Derivatives Flashcards by Gabriel Tishelman

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

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Market Value of the forwards contract on initiation date

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Value at expiration of a long position in a forward contract

S(T) - F(T)

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value at expiration of a short position in a forward contract

F(T) - S(T)

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Calculate value of forward contract based off given forward price and interest rate

Forward price given - initial price / (1+interest rate)^(T-t)

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After marking to market, the futures value of an existing contract

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Equities Futures Contract Price with Continuously Compounding Rate

Spot rate * e ^ (interest rate + carry costs - dividend yield)(T-t)

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Equities Futures Contract Price with Discrete Dividend Paid

(Spot Rate * (1+interest rate)^(T-t)) + carry costs - dividend

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Who is long in an FRA agreement

The fixed rate payer and floating rate receiver

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Forward Rate Agreement

An OTC forward contract in which the underlying is an interest rate

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When does fixed payer in FRA agreement profit

When the MRR rises

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Advanced Set

An arrangement when the interest rate is set at the time the agreement is made

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Settled in arrears

An arrangement when the interest payments are made at maturity

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Advanced Settle

An arrangement in which the FRA expires and settles at the same time

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Calculate FRA Fixed Rate

[(1 + LTT/360)/(1 + STT/360)-1] / (LT-ST/360)

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Forward pricing of contract under carry arbitrage model

F0 = FV ( S0 + CC0- CB0)

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Forward valuation of contract under carry arbitrage model

Vt = PV (Ft-F0)

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Fixed rate of interest rate swap

1 - present value of final year swap / sum of present values of all swaps

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Optimal hedge ratio

h = C+ - c- / s+ - s-

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Risk neutral probability of an up move

(1+ Rf - d) / (u-d)

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Put call parity

c = S - PV(X) + p

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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)

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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)

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2 Key Characteristics of Payments on Currency Swaps

1) Payment on each leg is in different currency
2) Payments are not netted

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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

Are there arbitrage opportunities in BSM model

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

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

gamma on a long call

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