Derivatives Flashcards
Carry arbitrage model
a no-arbitrage approach in which the underlying instrument is either bought or sold along with an opposite position in a forward contract
Market Value of the forwards contract on initiation date
0
Value at expiration of a long position in a forward contract
S(T) - F(T)
value at expiration of a short position in a forward contract
F(T) - S(T)
Calculate value of forward contract based off given forward price and interest rate
Forward price given - initial price / (1+interest rate)^(T-t)
After marking to market, the futures value of an existing contract
0
Equities Futures Contract Price with Continuously Compounding Rate
Spot rate * e ^ (interest rate + carry costs - dividend yield)(T-t)
Equities Futures Contract Price with Discrete Dividend Paid
(Spot Rate * (1+interest rate)^(T-t)) + carry costs - dividend
Who is long in an FRA agreement
The fixed rate payer and floating rate receiver
Forward Rate Agreement
An OTC forward contract in which the underlying is an interest rate
When does fixed payer in FRA agreement profit
When the MRR rises
Advanced Set
An arrangement when the interest rate is set at the time the agreement is made
Settled in arrears
An arrangement when the interest payments are made at maturity
Advanced Settle
An arrangement in which the FRA expires and settles at the same time
Calculate FRA Fixed Rate
[(1 + LTT/360)/(1 + STT/360)-1] / (LT-ST/360)
Forward pricing of contract under carry arbitrage model
F0 = FV ( S0 + CC0- CB0)
Forward valuation of contract under carry arbitrage model
Vt = PV (Ft-F0)
Fixed rate of interest rate swap
1 - present value of final year swap / sum of present values of all swaps
Optimal hedge ratio
h = C+ - c- / s+ - s-
Risk neutral probability of an up move
(1+ Rf - d) / (u-d)
Put call parity
c = S - PV(X) + p
Value of fixed rate interest swap at time T after initiation
NA * (current fixed swap - starting fixed swap rate) * (sum of present value factors for all time periods)
Value of receiver rate interest swap at time T after initiation
NA * (starting fixed swap rate - current fixed swap rate) * (sum of present value factors for all time periods)
2 Key Characteristics of Payments on Currency Swaps
1) Payment on each leg is in different currency
2) Payments are not netted
Notional amount on a currency swap
Initial amount * exchange rate
Is continuous trading available in BSM model
Yes
Is Short selling allowed in BSM model
yes
Are there brokerage costs in BSM model
no
Are there arbitrage opportunities in BSM model
No
Is the volatility of the underlying known in BSM model
Yes
Is the volatility constant in BSM model
yes
Are returns normally distributed in BSM model
Yes
Can prices randomly jump from one to another in BSM model
No
If puts are sold on stock, what is the appropriate delta hedge
Short sell shares of that stock
optimal number of hedging units
- (portfolio delta / delta of hedging instrument)
Delta
change in a given instrument fora given small change in the value of the stock, holding everything else constant
Gamma
change in delta for a given small change in the stock’s value
Effect of buying options on gamma
always increases gamma when options are purchased
Theta
change in a portfolio for a given small change in calendar time, holding all else constant
typically always negative
Vega
change in a given portfolio for a given small change in volatility, holding all else constant
very high when options are near the money
Rho
change in given portfolio for a given small change in the risk-free rate, holding all else constant
Implied volatility
Standard deviation that causes an option pricing model to give the current option price
Basis
the difference between the spot price and the futures price, as the maturity date of the futures contract nears, the basis converges towards 0
Expectations approach to value options
Discount at the risk-free rate the expected future payoff based on risk neutral probabilities
Settlement amount as pay-fixed party
NA * (MRR rate - FRA rate) * time /360 / (1+ discount rate) * time/360
Value of Swap contract time into the contract
1) Calculate the fixed swap rate (1-last pv) / sum pv
2) (og rate - rate from step 1) * sum pv * NA
Position to take when call option is overpriced
Sell the call, buy the shares, and borrow
value on a forward contract on index or stock
Present value of (price of forward contract when it was entered - price of forward contract today)
exp(-risk free rate * time)*(difference between price of forward contract when it was entered and price of forward contract today)
Equilibrium Futures Contract price based on the carry arb model
Q = (1/CF) * FV(B0+AI)-AT - FVCI
CF = conversion factor
B0 = clean price
AI = accrued interest
AT = would be accrued interest
Two rules of arbitrage
1) Do not use your own money
2) Do not take any price risk
Value additivity principal
The value of the portfolio is the sum of the values of the assets that make up the portfolio
Market value of a futures contract, prior to marking to market
(current price - previous close price) * multiplier
multiplier - set contract size
Reverse Carry Arbitrage
Short sale of the underlying and an offsetting opposite position in the derivative
Future value of asset adjusted for carry cash flows
FV (Spot + Carry costs - carry benefits)
Dividend index point
measure of the quantity of dividends attributable to a particular index
Carry arbitrage model for continuous compounding
F0 = S0 * e ^ [ (r+CC-CB)*T ]
Typical way that FRAs are settled at expiration
Advanced set, advanced settle
Conversion factor for a futures contract
Approximate decimal price at which a $1par of security would trade at if it had a 6% YTM
Two Generic Expressions in Carry Models
1) Forward Pricing - FV(B0 + CC - CB)
2) Forward Valuation - PV(Ft-F0)
Difference between a swap and a FRA
a swap hedges multiple periods, whereas FRA hedges one period
Typically the way that swaps are settled at expiration
Advanced set, settled in arrears
Value of a interest rate swap for fixed receiver
NA * (FSt - FS0) * PV (value of pay fixed swap)
Value of Currency Swap at Time t
[NA(a) * (payment rate of a * sum of PV(a) + PV of relevant spot rate(a))] - [exchange rate * NA(b) * (payment rate of b * sum of PV(b) + PV of relevant spot rate(b))]
Futures vs forwards
Futures are marked to market every day
Impact of dividends on call and put option, under black scholes model
dividends lower the value of the call option only
Impact of increased interest rates on call and put options
move the values further apart, calls increase, puts decrease
when is option gamma the highest
when the option is near or at the money
when can you not use the bsm model, using black model instead
when the underlying is costless to carry
Rho on call options vs Rho on put options
Rho on call options is positive
gamma on a long put
0
gamma on a long call
1
impact on d1 and d2 under black scholes model when dividends are introduced
both d1 and d2 are lowered because d2 has d1 in its formula
when is vega high
when an option trades near or at the money
