Derivatives Flashcards
Delta
Sensitivity of option value to change in underlying price
can be seen as a slope of a line plotting the value of hte option against the underlying price
positive for long calls, negative for long puts
|Delta| for one option varies between 0 and 1
More OTM = closer to 0
More ITM = Closer to 1
ATM = around 0.5
underlying itself (or a future or forward on the underlying) has a delta of 1
Gamma
change in delta per +1 change in underlying stock price
positive for long calls AND long puts
greatest when ATM, and close to expiry
can be seen as the CURVATURE of a line plotting the value of hte option against the underlying price (vs. Delta, which is the slope of the same line)
Position Delta
net delta of a position (sum of deltas of all constituents, careful of plus or minus)
eg for a covered call:
own one stock and one stock worth of call
net delta is 1 - call delta
it’s “ 1-“ bc delta for the underlying itself (and futures and fwds) is +1 if you’re long, and -1 if you’re short … the delta of an underlying is obviously 1
Protective put:
1 - absolute value of the put delta (just because long puts have negative delta)
effective of an option reduces delta below 1
a delta-netural position is constructed to have total delta of zero, no net exposure to the underlying
Calls Basics
Long call = unlimited potential profit
Short call = unlimited potential loss
Puts Basics
max profit = breakeven price
breakeven when value = premium paid, when stock price = strike - premium
Protective Put
long the underlying, buy put
buying downside protection
unlimited upside
equivalent to synthetic long call
Covered Call + its applications
long underlying, sell a call
limited upside, most of the downside remains
receive option premium
Applications:
Yield enhancement –> calls OTM, hope that options expire OTM and keep premium
Reducing a position at a favorable price –> happy to sell the stock –> calls ITM, so likely stock will be sold
Target price realization –> calls marginally OTM –> you think stock worth a bit more than current price, happy to sell at slightly higher price
Collar
combo: own the underlying, buy put on lower strike, and sell call on higher strike
Buying downside protection PLUS selling off upside
net premium could be a pmt (debit) or receipt (credit)
limits exposure to the range btwn the two strikes
zero-cost collar = strikes selected so that premiums net to zero
Straddle
(and reverse straddle)
(Long) Straddle = buy call + put @ same strike –> profit from high volatility –> ideal when you expect large movement but dont know which direction
Reverse Straddle = sell call and put @ same strike –> profit from low volatility –> ideal when you think prices will be stable, minor moves up or down
2 breakeven points
Disadvantage: more up-front premium (have to pay for both call and put)
Advantage: max loss = premium, benefit
Bull Spreads
General spread info:
memory trick: BUY LOW SELL HIGH
both the same type of option (eg buy low call sell high call)
Spreads = buy and sell equal # of options –> limited upside and downside
Money Spreads = options have diff strikes (but same underlying and expiry)
Debit spread = you pay more than you get in premiums
Credit Spread you receive more than you pay in premiums
Bull Spread = buy a low strike option and sell a higher strike option; profit if the underlying rises
pay more for low strike than you get for selling high strike for bull call (debit spread); for a bull put you receive more (credit spread) for selling the lower strike than you do to buy the higher strike, so you get a net INFLOW of premium
max loss = net premium
limited upside
both Bull call AND Put spread = buy low strike and sell high –> bull put means you’re long the put
Bear Spreads
General spread info:
Spreads = buy and sell equal # of options –> limited upside and downside
Money Spreads = options have diff strikes (but same underlying and expiry)
Debit spread = you pay more than you get in premiums
Credit Spread you receive more than you pay in premiums
Bear Spreads = Sell LOWER strike, buy Higher strike –> they’re just short bull spreads –> profiles are just reflections in the x-axis (flip over like turning back a page)
NOTE: If a question is ambiguous, use CALLs for BULL spread and PUTS for a BEAR spread
The Greeks: Delta and Gamma
The Greeks: Theta and Vega
Position Deltas
Delta of a Covered Call
Covered call = long underlying, sell call
Call Values Pre-Expiration
Put Values Pre-expiration
Delta of a Protective Put
Hedging a Short Position
Calendar Spreads
FOCUS ON WHERE YOU BENEFIT (in terms of underlying and volatility moves)
Long = buy a longer-dated option and sell a shorter-dated option at the same strike and underlying
Short = sell longer dated and buy shorter dated
both options the same (either two calls or two puts)
How to tell if you should be long or short: at expiry of the nearer-dated option, both will have the same intrinsic value, which cancels out (bc we’re both long and short) BUT the LONG position will also still have time value –> as long as time value > initial net premium paid, you gain
SO:
go LONG if close to or at ATM and expecting low price movement between now and the shorter-term expiry
go SHORT if expecting a big move in the underlying or a decrease in implied volatility
–> if your LT view on value of stock is bullish, want long call spread
Volatility Skew and Smile
two common observed patterns
Smile = further-from-ATM options have higher implied volatilities –> imagine plotting implied volatility vs. strike –> you’d see a u-shaped curve (smile)
Skew = implied volatility increases for more OTM puts, and decreases for more OTM calls –> more common than smile
–> because in a bear scenario, more demand from hedgers for OTM puts to protect their downside, bidding up value and therefore implied volatility
–> if you see a sharp increase in the level of skew plus a surge in implied volatility, means mkt is turning bearish –> little demand for OTM calls (hence why DECREASE in OTM call vol)
