Derivative Securities Princing - Appunti

Derivative securities pricing

Lecture 1

Derivatives trading process is getting very similar to the stock trading system. However, they remain very difficult and risky instruments. That is why the learning outcome is to understand what derivatives are and how they can be priced. Having a strong background in derivatives is fundamental for graduates in finance nowadays. As we have said, derivatives are risky, and a thorough knowledge is necessary in order to understand how they work. The user of the contract has to understand what a fair price could be, which is way more difficult than just looking at the market price on the internet. Estimating a value is complicated because various models and variables are taken into consideration. Users should, before signing a contract, have a complete picture of the risky factors which make it up.

Before estimating, we should introduce what a derivative is. A derivative is an instrument whose value depends on an underlying asset, therefore the value of a derivative is always dependent on another asset. In order to assess the value of a call option, we need to estimate the value of the underlying asset at the derivative's maturity by taking into account the multiple market factors which come into play at time T (the value of the call depends on the value of the asset at T, but we need to estimate it at time 0). Hence, the value of an option is not a value per se, it depends on the value of the underlying asset. The big difficulty is the estimation process because we do not know what the asset's price at maturity will be. In order to do that, we need to use models which can provide us with tools to assess whether the current option price is fair or not.

Derivatives are important because they can be used for transferring risks in the economy; they work similarly to insurance contracts. Companies sometimes cannot afford to carry risks in their books and thanks to these kinds of instruments, they manage to transfer risk to their counterpart. Transferring risk means finding an edging instrument that brings a profit when the portfolio incurs in a loss. This is known as the fundamental principle of edging.

Derivatives are important at a macro level (for the financial system) as well as at a micro level (for the individual investor). They are traded in different exchanges (such as the CBOE and CME). Otherwise, they can also be traded in the OTC market (bilateral trades) where corporations interact with each other directly.

The OTC market prior to 2008 presented various issues:

• Largely unregulated
• Banks acted as market makers (quoting bids and asks)
• Master agreements usually defined how transactions between two entities would be handled, although some transactions were already cleared through central counterparties (CCPs)

Pay attention to the correct meaning of "trade" and "clear". The latter happens when one party has to pay; this is the part of the process where we are able to understand whether the counterpart is defaulting or not. Here it comes the role played by the CCPs, because it knows if counterpart A or B are out of money and assesses the possibility of default (in order to reduce this possibility, it requires previous deposits).

The financial regulators wanted to increase transparency and reduce systemic risk, especially after the 2008 crisis. That is why the OTC market has been regulated since then. Now, both in the US and EU, standardized OTC products must be traded on Swap Execution Facilities. These are electronic platforms similar to regulated exchanges (ESMA systematically updates the widely used contract that must be traded on a trading-based system). Furthermore, there is a clearing obligation in order to reduce the probability of default (with this, we will know in advance whether or not a bank is in difficulty). Lastly, all trades must be reported to a central repository. In this way, authority can have a big picture of what is happening in the market. These are the main changes in OTC products trading process since the last financial crisis.

Lecture 2

We should not forget that a derivative (such as any financial instrument) requires two counterparts in order to be executed, which are the buy side and the sell side. The former can be represented by government, mutual funds, hedge funds, private equity funds, and wealth management offices. These players buy these kinds of financial products from the sales department through financial markets. Such office decides the strategy to implement then it communicates internally with the trading department, which is specialized in managing risks. Of course, within the scheme there are also advisory firms who can advise companies and banks on the most accurate action to make. Front sales offices and trading departments are the profit-driver player, whereas the structuring department is responsible for other stuff. Instead, the middle sales office is responsible for bureaucratic matters, indeed the salary is lower.

Derivatives were mostly traded in OTC prior to 2008. Since then, we are moving towards a more regulated OTC market. In 2008 derivatives were widely used by large banks, whose default risk was thought to be null, and no collaterals were asked for the subscriptions of such instruments. However, the story taught us the opposite. What are such instruments used for though? Derivatives are used for hedging risks and speculating. Hedging a risk fully can be pretty expensive and sometimes complicated as well, therefore some clients may require the team to hedge the risk partially. In addition to that, derivatives can be used to change the nature of a liability (the most common way: interest rate swaps). Industries desperately need hedging services; you would be surprised to know the actual use of derivatives. Just imagine a wooden deck maker which buys wood from its suppliers in advance. The mark-up for such a desk is tight and the company cannot afford to lose any of that because of the variation of wood prices. The only strategy is hedging this risk through a future or forward contract on wood.

