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

- Introduction
- Mechanics of Futures Markets
- Hedging Strategies Using Futures
- Interest Rates
- Determination of Forward and Futures Prices
- IR Futures
- Swaps
- Mechanics of Options Markets
- Later chapters

# 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)
- Day trader
- Position trader

- 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.

## Treasury stuff

- T-bills, are sold in terms ranging from a few days to 26 weeks. Bills are sold at a discount from their face value. For instance, you might pay $970 for a $1,000 bill. When the bill matures, you would get $1,000. The difference between the purchase price and face value is interest. It’s a kind of
*discount instrument*. - T-Notes earn a fixed rate of interest every six months until maturity. Notes are issued in terms of 2, 3, 5, and 10 years.
- Treasury bonds pay interest every six months until they mature. It used to be 10-30 years, but now it’s all under 10 years so there’s no difference between T-Notes.
- Treasury Inflation-Protected Securities, or TIPS, provide protection against inflation. The principal of a TIPS increases with inflation and decreases with deflation, as measured by the Consumer Price Index. When a TIPS matures, you are paid the adjusted principal or original principal, whichever is greater.
- TIPS pay interest twice a year, at a fixed rate. The rate is applied to the adjusted principal; so, like the principal, interest payments rise with inflation and fall with deflation.

## Margin

Margin in futures account is different from stock account, where you borrow money to buy stock. Investor deposits fund into a margin account. *Initial margine* is the amount that must be deposited at the time the contract is entered into. At the end of each trading day, the account balance is adjusted according to market price, i.e. *marking to market* the account. M2M actually moves money, e.g. if a futures price decreases, a broker of a long investor pays the exchange, which in turn pays a broker of a short investor. All futures account are marked to market daily (not for forward contract).

*Maintenance margin* is the minimum balance in the account, usually 75% of the initial margin. If balance falls below it, the investor gets a *margin call* to deposit extra fund to make up to maintenance margin, known as *variation margin*. If not, the broker will close out the position by selling the contract.

Bona fide hedger gets lower margin requirement than a speculator. *Day trade* (close out position on the same day) and *spread transaction* (take both a long position and another short position) get lower margin requriement than hedge transaction.

Clearinghouse members maintain a margin account with the clearinghouse known as *clearing margin*. There’s no maintenance margin. *Gross basis* = # all long positions + # all short positions. *Net basis* = long – short. Margin is calculated from *settlement price*, i.e. price at closing time. *Open interest* is # all long positions (= short) on one security.

Futures price increases for further maturity time in a *normal market*, decreases in an *inverted market*. Some futures such as stock indices are settled in cash. S&P 500 futures settlement price is the opening price of the index in the morning after the last trading day, to avoid chaotic trading and significant price movements toward the end of day when stock index futures, stock index options, and options on stock index futures all expire.

## Types of orders

- Market order: best market price
- Limit order: buy at price <= specified
- Stop/stop-loss order: sell once price <= specified
- Stop-limit order: combination
- Market-if-touched (board) order: sell once price >= specified
- Discretionary (market-not-held) order: market order at broker’s discretion (to wait for a better price)
- Time based: time-of-day for a particular period of time, open/good-till-canceled for indefinitely, fill-or-kill for immediately.

# 3. Hedging Strategies Using Futures

Someone who will sell an asset (or receive and own an asset) in the future can take a short futures position to offset asset price decrease. Someone who will buy can take a long futures position to offset price increase.

## Basis

*Basis* in a hedging situation = spot price of asset – futures price (sometimes the inverse on a financial asset). Basis *strengthens* when it increases, otherwise *weakens*. *Basis risk* is the uncertainty asscociated with the basis at close out time, which is in the future. For both long and short positions, the effective price for the asset at close out is S2 + F1 – F2 = **F1 + b2**, where S is spot price, F is futures price, b is basis. This would be same even if a different asset is used to hedge (cross hedging), because the formula can be rewritten as S2* + F1 – F2 + (S2 – S2*), where S2* is the spot price for the hedged asset. A short hedger’s position improves with strengthening basis (spot price is higher or futures price is lower), and a long hedge’s position worsens. If the close out time is indeed the contract time, then F2 = S2 so b2 = 0, so the effective price for both hedgers is indeed F1, the contract price.

Hedge expiration (close out) is usually chosen to be close but earlier than contract date because futures price gets more volatile as time gets closer to delivery.

