call parity formula and forward pricing formula, investors can apply the put call parity relationship to exploit mispricings.
		</div>
	
</ul>
<h2>Early Exercise of American Options on Forwards and Futures</h2>


<ul>
<li>Deep in the money American call options on futures behave similarly to the underlying futures contract, so it may make sense for the option holder to exercise the option and establish the futures position, in order to obtain interest from the margin account.</li>
<li>The same logic can be applied to American put options on futures contracts.</li>
<li>Recall that forward contracts will not pay any cash until expiration so this logic will not apply to options on forward contracts.</li>

<li>Therefore, an American call on a forward is the same as a European call on a forward.</li>
<li>However, because of the potential incentive to exercise early (i.e., to earn margin account interest) an American call on a futures contract is different from a European call on a futures contract and is thus more expensive.</li>

</ul>

Put-Call Parity for Options on Forwards

Swaps are private over the counter agreements that are largely unregulated; swaps can be highly customized to meet the needs of the counterparties.

A swap is a mutually agreed exchange of cash flows.

A swap has an initial value of zero; with the passage of time and change of market conditions, the swap may have negative value for one party and positive value for the other party. The party with positive value is exposed to default risk by the party with negative value.

No money is exchanged at swap initiation, with the exception of currency swaps.

Swaps can be categorized as: currency, interest rate, equity (including equity index), and “commodity &amp; other” swaps.

Swap Terminology
<ul>
	<li>Counterparties: The two swap participants</li>
	<li>Notional Principal: Money amount used as part of the calculation in determining the amount of the payment</li>
	<li>Standard (or Plain Vanilla) Interest Rate Swap: One counter party is pay fixed/receive floating and the other counter party is receive fixed/pay floating.</li>
	<li>Term/Tenor/Maturity/Expiration: Length of time over which payment exchanges will take place.</li>
</ul>
Early Exit of a Swap

A counterparty can make an early exit from a swap contract in one of the following ways:
<ul>
	<li>Negotiate a termination agreement with the other counterparty.  Note that the other counterparty may not be willing to cancel the swap.</li>
	<li>Locate a new counterparty and enter into an economically offsetting swap position with this other counterparty.  Note that while the new swap may offset market risk, the party seeking to terminate the initial swap now has counterparty risk exposure to two counterparties.</li>
	<li>Sell the swap to a new party.  Note that the original counterparty to the “would be” seller must approve the swap sale.  This is a rare exit strategy.</li>
	<li>Purchase a swaption that will enter into a swap that economically offsets the original swap.</li>
</ul>

Introduction to Swaps

In an interest rate swap, two parties will agree to: term, fixed rate, floating rate benchmark (commonly LIBOR), notional principal, and payment frequency.

The notional principal is not exchanged; rather it is used to calculate coupon payments.

“Plain vanilla interest rate swap” specifically refers to a fixed-floating agreement; the term “interest rate swap” may refer to plain vanilla or other variations.

<img class="aligncenter size-full wp-image-6422" title="Interest Rate Swap" src="https://financetrain.com/wp-content/uploads/ir-swap.png" alt="" />

As you can see in the above diagram, Party A is paying floating rate on its obligation, but wants to pay fixed rate. Party B is paying fixed rate, but wants to pay floating rate. They can enter into an interest rate swap, and the net result will be that each party can 'swap' their existing obligation for their desired obligation.

A common practice for swap agreements is that the floating rate is set in advance (at contract initiation and immediately following subsequent payment/reset dates) and paid in arrears (the coupon period precedes the actual payment).
<ul>
	<li>Because both the fixed and floating rate are known at initiation, both parties will know who has the larger payment on the first payment date.</li>
	<li>The parties can agree to payment netting, where the party with the larger payment simply pays the other party the difference.</li>
	<li>Fixed Payment = (Fixed Rate/Coupons per year) * Notional Principal</li>
	<li>Floating Payment = (Floating Rate/Coupons per year) * Notional Principal)</li>
	<li>Example: Notional equals $1 million.  At initiation, the fixed rate is 5% and the floating rate is 4%.  The parties have agreed that payment intervals are six months.  Thus, at the end of six months, the fixed rate payer/floating rate receiver will owe $50,000 and the floating rate payer/fixed rate receiver will owe $40,000.  A net payment of $10,000 can be made by the fixed rate payer to the fixed rate receiver.</li>
</ul>
A common motivation for entering into an interest rate swap is that an investor or a company wants to alter its interest rate exposure to better match its assets and liabilities.

