1.
2.
3.
4.
Margin calls : if the balance is less than maintenance margin, deposit money to initial margin.
Perfect hedge: The profit of assets for short hedger and the cost of assets for long hedger is fixed and certain at the contract date (t).
Short futures : The effective price = F(t1,T)+(P(t2)-F(t2,T)) / Long futures : The effective price = F(t1,T)+(P(t2)-F(t2,T))
Basis (基差) = Spot Price of Asset to be Hedged (St1) – Futures Price of Contract Used (Ft1 (t1,T))
1. Assume that you know that the asset will be purchased at time t2 and enter into a long futures contract at time t1 and close out the position on t2 ,
cost of Asset= St2 – (Ft2 – Ft1 ) = Ft1 + (St2 - Ft2 )= Ft1 + Basis
1. If t2=T, then St2 - Ft2 is also known at time t1 (ST – FT=0) we have a perfect hedge. That is, we have no basis risk.
2. If t2<T, then St2 - Ft2 is unknown at time t1 , we have an imperfect hedge. That is, we have basis risk.
3. Rule of thumb, choose a delivery month as close to but later than the expiration of the hedge
Cash-and-carry arbitrage : Actual forward price(FT, T) > No-arbitrage forward price / Reverse Cash-and-carry arbitrage : Actual forward price(FT, T) <
No-arbitrage forward price
5.
Trasaction
Time 0
Expiration Date T
Transaction
Time 0
Expiration Date T
Short Forward
0
FT, T - ST
Buy Forward
0
ST - FT, T
Buy 1 unit nondividend-paying stock
- non-dividend-paying
stock price
ST
Sell 1 unit non-dividendpaying stock
non-dividend-paying
stock price
- ST
Borrow zero coupon
bond
non-dividend-paying
stock price
- S0 erT
Lendzero coupon bond
- non-dividend-paying
stock price
S0 erT
Net cash flow
0
FT, T - S0 erT > 0
Net cash flow
0
S0 erT - FT, T > 0
6.
Discrete known dividends / Continuous Dividend :
Trasaction
Time 0
Short forward
0
Buy 1 unit stock
-S0
Lend dividend
Borrow zero coupon bond
S0
Net cash flow
0
7.
8.
9.
10.
11.
12.
13.
14.
Time t
Expiration date T
Trasaction
Time 0
Expiration Date T
F0, T - ST
Short Forward
0
F0, T - ST
D
ST
Buy e-qT unit stock
- S0 e-qT
ST
-D
D er(T-t)
Borrow zero coupon bond
S0 e-qT
- S0 e(r-q)T
- S0 erT
Net cash flow
0
F0, T - S0 e(r-q)T
0
F0, T + D er(T-t) - S0 erT > 0
Forward price for the commodity when there is a carry market : F0,T = S0e(r+u)T or F0,T = (S0+U)erT . Arbitrage :
1. For a commodity asset with F0,T >(S0+U)erT : Borrow an amount S0+U at the risk-free rate and use it to purchase one unit of commodity and to
pay storage costs. Short a forward contract on one unit of the commodity ; For a commodity asset with F0,T <(S0+U)erT : Sell commodity and
save the storage costs and invest the proceeds at the risk-free rate. Long a forward contract on one unit of the commodity
Commodity Futures Price : F0,T = (S0+U-Y)erT or F0,T = S0e(r+u–y)T or F0,T = S0e(c–y)T , cost of carry = (Interest cost + Storage cost(u) - Lease income) +
(Interest Cost - Dividend income), y = convenience yield
The commodity forward price is a biased estimate of the expected spot price, E0(ST ), with the bias due to the risk premium on the commodity, r–k :
F0,T = E0(ST ) e(r–k)T
European call option buyer : Payoff = max [0, spot price at expiration – strike price], Profit = Payoff – future value of option premium at expiration
date(premium*erT) / European call option seller : Payoff = - max [0, spot price at expiration – strike price], Profit = Payoff + future value of option
premium at expiration date / European put option buyer : Payoff = max [0, spot price at expiration – strike price], Profit = Payoff – future value of
option premium at expiration date(premium*erT) / European put option seller : Payoff = - max [0, spot price at expiration – strike price], Profit =
Payoff + future value of option premium at expiration date
Moneyless : In-the-money option (ITM): positive payoff if exercised immediately (S0>K for call, S0<K for put) / At-the-money option (ATM): zero
payoff if exercised immediately (S0=K) / Out-of-the money option (OTM): negative payoff if exercised immediately (S0<K for call, S0>K for put)
