3. Options and binomial trees

• a financial instrument that is based on the value of the underlying securities such as stocks

• 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

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

Intrnsic value is the max. betweeen 0 and the value it would have if exercised immediatly.

Time value difference between premium and intrinsic value.

• 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:

Having an option to buy the stock and enough cash is equivalent to having the stock and the option to sell it

• Bond + option (Principal protected note)

• Stock + option

• Two or more options of the same type (Spread)

• Two or moer options on different types (Combination)

  
  

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

• 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

Volatility is the same in the real world and the risk-neutral world.

A variable whose value changes over time in an uncertain way is said to follow a stochastic process

Future movements in a variable depend only on where we are, not the history of how we got there

Sequence of r.v.’s with constant expectation

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

• 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

• needed to account for real world stock price movement

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

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

• stock price follows a GBM

• no transaction costs

• no dividends

• no riskless arbitrage

• no taxes

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

• stock prices do not follow a GBM

• prices in the market do not reflect prices implied by the model

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

Delta —> take position in underlying

Gamma, Vega —> take position in other derivative

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

• leverage

• crashophobia

• volatility feedback