## Reading 56 – Derivative Markets and Instruments

Derivative securities are an asset class where they derive (hence the name) their value from an underlying asset. These underlying assets can be financial assets like equities or bonds or real assets like commodities. Derivatives allow investors to buy or sell exposure to these assets without having to buy the assets themselves. A derivative is a legal contract between the buyer and seller, who are referred to as the “long” and “short” (respectively) sides of the trade.

Derivatives are split into two classes: Forward Commitments (like futures and forwards contracts) provide the ability to lock in a purchase price of an asset at a future date and a specified price, and Contingent Claims, which are options (but not obligations) to transact in the future at a specified price. As forward commitments are contractual agreements to transact in the future, their payoff profiles are linear in nature and follow the price movements of the underlying asset. Contingent claims are not obligations, so their payoff profile is dependent on the actions of the buyers.

Investors can buy and sell derivatives either through an exchange or over-the-counter (OTC). Exchange-traded derivative contracts have the advantage of being more standardized and offer more transparency. OTC contracts offer more flexibility in customizing their terms through direct negotiations between buyers and sellers.

There are three primary types of forwards commitment: Forward contracts, Futures contracts, and Swaps. A Forward Contract is an agreement that obliges the buyer to purchase an asset at a specified priced at a later date. These contracts are traded OTC. The payoff profile for a forward contract is calculated as follows:

$$Buyer\quad payoff={ S }_{ t }-{ F }_{ 0 }\left( T \right)$$
$$Seller\quad payoff=-\left[ { S }_{ t }-{ F }_{ 0 }\left( T \right) \right]$$

In these formulas, St = the price of the underlying asset when the contract expires and F0(T) = the purchase price specified in the forward contract. As you can see, the buyer is profitable if the price they locked in when the forward was created is less than the asset’s market price at the time of contract expiration. During the life of the contract, its market value on the balance sheet of the buyer is that difference between the contractual purchase price and the current market value of the underlying asset.

Futures Contracts are very similar to forwards, in that they oblige the buyer to purchase an underlying asset at a specified price, but are much more standardized and typically trade on exchanges. Another important difference is that the daily gains and losses on the contract price over time (as the underlying asset changes in value) is settled with the exchange. If the buyer is “out of the money” because the asset price is below the contractual price, then the buyer must supply that difference in cash as collateral to the exchange. This process is known as the “mark-to-market” and is designed to prevent buyers and sellers from being exposed to counterparty risk in the event of a buyer or seller of these contracts going bankrupt while owing money. The payoff profiles are calculated using the same formulas as forwards.

Swap Contracts are essentially agreements wherein two parties exchange a series of cash flows. These are OTC contracts that are privately negotiated and highly customized. The most common swap (known as a “plain vanilla swap”) involves the exchange of fixed-rate payments for floating-rate payments. The contract will have a notional principal amount that will determine how large the payments will be, but the cash flows between the counterparties will only be the net difference between what is owed by each side. In a plain vanilla swap, if the floating rate exceeds the fixed rate during one payment period, then the only cash flow will be the fixed rate buyer paying the floating rate buyer the amount equal to how much their payment exceeds the payment from the floating rate buyer. There is no need for both sides to make the full payments since one side’s will always be smaller and therefore totally cancelled out.

The first type of contingent claim security is an Options Contract. As the name implies, an option is the right (but not the obligation) to buy or sell an asset at a predetermined price in the future. The right to buy a security is a Call and the right to sell is a Put. In an American-style option, the contract can be exercised any time before its expiration date, while a European-style contract can only be exercised at the expiration date. The predetermined price at which the option can be exercised is known as the Strike Price.

The payoff profile is similar to that of futures contracts but with an added consideration. Since the buyer will only exercise the option if it is “in the money”, then their payoff will either be the amount by which the market price of the underlying asset exceeds the strike price (or the opposite for a put option) or 0. Since the total profit from an options contract also includes the premium that the buyer paid to the seller to initiate it, the profit calculations are as follows (for a call contract):

$$Buyer\quad Profit=Max\left( 0,{ S }_{ t }-X \right) -{ c }_{ 0 }$$
$$Seller\quad Profit=-Max\left( 0,{ S }_{ t }-X \right) +{ c }_{ 0 }$$
$${ S }_{ t }=price\quad of\quad underlying\quad asset$$
$$X=strike\quad price$$
$${ c }_{ 0 } = premium$$

The worst that the buyer can do is lose the value of the premium they paid, but they have unlimited upside to their profit, while it’s the exact opposite for the seller. This calculation applies to call options. For put options, the calculation for the payoffs reverses the St and X values, since it is profitable for the buyer when the security they are selling is worth less than the strike price.

