## Derivatives

Options, Futures, and Other Derivative Securities by Hull (6th Ed)

# 1. Introduction

Market maker: one who quotes both a bid price and an offer price for some instrument.

## Basic types

• Forward contract
An agreement to buy/sell at a certain future time at a certain delivery price. Buyer has long position and seller has short position.Forward price is the delivery price that would make the contract have zero value (all contracts have zero value at the beginning).Range forward contract specifies a price range.
• Futures contract
Forward contract traded on an exchange. Delivery date is not exact–it refers to a month, the exchange specifies the period during the month when delivery must be made. For commodities it’s usually the whole month. Seller chooses the exact delivery time.

A practical difference between forward and futures is that futures is marked to market every day, since it’s exchange traded with a margin account. Thus a futures holding has daily cash flow, whereas forward is nothing more than a piece of paper before its maturity, when the only cash flow occurs.

• Options
Call option is the right to buy at a certain date for a certain price. Put option is the right to sell. Option has to be bought but does not have to be exercised–it’s “optional”, compared to forward/futures contract.Contract price = exercise/strike price
Contract date = expiration date/exercise date/maturityAmerican options can be exercised at any time up to the expiration date. European options only on the date. In US, one option is for 100 shares.Option seller takes short position or is said to write the option.

This page shows many option payoff plots, including those fancy ones mentioned in C10.

## Types of traders

• Hedger: reducing a risk that they already face. Hedging makes outcome more certain but does not necessarily improve it.
• Speculator: want to take a position, betting on a price change
• Scalper: very short-term (a few minutes)
• Arbitrageur: lock in a riskless profit by simultaneously entering into transactions in two or more markets. For examle, using differences in stock price and currency exchange in two markets

Hedge func strategies:

• Convertible arbitrage: long convertible bond, short underlying equity
• Distressed securities: buy securities from bankrupt companies
• Market neutral: buy overvalued securities and sell undervalued securities such that the exposure to the overall direction of the market is 0.

# 2. Mechanics of Futures Markets

In the futures market, leverage refers to having control over large cash amounts of commodities with comparatively small levels of capital. In other words, with a relatively small amount of cash, you can enter into a futures contract that is worth much more than you initially have to pay (deposit into your margin account). It is said that in the futures market, more than any other form of investment, price changes are highly leveraged, meaning a small change in a futures price can translate into a huge gain or loss.

A futures contract is usually closed out (sell the position) before maturity (actually before delivery period). If it does mature, the underlying asset is rarely actually delivered. Instead both long and short positions are closed out by settling in cash, then both sides can use a cash/commodity market to actually buy and sell the asset.

## Currency futures

Setup two portfolios to both worth one unit of foreign currency at T:

A. One long forward contract and Ke^{-r(T-t)} cash, where K is the contract price for exchange rate (amount of $per one unit of foreign currency, the reciprocal of the usual exchange rate) B. e^{-r_f(T-t)} cash in foreign currency, where r_f is the foreign risk-free interest rate. To equal them at time t, we have f + Ke^{-r(T-t)} = Se^{-r_f(T-t)}, where S is the current exchange rate. So the forward price (forward exchange rate) F = Se^{(r-r_f}(T-t)} which is the K that makes f = 0. It shows when foreign interest rate is larger, F is always less than S and decreases as time. Note that this is the same as a security with dividend yield of r_f. ## Commodity futures For stuff like gold/silver that is held by many people as investment, the previous equations hold such that no arbitrage condition will arise. Storage cost can be considered as negative fixed income (F = (S+U)e^{r(T-t)}) or negative yield (F = Se^{(r+u)(T-t)}). For stuff for consumption, investor may not be willing to sell the commodity to arbitrage (or there’s benefit to holding it, or simply they need to consume it so they cannot sell it), so F may be less than (S+U)e^{r(T-t)} (remember the arbitrage srategy for this case is to sell underlying security, save the money, and take long futures). The convenience yield y for holding the commodity under such condition is defined so that Fe^{y(T-t)} = (S+U)e^{r(T-t)}. The greater the possibility that shortages will occur during the life of the futures contract, the higher y. Cost of carry is the umbrella term for all scenarios: c = r, r-q, r-r_f, r+u, etc. If futures price increases with time, delivery is usually made at the beginning of the delivery period. ## Futures price and expected future spot price Capital asset pricing model says there’re two types of risk in economy: systematic and nonsystematic. Systematic risk is general market-wide factors that cannot be reduced by diversification. Nonsystematic (specific) risk is asset-specific factors that can be reduced by divesification. The fundamental equation for CAPM is: where Zs is stock or portfolio price, Zf is risk-free interest, Zm is market price. F = E(S_T) e^{(r-k)(T-t)} where E() is expected value, and k is the discount rate for the investment (expected return). If S_T is unrelated to stock market, k = r and so F = E(S_T) meaning futures price is the same as expected future sport price. If S_T is positively correlated, k > r (positive systematic risk) so F < E(S_T). # 6. IR Futures ## Basics Day count convention defines how interest accures over time, expressed as X/Y: X is how # days between the two days that we care about is calculated, Y is how # days in a reference period is calculated (e.g. actual calendar day, 30/day per month, 360/day per year). Interest earned between two dates = # calculated days between two dates / # calculated days in reference period * interest earned in reference period. ## Treasury bond/note futures Bond and note rate are quoted in unit of 1$/32. Cash (dirty) price = quoted (clean) price + accrued interest since last coupon date. Cash received by short party (seller) = quoted futures price * conversion factor + accured interest. Seller can choose the cheapest-to-deliver bond to deliver: min(quoted price – quoted futures price * conversion factor). Steps to calculate quoted futures price:

1. cash price S of cheapest-to-deliver bond = quoted price + accured interest
2. cash futures price F = (S – I)e^{r(T-t)}
3. quoted futures price = cash futures price – accured interest
4. adjusted quoted futures price = quoted futures price / conversion factor

A $100 Tbill with T* maturity has a present value of 100e^{-r* T*}. So the futures price for maturity T (current t=0) is F = 100e^{-r* T*} e^{rT} = 100e^{rT – r*T*} = 100e^{-r^hat(T* – T)}. Arbitrage condition arises when forward IR implied by Tbill futures price is different from the forward IR implied by Tbill’s own rates (r^hat = (r2T2 – r1T1) / (T2 – T1)). If Tbill forward rate is greater than futures forward rate, one can short futures, borrow money for T1, and invest it for T2 (Type 1 arbitrage). Otherwise one can long futures, borrow money for T2 and invest for T1. Another way to determine arbitrage is the implied repo rate, which is r1 in the r^hat equation (r1 = (r2T2 – r^hat(T2-T1)) / T1) with T2 – T1 = 90 days and T1 being the contract maturity time. If the rate is greater than the actual short-term Tbill rate, Type 1 can be used. Tbill is quoted as discount rate 360/n * (100-Y), where n is days to maturity, Y is cash price. It is the annualized dollar return as a percentage of face value. It is different from the rate of return, known as bond equivalent yield. Tbill futures is quoted as 100 – Tbill quote, so a 90-day Tbill futures is 100-4(100-Y). ## Eurodollar futures A Eurodollar is a dollar deposited in a bank outside US. Its IR is the rate earned on Eurodollars deposited by one bank with another bank (commercial lending), essentially the same as LIBOR. Eurodollar futures can be used to extend LIBOR zero curve beyond one year. Duration-based hedge ratio or price sensitivity hedge ratio N = S*D_S / (F*D_F), where S is value of asset and F the contract price of IR futures. # 7. Swaps Swap is private agreement between two parties to exchange cash flows in the future. It can be considered as a long position in one bond and a short in another, or a portfolio of forward contracts. ## Interest rate swap A and B can get different fixed and floating rate loans from financial institutions. A pays B fixed rate interest on some principal, and B pays A floating rate interest on the same principal. The purpose is to transform a fixed rate loan to a floating rate loan and vice versa (or to turn an asset earning fixed rate to earning floating rate). The floating rate is usually 6-month LIBOR (the rate 6-month before cash exchange time). The total gain for both parties is D_{fixed} – D_{floating}, where D is the difference between the IRs that the two parties get in the market. If the principal itself is not exchanged at the end of the swap it is called notional principal. Financial institutions can enter a swap without having an offsetting swap. Swap rate is the average of its bid and offer fixed rates. ## Currency swap A and B can get different fixed rate loans of different currencies from financial institutions. The principal amounts are exchanged at the beginning and at the end of the life of the swap. A diff swap is where a rate observed in one currency is applied to a principal amount in another currency. It is one type of quanto where the payoff is defined in one currency but actually paid in another currency. # 8. Mechanics of Options Markets ## Option types • Stock: one contract for 100 shares. Expiration within one year (every 3 months). Longer dated options is known as LEAPS. The spacing (unit) of strike price is$2.5 when stock price < $25,$5 for $25-$200, and \$10 otherwise.
• Foreign currency
• Index: one contract is to buy/sell 100 x index at a specified strike price, settled in cash.
• Futures: underlying futures matures shortly after the option. When a holder of a call option exercises, s/he acquires from the writer a long position in the futures plus a cash amount = futures price – strike price (to make futures contract have zero value). For a put option it’s a short position plus a cash amount.
• OTC: direct trade between parties.

## Stock options

• An option class is all call or put options (with different expiration and strike prices) on one underlying stock. An option series is one particular kind of contract (one expiration and one price).
• In/at/out of the money means potential gain/level/loss in the option. For a call option it means stock price >/=/< option strike price.
• Intrinsic value of an option = max(0, option value if exercised immediately).
• FLEX option is a nonstandard option traded on exchange.
• Exchange-traded options usually don’t adjust for cash dividend, but it does for stock dividend and splits.
• Initial margin for option trading is usually 50% of share value. Maintenance margin is 25%.
• Naked option is not combined with an offsetting position in the underlying stock, usually for short call (the contrary is covered call). The writer expects stock to fall so the buyer of the call option will not exercise it, so the writer will profit (only) from the option premium.
• Call options that lead to more shares being issued: warrant (usually added to bond issue), executive option, and convertible bond

# Later chapters

It gets into details of trading strategies and mathematical models.