The first and very easy kind of derivative we'll talk about is the forward. The forward price for a contract is the delivery price that would be applicable to the contract if it were negotiated today. At the time of subscription, the counterparts are already locked in the obligations that the contract implies, which will be finalized in some months' time. At subscription, the agreement price is discussed. At maturity, the payment must be made for the delivery of the underlying asset of the contract. The party that has agreed to buy is in the long position, whereas who has decided to sell is in the short position. The pricing formula is always the formula that does not contain arbitrage opportunities (between spot and future markets).

Investors may confuse forwards and futures, but they are actually quite different. Both refer to an obligation that has to be completed in the future, for which there are no outflows at the time of the agreement. Nevertheless, the futures are always traded on an exchange. That is why future are more detailed and fixed instruments (they are the same no matter the market they are exchanged from). The main difference from forward to future is the standardization within the contracts. Furthermore, future contracts are set daily, whereas forwards are set at maturity. Because of that, forward contracts may involve some credit risks. "Setting" means that the counterparts have to provide deposits through the CCPs. These deposits involve two main margins: the initial margin and the maintenance margin. Margins have to be made in order to be sure that the counterpart cannot default. Since there are no "daily setting" in forward contracts, they do not require any margins. Despite their differences, future and forward prices should converge to the same value as maturity gets closer.

Some terminology before proceeding:

• Volume: the volume of trades within a trading day
• Open interest: the total number of contracts outstanding

Hence, a stock may have high trading volume with low open interest at the same time, this is because there have been massive day trading activities (positions were opened and closed multiple times within the day). Another point that deserves to be pointed out is that closing a future requires entering into an offsetting trade and it must be closed before maturity if the counterpart does not want to proceed with the delivery of the underlying asset, which is physical or in cash (depends on the kind of the underlying asset traded).

Moreover, derivatives can be traded through CCPs or with bilateral transactions, which require collateralization in order to be sure that default is covered from something (credit support annex CSA).

We briefly talked about futures and forwards, now we will talk about options. Bear in mind that derivatives are priced with no arbitrage opportunities method.

There are two kinds of option:

• Call option: an option to buy a certain asset by a certain date for a certain price
• Put option: an option to sell a certain asset by a certain date for a certain price (strike price)

As we have said, an option represents an opportunity to buy or sell. Since no one would execute an option which is out of the money, it is impossible for an option to have a negative value (for definition). The buy side of an option, whether a call or a put, is the long position. Whereas the sell side of an option is in the short position.

Options can also be categorized in:

• European option: it can only be executed at maturity
• American option: it can be executed anytime

The course will provide pricing methods for both. Bear in mind that the main feature of options is that prices are more expensive as maturity increases because the underlying asset has more time to go up or go down. Leveraged strategies can be performed with options as well.

Lecture 3

The lecture of today will be focused on swap contracts, which are about interest rate. The basic definition of a swap contract is the "plain vanilla" contract, in which the counterparts swap fixed interest rate with floating interest rate (benchmark rate, provided by the market). The latter may be represented by the LIBOR (London Interbank Offer Rate), which is the main reference. It is the rate of interest at which an AA-rated bank estimates it can borrow money on an unsecured basis from another bank at 11am. As the definition states, LIBOR does not refer to actual transactions, but it is estimation-based. This interest rate depends on the currency and the maturity we are referring to (the European standard maturity is 6 months, whereas the American standard maturity is 3 months). Such difference is important because investors must relate to the most liquid market within a specific market, which has different maturity for Europe and United States.

The rate is now processed by central banks, whereas in the past it was administered by private banks. It is easily understandable why it could be corrupted in the past, since it was in private banks' interest to adapt the rate at their preferences. In addition, regulators plan to phase out LIBOR by the end of 2021 and replace it with rates based on transaction observed in the overnight market (less risky rate). Since they refer to overnight transactions and the market would need rates for longer maturity, the overnight rates will be aggregated according to specific needs. Such new reference rates will be SOFR (US dollar), SONIA (GBP sterling), ESTER (Euro short-term rate). The main feature of the new American rate is that it is calculated from "repos". Such instruments are "repurchase agreements", where A borrows money to B, who gives a collateral (govies) to the lender. In this way, if B defaults, A can keep the govies in order to cover the losses. Because of that, SOFR is said to be a secured rate which means that the rate is lower because the transactions are less risky for the lender. On the contrary, ESTER rate is calculated on the wholesale euro unsecured overnight borrowing costs of euro area banks. Before such rate, the interest rate overnight was EONIA, which was a lending rate (lending costs in the interbank market), whereas the ESTER is calculated on the borrowing costs of the wholesale market, and it is also based on more representative market data. Furthermore, as we have said, ESTER is a daily rate even though swaps may be 6 months long. Hence, it has to be compounded in order to obtain the 6 months rate.

