Derivatives

Question 1

Forward Contracts

Answer 1

Long forward position - party that agrees to buy

Short forward - agrees to sell

Typically no money is exchanged upfront

Question 2

Cash and Carry Arbitrage

Reverse Cash and Carry Arbitrage

Answer 2

Cash and Carry: If a forward/future is overpriced you:

sell forward –> borrow money –> buy spot asset

Reverse Cash and Carry: If a forward/future is underprice you:

buy forward –> borrow asset –> sell spot asset –> lend money

Question 3

Forward Rate agreement (FRA)

Answer 3

Purpose: agreement to borrow (long) or lend (short) at a fixed rate in the future

1. Based on Libor
2. Being Long = Pay fixed and receive floating
3. Being Short = Pay floating and received fixed

Question 4

Credit Risk in Forward Contracts

Answer 4

1. The winner is exposed to the credit risk
2. Marking-to-market will reduce credit risk (paying up multiple times)

Question 5

Futures are different from Forward….

Answer 5

1. Futures are marked to market at the end of each day
2. Futures are traded on exchanges (forwards are private contracts)
3. Futures are standard, forwards are customized
4. Futures use a clearing house as the counterparty
5. Futures contracts are regulated

Question 6

Futures Contracts Notes

Answer 6

1. Zero sum game. Settle up every day
2. Futures price will converge to spot price each day

Question 7

Value of a futures contract

Answer 7

current futures price - previous mark-to market price

Question 8

How Interest Rates affect Forward and Futures

Answer 8

If Interest rates and assets are positively correlated

1. Higher prices for futures (people want futures)
2. Higher reinvestment rates for gains
3. Lower borrow costs to fund losses

If interest rates and assets are negatively correlated

1. Higher price for forwards (people want forwards)
2. Avoid mark-to-market cash flows

Note: For bonds the short can deliver any treasury bond over 15 years

Question 9

Net Costs (NC)

Answer 9

storage costs - convenience yield

It makes the FP = S0 x (1 + Rf)^T + FV(NC)

FV (NC) = the future costs of holding the asset

This also will make contango occur

Question 10

Net Benefits (NB)

Answer 10

It makes the FP = S0 x (1 + Rf)^T - FV(NB)

FV (NB) = the future benefits of holding the asset

This also will make backwardation occur

Question 11

Eurodollar Deposits vs T-Bills

Answer 11

Eurodollar: U.S. dollar-denominated deposits outside the US

1. Priced off LIBOR, uses 360 days
2. Cannot be priced using no arbitrage models (cost and carry)

Eurodollars use ADD-ON interest (borrow $1M then pay back $1M plus interest)

T-Bills are discount instruments (only would borrow 980K)

Question 12

Types of Options

Answer 12

Call: the right to buy an asset

Formula: Max (0, S - X)

Put: the right to sell an asset

Formula: Max (0, X - S)

European option: Can only be exercised at expiration

American option: can be exercised at any time prior to expiration

Question 13

European Put-Call Parity

Answer 13

Formula: C0 + PV(X) = P0 + S0

Left side = fiduciary call

PV(X): zero coupon bond

Right side = protective put

Remember: PV of bond is X / (1 + R)^T

Question 14

Synthetic call

Answer 14

Formula: C0 = P0 + S0 - PV(X)

Breakdown: Long put, long stock, short bond

Question 15

Synthetic Stock

Answer 15

S0 = C0 - P0 + PV(X)

Long call, short put, long bond

Question 16

Synthetic Put

Answer 16

P0 = C0 - S0 + PV(X)

Long call, short stock, long bond

Question 17

Synthetic Bond

Answer 17

PV(X) = P0 - C0 + S0

Long put, short call, long stock

Question 18

Binomial Model: Calls and Puts

Answer 18

Propability of up-move = 1 + Rf - D / U - D

D and U are opposites of each other. Up move of 1.15 means 1/1.15 is a down move of 0.87.