## Hedge ratio

*Hedge ratio* = size of position in futures contract / size of exposure (actual buy/sell). When the hedger is long the asset (has asset, will sell) and short futures, the change in the value of the hedger’s position during the life of the hedge is dS – h * dF, where dS is change in spot price, dF is change in futures price. For a long hedge it’s the inverse. The variance of the position is \sigma_S^2 + h^2\sigma_F^2 – 2h \rho \sigma_S \sigma_F, where \sigma is standard deviation, \rho is correlation coefficient between dS and dF. Thus the optimal h that minimizes position variance is **h* = \rho \sigma_S / \sigma_F**. Optimal number of contracts is thus h* * N_A / Q_F, where N_A is size of position being hedged and Q_F is size of one futures contract. *Hedge effectiveness* is the proportion of the position variance that is eliminated by hedging = h*^2 \sigma_F^2 / \sigma_S^2 (?).

## Hedging an equity portfolio

Optimal hedge ratio h* = \beta, the correlation between portfolio return and market (index) return. So the number of short index futures contracts to hedge is \beta * P / A, where P is the portfolio value, A the current value of the underlying index per contract (index number * multiplier for contract). The hedge removes the risk arising from market moves and leaves the hedger exposed only to the performance of the portfolio relative to the market.

To change the beta of a portfolio to a smaller beta*, take a short futures position of (b – b*)S/F contracts. To a larger beta*, take a long position of (b* – b)S/F contracts.

When there’s no liquid futures contract that matures later than the expiration of the hedge, a strategy known as *rolling the hedge forward* can be used, in which a sequence of futures contracts are entered. When the first contract is near expiration, it’s closed out and a second contract is entered, and so on.

# 4. Interest Rates

- LIBOR

London Interbank Offer Rate is quoted by banks as IR for loans to others for 1, 3, 6, 12-month (the payment frequency is called tenor). It’s regarded as short-term opportunity cost of capital, and usually used as risk-free rate. - Repo

A repurchase agreement means the owners of securities sell them to a counterparty and buy them back later at a slightly higher price. It’s just like the counterparty gives a loan that earns interest. Repo rate is a bit higher than T-bill. The most common type is*overnight repo*. Longer types are called*term repo*. - Continuous compounding of interest

The amount of annual interest compounded m times per annum is (1 + R/M)^{mn}, where R is the an annual interest rate. Continuous compounding means m goes to infinity, when the value becomes e^{R*n}. Let Rc be a rate with continuous compounding and Rm for compounding m times per annum, then for the interest amount to be the same we have Rc = m ln(1 + Rm/m) and Rm = m(e^{Rc/m} – 1). - Zero rate

n-year spot rate is called*n-year zero-coupon yield*(coupon is interest/dividend payment. Zero coupon means all interests are paid at maturity). - Bond pricing

A bond’s principal (par value, face value) is received at the end. The theoretical bond price = \sum coupon value * e^{-zero rate at coupon payout term * coupon payout term} = \sum coupon value * e^{-bond yield * coupon payout term}. Bond yield can be solved iteratively by Newton-Raphson. Par yield is the coupon rate that makes bond price = par value(?). - Forward rate

Forward interest rate is the rate implied by current zero rate for a period of time. For a period of time between T1 and T2 where T1 < T2 is: R_F = (R2T2 – R1T1) / (T2 – T1), where R1 and R2 are spot rate of interests for T1 and T2 years. The simplest case is T1 = 1, T2 = 2, where R_F = 2R2 – R1, i.e. R2 is the mean of R1 and R_F, which is the result of continuous compounding: e^{R1 + R_F} = e^R2.The above equation can be rewritten as R_F = R2 + (R2-R1)T1/(T2-T1), so if R2 > R1 (yield curve is upward sloping) then R_F > R2 > R1, so R_F(t) curve is always above R(t), which is above coupon-bearing bond yield curve. If R2 < R1 then R_F < R2 < R1, R_F(t) < R(t) < R_coupon(t). Take limit as T2 -> T1, we get R_F = R + T dR/dT, the*instanteous forward rate*.A forward rate agreement (FRA) is an OTC agreement that a certain IR will apply to either borrowing or lending a cetain principla during a specified future period of time. - Duration