Example: A borrower is locked into fixed rate debt and anticipates that interest rates will fall.  The borrower can enter a swap as the floating rate payer/fixed rate receiver.  By receiving a fixed rate, the borrower’s debt is offset and if interest rates fall as anticipated then the floating rate payment on the swap will be lower than the fixed rate payment on the debt.  In a situation such as this the notional principal of the swap will need to equal the amount of the borrower’s debt.

Plain Vanilla Interest Rate Swap

An equity swap is a derivative contract in which two counterparties agree to exchange a set of their future cash flows on preset dates in the future. Structurally, equity swaps are very similar to <a title="Plain Vanilla Interest Rate Swap" href="https://financetrain.com/plain-vanilla-interest-rate-swap/">plain vanilla interest rate swaps</a>: there is a single notional principal and the structure is pay fixed/receive floating.

One party’s payments are based on the cash flows related to the performance of stock or an index. This is called the equity leg. The other payment is based on fixed income cash flow, such as a benchmark rate. However, there are many variations as to what constitutes the other leg.

Equity Swap Motivation: Commonly, a party will enter into an equity swap with the objective of either obtaining equity return exposure for a period of time or hedge existing equity risk exposure for a period of time.

Example of a simple equity index swap:

Party A swaps $1 million at LIBOR + 0.10% against $1 million (S&amp;P to the $1 million notional).

In this case Party A will pay (to Party B) a floating interest rate of LIBOR +0.10% on the $1 million notional and would receive from Party B any increase in the S&amp;P equity index based on the $1 million notional.

In this example, assuming a LIBOR rate of 6% p.a. and a swap tenor of 180 days, the floating leg payer/equity receiver (Party A) would owe (6%+0.10%)*$1,000,000*180/360 = $30,500 to the equity payer/floating leg receiver (Party B).

At the same date (after 180 days) if the S&amp;P had appreciated by 10% from its level at trade commencement, Party B would owe 10%*$1,000,000 = $100,000 to Party A.

Negative Equity Returns: When analyzing the payments on an equity swap it is important to note that while interest rates are almost never negative, equity returns regularly experience periods of negative returns.

Example: In our example, with the equity index referenced in a swap is the S&amp;P 500, with a notional principal of $1 million, if the index returned a negative 5% for the payment period, then the floating rate (equity index) payer/fixed interest rate receiver would actually receive $50,000 for the negative equity return on top of the fixed interest coupon payment.

Equity Swaps

<ul>

<li>Currency swaps have some key differences from <a href="https://financetrain.com/plain-vanilla-interest-rate-swap/" title="Plain Vanilla Interest Rate Swap">interest rate swaps</a>:</li>
	<ul>
		<li>There are two notional principal amounts;</li>
		<li>Each notional is in a different currency;</li>
		<li>The notionals are exchanged at the beginning of the swap and then again at the end of the swap.</li>
		<li>Periodic cash flow payments are made in different currencies.</li>

	</ul>
	
<li>Currency Swap Steps:</li>
	<ol>
		<li>Initial exchange of notionals.</li>
		<li>Periodic exchange of coupon payments; given the different currencies, netting may not be feasible in currency swaps like it is for plain vanilla interest rate swaps.</li>
		<li>Return of notional principals at swap expiration.</li>
	</ol>
	
<li>Motivation: Because a company may have a borrowing advantage in one currency but wishes to have the debt exposure in another currency, the company may wish to enter a currency swap.</li>
	<ul>
		<li>Example: a U.S. company with high quality credit can borrow cheaply within the U.S. The company wishes to build a factory in Colombia, but does not have the credit history to borrow at comparable rates. The company may opt to finance the project with U.S. denominated debt and enter into a currency swap to receive U.S. and pay Colombian pesos.</li>
		<li>Common users of currency swaps are borrowers who wish to convert their debt into a different currency.</li>

	</ul>	
</ul>
Swaps are can be referred to as an "exchange of borrowings" because the parties have independently borrowed financial capital and through the use of a swap have agreed to exchange the proceeds.

