Derivatives Flashcards
FPt = S0 * (1+Rf)^t
For an arbitrage opportunity of an:
1) overpriced asset - borrow money > buy the spot asset > short the asset in the forward market
2) underpriced asset – borrow asset > short the asset > lend the money > lend the forward
No arbitrage price of a forward contract: Equity with a discrete dividend
FP = (S0 -PVD) * (1+Rf)^t
= (S0 * (1+Rf)^t) - FVD
Value of a long position in a forward contract on a dividend paying stock
V(long position) = [St - PVDt] - [FP / (1+Rf)^T-t]
V(long) = V(short)
Taking S0 and removing PVD because the long position in a forward contract on a dividend paying stock.
Price of Equity Index Forward Contracts
FP = S0 * e^(Rfc - deltac)*T
where,
Rfc = ln (1+Rf)
No arbitrage forward price on coupon paying bond
FP = (S0-PVC) * (1+Rf)^T
= S0* (1+Rf)^T - FVC
Value of forward contract prior to expiration
V(long position) = [St-PVCt] - [FP / (1+Rf)^T-t]
AND
FP = [(full price) * ( 1+Rf)^t - AIt - FVC]
where,
AI = Accrued Interest = (days since last coupon payment / days between coupon payments) * coupon amount
Full price = clean price + accrued interest
Quoted Futures’ price
= FP/CF = [(full price) *(1+Rf)^t - AIt - FVC] * (1/CF)
x by y FRA notation
contract expires in x months and the loan is settled in y months
The value of the FRA is the interest savings due to a lower rate, discounted by the time it takes
Currency Forward Pricing
FP = S0 * [ (1+Rp)^T/(1+Rb)^T]
where,
Rp = price currency interest rate
Rb = base currency interest rate
After initiation forward contract pricing
Vt = [(FPt - FP) * (contract size)] / (1+Rp)^T-t
Value of a futures contract
= current futures price - mark to market price
Discount factor for LIBOR rates(z)
z = 1/ [1 +(LIBOR * days/360)]
Periodic swap rate
SFR = (1 - last discount rate) / sum of discount factors
For annual,
SFR(periodic) * # of settlement periods per year
Interest Rate Swap
Value to payer = Sum(z) * (SFRnew-SFRold) * (days/360) * notional principal
Equity Swaps
SFR(periodic) = (1 - last discount factor) / sum of discount factors
Binomial Probability of an Up Move
P(u) = (1 + Rf - D) / (U-D) P(d) = 1 - P(u)
Put-Call Parity
Fiduciary call is equal to the value of a protective put
where, a fiduciary call is a long all plus investment in a zero coupon bond with the face value equal to the strike price.
S0 + P0 = C0 + PV(x)
C0 = S0 + P0 + PV(x)
Hedge Ratio
C+ + C- / (S+ + S-) = the # of shares in an arbitrage trade
Interest Rate Call Option Payoff
If the i.r. ^^, the value of the call option ^^ and there is a decrease in the value of the put option
If the i.r. decreases, the value of the call option decreases and there is an ^^ in the value of the put option.
Interpretations of the Black Scholes Merton model
- The BSM values the PV of the expected option payoff at expiration
- Calls are a leveraged stock investment where N(d1) units of stock are purchased using (e^-rT * X * N(d2)) of borrowed funds.
- N(d2) is interpretted as the risk neutral probability that a call option will expire in the money. N(-d2) or 1 - N(d2) is the risk neutral probability that a put option will expire in the money.
Delta
A change in the asset prices vs. A change in the options’ price.
Put option delta increases from -e^-deltat to 0
Call option delta increases from 0 to e^-deltat
Gamma
the rate of change in delta vs the change in stock prices
long calls and puts have a positive gamma
Gamma is highest at-the-money
Vega
the sensitivity of options price changes to changes in the volatility of returns of the underlying asset.
Call and put options ^^ in price as volatility ^^
Vega gets larger as the options’ prices get closer at the money
Rho
Measures the sensitivity of an options price to changes in the risk-free rate
Call options ^^ with an ^^ in Rf
Put options decrease with an ^^ in Rf
Theta
The sensitivity of an options price to the passage of time
As time passes, the call options approaches maturity, decreasing its speculative value.
Covered Call
Long stock & Short Call
Investment at inception = S0 - C0 Value at expiration = St - max(St-X,0) Profit at expiration = min(X-St,0) - (S0-C0) Max Gain = X - S0 - C0 Max Loss=S0-C0 breakeven point = S0-C0
Main point of a covered call is to generate income on the premiums
Protective Put
Long stock and Long put
Use to minimize downside risk
Investment at Inception = S0+P0 Value at Expiration = Max(St,X) Profit at Expiration = Max(St,X) - (S0+P0) Max Gain = St - (S0+P0) Max Loss = (S0-X) + P0 Breakeven Point = S0+P0 Policy Deductible = S0-X
Bull Call Spread
Buy a call with a lower exercise price, Xl & subsidizing that purchase by selling a call with a higher exercise price, Xh
Bear Call Spread
Selling a call with a low, Xl, and buy a call with a high Xh.
Bear Put Spread
Buy a higher, Xh & sell a Xl.
Profit = max(0,Xh-St) - max(0,Xl-St)- Ph0 - Pl0
Max Profit = Xh - Xl - Ph0 -Pl0
Max Loss = S0 - Xl + (P0-C0)
Breakeven Price = S0 + (P0-C0)
Straddle
Long call and long put with the same strike price on the same security
As if the investor is betting on higher levels of volatility
Profit = Max(0, St-X)
Max Profit = St-X - (C0+P0)
Max Loss = C0+P0
Breakeven Price = X - (C0+P0) OR X + (C0+P0)
Annual Volatility of security X
252 trading days a year
Sigma = %changeP * SqRt(252/ trading days @ maturity)
where,
%changeP = (breakeven price - current price) / current price