Swap option is an agreement to exchange cash flows at specified future times according to certain specified rules. Plain vanilla is the most common contract (fixed rate vs floating rate). Once the basic elements of the contract are established (reference rate, future dates), the valuation of the counterpart defaulting risk has to be considered (it depends on the identity and reliability of the counterparts). At the beginning, the contract must be based on a risk-free rate, for which OIS rate is a good proxy (max one year). Unfortunately, since OIS rate does not have any kind of maturity we will have to choose another rate to use as a proxy for longer maturity. In this case, we should refer to bond rates which, for longer maturity, are mostly coupon bonds. Since their rates are influenced by previous cash flows, zero rates bonds would be the perfect choice but they may be not available for the needed maturity. For such cases, the bootstrap method can provide us with zero rates when they are not available.

How can we use an interest rate swap? We could use it in order to convert a liability or an asset. Whenever an individual owns a liability, he has to pay an interest. On the contrary, if he owns an asset, he will then receive an interest. When using a swap, first, the investor has to offset his liability (he receives the corresponding inflow of outflow). Once he does that, the outflow of the loan will be offset by the inflow of the swap, and he will be left with the fixed rate of the swap. In an ideal world, the offsetting is precisely equal, but in our world, it may not happen because the differences in compounding calculation may result in different values of the hedging position. Bear in mind that in the interest rate market, the fixed rate is based on the observed value of floating values at the beginning of the period.

Lecture 4

This lecture is about swap contracts and how they are traded amongst commercial firms and banks. Bear in mind that financial institutions are always involved in the transaction because they act as dealers, which means that they stay between the two counterparts. Indeed, a transaction between two firms requires the installment of two contracts with the two firms, with opposite sign. The bank's profit is generally given by commissions, which in this case is the bid-ask spread. The former is the price for which the bank is willing to buy swaps, and the latter is the price for which the bank is willing to sell swaps. In this way, since they sign two opposite equal contracts, they are hedged for the market risk relative to the interest rate behavior. Although they are covered against such risk, the counterparts' default risk remains, but it can be hedged with credit default swaps (CDS). At the beginning of the contract, it is possible to actualize the future cash flows, but we have to pay attention to multiple issues, namely, the correct day count and the "continuous vs discrete" matter. The former refers to the number of days to use when performing a formula because the two counterparts would usually opt for opposite conventions. For example, the buy side would prefer using the "act/365" because the result would be lower, whereas the sell side prefers the "act/360" because the result would be higher. That is why the contract conditions have to be clearly stated within the contract. In fact, confirmations specify the terms of transactions, and the ISDA (international swap and derivatives association) has developed Master Agreement that are widely used within the market. Instead, CCPs are used for standard swaps between financial institutions; such instruments are also the most liquid among derivatives. The latter refers to the problem of correct actualization, since discrete and continuous time-based rate have different present value. Furthermore, it is common usage not using continuous time in actualizing the future cash flow because it would not correctly reflect the reality.

Next, we are going to talk about swap contracts valuation methods. Initially, the contract is worth zero, which means that the expected value of the "floating leg" transaction is equal to the expected value of the "fixed leg" transaction. Although theoretically the price should be as just described, the real-world swap prices are close to zero. This is because the bid-ask spread of the financial institution who acts as a dealer. In order to proceed rationally towards a correct valuation, it is important to understand thoroughly the future cash flows of each counterpart. Bear in mind that the buyer of a swap always wants to trade its floating rate with a fixed rate. From such perspective (fixed), the swap value is:

• Positive, if the floating rates increase
• Negative, if the floating rates decrease

We have now established how the swaps' value moves according to the behavior of interest rates, now we will assess how large that movement can be. Just think of a swap contract as a multiple forward rate agreement (FRAs), thanks to which investors can decide at the present what rate they will pay in the future. In order to assess the value of the swap, we will therefore calculate the value of each forward contract within the swap. Once done that, we will sum these values up. Although calculating the values of each transaction ex-post is quite easy, calculating them ex-ante can be tricky and complicated. In order to do that, we have to assume that the future rates will be equal to the forward rates. With these, we are going to calculate the cash flow generated by the floating rate leg. After that, it will be possible to actualize the future cash flow and obtain in such a way the current value of the swap contract. This is how swaps are valuated. As we have said earlier, the continuous compounded calculations are perfect for discount rate, whereas it cannot be applied to cash flows calculations (professionals use discrete compounded rate). Since there may be continuous compounded rates within the formula, it has to be converted.