Stochastic Calculus Flashcards
Differential of cdf standard normal
pdf of standard normal
What is the integrating factor trick
Used to get expressions so can integrate easily and analytically. usually exp(lamda*t) or something along those lines. What function (integrating factor) can i multiply something by to get something i can integrate and you can use the product rule of differentiation.
Why is there no ito correction term sometimes
If using the product rule and both functions are finite variation and continuous and can be differentiable then no need for correction term
Describe qualitatively the behavior - look at Practice Qs 176
How to prove a martingale
Es(xt) = xs
or Es(xt)-xs=0 ie Es(Mt-MS)=0
How to prove local martingale
Show that the process can be written as a stochastic integral with respect to a martingale. AKA differentiate
Add in from 179 onwards
4 qualities of brownian motion
Starts at t=0
Is continuous in time not differentiable
Increments are normall distirbuted ~N(0,s-u) where s-u is time length
Increments are independent of what happened previosly
Name three basic strategies determining the holding function for investors share holdings
Passive, linear, constant dollar
Explain passive strategy
Hold same value throughout
Explain linear strategy
Proportional holdings to the share price
Explain constant dollar stratgey
Have same dollar exposure to shares at all times - inversely proportional to share price.
What’s the delta and gamma greek
The delta of an option reports how much an option will change in value for every $1 move in the underlying security
Gamma represents the rate of change between an option’s Delta and the underlying asset’s price. Higher Gamma values indicate that the Delta could change dramatically with even very small price changes in the underlying stock or fund.
What’s the theta greek of a stock
Theta is generally expressed as a negative number for long positions and a positive number for short positions. It can be thought of as the amount by which an option’s value declines daily.
Its the rate of decline in value of an options due to passage of time - time decay
What’s the vega greek of a stock
Vega is the measurement of an option’s price sensitivity to changes in the volatility of the underlying asset. Vega represents the amount that an option contract’s price changes in reaction to a 1% change in the implied volatility of the underlying asset.
What’s the relationship between theta and gamma
If theta +ve gamma must be negative and vice versa cant get a win win
Theta is roughly proportional to volatility^2 times gamma (form a straight line)
What’s the time decay for the three holding strategies
Linear: Negative
Constant dollar: positive
Passive: 0
Traits of a levy process
L0=0
Independence of increments
stationarity of increments: they all have the same probability distribution
Continuity in probability
NOT ALWAYS DIFFERENTIABLE
Poisson process
Process defined by sequence of jumps of 1 at random times separated by independent time intervals whose distirbution is exponential.
Compound poisson process, how it differs to poisson process
Jump time involves process jumping by a random number taken from some probability distribution g(x) jump amounts are idnependent from each other and form the jump times. Count of jumps is a poisson process rate from original rpocess multiplied by proportion of the jump distribution in range of interest
Let mt = Max brownian motion what is P(mt>a)
=2*P(Bt>a)
Relate gamma process to a compound poisson process
Its a compound poisson process in which rate of jumps is infinite but there are only finitely many jumps in excess of a certain size
What is the drift an dvolatility for wiener process
Drift 0 volatility 1
What levy process is a martingale?
If levy process Lt has drift mew and volatility sigma then
Lt-Mewt is a martingale
(Lt-Mewt )^2-Sigma^2t is a martingale assuming 0 drift
Squaring a martingale?
Not anther martingale because of jensens, unless its a constant
Quadratic variation of a poisson process
Quadratic variation is equal to the process itself as each increment is 0 or 1, both of which are unchanged by squaring. Quadratic variation for compound poisson is another compound poisson process with same jump rate but squared jump size
Quadratic variation of a weiner process
dt
Is quadratic variation a martingale
Quadratic variation integrated from 0 to t of levy process is still a levy process. if sigma is volatility of Lt then inetgral 0,t of dLs^2 - sigma^2t is a martingale as before.
Taylors expansion
df=df/dtdt+1/2d2f/dt^2*dt^2
ito integral in terms of sums
Its the limit of a riemann sum when always taking the left point of the intervals
What is an adapted process
One that only relies on historical data, doesn’t peek into the future
What is the ito isometry concept
Process that is integral of change in t adapted to change in Brownian motion has a normal distribution at all times
What are the rules of preservation of local martingales
If process x is adapted and multiplied by a known local martingale then integral is also a local martingale.
Preservation of martingales applies to process with zero drift. A proven local martingale through integrals is also a local martingale.
Martingales squared or differences
Difference of two martingales is also a martingale
Square is not
Whats a OU process
Ornstein Uhlenbeck process has a certain form
Its a mean reverting process where alpha (constant multipled by bracket) is speed of mean reversion and sigma is the volatiltiy of Xt
Ways to solve an SDE which doesnt have a closed form
Eulers approximations
Finite difference method
Monte carlo simulation
Tree method
Product rule
d(XtYt)=YtdXt +XtdYt+d(XtYt)
d[Pt,Xt] where pt is differentiable
0 because P has finite variation and is continuously differentiable