Currency Swaps

<ul>
		<li>Pricing: The determination of initial swap terms at the start of the swap's life</li>
		<ul>
			<li>Interest Rate Swap Price = interest rate paid by fixed rate payer</li>
		</ul>
		<li>Valuing: Calculating the market value of a swap at any point in its life.</li>
		<ul>
			<li>A swap's value at initiation is set to zero.</li>
		</ul>
		<li>Key Swap Valuation Concepts</li>
		<ul>
			<li>Analysts must recognize that a swap's cash flows can be replicated by the cash flows from a portfolio of other financial instruments.</li>
			<li>The value of a floating rate instrument is par (1.0) at its start and on all coupon days.</li>
				<ul>
					<li>For example a floating rate bond's value will change in the time between issuance and the coupon payment date, but when the bond's next coupon payment is made, the rate will be reset and the bond value will return to par.</li>
				</ul>
		</ul>
	</ul>

Swap Pricing vs. Swap Valuing

<ul>
	<li>The price of a <a href="https://financetrain.com/plain-vanilla-interest-rate-swap/" title="Plain Vanilla Interest Rate Swap">plain vanilla interest rate swap</a> is quoted as the fixed rate side; never forget that the value of a swap is not the same as the price.</li>
	<li>In order to find the appropriate fixed rate for the interest rate swap's price, the swap can be viewed as a combination of bonds.</li>
</ul>
<h2>Pricing an Interest Rate Swap</h2>
		<ul>
			<li>The coupon rate that equates the value of the fixed rate bond to the value of the floating rate bond must be calculated in order to set the swap value equal to zero at inception. </li>
			<li>The value of the floating rate bond will be par at inception and at each coupon reset date.</li>
			<li>Calculating the fixed rate that will set the initial value of the swap to zero:</li>
				
			<div class="formula1">
			FS(0,n,m) = 1.0 - B0(hn) / SB0(hj)
			
			<ul>
				<li>FS(0,n,m) = The fixed rate on the swap</li>
				<li>B0(hn) = The present value factor for the hypothetical notional principal payment of 1.0.</li>
				<li>B0(hj) = The present value factor for each interest rate payment; this factor is based on the expected floating rate payments in the future.</li>
				<li>The interest rate calculated by this formula will need to be annualized in order to get the swap's price (i.e., if the calculation for a swap that pays every six months generates an answer of 0.025, then the swap's price is 5%)</li>
				<li>This method must be practiced. Candidates can start by walking through and/or practicing problems open book to obtain better understanding and then gradually easing to closed book practice. CFAI may require candidates to price the fixed rate on a plain vanilla interest rate swap on the exam.</li>
			</ul>
			
			</div>
			
		</ul>
<h2>Valuing an Interest Rate Swap</h2>

<ul>
	<li>Most likely, the value of a plain vanilla interest rate swap will only equate to zero at initiation, as interest rates will change over the life of the swap.</li>
	<li>In order to value the swap, an analyst will need to value corresponding fixed and floating rate bonds based on current market place interest rates.</li>
	
	<div class="formula1">
	Value of a Swap (fixed rate receiver) = Value of Equivalent Fixed Rate Bond - Value of Equivalent Floating Rate Bond
	 
	Value of a Swap (floating rate receiver) = Value of Equivalent Floating Rate Bond - Value of Equivalent Fixed Rate Bond
	 
	It is possible to use $1 as the notional principal for valuing the bonds and ramp the results based on the swap's actual notional principal; this may be the easier approach if the swap's notional value is either a high number or an awkward number.
	