Example: S0 = $30, up factor is 1.15 and Rf = 7%, strike is $30

Step one: Up move is 30 * 1.15 = 34.5
Step two: Down move is 30 * .87 = 26.1
Step three: A call option would be worth $4.50 in the up state and $0 in the down
Step four: Probability of up state = 1.07 - .87 / 1.15 - .87 = 0.0.7143
Step five: C0 = (4.5 * .715) + ($0 * 0.285) = $3.00

Question 19

Black-Scholes-Mertan Model

Answer 19

Only works for European options

Assumptions

1. Rf and volatility are constant
2. Markets are frictionless
3. Underlying asset has no cash flow and is normally distributed

Limitations

1. Cannot price options on bonds or interest rates
2. Less useful with taxes and transaction costs

Question 20

Delta

Answer 20

Definition: change in the price of an option for a 1-unit change in the price of the underlying stock (speed)

Call Range from 0-1

Out-of-the-money is closer to 0

In-the-money is closer to 1

Put Range is from -1 to 0

Out-of-the-money is closer to 0

In-the-money is closer to -1

Question 21

Gamma

Answer 21

Definition: rate of change in delta as stock price changes (acceleration)

1. Largest when option is at-the money and close to expiration
2. Small for deep in-the-money and deep out-of-the-money options not close to expiration
3. Low gamma would reduce trading cost

Question 22

Swaps

Answer 22

1. zero at initiation

Swap value: difference in the value of the fixed payments and floating payments

Question 23

Replicating Swaps: Fixed-Rate Side

Answer 23

Replicated by:

1. Issuing fixed rate bonds (match maturity and payment dates)
2. Using proceeds to purchase floating rate notes at LIBOR
3. Vpayer = Vfloating - Vfixed

Important:

1. Bonds have principal payments, swaps DO NOT

Question 24

Replicating Swaps: Off-market FRAs

Answer 24

1. Swap rate is constant
2. Off-market FRAs have forward rates that do not yield a zero value at initiation

Question 25

How to Replicate an FRA with Options

Answer 25

An FRA can be replicated with two options Long interest rate call + short interest rate put If rates increase, the call wins. Rates fall the put loses

Question 26

Replicating Swaps: Floating-Rate Side

Answer 26

On each settlement date, the value of a floating rate note will always reset to par Floating coupons are paid in arrears (based on LIBOR at the start) Example: $100 par value, 3-year annual-pay FRN, and 12-month (forward) LIBOR of 4%, 5%, and 3% PV = 104 / 1.04 + 105 / 1.05 + 103 / 1.03

Question 27

C Formula

Answer 27

C = (1 - Z4) / Z1 + Z2 + Z3 + Z4 Pay attention to the time frame: monthly, quarterly, semi-annually, etc.

Question 28

Z discount factor formula PV of Payment

Answer 28

Each Z is a discount factor **Discount factor:** 1 / [1 + ( r \* days/360)] **PV of payment:** r \* (days/360) **then** Par / (1 + x) Example: Find the PV of a $100 payment, 180 days away. 5% LIBOR Step one: .05 \* 180/360 = .025 Step two: PV = 100 / (1 + 0.025) = 97.561

Question 29

Valuing a Currency Swap

Answer 29

Principal amounts are exchanged at initiation PV (Euro fixed "bond") - PV($ fixed "bond") Use yield curve for each currency

Question 30

Equity Swaps

Answer 30

Equity return payer Pays any positive return on equity Receive fixed rate payment plus any negative equity Fixed-rate payer Pays fixed rate plus negative equity return Receives positive equity return

Question 31

Payer Swaption Receiver Swaption

Answer 31

Payer swaption: right to enter swap as fixed-rate payer (wins if rates increase) Receiver swaption: Right to enter swap as fixed-rate receiver (wins if rates fall)

Question 32

Swaption Uses

Answer 32

1. Hedge anticipated floating rate exposure in the future 2. Speculate on interest rate changes 3. Terminate existing swap

Question 33

Swaption Credit Risk

Answer 33

1. Current credit risk: payment due 2. Potential credit risk: future obligations 3. Highest in the middle of the life of a swap Reduce by: 1. Marking-to-market 2. Payment netting

Question 34

Caps and Floors Similar too.....