## 9. Properties of Stock Options

c/C is European/American call option price, p/P for put. Bounds on option price to avoid arbitrage with D is amount of future dividends:

max(S0 – D – Ke^{-rT}, 0) <= c <= S0
max(Ke^{-rT} + D – S0, 0) <= p <= Ke^{-rT} (K – S0 < = P <= K)

Put-call parity is the relationship between put and call option prices. A portfolio of long call and short put is equivalent to a forward, because both options will always be exercised, therefore the value of the portfolio is the value of forward contract:

c – p = S0 – D – Ke^{-rT}

For American options we only have bounds:

S0 – D – K <= C – P <= S0 – Ke^{-rT}

Effect of factors on option prices (both American and European), with factor increasing:

S0 K T \sigma r D + – + + + – – + + + – +

\sigma is asset volatility. Note that if dividend is involved, relationship between T and European options may be uncertain.

When D=0, it’s always better to exercise American put early, but not for American call, for two reasons: the insurance that option provides and time value of money. For call, exercise means spending money to buy the asset, which loses the insurance provided by option against asset devalue, and loses the interest the money would earn. Even selling off the asset immediately won’t be as good as reselling the option, or hold the option and short asset. For put, it’s better to exercise early and earn interest on the money.

## 10. Trading Strategies Involving Options

• Bull: long call, short call at a higher strike price. Profit when stock price rises.
• Bear: long put, short put at a lower strike price. Profit when stock price lowers.
• Box: bull + bear at same strike prices
• Butterfly: 3 strike prices. Profit when stock price doesn’t change too much.
• Calendar: options with same strike price but different expiration dates.
• Diagonal: different strike prices and different dates.

Combinations

• Straddle: long call and put with the same strike price and date. Profit when stock price moves a lot either way.
• Strip: long one call and two puts when price decrease is more likely.
• Strap: long two call and one put when price increase is more likely.
• Strangle: long call and put with different strike price and same dates

## 15. The Greek Letters

They represent risk of a portfolio.

• \Delta = d\Pi / dS, \Pi is portfolio value, S is price of underlying asset
• \Theta = d\Pi / dt, t is passage of time
• \Gamma = d(\Delta) / dS = d^2\Pi / dS^2
• \Vega = d\Pi / d\sigma, \sigma is variance (volatility) of the underlying asset
• \Rho = d\Pi / dr, r is interest rate

From the Black-Scholes-Merton equation derived in C13, we have:

\Theta + r S \Delta + 1/2 \sigma^2 S^2 \Gamma = r \Pi

## 18. Value at Risk

VaR is a single value that describes the total risk in a portfolio. We want to be able to say that “we are X percent certain that we won’t lose more than VaR dollars in the next N days”, i.e. VaR is the loss level, which is the (100-X) percentile of the probability distribution of the portfolio value change, over the next N days. Banks are required to hold capital = k * VaR(N=10, X=99), with k >= 3 and chosen by regulators for each bank.

## 20. Credit Risk

Unconditional default probability is for a particular year from now. Conditional (default intensity or hazard rate) for year N means there’s no earlier default from now to year N-1. Recovery rate for a bond is its market value immediately after a default as a percent of its face value, usually between 20-50% depending on the class of the bond.

Asset swap is a combination of a defaultable bond with a fixed for floating interest rate swap. The bond coupon is swapped into a floating rate based on LIBOR plus a spread. The asset swap spread is the spread over LIBOR that the fixed rate (bond coupon payer) receives. This spread consists of two parts: one is from the difference between the bond coupon and the par swap rate; the other is from the difference between the bond price and its par value.

## 21. Credit Derivatives

Credit default swap is a contract that insures against the risk of default (a credit event) of a company (a reference entity). Buyer of CDS pays seller periodically (like insurance premium) until the end of CDS or a credit event, at which point buyer and can sell bond at face value to the seller. Total face value of bond is CDS’s notional principal. Total amount paid per year as a percent of the notional principle is CDS spread.

Binary CDS means the payoff is a fixed dollar amount.

Total return swap exchanges all returns on a bond (or any asset) for LIBOR + spread.

Collateralized debt obligation creates new types of securities (tranches) with widely different risk characteristics from a portfolio of debt instruments, as a way to increase the quality of debt.

## 22. Exotic Options

• Package: combination of call, put, forward, cash, and underlying asset.
• Nonstandard American: Bermudan option is exercisable on certain specified days of its life.
• Compound: option on option
• Chooser: for a period of time the holder can choose whether the option is call or put
• Barrier: payoff depends on whether the underling asset’s price reaches a certain level (barrier) during a certain period of time
• Binary: pays something (cash or asset) or nothing
• Lookback: payoff depends on the max or min aset price reached during the life of the option
• Shout: lookback to a certain “shout” time
• Asian: payoff is defined as the average value of the underlying asset during a certain time.
• Exchange: exchange one asset for another
• Rainbow/basket: involves multiple assets