Credit Derivatives are another common form of contingent claim. These are used to provide protections to buyers in the event of a credit event. Total Return Swaps are commonly used wherein the buyer agrees to pay a specified interest rate while the seller pays the total return (interest and capital) of a given bond security. The seller must continue to make these payments even if the bond defaults, so the buyer is now exposed to the returns from that bond with no risk of losing out in the event of default. Credit Spread Options are similar to typical options except that they are based on the credit spread of a specified bond, rather than the cash price. The credit spread of a bond is the amount by which its yield exceeds a given rate, which is usually a Treasury rate of the same maturity. Credit Default Swaps act like insurance policies for default events. In exchange for a series of regular payments from the buyer, the seller agrees to pay a specified amount to the buyer if the underlying bond goes into default.

Asset-backed Securities, which you should remember from the fixed income section, are also a form of contingent claim contract. They divide up the cash flows from a pool of assets into different levels (“tranches”) based on the different levels of risk that buyers want to take on. The riskiest tranches have the highest yields but are also the first to lose principal value if there are defaults or prepayments in the pool of assets. ABS securities made up of mortgage loans are known as Collateralized Mortgage Obligations (CMO) and ones made up of bonds are Collateralized Bond Obligations (CBO) or Collateralized Loan Obligations (CLO), both of which fall under the broader classification of Collateralized Debt Obligation (CDO).

There are a number of benefits and issues with the derivatives market. One positive aspect is the way in which futures and options prices give insight into their underlying assets. Since both types of contract are very liquid and cheap to purchase (relative to their underlying assets), the prices at which they trade provides information about the expected price and volatility of those assets. In general, derivatives have lower transactions costs, greater liquidity, and are easier to gain short exposure than the assets to which they correspond.

On the negative side, many believe that derivatives markets are so easy to use that they make it too easy for speculators to make large short-term bets that are closer to gambling activities than investing. Since derivatives make it easy to take on leveraged positions (exposure to larger asset amounts than one could purchase), this speculative activity could have a destabilizing impact on the markets. Another concern is that the complexity of some derivative assets makes them more volatile because the investors using them may not totally understand the risks to which they are being exposed.

## Reading 57 – Basics of Derivative Pricing and Valuation

Arbitrage plays an important role in keeping markets stable and functioning properly. The Law of One Price states that an asset should trade at the same price regardless of venue, and arbitrage is the mechanism by which this is enforced in the market. In markets with low transactions costs and good information flow, investors can spot pricing discrepancies and use trading activity to take advantage. If a security is trading at a higher price in one market over another, an investor can quickly buy the security at the lower price and sell at the higher price. This activity provides a return without the investor having to take on any risk since their net position is zero after completing both transactions. As these trades occur, the prices in the two markets will converge on one value.

One difference between derivative valuation and other asset classes is that the calculations for derivatives assume a risk-neutrality, rather than risk aversion, among investors. Since the arbitrage mechanism means that any investor can earn the risk-free market return, we can use the risk-free rate and not have to add a risk premium when discounting to find present or future values. The price of derivatives is found by assuming that there are no arbitrage opportunities available in the market, which is referred to as The Principle of No Arbitrage.

Derivative securities are also treated differently than other assets because the verbiage used is not the same. For equity securities, an analyst’s goal is to determine the correct fundamental value of the security and either buy and sell it depending on how that value corresponds to the current market price. In the case of futures, forwards, and swaps, however, the value of the contract becomes positive or negative in response to changes in the price of the underlying asset. We can not refer to the “price” of the contract in that sense, because the price is the original cost of the contract at the time it was created.

Forward contracts are valued in based on how the price of the underlying securities changes over the life of the contract. The “price” of the contract is a fixed value, since it corresponds to the agreed upon price at inception of the contract. The value of the contract is what fluctuates over time. At the time the contract is created, its value is equal to 0 and the forward price is equal to the spot price discounted to a future value at the risk-free rate, as written out below:

$${ V }_{ 0 }\left( T \right) =0$$
$${ F }_{ 0 }\left( T \right) ={ S }_{ 0 }{ \left( 1+r \right) }^{ T }$$

During the life of the contract, the value is the spot price of the underlying asset minus the present value of the forward price (discounted by the remaining time left):

$${ V }_{ t }\left( T \right) ={ S }_{ t }-{ F }_{ 0 }\left( T \right) { \left( 1+r \right) }^{ -\left( T-t \right) }$$