	</div>
	<li>One swap party will have a positive value and the other party will have an equivalent negative value.</li>
	<li>As market rates change, the values of the bonds change.</li>
<table class="mytable">
<tr>
<th scope="colspan">INTEREST RATES</th>
<th scope="colspan">SWAP PARTY</th>
<th scope="colspan">VALUE</th>
</tr>
<tr>
 <th scope="row" class="spec">Rise</th>
 <td>Fixed Rate Payer</td>
 <td>Increases</td>
</tr>
<tr>
 <th scope="row" class="specalt">Rise</th>
 <td class="alt">Fixed Rate Receiver</td>
 <td class="alt">Decreases</td>
</tr>

<tr>
 <th scope="row" class="specalt">Fall</th>
 <td class="alt">Fixed Rate Payer</td>
 <td class="alt">Decreases</td>
</tr>
<tr>
 <th scope="row" class="spec">Fall</th>
 <td>Fixed Rate Receiver</td>
 <td>Increases</td>

</tr>
</table>
</ul>

Pricing and Valuing a Plain Vanilla Interest Rate Swap

<ul>
<li>Four types of <a href="https://financetrain.com/currency-swaps/" title="Currency Swaps">currency swaps</a> exist:</li>
	<ul>
		<li>Pay one currency at a fixed rate, receive another currency fixed rate.</li>
		<li>Pay one currency at a fixed rate, receive another currency at a floating rate.</li>
		<li>Pay one currency at a floating rate, receive another currency at a fixed rate.</li>
		<li>Pay one currency at a floating rate, receive another currency at a floating rate.</li>
	</ul>
<li>A currency swap will have some (not all) similarities to an interest rate swap:</li>
	<ul>
		<li>Value is zero at initiation.</li>
		<li>For currency swap types 1-3, a fixed rate must be priced that so the present values to each party are equal.</li>
		<li>Regarding type 4, since both rates are floating, a fixed rate does not need to be found.</li>
	</ul>
<li>Currency Swap Differences to Interest Rate Swaps</li>
	<ul>
		<li>For currency swaps, an interest rate must be priced for each currency.</li>
		<li>Each side of the currency swap has its own notional principal in its own currency. Therefore, if one side of the swap has a notional set to 1, then the notional for the other party will be 1/exchange rate.</li>
	</ul>
<li>For currency swap type 1 (fixed-fixed), the rate on the domestic side is the fixed rate on a plain vanilla interest rate swap in the home country. The rate on the foreign side is the fixed rate on a plain vanilla interest rate swap in the foreign country.</li>
<li>For swap types 2 and 3, (fixed-floating, floating-fixed), the fixed rate sides are determined by the fixed rate on a local swap and the floating rate will be based on the local short term rates.</li>
<li>For swap type 4 (floating-floating), the floating rates are based on local market floating rates.</li>
<li>Given the incremental complexity of pricing currency swaps over pricing interest rate swaps, candidates are encouraged to obtain proficiency in successfully pricing plain vanilla interest rate swaps before pricing currency swaps.</li>
<li>Valuing Currency Swaps</li>
	<ul>
		<li>The same approach to valuing interest rate swaps applies to valuing currency swaps, with the additional complexity of moving exchange rates. This will make calculating the present values to each swap party a little more complicated.</li>

