Derivatives Flashcards
Forward rate is less than spot rate (carry arbitrage)
Under carry arbitrage, when interest rates fall, forward prices decline.
Carry benefits vs. cost
Benefits (dividends, foreign interest, bond coupon payments)
represented by lowercase gamma
Costs (zero for financial instruments, but waste, storage and insurance on commodities)
Represented by lowercase theta
Value of a Forward Contract at Expiration
V(Long) = Spot - Forward (buying forward contracts)
V(Short) = FWD contract - Spot price (selling forward contracts)
Arbitrageur rules
- Borrow funds to purchase the underlying asset on a forward for futures contract
And invest proceeds from short selling transactions at the risk-free interest rate
- Don’t take price risk. Only focus is market price risk of the underlying or derivative.
FV of spot price using continuous compounding rate on a forward contract
FV(S0) = S0(underlying) x e^(rate x T)
FV spot price using annual compounding on forward contract
FV(S0) = S0 x (1+r)^T
Cash flows on Forward Contract
Done in parallel (not sequential)
@T=0
- Borrow funds to purchase underlying asset (+) plus financing using RFR
- Purchase underlying asset (-)
@ T=1
- Sell forward contract for delivery of underlying asset at T=1
FWD = FV(S0) - no arbitrage
Market FWD Price > Carry Model FWD Price
- Sell forward contract
- Borrow arbitrage profit at T=0
And pay off loan at contract expiration
Reverse Carry Arbitrage
Opposite transactions than Carry Model. Relevant when:
Market FWD Price < Carry Model FWD Price
- Buy forward contract
- Short the underlying
- Lend the short sale proceeds
- Borrow arbitrage profit
Valuation of Forward Contract
Forward value is the prevent value of the difference in forward contract prices PV(new - old)
Vt(T) = PV of differences in forward prices
Vt(T) = PVt,T[Ft(T) - F0(T)]
FV of underlying adjusted for carry cash flows
FV(0,T) = S0 + theta - gamma
I.e. spot price + carry costs - carry benefits
As carry benefits increase, forward price decreases (carry benefits lessen the burden of “carrying” the underlying by reducing the cost to carry the asset)
Continuous Dividend Yield
Assumption that dividends accrue continuously over the period in question rather than on specific discrete dates
Interest rate compounded annually equivalence as continuously compounded
(1+r)^T-t = ln(1+r)
Future value of the underlying adjusted for carry (continuous dividend yield)
FWD(T) = Spot x e^(r+theta-gamma)
Price of an equity forward contract
Forward Price=
S0 - PVD) x (1+r)^(T-t/365
Value of an Equity Forward Contract
The value of the long position in a forward contract on a stock at time t =
[S - PVD] - [FP / (1+r)^(T-t)
Price of a Fixed Income Forward Contract
Forward Price =
(S - PVC) x 1+r^(T-t)
Note: for treasury bonds, coupons are paid semi-annually
Value of a Fixed Income Forward Contract
Forward Price Value =
[S-PVC] - FP / (1+r^T-t)
Option valuation key assumptions
- Replicating instruments are identifiable and investable
- No market frictions such as transaction costs and taxes
- Short selling is allowed
- Underlying instrument follows a known statistical distribution
- Borrowing/Lending at RFR
Value of a call option at expiration
Call Value =
Max (0, MP - Strike)
Note: Unlimited upside and downside limited to zero
Value of Put Option at expiration
Value of Put option =
Max(0, Strike - Market Price)
Note: Upside limited (price doesn’t fall below zero)
Assumptions of the BSM model
- Implies that the continuously compounded return is normally distributed
- Price of underlying instrument moves smoothly from value to value
- Underlying instrument is liquid
- Continuous trading is available
- Short selling permitted
- No market frictions
- No arbitrage
- All options are European
- RFR is known and constant/borrowing and lending allowed at RFR
- Volatility of return on the underlying is known and constant
- Yield on the underlying is continuous known and constant yield at an annualized rate
BSM Model
PV of the expected option payoff at expiration
Described as having two components: stock and bond
Stock and Bond Components - Call Option
SN(d1) = stock component
e^(-rT) * XN(d2) = bond component
Call option = stock component - bond component
= SN(d1) - e^(-rT) * XN(d2)
Stock and Bond Components - Put Options
Stock component =
SN(-d1)
Bond component =
e^(-rT)*XN(-d2)
Bond - Stock
e^(-rT)*XN(-d2) - SN(-d1)
Replicating strategy cost (Call Option)
Purchase underlying stock by borrowing (zero-coupon bond)
Ns(price of stock) + Nb(price of bond)
Where Ns = N(d1)
Where Nb = -N(d2)
Where price of zero-coupon bond B = e^(-rT)X
Replicating strategy cost (Put option)
Ns(stock) + Nb(bond)
Where Ns = -N(-d1)
Where Nb = N(-d2)
Price of a zero coupon bond B = e^(-rT)X
Determine contracts rate on Forward Rate Agreement
E.g. 3 x 5 FRA
Where 90-day LIBOR is 5%
Where 150-day LIBOR is 6%
Step 1. De-annualized each rate
.05 x (90/360)
.06 x (150/360)
Step 2. Calculate the period rate
[Deannualized 150-day LIBOR/ Deannualized 90-day LIBOR] - 1
Step 3. Annualize the period rate
FRA Price x {360/[150-90]}
Option Gamma
change in a given option’s delta for small change in stock’s value (i.e. rate of change of delta as stock changes)
**Change in delta / change in value of underlying
Measure of the curvature in the option price in relationship to the stock price
Gamma of call = gamma of put
Always non-negative, gamma is zero on a stock
Buying options will always increase net gamma
Considered analogous to Convexity, in that it measures LARGE changes in delta (second-order effect)
Delta
Most fundamental risk of an option is sensitivity to the price of the underlying.
