Derivatives Flashcards
Forward Commitments/Contracts (Forward Rate and Valuation)
Forward Contract: price will be specified by both sides which will decide value of the forward rate contract, value is zero sum game, one side wins while the other side losses
Valuation at initiation = 0
Forward Price (FP) = S_0 * (1+R_f)^T
Value during life of contract = Spot_t - (FP/(1+R_f)^T-t), think of this as spot rate at time t vs the FP at time t (FP is calced for end of period so we need to discount it back to time t)
Value at end of contract = Spot_T - FP, we have reached time T
All forward contracts are expected to grow at the risk free rate naturally
Forward contract on stock: FP = (S_0 - PV(Divs)) * (1+R_f)^T = (S_0 * (1+R_f)^T - Future Value of Dividends. Need to factor in the negative cashflow that a dividend has a the price of the stock
Value of long position on stock = (Spot_t - PV(Divs)_t) - (FP/(1+R_f)^T-t)
Forward contract on bond: FP = (S_0 - PV(Coupons)) * (1+R_f)^T
Value of long position on bond = (Spot_t - PV(Coupons)_t) - (FP/(1+R_f)^T-t)
Forward contract on equity index will view divs are continuous so use e, FP = S_0 * e^ (R_f - Continuous Divs Rate) * T
Future Contract on Fixed Income, you may need to deliver the bond if you are short so the FP adjusts to QFP = FP/CF, basically take out cash flows received and adjust for accrued interest
Forward Rate Agreements (FRAs) & Swaps (Interest Rate, Equity, Currency)
Forward Rate Agreement (FRA): Notion is 2x3, agreement to enter a loan in 2 months for 1 month. Agreement expires when the loan begins, we are interested in the 30 day spot rate in 60 days, if our spot rate < the actual spot rate than we will be happy as we are paying less interest than others. As time gets closer we are still interested in those dates (use 60 day and 90 day to start but as time moves closer decrease the number). Make sure to adjust rates from annual amount to days needed. If 60 day spot is 2% and 90 day spot is 3% than we need to calc what the 30 day spot is in between. In this case we can use trick 2+2+x = (3*3), x=5, the forward rate will be 5%
Vanilla Interest Rate Swaps: Have value if interest rates move in your direction, you have to pay if they move in the other direction, factor notional and days into payment when finding difference and any price payments
Currency Swap: Same deal except one person is paying fixed exchange rate while other is paying floating
Equity Swap: Same thing except one is paying set amount other gets price movement. Fixed payer usually will get the divs
Usually to enter into a swap a small % interest is paid by the party entering the swap with the broker
Put/Call Parity, Option Pricing via Binomial Model & Interest Rate Options
Put Call Parity: Call + Risk Free Bond = Put + Stock
Synthetic Call = Put + Stock - Risk Free Bond
Synthetic Put = Call + Risk Free Bond - Stock
Option Pricing Model: Generates paths however chance of each path depends on up and down move and R_f, not just 50/50.
Chance of up move = 1 + R_f - D/(U - D) = 1 + Risk Free Rate - Down Move / (Up Move - Down Move). Up and Down moves are value after moves, if up 33% and down 25% than U = 1.33 and D = .75
Use these weights to find expected values and make sure to discount
Interest Rate Options (think Bond, not option): Risk neutral probabilities are equal, 50/50, and discount depends on rates within the tree
Black-Scholes Model Assumptions and Limitations
Assumptions:
Price is lognormal and price change is smooth
Risk free rate & volatility & yield of underlying are constant and known
Markets are frictionless
Options are European (can only be exercised at expiration)
Interpretations:
Call are leveraged investment in N(d1) worth of stock for every (e^-rT)*N(d2) worth of borrow funds
Puts are N(-d2) worth of bond for every short position in N(-d1) in stock
N(d2) is probability that call ends in the money
N(-d2) is probability that put ends in the money
Div paying stock will have benefit of carry (div yield) which will outweigh cost of carry and will cause call price to go down and put price to go up (when div is paid stock price falls)
Options in foreign currency will have carry yield of that foreign currency
Call = Long futures on stock + short bond (Borrowing money at R_f to long)
Put = Short futures on stock + long bond (Taking money from shorted stock and getting R_f)
Equivalent Interest Rate Contracts & Swaptions
Long interest rate call + short interest rate put = Long FRA (paying fixed, receiving floating, hoping that interest rates go up)
Short interest rate call + long interest rate put = Short FRA (paying floating, receiving fixed, hoping that interest rates go down)
If you set calls or puts at the same exercise price at different maturities you can create a price cap or price floor for rates over time
Long cap + short floor = payer swap (you pay fixed, you want rates to go up)
Short cap + long floor = receiver swap (you receive fixed, you want rates to go down)
If cap and floor exercise prices are equal than they are equal to each other
Swaptions
Payer Swaption: Gives you the right (for a premium) to pay fixed and receive floating (you want rates to go up)
Receiver Swaption: Give you the right (for a premium) to receive fixed and pay floating (you want rates to go down)
If you long payer swaption and short receiver swaption than you have a payer swap
If you short payer swaption and long receiver swaption than you have a receiver swap
Callable Bond = Straight Bond + short receiver swaption (loss when interest rates go down)
Greeks (Delta, Gamma, Vega, Rho, Theta)
BSM Model has 5 inputs, each Greek tracks something: Asset price, exercise price, asset price volatility, time to expiration, risk free rate
Delta (△ asset price): Calls have positive Delta, Puts have negative Delta
Gamma (△ Delta): More volatile delta the better for both call/put
Vega (△Volatility): More volatile the asset the better for call/put
Rho (△Rates): If rate up than call price is up and put price down because of put/call parity, if rates go up than bond prices will go down so either calls need to go up to balance or puts need to go down to balance
Theta (△Time): Theta value goes to zero as expiration is closer, negative for deep in the money puts that are European because the stock can still bounce back but it can only go down to 0, can’t go down beyond that
Exercise Price: If exercise price up than call value down, put value up
Delta is one unit change for e^-aT * N(d_1) for call or -e^-aT * N(d_1) for put
Delta Neutral Hedge: Cancel out delta via selling calls and holding stock,
- Sold Calls = Shares/Delta of call
- Only works with small movements, Gamma is measure of how good a delta hedge will be or not
- Implied volatility can be drawn by looking at prices of options