Bond price B = \sum_{i=1}^n c_i e^{-y t_i}, where the bond pays c_i at t_i and the yield is y.The duration of bond is D = \sum_{i=1}^n t_i c_i e^{-y t_i} / B = \sum_{i=1}^n t_i [c_i e^{-y t_i} / B]. The [] term is the ratio of the present value of the payment at t_i to the bond price, so D is a weighted average of the times of payments.dB/dy = – \sum_{i=1}^n t_i c_i e^{-y t_i} = – B * D, so dB/B = – D * dy, meaning percentage of change in bond price is its duration multiplied by the size of parallel shift in yield curve.The above y assumes continuous compounding. If instead with discrete compounding of m times per year, then dB = – B D dy / (1 + y/m). Define*modified duration*as D* = D/(1 + y/m) then we get back dB/B = – D* * dy. - Term structure of IR
- Expectations theory: long-term interest rates reflect expected future short-term rates, so forward rate should be the same as expected future spot rate of the same period.
- Market segmentation theory: there’s no relationship between short, mid, long term rates.
- Liquidity preference theory: forward rate is always higher than expected future spot rate. Investor likes to invest for short terms to preserve liquidity, while borrower likes to borrow for long term. If long and short term rates equal, banks would have large risks financing long term loans by short term deposits. So long term rate is usually higher to discourage borrowing and encourage saving.

# 5. Determination of Forward and Futures Prices

Short selling: Investor let broker borrow shares and sell them, then later buy back the shares to close out the position. If the broker runs out of shares to borrow the investor must close out immediately (short squeeze). It is only allowed on an uptick (latest share price increased). Investor pays the broker any income on the short position, which will be transfered to the account from which the shares are borrowed.

## Value of forward contract

In order not to create arbitrage opportunities, forward price F (changes with t) and spot price S for a no-income security must be F = S e^{r(T-t)} where r is risk-free interest rate, T is forward contract maturity date, t is current time (T and t are in years). If F is larger, one can borrows S as loan to buy the asset and take a short forward contract at F. When T comes the contract is exercised, and one profits F-Se^{r(T-t)}. If F is smaller, one can short sell the asset, save the proceed for r, and take a long forward contract at F. When T comes one takes out the saving to buy the asset at F to close out the short selling, and profits Se^{r(T-t)}-F.

Compare two portfolios, A with a long forward contract and an amount of cash Ke^{-r(T-t)} where K is the contract price (the present vale of the cash to exercise the contract) so at time T the contract is exercised and the porfolio has one unit of security. Portfolio B always has one unit of security. The two portfolio should worth the same at time t as well, so the value of the contract is f = S – K e^{-r(T-t)}. F is the value of K that makes f=0, which is the same as above. The value of a short forward contract is the negative. This comparison method can be used for the more complicated cases in the following.

If the security provides fixed income (dividend per share) I as the present value discounted using risk-free interest rate, the forward price formula becomes F = (S – I) e^{r(T-t)}. Similarly for contract value f = S – I – K e^{-r(T-t)}. Portfolio B has one unit of security and borrowed amount I (which will be repaid by the income on the security).

If the security provides dividend yield (% of security) q per annum, F = S e^{(r-q}(T-t)}. Portfolio B has e^{-q(T-t)} amount of the security (to make one unit at time T).

In general f = (F – K)e^{-r(T-t)}, otherwise one can buy both a long and a short contracts for time T, gaining positive present value at the amount of the difference between the two terms.

When security is strongly positively correlated with interest rate, long futures price is usually higher than forward price because futures account settles daily, and the daily gain can be invested at a high-than-average interest rate, loss be financed by a lower-than-average rate.

## Stock index futures

If an index pays dividend yield q, its futures price is the same as above F = S e^{(r-q}(T-t)}. If F is larger, one can profit from buying stocks underlying the index and shorting futures. If F is smaller, one can profit from shorting/selling stocks and taking long futures.

The return on a particular portfolio of stocks D1 (on $1) and the return on the market D2 is approximately D1 = alpha + beta * D2, where alpha is a constant and beta a parameter. The change in portfolio value is S*D1, where S is the starting portfolio value. The change in one index futures contract is approximately F*D2. The uncertain component of the change in the value of portfolio is beta * S / F (why???), which is also the number of short futures contract to hedge with (see C3).

## 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:

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

## Treasury bill futures

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 | |
---|---|---|---|---|---|---|

Call | + | – | + | + | + | – |

Put | – | + | + | + | – | + |

\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

Spreads:

- 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

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