	</ul>
</ul>

Pricing and Valuing Currency Swaps

<ul>
<li>Three common types of <a href="https://financetrain.com/equity-swaps/" title="Equity Swaps">equity swaps</a> are:</li>
	<ul>
		<li>Pay Fixed Interest Rate, Receive Return on Equity</li>
		<li>Pay Floating Interest Rate, Receive Return on Equity</li>
		<li>Pay Return on One Equity, Receive Return on a Different Equity</li>
	</ul>
<li>Pricing Equity Swaps</li>
	<ul>
		<li>The same formula used to find the fixed interest rate when pricing a <a href="https://financetrain.com/plain-vanilla-interest-rate-swap/" title="Plain Vanilla Interest Rate Swap">plain vanilla interest rate swap</a> or a currency swap to obtain an initial swap value of zero is applied.</li>
		<li>The market value of a pay floating-receive return on equity swap is automatically zero at swap initiation since the floating rate portion of the swap equals 1.0 (making the numerator of the rate pricing equation zero, because it is 1 minus 1).</li>
		<li>Pricing an equity for equity swap can be done by going long on one stock and short on the other. Like other swaps, this swap is valued at zero on initiation.</li>
	</ul>
<li>Valuing Equity Swaps</li>

		The market value of an equity swap at any day "t"
		<div class="formula1">
		= (St/S0) - Bt(hn) - (FS(0,n,m)*&Sigma;Bt(hj)) 
		<ul>
			<li>St = stock (or index) price at time t</li>
			<li>S0 = stock (or index) price at time 0</li>
			<li>Bt(hn) = present value of $1 to be repaid at swap's expiry time "n"</li>
			<li>FS(0,n,m) = fixed rate on the swap</li>
			<li>Bt(hj) = present value factor for each interest rate payment, based on the current structure of interest rates; these are calculated at time t and summed.</li>
			
		</ul>
		
		</div>

</ul>

Pricing and Valuing Equity Swaps

<ul>
	<li><a href="https://financetrain.com/plain-vanilla-interest-rate-swap/" title="Plain Vanilla Interest Rate Swap">Interest Rate Swaps</a> as a Series of Forward Rate Agreements</li>
		<ul>
			<li>An interest rate swap can be recreated by a series of forward rate agreements (FRAs).</li>
			<li>A difference between an interest rate swap and a series of FRAs is that the swap will have a single fixed rate, but the forward contracts will be priced at different interest rates based on the slope of the yield curve (unless, of course the yield curve is flat, then a single rate would hold for the series of FRAs).</li>
		</ul>
	<li>A plain vanilla interest rate swap can be viewed as a combination of long and short positions in <a href="https://financetrain.com/introduction-to-futures-contracts/" title="Introduction to Futures Contracts">interest rate futures contracts</a>, where each futures contract would have successive maturity dates.</li>
	<li>A <a href="https://financetrain.com/currency-swaps/" title="Currency Swaps">currency swap</a> replicates the act of issuing fixed or floating rate debt in one currency and using the funds raised to purchase a fixed rate or floating rate bond in another currency.</li>
	<li>An <a href="https://financetrain.com/equity-swaps/" title="Equity Swaps">equity swap</a> is akin to issuing one security and using the funds raised to purchase another security.</li>

</ul>

Swaps as Theoretical Equivalents of Other Derivatives

<ul>
	<li>Swaption provides option holder the option to enter into a swap.</li>
	<li>Payer vs. Receiver</li>
		<ul>
			<li>Payer Swaption: The holder can enter into a swap as the fixed rate payer/floating rate receiver</li>
			<li>Receiver Swaption: The holder can enter into a swap as the floating rate payer/fixed rate receiver.</li>
			<li>Parties who expect the need for a swap in the future and want to lock in the swap rate now are common users of swaptions.</li>
			<li>Swaptions provide flexibility to not enter a swap or postpone swap entry for a more desirable rate.</li>
			
		</ul>
	<li>Interest Rate Swaptions - Payoffs and Cash Flows</li>
	The holder of a payer swaption with positive value can realize this positive value in three ways (note the swaption holder will be in a situation where the floating rate received exceeds the fixed rate paid):
	
		<ol>
			<li>Exercise the swaption and enter into a pay fixed-receive floating interest rate swap; note that this strategy entails risk as interest rates could change and thus change the floating payment received.</li>
			<li>Exercise the swaption and enter another pay floating-receive fixed interest rate swap at current rates. The income and outgoing swaps will offset and the swaption holder has created an annuity for him/herself.</li>
			<li>The swaption holder may be able to arrange to receive a lump sum payment equal to the present value of the annuity created in approach #2.</li>
		</ol>

