Derivatives Flashcards
Forward Contracts
Long forward position - party that agrees to buy
Short forward - agrees to sell
Typically no money is exchanged upfront
Cash and Carry Arbitrage
Reverse Cash and Carry Arbitrage
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
Forward Rate agreement (FRA)
Purpose: agreement to borrow (long) or lend (short) at a fixed rate in the future
- Based on Libor
- Being Long = Pay fixed and receive floating
- Being Short = Pay floating and received fixed
Credit Risk in Forward Contracts
- The winner is exposed to the credit risk
- Marking-to-market will reduce credit risk (paying up multiple times)
Futures are different from Forward….
- Futures are marked to market at the end of each day
- Futures are traded on exchanges (forwards are private contracts)
- Futures are standard, forwards are customized
- Futures use a clearing house as the counterparty
- Futures contracts are regulated
Futures Contracts Notes
- Zero sum game. Settle up every day
- Futures price will converge to spot price each day
Value of a futures contract
current futures price - previous mark-to market price
How Interest Rates affect Forward and Futures
If Interest rates and assets are positively correlated
- Higher prices for futures (people want futures)
- Higher reinvestment rates for gains
- Lower borrow costs to fund losses
If interest rates and assets are negatively correlated
- Higher price for forwards (people want forwards)
- Avoid mark-to-market cash flows
Note: For bonds the short can deliver any treasury bond over 15 years
Net Costs (NC)
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
Net Benefits (NB)
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
Eurodollar Deposits vs T-Bills
Eurodollar: U.S. dollar-denominated deposits outside the US
- Priced off LIBOR, uses 360 days
- 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)
Types of Options
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
European Put-Call Parity
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
Synthetic call
Formula: C0 = P0 + S0 - PV(X)
Breakdown: Long put, long stock, short bond
Synthetic Stock
S0 = C0 - P0 + PV(X)
Long call, short put, long bond
Synthetic Put
P0 = C0 - S0 + PV(X)
Long call, short stock, long bond
Synthetic Bond
PV(X) = P0 - C0 + S0
Long put, short call, long stock
Binomial Model: Calls and Puts
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
Black-Scholes-Mertan Model
Only works for European options
Assumptions
- Rf and volatility are constant
- Markets are frictionless
- Underlying asset has no cash flow and is normally distributed
Limitations
- Cannot price options on bonds or interest rates
- Less useful with taxes and transaction costs
Delta
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
Gamma
Definition: rate of change in delta as stock price changes (acceleration)
- Largest when option is at-the money and close to expiration
- Small for deep in-the-money and deep out-of-the-money options not close to expiration
- Low gamma would reduce trading cost
Swaps
- zero at initiation
Swap value: difference in the value of the fixed payments and floating payments
Replicating Swaps: Fixed-Rate Side
Replicated by:
- Issuing fixed rate bonds (match maturity and payment dates)
- Using proceeds to purchase floating rate notes at LIBOR
- Vpayer = Vfloating - Vfixed
Important:
- Bonds have principal payments, swaps DO NOT
Replicating Swaps: Off-market FRAs
- Swap rate is constant
- Off-market FRAs have forward rates that do not yield a zero value at initiation
How to Replicate an FRA with Options
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
Replicating Swaps: Floating-Rate Side
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
C Formula
C = (1 - Z4) / Z1 + Z2 + Z3 + Z4
Pay attention to the time frame: monthly, quarterly, semi-annually, etc.
Z discount factor formula
PV of Payment
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 $100 payment, 180 days away. 5% LIBOR
Step one: .05 * 180/360 = .025
Step two: PV = 100 / (1 + 0.025) = 97.561
Valuing a Currency Swap
Principal amounts are exchanged at initiation
PV (Euro fixed “bond”) - PV($ fixed “bond”)
Use yield curve for each currency
Equity Swaps
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
Payer Swaption
Receiver Swaption
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)
Swaption Uses
- Hedge anticipated floating rate exposure in the future
- Speculate on interest rate changes
- Terminate existing swap
Swaption Credit Risk
- Current credit risk: payment due
- Potential credit risk: future obligations
- Highest in the middle of the life of a swap
Reduce by:
- Marking-to-market
- Payment netting
Caps and Floors
Similar too…..
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)
Cap Payoff
Floor Payoff
Interest rate collar
Zero-cost collar
- 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)
- Zero-cost collar: cost of cap purchased = cost of floor sold
How Changes in Interest Rates affect Options
Credit Default Swaps (CDS)
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
Single-Name CDS Payoff for Bonds
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
Index CDS
Covers multiple issuers and is equally weighted
Formula: Payoff: (notional / # of entities) - (par * share of notional)
Credit correlation is important factor in pricing
CDS coupon
- CDS coupon does not = CDS spread
- ISDA standard: 1% investment grade, 5% high yield
- Can settle by physical or cash
Physical vs Cash CDS Settlement
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
Loss Given Default
Expected Loss
Loss given default (LGD): 1 - recovery rate
expected loss = hazard rate * loss given default
Upfront Payment for CDS
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
CDS Profit Formula
Profit = change in spread * duration * notional
Types of CDS
- Naked CDS: Betting on default (doesn’t hold underlying)
- Long/short trade: buy protection for one and sell on another (betting difference in spreads)
- Curve trade: a long/short trade on the same position with different maturities (expects the YC to flatten)
- Curve-steepening trade: going short l/t and long s/t on the same entity (expects the YC to steepen)
Calculate Swap Rate and Fixed Payment
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
Delta Neutral Hedging
of calls for delta hedge = # shares of stocks / delta of call option
- Only works for very small changes
- Must be frequently rebalanced
Options on Futures
- Benefit to early excercise of deep-in-the-money options
- American options are more valuable than European
3.
Greek Risk (BSM Model) and Relation to Calls/Puts
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
