Opties Flashcards
How do calls and puts respond to dividends
Call prices will be lower, as cash dividends will decrease stock price increases.
Vice versa puts will be higher, as price increases are lower.
Stock-dividends are more complex, and accounted for in the contract.
Why may options be preferred over stocks?
They create a levered payout and can thus be useful if you want to achieve something with limited upfront payment. Conversely, risk is higher.
Through leverage, we can achieve insurance value. A relatively cheap options can insure against a disaster.
Name three main goals of options
hedging
Speculation
Arbitrage
Name some advantages of options
-Redundant assets (synthetically) but make risk transfer and speculation cheaper and more effective due to
-Reduced TX fees
-Lump many transactions together
-Provide ways to make leveraged bets
-Regulatory arbitrage: sometimes options can circumvent regulatory restrictions
Describe the covered call and its use
Long stock + short call
Reduces volatility by capping upside profits and reducing downside losses. To make money, you should correctly anticipate future stock volatility to be lower than market’s expectation.
Describe a protective put
Long stock, l;ong put
Strongest performane during bear markets. Protects the portfolio from decline, but you can capitalize on portfolio returns if the market performs very well.
Describe a collar
Long stock, long put, short call (different strikes)
Brackets the portfolio betweeen two bounds and reduces the cost of the protective put at the expense of giving up some profit potential.
Describe straddles and strangles
Bets on volatility. Pay off is a V-shape for a straddle, or a V shape with space in between for a strangle.
Straddle is better, but more expensive.
Payoff is achieved for both upward and downard volatility.
What is the put-call parity formula and intution
c + Ke^r-t = p + s0
Use it it to calculate prices, or find out arbitrage opportunities if violated
For European options, describe the impact of these factors on call & put option prices, so negative or positive relation
- Price of underlying
- Strike price
- Time to expiration
- Volatility of underlying
- Interest rate
- Dividends
Price of underlying: positive for call, negative for put
Strike price: negative for call, positive for put
Time to expiration: unknown
Volatility: both positive
Interest rate: positive for call, ngative for put
Dividends: negative for call, positive for put
so meaning: higher strike price is lower value for calls, since the relationship is negative.
For interest rates:
European Call Options:
Increase in Interest Rates: When interest rates increase, the present value of the strike price (which is paid at expiration) decreases, making it less costly to carry the position until expiration. Therefore, it’s cheaper to buy the stock at a later date for a fixed price, which increases the value of the call option.
Decrease in Interest Rates: Conversely, if interest rates decrease, the present value of the strike price increases, making the option less attractive since it becomes more expensive to buy the stock at the strike price in the future. Hence, the value of the call option decreases.
European Put Options:
Increase in Interest Rates: An increase in interest rates makes it more attractive to sell the stock at the strike price in the future since the proceeds can be reinvested at the higher rate. Additionally, the cost of holding cash (the proceeds you’d get if you sold the underlying asset now and held the cash until the option expiration) is higher. This makes the put option less valuable.
Decrease in Interest Rates: A decrease in interest rates reduces the future value of the cash you would get from selling the underlying asset at the strike price, making the put option more valuable because the alternative (holding cash) is less attractive.
What is the hedge ratio
The number of options that need to be held to hedge a stock..
(Payoff up - Payoffdown)/(Return up - Return down)
Probably not a formula that needs to be known. The intuition is that you would need 3 options to hedge 5 shares for example. Then the hedge ratio would be 3/5th (or -3/5th?)
What is the intuition behind the risk-neutral world for option pricing
Risk-neutral world is a hypothetical world in which investors do not require premium for risk, the expected return on all assets is the risk-free rate, but the option price computed in the risk-neutral world equals the option price in the real world.
Name the assumptions underlying Black-scholes
- No TX costs /taxes
- Riskless borrowing and lending and short selling are possible
- No arbitrage
(These assumptions imply perfect markets) - Underlying asset prices follows brownian motion -> lognormal distribution and continuous asset price ( no jumps)
- No dividends during life of derivative
- security trading is continuous
- Rf rate and vol are constant
If vol is time varying, asset price no longer has a lognormal distribution so returns are no longer normal.
intuition:
If we cut δt small enough and add enough time steps, binomial tree converges to distribution behavior of geometric Brownian motion.
If stock price follows geometric Brownian motion, option price and stock price depend on the same underlying source of risk.
So also, in continuous time we can set up a riskless portfolio consisting of stock and option
Hedging argument: choose delta such that the delta of the stock - f is risk-free rate → Continuous rebalancing needed to keep portfolio risk-free
Portfolio is riskless (in this small-time interval) and must earn risk-free rate → Derive BS formula by solving PDE
Magic → Only volatility matters, we do not need to worry about risk and risk premium if we can hedge away the risk completely (in all states).
Under black scholes, does the option value depend on the expected rate of return on a stock?
No, this information is already built into the formula with the inclusion of the stock price, which already reflects the stock’s return and risk characteristics.
Given the BS formula: c = e^(-rT)[S_0 N(d1)e^(rT) - KN(d2)]
Describe N(D1) for calls or n(-d1)
and N(d2)
N(d1) is the delta for call and N(-d1) is the delta for a put. So appreciation per appreciation of 1 of the underlying.
N(d2) is risk-neutral probability that a call will be exercised. K*N(d2) is the expected cost of exercise in the risk neutral world.
