Derivative Securities Pricing - Riassunto

Derivatives

Call X=10 S Opt11 +112 +213 +3A derivative is an instrument whose value depends on, or is derived from, the value of another asset. Derivatives play a key role in transferring risks in the economy. (I have an exposure on an asset on a long position, and then I buy a put option, this is the example of before) If I only buy the put option, it is a speculative tool: I have no underlying asset to cover the risk (I make money when the price drops).

How derivatives are traded

Two main ways to trade are:

• Through Exchange
• In the over-the-counter (OTC) market where traders contact each other directly

Size of OTC and exchange-traded markets

OTC – Swap and CDS

The Lehman bankruptcy

Lehman’s filed for bankruptcy on September 15, 2008. This was the biggest bankruptcy in US history. Lehman was an active participant in the OTC derivatives markets and got into financial difficulties because it took high risks and found it was unable to roll over its short-term funding.

Uses of derivatives

There are three uses:

• Hedging: I use derivatives because I want to offset the risk. The derivative is an edging tool, it will lower the risk (I have an exposure on an asset on a long position, and then I buy a put option, this is the example of before).
• To speculate: If I only buy the put option, it is a speculative tool: I have no underlying asset to cover the risk (I make money when the price drops).
• To arbitrate: it is a riskless transaction that can occur when the prices of two assets are not efficiently aligned. If there is this disalignment, we can exploit it with a riskless transaction (e.g., the same asset has different prices in Milan and in London, I can buy it in Milan and sell it in London).
• To change the nature of a liability

FRN (floating rate note) + SWAP = Fix rate bond

In the end, the final outcome is that we have a fix rate bond.

Forward price

The forward price for a contract is the delivery price that would be applicable to the contract if were negotiated today.

Terminology

• The party that has agreed to buy has what is termed a long position (positive sign).
• The party that has agreed to sell has what is termed a short position (negative sign).

Cash price: I set the price today and I receive.

Future price: I set the price today, but I receive the goods in the future. If the price goes up and I am in a long position, I will have a profit (I will pay $50 for something that now costs $75). On the other side, if the price goes down I will have a loss.

Payoff graph

Profit from a long forward position:

Price K is the delivery price 50 (the price that we agreed).

Profit from a short forward position:

If we are in a short position, we make money when the price goes down (e.g., we receive 50 when the actual value is 25).

Futures contract

Agreement to buy or sell an asset for a certain price at a certain time. They are similar to forward contracts. Whereas a forward contract is traded OTC, a futures contract is traded on an exchange.

Examples of futures contracts:

• Buy 100 oz. of gold @ US$1400/oz. in December
• Sell £62,500 @ 1.4500 US$/£ in March
• Sell 1,000 bbl. of oil @ US$90/bbl. in April

Gold

How to set the price for a forward contract

Suppose that:

• The spot price of gold is US$1,400
• The 1-year forward price of gold is US$1,500
• The 1-year US$ interest rate is 5% per annum

Is there an arbitrage opportunity? We have two opportunities:

• Buy now and keep it at 1400
• Buy forward at 1500

The problem is that I have to pay 1400 now. If I don’t have the money and I borrow it, in a year's time (as the interest rate is 5%) I will have to pay 1470 (the fair forward price). There is an arbitrage opportunity: I will sell the overvalued asset (the forward price) and I will buy the undervalued asset. If there is a contrast between fair price and market price, there is an arbitrage opportunity. We should also include the cost of carry in the fair forward price: it is the cost of bringing the asset from time 0 to time T (e.g., how much does it cost for the storage?). Carry trade: trade in which I buy the spot and sell the forward.

Oil

Suppose that:

• The spot price of oil is US$100
• The quoted 1-year futures price of oil is US$125
• The 1-year US$ interest rate is 5% per annum
• The storage costs of oil are 2% per annum

The fair price is 100 * 7% = 107 We have a carry trade in this case as well: buy the asset on the spot and sell the asset in the forward.

Suppose that:

• The spot price of oil is US$100
• The quoted 1-year futures price of oil is US$80
• The 1-year US$ interest rate is 5% per annum
• The storage costs of oil are 2% per annum

We have a reverse carry trade: I will buy the asset in the forward and I will sell the asset on the spot. I can’t buy only the forward asset, because if the price goes up, I will lose money. If I don’t buy both transactions, the trade will be risky. I can calculate the fair price in two different ways: F* (fair price) = S * e^rt -> the correct price is the short price S* (fair price) = F * e^-rt -> the correct price is the forward price.

Options

• A call option is an option to buy a certain asset by a certain date for a certain price (the strike price).
• A put option is an option to sell a certain asset by a certain date for a certain price (the strike price).

An American option can be exercised at any time during its life. A European option can be exercised only at maturity. A Bermudan option can be exercised on specified dates during its life.

Options vs futures

A futures/forward contract gives the holder the obligation to buy or sell at a certain price. An option gives the holder the right to buy or sell at a certain price.

