Options Flashcards
Difference dynamic and static replication strategy
static replication: no need to adjust PF before maturity, e.g. put call partiy: buy put, stock and borrow PV of the strike price
dynamic replication: you need to adjust PF as time passes and stock changes.
Payoffs Options
The payoff from a long position in a European call option is max(S_T-K,0)
The payoff from a short position in a European call option is min(K-S_T,0) // -(S – K)
The payoff from a long position in a European put option is max(K-S_T,0)
The payoff from a short position in a European put option is min(S_T-K,0) // -(K – S)
What is a protective put?
A long European put on a stock and the stock itself
What is a covered call?
A short European call on a stock and the stock itself
What is a Collar option strategy?
A short OTM call and a long OTM put on a stock and the stock itself
What is a bull spread?
An investor entering a bull spread hopes the stock price will increase. It limits the upside and downside risk. You buy a European call option on a stock with a certain strike price K_1 and sell a European call option on the same stock with a higher strike price K_2, where K_2>K_1.
What is a bear spread?
An investor entering a bear spread hopes the stock price will decline. It limits the upside po-tential and the downside risk. Created by buying a European put with one strike price and sell-ing a European put with another strike price. The strike price of the option purchased is great-er.
What is a box spread?
A box spread is a combination of a bull call spread with strike prices K_1 and K_2 and bear put spread with the same two strike prices. The payoff is always K_2-K_1, so the value of a box spread is therefore always the present value of this payoff, or (K_2-K_1 ) e^(-rT)
What is a butterlfy spread?
A butterfly spread is positions in options with three different strike prices. Created by buying a European call option with a low strike price K_1, buying a European call option with a high strike price K_3, and selling two European call options with a strike price K_2 that is halfway between K_1 and K_2. The strategy is appropriate when significant price movements are unlikely.
What is a straddle?
If you believe the stock price will move a lot, but you do not which direction, you can create a straddle. It involves buying a European call and put with the same strike price and expiration date. If there is a sufficient large move in either direction, a significant profit will result.
Factors influencing the value of a european call
→ “Intrinsic value”: how deep in the money the option is
→ Time value of strike: benefit of delaying paying the strike price
→ Present value of dividends (options not dividend protected)
→ Insurance value: insurance that final payoff on European call is max(ST −K,0) rather than ST −K
Early Exercise American Options
Call on non-dividend paying stock: it is never optimal to exercise the call early if r >= 0 American vs European call, we must have C>c, hence C>S_0-Ke^(-rT)
American Call on dividend paying: stock It may be optimal to exercise early. If optimal so, then it is just before the dividend. Early exercise is more likely to be optimal if dividends are high.
American Put with no dividends: early exercise may be optimal (P_A≥P_E). Then give up insurance value and collect strike price earlier. Early exercise is optimal when the option is deep ITM
With dividends, early exercise may be optimal (P_A≥P_E). Give up insurance value and collect strike earlier. Dividends reduce incentive to exercise early. If exercise early, more likely after a dividend
Self financing Dynamic Replication:
Self-financing dynamic strategy is a strategy that does not require any infusion or withdrawal of money after the initial purchase of assets at time 0. Any purchase of new assets after time 0 must be financed by the sale of old ones.
The cost of achieving the payoff at time T from such a strategy is the time 0 cost of setting up the strategy
In a multi period model we need to rebalance the replicating portfolio at each node.
Replicating Portfolio Formula
ΔuS+B(1-r) = C_u,
ΔdS+B(1+r)=C_d
Δ=(C_u-C_d)/(uS-dS)
B=(C_d-ΔdS)/(1+r)=(C_d-ΔdS)/exp(-rT)
C=ΔS+B
Risk neutral valuation formulas
q_u=(e^rT-d)/(u-d) q_d=1-q_u
Binomial tree: C=e^(-rT) [q×C_u+(1-q)×C_d ]
P=e^(-rT) [q×P_u+(1-q)×P_d ]