Answer 34

Buyer of cap: benefits when interest rates rise (similar to a call on LIBOR or long put on a bond) Buyer of a floor: benefits when interest rates decrease (similar to a put on LIBOR or call on a bond)

Question 35

Cap Payoff

Answer 35

Question 36

Floor Payoff

Answer 36

Question 37

Interest rate collar Zero-cost collar

Answer 37

1. Interest rate collar: buy a cap and floor over the same time period Type 1: purchase a cap and sell a floor (makes the cap cheaper) Type 2: purchase a floor and sell a cap (makes the floor cheaper) 2. Zero-cost collar: cost of cap purchased = cost of floor sold

Question 38

How Changes in Interest Rates affect Options

Answer 38

Question 39

Credit Default Swaps (CDS)

Answer 39

Purpose: Insurance contract If an event occurs the buyer (short credit risk) gets compensated from the seller (long credit risk) Note: CDS spread: the premium paid

Question 40

Single-Name CDS **Payoff** for Bonds

Answer 40

**Formula:** Payoff = notional - (% of par \* notional) Payoff on a single-name CDS is based on the cheapest-to-deliver obligation **THINK:** Bascially they give back the partial bond for a full one

Question 41

Index CDS

Answer 41

Covers multiple issuers and is equally weighted **Formula:** Payoff: (notional / # of entities) - (par \* share of notional) Credit correlation is important factor in pricing

Question 42

CDS coupon

Answer 42

1. CDS coupon does not = CDS spread 2. ISDA standard: 1% investment grade, 5% high yield 3. Can settle by physical or cash

Question 43

Physical vs Cash CDS Settlement

Answer 43

Physical: Swap buyer delivers the asset, swap seller gives cash at par value Cash: Swap seller gives cash at par value less current value Example: value is now 20 and par is 100. Swap seller pays 80

Question 44

Loss Given Default Expected Loss

Answer 44

Loss given default (LGD): 1 - recovery rate expected loss = hazard rate \* loss given default

Question 45

Upfront Payment for CDS

Answer 45

Upfront payment: (CDS spread - CDS coupon) \* duration Protection leg is the expected cash flows paid by seller Premium leg is the expected cash flow payments from the buyer

Question 46

CDS Profit Formula

Answer 46

Profit = change in spread \* duration \* notional

Question 47

Types of CDS

Answer 47

1. Naked CDS: Betting on default (doesn't hold underlying) 2. Long/short trade: buy protection for one and sell on another (betting difference in spreads) 3. Curve trade: a long/short trade on the same position with different maturities (expects the YC to flatten) 4. Curve-steepening trade: going short l/t and long s/t on the same entity (expects the YC to steepen)

Question 48

Calculate Swap Rate and Fixed Payment

Answer 48

Example: Calculate the swap rate and the fixed payment on a 1-year, quarterly settlement swap with a notional principal of $10MM Maturity Annalized Rate Discount Factor 90 day 4.5% 0.98888 180 day 5.0% 0.97561 270 day 5.5% ? 360 day 6.0% 0.94340 270 day = 100 / 1 + (0.055 X 270/360) Now the Z formula: 1 - 0.94340 / 0.98888 + 0.97561 + 0.96038 + 0.94340 = 0.0146 (Remember this is QUARTERLY for this question. Need to annualize. **1.46% x (360 / 90) = 5.84%** **Value of payment would be $10M x 0.0146**

Question 49

Delta Neutral Hedging

Answer 49

of calls for delta hedge = # shares of stocks / delta of call option ## Footnote 1. Only works for very small changes 2. Must be frequently rebalanced

Question 50

Options on Futures

Answer 50

1. Benefit to early excercise of deep-in-the-money options 2. American options are more valuable than European 3.

Question 51

Greek Risk (BSM Model) and Relation to Calls/Puts

Answer 51

**_Input Calls Puts_** Delta Asset Price Positive Negative Vega Volatility Positive Positive Rho Rf Positive Negative Theta Time to exp Value moves to zero closer to maturity Price Exercise Price Negative Positive