HOW TO PROFIT FROM IT IN PHOTO:
Interest Rate Swap Notional Principal Formula
MEMORIZE FORMULA AND BOTTOM BOX OF SLIDE (positive D etc)
used to amend portfolio duration
Ns = notional swap principal
MDt = target MD
MDp = current portfolio MD
MDs = MD of the swap
MVp = MV of portfolio
SWAP DURATION = net of the MDs of equivalent positions in fixed-and-floating rate bonds
–> D or what you receive minus D of what you pay –> eg MDpayfixed = MDfloat = MDfixed –> negative bc fixed bond duration > that of FRN
How to Hedge against ST Interest Rate Changes
for futures: if fear rates will rise, then quote will fall, so hedge by selling futures (sell high, buy cheap, if rates do rise)
–> # of contracts for hedge = principal for actual loan / reference deposit
Fixed Income Futures
FI:
changes in the price of a FI future are driven by changes in the TD bond price –> work in terms of BPv
NOTE: DONT FORGET to MULTIPLY BY CONVERSION FACTOR
Cross-Currency Basis Swap
aimed at adjusting FX exposure
NP’s exchanged
used to convert debt in one currency to debt in another currency
usually floating for floating
Other forms of currency swap may omit the exchange of principals
Basis quoted on non-USD leg of swap –> party “paying” the basis pays non-USD reference rate PLUS the basis, recieves USD reference rate
FORWARDS are better for hedging one-off currency exposure because they’re flexible
FORMULA: (equivalent to the one for FI futures)
Notional principal needed to hedge (per $1 of notional principal) =
change in BPV / BPVswap
Equity Swaps
Equity Futures
Equity:
Same as FI futures, but work in terms of beta (want to change your beta exposure)
Effective (ex-post) beta is the beta you actually end up with –> results can differ from expectations because of basis risk –> exists when hedge is not perfect and the price relation btwn the contract and the hedged item changes in unexpected ways
Effective beta =
(% change in value of portfolio) / (% change in value of the index)
Equitization =
situation where a cash balance is held and the aim is to replicate the return on a target index –> so S = cash, Beta(s) = 0, where target beta = 1 (and usually Futures beta =1, which simplifies the formula to just S/F)
Trading Volatility & Formulas
NEED TO KNOW ALL THESE FORMULAS
note that variance notional will likely be given in vega, so need to convert to variance notional by dividing by 2x the strike (bottom right box below)
Probability of a change in fed funds rate and how to Infer Market Expectations
Target fed funds rate = midpoint of the target range
FFE RATE IMPLIED BY FUTURES CONTRACT PRICE
= 100 - the quoted future contracts price
(quoted price will likely be given)
NOTE: when calculating the bottom left part of the equation (fed funds rate assuming rate hike), add the assumed hike bps to bookends of the current RANGE, then AVERAGE, just like you did to find the current target fed funds rate.
Roll Yield
convergence = the price of a futures contract (or fwd) will equal spot price at maturity
p. 99 of mind maps for detail
if a hedger is short (long) the underlying (Spot) asset and long (short) a corresponding future, and holds to maturity –> convergence means youll have a predictable + or - return, regardless of whether prices rise or fall
–> since change in price is hedged, the return will be purely reflective of RELATIVE change in the futures price vs the spot
for a market in contango:
contango = fwd > spot (when enter the hedge)
–> F = S at maturity, so fwd price has fallen in a relative manner (by initial basis)
–> whoever is short the spot and long fwd will have a loss (negative roll yield)
–> whoever is long the spot and short the fwd will have a gain (positive roll yield)
–> G/L reversed if backwardation
Foreign Asset Returns and Variance
Rfc = foreign asset return in its own currency
variance formula is the same as 2-asset variance formula
Currency Strategies
Overlay mgr = one who specializes in hedging away currency risk
carry trade = assume fx risk bc unhedged
Volatility trade = only tading on vol; aim to be delta-hedged (no exposure to underlying)
Currency Hints
Issues w Fwds/Futures Hedging + MVHR formula
Cross hedge note: hedged item and hedged vehicle dont move preciesly in lock-steop –> introduces risk to the hedge
macro hedge = using one currency to hedge all the others in the portfolio (hedge portfolio-wide risk)
MVHR = a way of adjusting the amt of futures contract you use to hedge to account for the interdependence of the fx return risk and the return of the asset
–> if Rfc and Rfx are positively orrelated, need to hedge more
–> if negatively correlated, there’s lower overall currency volatility –> don’t need to hedge as much
NOTE: Forwards can’t completely hedge portfolio value because of the risk of fluctuations in value; you can’t know the actual return of your portfolio over the future 12 months, so won’t know at the beginning how many to sell forward, so might lead to over or under hedging
Benefits / Disadvantages of hedging with Fwds/Futures vs. Options
Quotation of Currency Options
might be described in terms of delta
eg, a 25-delta call means a call with delta of 0.25, while a 75-delta put means a put with delta of -0.75 (put deltas always negative)
in this convention ATM means 50-delta (bc ATM options have a delta of approx 0.5 (+ for calls, - for puts))
Nondeliverable Forwards (just know what they are)
Calculate Effective Interest Paid on Floating Rate Loan
1) calculate effective interest
Effective interest =
[Loan amount x (reference rate + spread) x (days in settlement period / 360)]
MINUS:
[(notional value of the rate cap or the loan amount) x (max(0, reference rate - exercise rate) x (days in settlement period / 360)]
2) use that to get to the effective rate:
effective rate =
([(loan amount + effective interest paid PLUG from above) / Loan amount] ^ (360/days in settlement period) ) - 1