= change in value of option / change in value of the underlying
Call delta ranges from a value of 0 to 1
Put delta ranges from a value of 0 to -1
Considered analogous to duration in that it measures SMALL changes
Vega
An important sensitivity measure for options; reflects the relationship between the option price and the volatility of the underlying
Change in value of option / change in volatility of underlying
Duration of Option (Delta)
Delta can be used to approximate the new price of an option as the underlying changes
C + 🔺C x 🔺S (call option)
P + 🔺P x 🔺S (put option)
Convexity of Option (Gamma)
Using delta and gamma, the new call price is:
C + 🔺C x 🔺S + {[call gamma (🔺S)^2] / 2}
Delta Neutral Hedge
Determine the number of option contracts needed to create a delta neutral hedge =
of shares in stock / delta on option
Delta Put Option
Delta of call option -1
Forward Rate Agreement (FRA)
An OTC forward contract in which the underlying is an interest rate on a deposit
Fixed receiver - short FRA
Floating Receiver - long FRA
No initial exchange of cash flows & FRA value is 0 @ initiation
What is advanced set?
Reference interest rate that is set at the time the money is deposited to determine the interest accrued over the life of the deposit.
This convention is almost always used because most issuers and buyers of the financial instrument want to know the rate on the instrument while they have a position in it.
Advance Set, Advanced Settled
The convention used for settling FRAs. The settlement of interest accrued is paid at the time of FRA expiration
Formula for calculating interest accrued on bank deposit (aka LIBOR spot)
Notional Principal x (1+LIBOR(actual/360)
How to compute accrued interest for fixed income forward or futures contract?
Accrued interested =
Accrual period x periodic coupon payment
OR AI = [(days since last coupon payment / days between coupon payments) * coupon amount]
What is the purpose of a conversion factor for a fixed income futures contract?
Fixed income futures contracts halve more than one bond that can be delivered by the seller.
Bonds trade at different prices based on maturity and stated coupon
Conversion factor is an adjustment used to make all deliverable bonds roughly equal in price
When settling a futures contract, what is cheapest-to-deliver?
Seller will deliver the bond that is least expensive after adjusting bonds with a conversion factor.
Quoted price of a bond vs. Full Price
Quoted price is also known as the clean price and doesn’t include the interest accrued since the last coupon date
Full Price is also known as the dirty price and includes the interest accrued since the last coupon date
How do you solve for the quoted futures price based on carry arbitrage model?
Quoted Futures Price = Conversion factor adjusted future value of the underlying adjusted for carry =
(1/CF) * [((1+r)^T)* (S0 + AI(0) - AI(T) - FVCI]
Define a currency swap
Two counterparties agree to exchange future interest payments in different currencies using a fixed or floating interest rate
Requires the exchange of notional amounts at both initiation and expiration
Define interest rate swap and explain how it is derived
One party agrees to pay floating and receive fixed or vice-versa
Derived from the LIBOR curve
Define the steps to calculate the swap fixed rate
Step 1. Calculate the DFs for each period
DFs = 1 / (1 + (LIBOR x days/360))
Step 2. Calculate the periodic SFR
SFR(periodic) = 1 - final DF / sum of DFs
Step 3. Calculate the annual SFR
SFR(annual) = SFR(periodic) x # of settlement periods
What is the formula to calculate the MV of an IRS?
Value to the payer =
Sum of DF x (SFRnew - SFRold) x (days / 360) x Notional principal
Where days = # of settlement periods
Valuation of a currency swap
Value of swap (currency A)
= NP x {SFR(periodic) x Sum of PV Discount Factors + Final DF}
Value of swap (currency B translated to currency A)
=[NP x {SFR(periodic) x Sum of PV Discount Factors + Final DF}] * exchange rate
VA - VB
