Derivative Securities Princing - Appunti

Options

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 having 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

A 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 transaction and the market would need rates for longer maturity, the overnight rates will be aggregated according to specific needs. Such new references 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 agreement", where A borrows money to B, who gives a collateral (govies) to the lender.

In this way, if B defaultsA can keep the govies in order to cover the losses. Because of that, SOFR is said to be a secured ratewhich means that the rate is lower because the transactions are less risky for the lender. On thecontrary, ESTER rate is calculated on the wholesale euro unsecured overnight borrowing costs ofeuro 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 ofthe wholesale market, and it is also based on more representative market data. Furthermore, as wehave said, ESTER is a daily rate even though swaps may be 6 months long. Hence, it has to becompounded in order to obtain the 6 months rate.

Swap option is an agreement to exchange cash flows at specified future times according to certainspecified rules. Plain vanilla is the most common contract (fixed rate vs floating rate). Once the basicelements of the contract are established

(reference rate, future dates), the valuation of the counterpartdefaulting risk has to be considered (it depends on the identity and reliability of the counterparts). Atthe beginning, the contract must be based on a risk-free risk, for which OIS rate is a good proxy (maxone year). Unfortunately, since OIS rate does not have any kind of maturity we will have to chooseanother 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 anasset, he will then receive an interest.

When using a swap, first, the investor has to offset his liability(he receives the corresponding inflow of my 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 require the instalment 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 institution, 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 use 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 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 act 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 values 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), thank 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 flow calculations (professionals use discrete compounded rate). Since there may be continuous compounded rates within the formula, it has to be converted in discrete time in order to be financially equivalent, because 10% returns in discrete time do not equal 10% returns in continuous time. In order to do that, we need to apply specific formulas.

Once done that, we will obtain two rate that are notionally different but financially equivalent because they are simply compounded differently.

Lecture 5

This lecture will be the last one of the review sessions. We will further analyze the option instrument, in which investors have the faculty to exercise the option or not. Their decision will be based on the market condition at maturity. Forwards and swaps, as we have seen, had symmetrical payoff because the investors are obliged to respect the contract agreement at maturity (the amount of loss equals the amount of gain). On the contrary, options have asymmetrical payoff because investors can either choose to exercise it or let it expire. Since options incorporate a right to decide in the future, they have a premium price to pay in advance, whereas forwards and swaps have no such obligation. In fact, the seller of the option requests an upfront payment (called premium) in order to let the investor decide whether to exercise the right or not in.

The future. Bear in mind that option sellers are mostly financial institutions, which hope the underlying asset's price does not reach the strike price of the contract.