</ul>
<h3>Value of an Interest Rate Swaption at Expiration</h3>

<ul>
<li>Payer Swaption payoff at expiration (based on $1 notional) =</li>

	<div class="formula1">
	 = Max[0,FS(0,n,m) - x] &Sigma;B0(hj) 
	
		<ul>
			<li>FS(0,n,m) = Market rate on the underlying swap at swaption expiration.</li>
			<li>X = The exercise rate that the payer would pay under swaption terms</li>
			<li>B0(hj) = Present value factor for each interest payment, based on the term structure at the expiration of the swaption</li>
		</ul>
	</div>

<li>Receiver Swaption payoff at expiration (based on $1 notional) =</li>

	<div class="formula1">
	 = Max[0, x - FS(0,n,m)] &Sigma;B0(hj) 
	
		<ul>
			<li>FS(0,n,m) = Market rate on the underlying swap at swaption expiration.</li>
			<li>X = The exercise rate that the receiver would receive under swaption terms</li>
			<li>B0(hj) = Present value factor for each interest payment, based on the term structure at the expiration of the swaption</li>
		</ul>
	</div>


</ul>

Swaptions and their Valuation

<ul>
	<li>Current <a href="https://financetrain.com/overview-of-credit-risk/" title="Overview of Credit Risk">Credit Risk</a>: The situation where one swap party is owed a payment now and the other party cannot make the payment.</li>
	<li>Potential Credit Risk: The possibility that the other party may default in the future.</li>
	<li>The amount at risk for default is equal to the swap's market value at any given point in time.</li>
	<li>While swaps are over the counter, the parties can agree to a marking to market method for controlling credit risk.</li>
	<li>Swap spread is the difference between a swap's fixed rate and an equal maturity risk free rate, such as the yield on government debt with a same maturity.</li>
	<li>Swap rates can be quoted as a spread over equivalent risk free rates; this spread can be interpreted as a gauge of credit risk in the general market.</li>
		<ul>
			<li>During times of economic turmoil, swap spreads tend to widen.</li>
		</ul>

</ul>

Swap Credit Risk and Swap Spread

<ul>
	<li>Interest rate caps and floors are option like contracts, which are customized and negotiated by two parties.</li>
	<li>Caps and floors are based on interest rates and have multiple settlement dates (a single data cap is a "caplet" and a single date floor is a "floorlet").</li>
	<li>Like other options, the buyer will pay a premium to purchase the option, so the buyer faces credit risk.</li>
	<li>Caps are also called ceilings because the buyer is protected from interest rates rising above the strike rate.</li>
		<ul>
			<li>The payment to the option holder when rates rise above the strike rate is the difference between the market rate and the strike rate, multiplied by the notional, and divided by the number of settlements per year.</li>

		</ul>
	<li>Floors set a minimum interest rate payment because if interest rates fall below the strike rate the floor holder is protected; payments are calculated the same as caps.</li>
		<ul>
			<li>Floors are commonly employed by floating rate bond holders to protect their rates from falling below a certain level.</li>

		</ul>
		<div class="formula1">
		Cap Payment = Max[0; Notional &times; (Index rate - Cap strike rate) &times; (Days in settlement period / 360)]
		 
		Floor Payment = Max[0; Notional &times; (Floor strike rate - Index rate) &times; (Days in settlement period/360)]

		</div>
</ul>
<h3>Cap and Floor Payoffs and Interest Rate Collars</h3>

<ul>
	<li>An interest rate collar can be created by buying a cap and selling a floor.</li>
	<li>This creates an interest rate range and the collar holder is protected from rates above the cap strike rate, but has forgone the benefits of interest rates falling below the floor rate sold.</li>
	<li>When the cost of the floor sold equals the cost of the cap purchased, it is called a "zero cost collar".</li>
	