At the expiration of the contract, the value is simply the spot price of the underlying asset minus the forward value (with no discounting):

$${ V }_{ t }\left( T \right) ={ S }_{ t }-{ F }_{ 0 }\left( T \right)$$

There are some costs, both implied and explicit, involved with holding futures and forwards contracts in addition to the contractual expenditures. The Opportunity Cost is already accounted for in the present value calculation but represents the potential loss or gain that could have been incurred by spending money on these contracts compared to another investment. There are Carrying Costs associated with any position that is held for a length of time. Fortunately, the carrying costs of a futures or forwards contract is necessarily lower than that of the equivalent cash security. The Convenience Yield is the biggest non-monetary benefit of using a derivative security. It is the benefit of holding the derivative over directly owning its underlying asset. This can be quite large, especially when looking at securities that represent physical assets that could require significant holdings costs like crude oil.

Forward Rate Agreements are a type of forward security wherein the investor agrees to make a fixed interest payment at a future date in exchange for an unknown future interest payment they will receive. It behaves similar to a swap agreement, except it is typically for a single payment. Like a swap, it is an OTC security.

Forwards and futures shares many features in their basic function, but have a few differences that are important to note. The primary difference is that futures are traded over an exchange while forwards are OTC. Since a forward is traded directly between two counterparties, there is a greater risk of one side failing to make its obligated payments or deliver the underlying security compared to a future. This is known as default risk. In addition to the decreased counterparty risk, and because it is traded through an exchange, futures are also less risky in this regard because of the mark-to-market process. The most unrealized gain or loss that a futures investor is at risk of losing is the current day, rather than all that has accumulated since the contract inception or last payment date.

Swap Contracts can be more complicated to value properly compared to futures and forward securities because they tend to involve a series of cash flows, rather than a single transaction. A swap can be looked at as a series of forward contracts where we calculate a present value for each one to find the total value of the swap. Similar to other derivatives, the price of the swap is determined at its inception, but the value will change over the life of the contract. A simple way to visualize the value of the swap is to think of each series of payments as a bond with either a fixed or floating interest rate, and the value of the swap is the net difference of the two. If we were buying the fixed payments, our value would be equal to the value of the fixed rate bond minus the value of the floating rate bond.

We’ve already covered the payoff and profit calculations of options securities, but there are several additional considerations for understanding their values. The Exercise Value differs between American and European options because American options can be exercised at any point during the life of the contract, while European option exercise value is simply the value of the contract at the expiration since that is the only time they can be exercised. The Time Value of options is simply the fact that the more time there is until expiration, the more valuable the security is. Moneyness is the measurement of where the option strike price is in relation to the underlying security price. If the strike price is higher than the underlying, it is out of the money and won’t be exercised, while if it is lower, than the option is in the money and could be exercised.

Other factors that affect the value of options securities include the current risk-free rate. The holder of a call option can factor in the risk-free return they can get on the capital that they would spend to exercise their option since they know they retain the choice as long as they hold the contract. The higher the risk-free rate, then the higher the value of the call option. The opposite is true of the put option since the contract holder is missing out on the risk-free return they could get with the proceeds of the sale. Volatility is another major factor. Since increased volatility means an increased chance that the underlying security price goes into the money, higher volatility means a higher value for the options contract. Payments like dividends and interest of the underlying security are not paid to the holder of an option, so these are seen as a negative factor for call options and a positive for put options.

Options contracts can never be worth less than 0 because the holder of the contract doesn’t have to exercise them if they are out of the money. The lowest value is calculated as the maximum of either 0 or the present value of the underlying minus the strike price. For put options, it is between 0 and the strike price minus the PV of the underlying.

The concept of put-call parity is an important one when looking at options contracts. It relies on the assumption that there are no arbitrage profits to be made because market participants would immediately take advantage of any that appear and they would not last. This relationship allows the creation of synthetic put options since a fiduciary call option and protective put option must have the same value at inception. We can create the same exposure as a put by owning long call, a short in the underlying, and a long position in a risk-free bond. These relationships must hold true or there would be arbitrage opportunities to exploit. A similar approach exists for put-call forward parity since we can create a synthetic forward using a long call position, short put position, and a zero-coupon bond with a face value equal to the fair value of the total position.

A binomial tree is a useful tool for trying to model option contract values over a period of time. It is essentially a series of decision points that allows you to get a weighted average value for future points in time. The binomial tree splits at each outcome point, at which step you calculate the value of the contract given that particular outcome and weight that by the probability of that outcome. The primary outcome occurring is whether the value of the underlying asset goes up or down. Since each point only has two outcomes, the probabilities for them are always x and 1-x. Because of the manual nature of calculating each step, the exam problems using this method typically only include 2-3 steps in a binomial tree.

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