Insurance buying : Floors(保護性賣權) : Asset + Put(怕跌) / Caps(保護性買權) : - Asset + Call(怕漲)
Insurance selling : Covered call writing(掩護性買權) : Asset - Call(看漲) / Covered Put Writing(掩護性賣權) : - Asset - Put(看跌)
Bull spread (多頭價差 : 買低賣 ) : bet that the price of the underlying asset will increase but not very far
1. Payoff of bull spread using calls : C(K1)-C(K2) > 0, K1<K2 / Payoff of bull spread using puts : P(K1)-P(K2) < 0, K1<K2
Stock price
range
Payoff from
the first long
call
Payoff from
the second
short call
Total
payoff
Total profit
Stock price
range
Payoff from
the first long
put
Payoff from
the second
short put
Total
payoff
Total profit
ST<K1
0
0
0
0- [C(K1)C(K2)]<0
ST<K1
K1-ST
-(K2-ST)
K1-K2 >0
K1-K2 -[P(K1)P(K2)] <0
K1≤ST<K2
ST-K1
0
ST-K1 >0
ST-K1 - [C(K1)C(K2)]
K1≤ST<K2
0
-(K2-ST)
-(K2-ST)<0
-(K2-ST)-[p(K1)p(K2)]
K2≤ST
ST-K1
K2-ST
K2-K1 >0
K2-K1 -[C(K1)C(K2)] >0
K2≤ST
0
0
0
0- [P(K1)P(K2)]>0
15. Bear spread (空頭價差 : 買 賣低) : bet that the price of the underlying asset will decrease
1. Payoff of bear spread using calls : C(K2)-C(K1) < 0, K1<K2 / Payoff of bear spread using puts : P(K2)-P(K1) > 0, K1<K2
Payoff from
the second
short call
Total
payoff
Total profit
Stock price
range
Payoff from
the first long
put
Payoff from
the second
short put
Total
payoff
Total profit
ST<K1
0
0
0
0- [C(K2)C(K1)]>0
ST<K1
K2-ST
-(K1-ST)
K2-K1 >0
K2-K1 -[P(K2)P(K1)] > 0
K1≤ST<K2
0
K1-ST
K1-ST < 0
K1-ST- [C(K2)C(K1)]
K1≤ST<K2
K2-ST
0
K2-ST>0
K2-ST-[p(K2)p(K1)]
K2≤ST
ST-K2
K1-ST
K1-K2 < 0
K1-K2-[C(K2)C(K1)] < 0
K2≤ST
0
0
0
0- [P(K2)P(K1)]<0
高
Payoff from
the first
long call
高
Stock price
range
16. Butter y spread(蝶狀價差 : 買低 賣中*2) : bet that a signi cant stock price move in either direction is unlikely. Payoff f of butter y spread using
calls : C(K1)+C(K3)-2C(K2) > 0, K1<K2<K3 / Payoff of butter y spread using puts p(K1)+p(K3)-2p(K2) > 0, K1<K2<K
Stock price
range
Payoff from
the first
long call
Payoff from
the second
long call
Payoff
from the
short call
Total payoff
ST<K1
0
0
0
0
Total profit
Stock price
range
Payoff from
the first
long put
Payoff from
the second
long put
Payoff from
the short
put
Total payoff
ST<K1
K1 - ST
K3 - ST
-2(K2-ST)
K1+K3-2K2=
0
0[C(K1)+C(K3)-2
C(K2)]<0
K1≤ST<K2
ST-K1
0
0
ST-K1>0
K1≤ST<K2
0
K3 - ST
-2(K2-ST)
STK3-2K2>0
K2≤ST<K3
ST-K1
0
-2(ST-K2)
2K2-ST-K1>0
K2≤ST<K3
0
K3 - ST
0
K3-ST>0
K3≤ST
ST-K1
ST-K3
-2(ST-K2)
2K2-K1-K3=0
K3≤ST
0
0
0
0
17. Straddle combination (買進跨式部位(下跨式)) : Involves buying a call and a put with the same strike price and the same expiration date, bet that
volatility will be high relative to the market’s assessment. Payoff from A Straddle Combination (K2=S0): ATM CALL+ATM PUT (C(K2)+P(K2))
Stock price range
Payoff from the long call
Payoff from the long put
Total payoff
Total profit
ST<K2
0
K2-ST
K2-ST>0
K2-ST-[C(K2)+P(K2)]><=0
K2≤ST
ST-K2
0
ST -K2>0
ST-K2-[C(K2)+P(K2)]><=0
18. Strip : buying a call and two puts with the same strike price and the same expiration date, betting that there will be a big price move and considers a
decrease to be more likely. Payoff from a Strip Combination (K2=S0): 1ATM CALL+2ATM PUT (C(K2)+2P(K2))
19. Strap : buying two calls and a put with the same strike price and the same expiration date, betting that there will be a big price move and considers a
increase to be more likely. Payoff from a Strap Combination (K2=S0): 2ATM CALL+1ATM PUT (2C(K2)+P(K2))
Stock price
range
Payoff from
the long call
Payoff from
the long put
Total
payoff
Total profit
Stock price
range
Payoff from
the long call
ST<K2
0
2(K2-ST)
2(K2ST )
2(K2-ST) [C(K2)+2P(K2)]><=0
ST<K2
0
K2≤ST
ST-K2
0
ST-K2
ST-K2[C(K2)+2P(K2)]><=0
K2≤ST
2(K2-ST)
Payoff from
the long put
K2-ST
0
Total payoff
Total profit
K2-ST
K2-ST[2C(K2)+P(K2)]><=0
2(K2-ST)
2(ST-K2) [2C(K2)+P(K2)]><=0
20. Strangle (混合價差) : buying a call and a put with the same expiration date but different strike prices, betting that volatility will be high relative to
the market’s assessment. Payoff from a Strangle Combination: (K1<K2<K3, S0=K2): OTM CALL+OTM PUT (C(K3)+P(K1))
21.
