Derivatives Flashcards
Forward Contract: Vt(O,T) =
Forward Contract: V_t (0,T)=S_t-F(0,T)/〖(1+r)〗^((T-t)) , value of long, if
Equity Forward Contract: F(0,T), F(0,T) continuous, Vt(0,T) continuous
Equity Forward Contract: (0,T)=[S_0-PV(D,0,T)]〖(1+r)〗^T , D-dividends; F(0,T)=(S_0 e^(-δ^c T) ) e^(r^c T); V_t (0,T)=S_t e^(-δ^c (T-t))-F(0,T)e^(-r^c (T-t))
Fixed Income Forward Contract: F(0,T)
Fixed Income Forward Contract: (0,T)=[B_0^c (T+Y)-PV(CI,0,T)]〖(1+r)〗^T , CI-coupon interest
FRA
FRA: (0,h,m)=(1+L_0 (h+m)((h+m)/360))/(1+L_0 (h)(h/360) )-1 , where h - # days until FRA starts, m – length of FRA (# of days)
Forward Contract on Currency: F(0,T), F(0,T) continuous
Forward Contract on Currency: (0,T)=[S_0/(1+r^f )^T ] 〖(1+r)〗^T ; F(0,T)=(S_0 e^(-r^fc T) ) e^(r^c T)
Credit risk in a forward contract arises when
Credit risk in a forward contract arises when the counterparty that owes the greater amount is unable to pay at expiration
The value of a futures contract just prior to marking to market is
The value of a futures contract just prior to marking to market is the accumulated price change since the last mark to market. The value of a futures contract just after marking to market is zero.
Futures Price: fo(T) =
Futures Price: f_0 (T)=S_0 〖(1+r)〗^T
Futures Price with storage costs, cash flows, convenience yield:
Futures Price with storage costs, cash flows, convenience yield: f_0 (T)=S_0 〖(1+r)〗^T+FV(storage costs)-FV (cash flows)-FV(convenience yield) [or between each]
Futures Price =
Futures Price = Spot Price – Risk Premium
Stock Index Futures Price: fo(T) =
Stock Index Futures Price: f_0 (T)=S_0 〖(1+r)〗^T-FV(Dividends)
co(T1),co(T2) , po(T1), po(T2)
co(T2) ≥ co(T1) ; earlier expiration T1, later expiration T2; p0(T2) can be either greater or less than p0(T1)
Put-Call Parity
Put-Call Parity: c_0+X/(1+r)^T =p_0+S_0
Binomial Model (same for put)
Binomial Model (same for put): c=(πc^++(1-π)c^-)/(1+r), π=(1+r-d)/(u-d), c^+=(πc^(++)+(1-π)c^(+-))/(1+r), c_T=Max(0,S_T-X)
Black-Scholes-Merton model assumptions (6)
Black-Scholes-Merton model assumptions: the underlying asset follows a lognormal distribution, the risk-free rate is known and constant, the volatility of the underlying asset is known and constant, there are no taxes or transaction costs, there are no cash flows on the underlying, and the options are European
↑risk-free rate [effect on call and put prices]
↑risk-free rate ↑call option prices, ↓put option prices
Gamma is larger when
Gamma is larger when there is more uncertainty about whether the option will expire in or out of the money
Hedge Ratio, call/put overpriced (underpriced)
Hedge Ratio: =(c^+-c^-)/(S^+-S^- ) , n=(p^+-p^-)/(S^+-S^- ) , call overpriced (underpriced) sell (buy) call, buy (short) stock n * # calls = # shares, put overpriced (underpriced) sell (buy) put, short (buy) stock n * # puts = # shares
Annualized Fixed Rate on a Swap today:
Annualized Fixed Rate on a Swap today: Calculate B0(hj) for each interest payment: B_0 (h_j )=1/(1+L_0 (h_j)(h_j/360)) , where hj – interest payment dates, Li(m) – m-day Libor on day i Calculate the fixed swap interest payment rate: FS(0,n,m)=(1.0-B_0 (h_n))/(∑_(j=1)^n▒〖B_0 (h_j)〗) , where n – number of interest payments, m – number of days between each payment, hn-expiration date of swap, $1 hypothetical notional Calculate the annualized fixed rate: Annualized = FS(0,n,m) * (360/m) *notional
Market Value of the swap x days later:
Market Value of the swap x days later: Calculate Bx(hj) for each interest payment: B_x (h_j )=1/(1+L_x (h_j)(h_j/360)) PV remaining fixed payments plus $1 hypothetical notional =FS(0,n,m)*∑_(j=1)^n▒〖B_x (h_j ) 〗+$1(B_x (h_n )) PV remaining floating payments plus $1 hypothetical notional =[L_0 (h_1 )*(m/360)+1]*B_x (h_1) Market Value (=what you receive – what you pay)
Cap Payoff =
Cap Payoff = max (0, reference rate – cap rate)/# periods * notional
Floor Payoff =
Floor Payoff = max (0, floor rate – reference rate)/# periods * notional
A CDS (credit default swap) is
A CDS (credit default swap) is protection against losses from the default of a borrower