Black scholes Flashcards
what is the black scholes formula?
A limiting case of the binomial formula for option pricing
elaborateon the assumptions of black scholes formula
Two categories of assumptions.
1) Regarding stock
2) Regarding economic environment
Stock: contniuously compounded returns are normally distributed and independent over time.
Volatility is known and constant.
Future dividends are known
Economic environment:
risk free rate is known and constant
No transacation costs or taxes
it is possible to short sell costlessly and to borrow at the risk free rate.
what does black scholes formula in its basic form compute
european option price considering continuous dividends.
what can we do if we want to use black scholes on things like currencies and futures etc?
Convert it to use prepaid forward instead
how can we use black scholes formula wiht discrete dividends?
WE convert it to the prepaid form as before. Then we use the fact that prepaid forward + pv(div) is equal to stock price. So we interchange these
what are option greeks
formulas that express change in the option price when an input to the formula changes, taking as fixed all the other inputs.
what is the purpose of greeks, mainly?
Risk exposure anlysis
key to remember regarding the greeks?
Looks and considers only one changing variable at a time. in real life, all of them are likely to change at the same time.
elaborate on all the greeks
delta: option price change vs 1 unit change in stock
gamma: change in delta from one unit stock change
vega : change in option price from 1 percent change in volatility of the stock
theta: change in option price when there is a change in 1 day in time to expiration
rho : change in option price when interest rate change 1 percentage point
psi : change in option price when there is a 1% change in dividend yield (continuous)
elaborate on delta
defined as the number of shares in the replicating portfolio. for calls, delta is positive. for puts, delta is negative.
delta is the sensitivity of the option relative to the stock.
ITM options have higher deltas than OTM.
relate black scholes formula to replicating portfolio
BS consist of two parts. These are literally the share position and the bond position.
∆ is given by e^(-∂t)N(d1), and we set S_0 on it as well to get ∆S_0.
Same thing for hte bond posiiton.
elaborate on gamma
change in delta as the underlying change one unit.
always positive, regardless of put or call.
because of put call parity, gamma is the same for puts and calls when the terms are equal.
Deep ITM and deep OTM have deltas close to 1 or 0. In other words, they are very little affected by stock price movements. This reflect the behavior of gamma. Gamma must be small in these outer regions to make this happen. Gamma is largest for ATM options.
elaborate on the “greek measures of a portfolio”
important concept. The greeks of a portfolio is equal to the weighted sum of individual asset greeks. As a result, the portfolio greeks are easy to compute. Therefore, the risk is easy to assess.
For instance:
∆_{portfolio} = ∑w_i ∆_i
w_i is the quantity of the given option.
This relationship holds for all greeks.
how do we find the volatility of an option?
volatility of an option is equal to the elasticity of the option multiplied by the stock volatility
what is option elasticity?
%change in option price / %change in stock price
It is extemely easy to compute, since the formula end up being:
S∆ / C
what can we say about the beta of an option?
ß = ß_asset x elasticity
elaborate on payoff and profits of options
Important distinction: we earlier talked about replicating portfolio, and the cost of this portfolio. This is not what we talk about now.
Now, we consider the option price/premium as the cost. Then we need to compound it to align the payoff and cost (in the case of profit).
The PAYOFF at some point in time is equal to the price of the option at that point.
Then, we can find the profit by considering what we payed for it, and future value it to the point in time we considered the payoff at.
what can we say about the slope of option profit
Same as payoff, shifted.
The slope becomes greater and greater, eventually approaching 1 (for a call, opposite for a put).
This refelct the delta.
The gamma is the slope of the delta, and it change differently. Delta is greatest at the ATM level, so this is where the second derivative of the price is steepest.
what is a calendar spread?
options with differnet time to maturity