Types of traders

• Hedgers: that have a covered position
• Speculators: that have a naked position
• Arbitrageurs: the arbitrageur spots that the price is wrong and trades this difference to gain a profit

Hedging example

An investor owns 1,000 Microsoft shares currently worth $28 per share. A two-month put with a strike price of $27.50 costs $1. With the put option I have the right to sell at a price that is higher than the spot price (if the price goes down). In red, we have the put option.

Speculation example

An investor with $2,000 to invest feels that a stock price will increase over the next 2 months. The current stock price is $20 and the price of a 2-month call option with a strike of 22.50 is $1. Stock: investor buys 100 stocks @$20 → If the final price is $27, gain $700 → if $19, loss $100. Call: investor buys 20 calls (2000/1*100) and enters the right to buy 2000 stocks @$22.5 → If the final price is $27, gain $7000 ((27-22.5)*2000) → if $19, loss $2000.

Arbitrage example

A stock price is quoted as £100 in London and $140 in New York. The current exchange rate is $1.43 per pound (or 1.43 $/£). There is an arbitrage opportunity because in New York, as the exchange rate is 1.43, the price should be 143.

Dangers

Traders can switch from being hedgers to speculators or from being arbitrageurs to speculators. It is important to set up controls to ensure that trades are using derivatives for their intended purpose.

Future contracts

Regarding futures, the important part is that everything has to be standardized (the size, the price has to be the same). A forward, on the other hand, has no standardization procedure. The futures have to be settled daily (if a person owes me $1M, the risk that the money will be delivered, if they are settled daily is 0, if it is settled in a month's time, the $1M will rise due to the interest).

The “a” image pictures the usual behavior of the future and spot price. Futures worth more than spot because of interest expenses to maintain the trade. In the spot, we pay at the beginning and we receive money at the end. In the futures, there is no movement of money at the beginning, because we borrow money.

Contango: a situation where we have more offer than demand. We can hold the commodity in warehouses and wait to deliver it at maturity.

Backwardation: a situation where we have more demand than offer.

Collateralization in OTC markets

It is becoming increasingly common for transactions to be collateralized in OTC markets. Consider transactions between companies A and B. These might be governed by an ISDA Master agreement with a Credit Support Annex (CSA). The CSA might require A to post collateral with B equal to the value to B of its outstanding transactions with B when this value is positive. If A defaults, B is entitled to take possession of the collateral. The transactions are not settled daily and interest is paid on cash collateral. Following the 2007-2009 crisis, there has been a requirement for most standardized OTC derivatives transactions to be cleared centrally through clearing houses.

Clearing houses

Is way safer, my collateral will be used only if I’m defaulted. The only risk is that the clearing house defaults, which is way smaller.

Forward vs futures

An idea of leverage

We are considering investing $100 at the beginning of 2018 in several assets. The idea is to short Greece, but we don’t know what asset to choose. First idea: short Greek bonds: If we short the index, we can already see that the leverage gives us. If we consider the CDS. If we keep reinvesting the money in the Greek.

Swap

Is an agreement to exchange cash flows at specified future times according to certain specified rules. A “Plain Vanilla” is the most common swap, and the widest in terms of use and quantity in the market. The main feature is that there is an agreement to receive floating (uncertain) and to pay fixed (e.g., an agreement by Microsoft to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years). I will receive the rate that was in the market at the beginning of the period. 4.20 * 100 / 2 = 2.10. If I buy a swap and the market rates are going up, I’ll receive increasing cash flows. N.B.) this is an ex post representation: when a contract is signed, the market rates are unknown.

Typical uses of an interest rate swap

Converting a liability from → fixed rate to floating rate → I have a liability fixed (-C ) I sell the swap so that I pay floating and receive the fixed Fix- C + Fix - Floating Fix → floating rate to fixed rate → I have a liability floating (-C ) I buy the swap so that I pay fixed and receive the floating Fl- C + Floating - Fix Fl

Converting an investment from → fixed rate to floating rate I have an investment fixed (+C) → I buy the swap so that I pay fixed and receive the floating Fix+ C – Fix + Floating Fix → floating rate to fixed rate → I have an investment floating (+C ) I sell the swap so that I pay floating and receive the fixed Fl+ C - Floating + Fix Fl

➢ The bid means that the bank buys the swap at 6.03 ➢ The offer means that the bank sells the swap at 6.06 (this means that, if the bank receives 6.06 fixed, it will have to pay floating)

Day count

A day count convention is specified for fixed and floating payment (e.g., LIBOR is likely to be actual/360). In our class, we won’t take care of the day count.

Confirmations

They are the agreements that specify the terms of a transaction. The International Swaps and Derivatives has developed Master Agreements that can be used to cover all agreements between two counterparties.

Swap rate

A LIBOR rate is a short-term rate for loans with the maturity of 6 months. The 5-year swap rate is not a long-term rate. It has a risk corresponding to a loan that lasts 6 months, continuously refreshed. Swap rates can also be used to build the Interest Rate curve. This is an interest rate curve built on Italian Government Bonds, but they are risky bonds: we are including the default risk compensation. If we need a risk-free interest rate curve, we might rely on LIBOR markets. Nowadays we rely on the OIS rates (Overnight Loans): in which the probability that the counterparty defaults is really low.