Stock price range
Payoff from the long call
Payoff from the long put
Total payoff
Total profit
ST<K1
0
(K1-ST)
(K1-ST)>0
(K1-ST)-(C(K3)+P(K1))>=<0
K1<ST<K3
0
0
0
0- (C(K3)+P(K1)) <0
K3≤ST
(ST-K3)
0
(ST-K3)>0
(ST-K3)- (C(K3)+P(K1))>=<0
Boundary condition for European call and put under no dividends : S0 ≥ c(S0,K, T) ≥ S0 –Ke-rT / K ≥ p(S0,K, T) ≥ Ke-rT-S0
26.
27.
fl
‑
28.
29.
fl
25.
fi
23.
24.
fl
22.
Arbitrage opportunity : If c > S0 : sell the call and use the proceeds to buy the stock. If get exercised, get K / If c < S0 –Ke-rT : purchase of a call
and short sell stock and lending, get S0 –Ke-rT - c / If p < Ke-rT-S0 : purchase of a put and stock and borrowing, get Ke-rT - S0 - p / If p > S0 : sell
the put and the stock. If get exercised, get K
Put Call Parity for European Option : S0 + p0 = Ke-rT + c0 : If p > Ke-rT +c- S0 : buy synthetic put and sell put option / If p < Ke-rT +c- S0 : sell
synthetic put and buy put option
American option is worth at least as much as the corresponding European option : C(S0, K, T) ≥ c(S0, K, T), P(S0, K, T) > p(S0, K, T)
Boundary on the American call option price : C(S0,K, T) ≥ S0 – K, never Exercising an American Call Early (No Dividends), European and American
call options on a non-dividend-paying Stock have the same value. C(S0,K, T) = c(S0,K, T)
Boundary on the American put option price : K-S0 ≥ P(S0,K, T), exercising an American Put Early (No Dividends ), American put options is greater
than European put options on a non-dividend-Paying Stock : P(S0,K, T) > p(S0,K, T). If c(S0,K, T)<(K-Ke-rT), i.e. K-S0 ≥ P(S0,K, T) : early exercise ;
if c(S0,K, T)>(K-Ke-rT), I.e. K-S0 < P(S0,K, T) : not early exercise
American Option inequality relation between calls and puts : S0-K ≤ C-P ≤ S0-Ke-rT , c+Ke-rT-S0 ≤ P(S0,K, T) ≤C+K-S0 (under no cash dividends) /
c+Ke-rT-S0 ≤ P(S0,K, T) ≤C+K- (S0-PV(D)) (under cash dividends)
Effect of dividend : c > (S0 - De-rt) - Ke-rt , p > Ke-rt - (S0 - De-rt)
1. Generalized Parity Relationship for European Options with Dividend : c(S0, K, T) – p(S0, K, T) = S0 - De-rt - Ke-rT
2. Early exercise is NOT optimal for European call option if interest savings exceed dividend lost : c(St,K, T-t)= (St-D-K)+ p(St,K, T) +(K- Ke-r(T-t))
≥ St–K, if K - Ke-r(T-t) + p > D / c(St,K, T-t)= (St-D-K)+ p(St,K, T) +(K- Ke-r(T-t)) ≥ St–K, if K - Ke-r(T-t) + p < D
3. American call option is not exercised if the dividends are small in general : Benefits of early exercise : Receives the stock and thus receives
dividends / Costs of early exercise : Pays the strike price prior to expiration (this has an interest cost), and loses the insurance implicit in the call
option
Slope Restriction on Option Prices : C(K1) > C(K2), P(K2) > P(K1), C(K1) – C(K2) < K2 – K1 ,│P(K1) – P(K2)│< K2 – K1 , K2 > K1
Convexity Restriction on Option Prices : (C(K1) – C(K2)) / (K2 – K1) > (C(K2) – C(K3)) / (K3 – K2) ; (P(K2) – P(K1)) / (K2 – K1) > (P(K3) – P(K2)) /
(K3 – K2)
高
1.