What is an option? Explain the graphs of the four different options.
a financial instrument that is based on the value of the underlying securities such as stocks
Valuating options with binomial trees. Define probability of up / down movements. What is the difference whilst valuating European or American Options?
for European options we take the discounted value at each node
for American options we take either the discounted valued of the option at the node or the value if option excercised immedialty
of course we take higher value of both
Explain the following terms:
premium
moneyness (in/at/out of the money)
Premium: current market price of option
In-the-money: would lead to positive cashflow if exercised immediately
At-the-money: … zero …
Out-of-the-money: … negative
What do we understand as the intrinsic value and the time value of options?
Intrnsic value is the max. betweeen 0 and the value it would have if exercised immediatly.
Time value difference between premium and intrinsic value.
What are the bounds on the value of options?
the call-option can not be worth more than underlying itself
the put-option can not be worth more than strike price
American options are worth at least as much as European options
lower bound fur european:
What is the put call parity?
Having an option to buy the stock and enough cash is equivalent to having the stock and the option to sell it
Name four trading strategies with options
Bond + option (Principal protected note)
Stock + option
Two or more options of the same type (Spread)
Two or moer options on different types (Combination)
Explaing the trading strategy stock + option.
Explain the Bull Spread.
Explain the Bear Spread:
How do you set up a riskless portfolio?
Combination of Shares and Options:
The sum of increase of the total share value & option payoff is equal to the sum of decrease of total share value & option payoff
—> can be solved for option price
What does risk neutral valuation mean?
value of a derivatice is its expected payoff in a risk-neutral world discounted at the risk-free rate
Process: Calculate p (prob. of up- or downward movement) —> calculate expected payoff —> discount at r —> = price
What is the Girsanov’s Theorem?
Volatility is the same in the real world and the risk-neutral world.
What is a stochastic processes?
A variable whose value changes over time in an uncertain way is said to follow a stochastic process
What is a Markov process?
Future movements in a variable depend only on where we are, not the history of how we got there
What are Martingales?
Sequence of r.v.’s with constant expectation
What is a Wiener Process / Brownian Motion
A variable z follows a wiener process if
it starts in 0, has a mean 0
changes in z correlate to changes in standard deviation
Value changes of z for any two different short amounts of time are independent
What is a Generalized Wiener Process?
normal Wiener Process has drift rate 0 and variance rate of 1
Generalized Wiener Process has an expected drift rate of a and a variance of b^2
What is the Geometrical Brownian Motion? (GBM)
needed to account for real world stock price movement
What are the characteristics of Itôs process and Itôs Lemma? Why do we use it?
Itôs process: - we take the drift and variance rate for the Gemoetrical Brownian Motion as function of time t and the underlying x
Itôs Lemma: if we know the stochastic process followed by x, it gives us the stochastic process followed by some function G(x,t)
—> we use it to model the stock prices
What is the BSM model and what are its principles?
The BSM is a model for pricing European options on stocks.
option price and stock price depend on the same source of uncertainty
in any short period movements of stock and derivative are perfectly correlated
by forming a portfolio of stock + option we eliminate the source of uncertainty
—> instanteously riskless & earning the risk free rate
What are assumptions of the BSM?
stock price follows a GBM
no transaction costs
no dividends
no riskless arbitrage
no taxes
What is the BSM differential equation?
BSM pricing formulas for European options:
How are the risk-neutral valuation and the BSM connected?
Variable (mu) does not appear in BSM diff. equation
—> independent of all variables affected by risk preference
—>solution is the same in risk-free and real world
—> no need to assume risk neutrality of investor
What are the problems with the BSM today?
stock prices do not follow a GBM
prices in the market do not reflect prices implied by the model
What are the greeks? Explain them.
Delta: rate of change of option w.r.t. price of underlying asset price
Gamma: rate of change of Delta w.r.t. price of underlying
—> high if at the money
Theta: rate of decline in the value of an option over time
—> usually negative
Vega: rate of change of the value of a derivative w.r.t. volatility
—> long = positive, short = negative
Rho: rate of change of the value of a derivative w.r.t. interest rate
How can we hedge Delta, Gamma or Vega?
Delta —> take position in underlying
Gamma, Vega —> take position in other derivative
What is the volatiliy smile? What are volatility surfaces?
Smile: Relationship between implied volatility and strike price for options with a certain maturity
Surface: Implied volaitility as a function of strike price and time to maturity
What are the reasons for negative correlation btw. equity prices and volatility?
leverage
crashophobia
volatility feedback
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