Bootstrap an interest rate curve

I’d like to get the rates for a specific maturity. When we build an interest rate curve, we need to use zero rates: rates associated with zero coupon bonds (that don’t have middle payments). Bootstrapping is a procedure in which I have the rates at 6 months, 12 months, and 18 months and I determine the one at 24 months. I set up an equation like this: Payments Discount Rate unknown (* 2 years). Using the same process, the bootstrapping procedure, we can calculate the 3 years rate, the 4 years rate … (all of these rates will be zero coupon rates).

Valuation of an interest rate swap

If I sign the contract, initially the value is 0, and later on, due to market condition, it can be different from 0. The Interest Rate Swap can be valued as:

1. The difference between the value of a fixed-rate bond and the value of a floating-rate bond.
2. A portfolio of forward rate agreements (FRAs).

In terms of bonds

If I look at a balance sheet in which I have a fixed rate bond and a floating rate bond, and I try to understand what are the cash flows. At the end of the year, in terms of cash flows, I will have:

• - Fixed
• + Floating

If I have this situation, the cash flows will collapse and the cash flows (long floating, short fixed) will be exactly equal to buying a swap. Then, if I have to calculate the value of the swap, I’ll calculate the value of the bond with fixed rate and the value of the bond with floating rate and I’ll take the difference.

How to calculate the fixed rate bonds

If the coupon is fixed is just a discounted cash flow formula → The coupon is fixed, the bond is worth 100 r = 5% → If the bond is worth > 100 r < 5%. If the market rate (r) is below the coupon rate, the bond is particularly attractive because it pays more. → Vice versa, if the bond is worth < 100 r > 5%

How to calculate the floating rate bonds

In a floating rate bond, we’ll receive the market rate. The coupons will be calculated using the market rate at the beginning of the period (we can’t adjust the rate in the middle of the period). On the first payment date, the value of the floating rate bond is equal to par (Value = L). However, if my valuation date is before the payment date, it’s not perfectly floating: the value will be L + K*. K* is the payment of the first rate that will be paid in time T*. The rate of the floating bond is calculated in this way: on the first payment date it will be equal to par, and we will add up the par component with the first coupon, that might not be aligned with the market rate. The fix rate is the price that we’re willing to pay to receive the floating rate. The value of these two legs must be the same at the beginning of the contract (the discounted payments must be the same). As time passes, the floating rate yield curve changes, and the value of the swap changes. If the value of the stock was 0 using a swap rate of 2%, if I charge you a swap rate of 2.5% instead, how much should the value of the stock be? From my perspective, the swap has negative value. To go back to a swap that has value 0:

• Change the structure of LIABOR
• If the swap rate is different from the kind of swap rate that gives value 0, it can be adjusted by providing a sum of money

Example

Discount factor = e^(-time*spot rate) Cash flows fixed leg = principal * fixed rate NPV (CFs) = cash flows discounted = discount factor * cash flows Cash flows floating leg = 100 + 2.9/2. The value of the floating bond is equal to the notional plus whatever amount of interest is own just before the payment NPV (CFs) = cash flows discounted = discount factor * cash flows Swap value = difference between the value of the fixed leg and the floating leg In this case we have a Forward Rate Agreement (FRA) The first row is easy to understand:

• The fixed leg has a cash flow of 1.5 which becomes 1.490 when discounted
• The floating leg has a rate of 2.9% (the same as before) The cash flow is 2.9% * 100 * 0.5 which is also discounted
• The FRAs value is the difference between the two NPV

We don’t know the payment of the floating leg in the future: we calculate the expected value. We have the yield structure of the spot rate → 0.25 years 2.8% → 0.75 years 3.2% → 1.25 years 3.4%. We can use this yield structure to figure out what is the expected LIBOR. The implied forward rate is the rate that gives the same amount of money if:

• I invest in LIBOR for 9 months at a rate of 3.2%
• I invest in LIBOR for 3 months at a rate of 2.8% and then reinvest that amount at the implied rate

PROBLEM: the rates that we obtain are discrete compounded rates, so we have to turn them into continuously compounded rates. The formula that we used in the graph is Rm = m(e^(Rc/m) – 1) Where:

• m is the number of payments in the year
• Rc is the discrete compounded rate

We can calculate all the cash flows, the NPV, and the FRAs value.

Overnight indexed swaps

Should OIS rate (Overnight Indexed Swap) equal the LIBOR rate? The LIBOR rate is the rate that is applied when a bank borrows money to another bank (e.g., for 3 months). The risk is that the bank will default in 3 months’ time. The OSI rate is even more risk-free than the LIBOR because the only risk is that the borrower defaults during the night. If we lend money using OSI and LIBOR for 3 months there is a big difference:

• LIBOR: we’ll see the money only after 3 months
• OSI: every night I lend the money and I get it back in the morning

Swaps and forwards

A FRA is an agreement that involves paying in a specific point in time a floating rate and receiving a fixed rate. When we are valuating the value of the swap, the value of the swap is the fair.