	
</ul>

Interest Rate Derivatives - Caps and Floors

<h2>What are Credit Default Swaps (CDS)?</h2>
<ul>
	<li>A Credit Default Swap is an agreement between two parties in which a protection buying party pays a premium to a protection selling party; in return for this premium the protection selling party will pay the protection buying party a specified notional amount if a specified credit event takes place in a specified time period affecting a specific debt instrument (called the reference entity).</li>
		<ul>
			<li>Credit events include bankruptcy, bond default, or debt restructuring.</li>
		</ul>
	<li>Buying a CDS means buying credit protection and paying the premium.</li>
	<li>Selling a CDS means selling credit protection and receiving the premium.</li>
	<li>Physical CDS Settlement: CDS buyer delivers the referenced entity (ex. transfers bond ownership) to the CDS seller and the CDS seller pays the notional amount to the CDS buyer.</li>
	<li>Cash CDS Settlement: CDS seller pays the net difference between the notional amount and the market value of the referenced entity (ex. the defaulted bond may be trading for some small fraction of par on the market).</li>

</ul>
<h2>Advantages of CDS</h2>

<ul>
	<li>Credit default swaps provide a method for investors to independently manage credit risk and interest rate risk on fixed income securities.</li>
	<li>CDS allow investors to profit from credit events without actually holding a short position in the underlying security.</li>
	<li>When the bond market does not offer an investor the desired maturity, currency, and credit exposures, swaps and CDS can be used to create the desired exposure.</li>
	<li>CDS can allow a loan portfolio to transfer credit risk confidentially.</li>
</ul>

<h2>Users of CDS</h2>

<ul>
<li>Banks: Can use CDS to hedge loan portfolio risk and manage regulatory capital requirements.</li>
<li>Investment Banks: Can use CDS to hedge the credit risks on bond inventories for bonds in which they are market makers.</li>
<li>Hedge Funds: Can use CDS to execute strategies.</li>
<li>Insurance Companies: Sell CDS to generate income and diversify their portfolios (but this can be risky business, ask AIG).</li>
<li>Investment Managers: Can use CDS to attempt to exploit market mispricings to generate excess returns.</li>
<li>Corporations: Can use CDS as part of their risk management strategies.</li>
</ul>

Credit Default Swaps (CDS)

<ul>
	<li>Basis Trades: Made based on the difference between a bond's yield and the <a href="https://financetrain.com/credit-default-swaps-cds/" title="Credit Default Swaps (CDS)">CDS</a> premium.</li>
	<li>Curve Trades:</li>
		<ul>
			<li>Flattener: Buy the short term CDS and sell the long term CDS</li>
			<li>Steepener: Sell the short term CDS and buy the long term CDS</li>
		</ul>
	<li>Index Trades: An investor can buy or sell a credit index</li>
	<li>CDS Basket Trades (or Correlation Trades): Credit protection can be sold on an entire portfolio of bonds; the seller pays only for the individual issue that defaulted and the basket continues.</li>
		<ul>
			<li>First to Default Baskets: seller will deliver the entire notional amount of the basket once the first default takes place.</li>
		</ul>
	<li>CDS Options:</li>
		<ul class="list2">
			<li>Payer Options: Provide the option holder the right to buy credit protection in the future.</li>
			<li>Receiver Options: Provide the option holder the right to sell credit protection in the future.</li>
		</ul>	
	<li>Capital Structure Trades: Designed to exploit mispricings in a company's capital structure.</li>
		<ul>
			<li>For example, if a trader expects a company to increase its leverage and does not feel that this has been appropriately priced in the market, then he/she may buy a CDS and buy a long stock call option (in order to make money if the stock rises).</li>
			<li>Other capital structure trades may involve taking a long position on the CDS for one debt issue by a company and taking a short position in another debt issue by the same company.</li>
		</ul>		
</ul>

Credit Derivative Trading Strategies

Derivatives Part 2

This frontmatter description will appear below the title

COURSE

Derivatives Part 2

Lessons