Stochastic Calculus

Question 1

Differential of cdf standard normal

Answer 1

pdf of standard normal

Question 2

What is the integrating factor trick

Answer 2

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.

Question 3

Why is there no ito correction term sometimes

Answer 3

If using the product rule and both functions are finite variation and continuous and can be differentiable then no need for correction term

Question 4

Describe qualitatively the behavior - look at Practice Qs 176

Question 5

How to prove a martingale

Answer 5

Es(xt) = xs
or Es(xt)-xs=0 ie Es(Mt-MS)=0

Question 6

How to prove local martingale

Answer 6

Show that the process can be written as a stochastic integral with respect to a martingale. AKA differentiate

Question 7

Add in from 179 onwards

Question 8

4 qualities of brownian motion

Answer 8

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

Question 9

Name three basic strategies determining the holding function for investors share holdings

Answer 9

Passive, linear, constant dollar

Question 10

Explain passive strategy

Answer 10

Hold same value throughout

Question 11

Explain linear strategy

Answer 11

Proportional holdings to the share price

Question 12

Explain constant dollar stratgey

Answer 12

Have same dollar exposure to shares at all times - inversely proportional to share price.

Question 13

What’s the delta and gamma greek

Answer 13

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.

Question 14

What’s the theta greek of a stock

Answer 14

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

Question 15

What’s the vega greek of a stock

Answer 15

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.

Question 16

What’s the relationship between theta and gamma

Answer 16

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)

Question 17

What’s the time decay for the three holding strategies

Answer 17

Linear: Negative
Constant dollar: positive
Passive: 0

Question 18

Traits of a levy process

Answer 18

L0=0
Independence of increments
stationarity of increments: they all have the same probability distribution
Continuity in probability
NOT ALWAYS DIFFERENTIABLE

Question 19

Poisson process

Answer 19

Process defined by sequence of jumps of 1 at random times separated by independent time intervals whose distirbution is exponential.

Question 20

Compound poisson process, how it differs to poisson process

Answer 20

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

Question 21

Let mt = Max brownian motion what is P(mt>a)

Answer 21

=2*P(Bt>a)

Question 22

Relate gamma process to a compound poisson process

Answer 22

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

Question 23

What is the drift an dvolatility for wiener process

Answer 23

Drift 0 volatility 1

Question 24

What levy process is a martingale?

Answer 24

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

Question 25

Squaring a martingale?

Answer 25

Not anther martingale because of jensens, unless its a constant

Question 26

Quadratic variation of a poisson process

Answer 26

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

Question 27

Quadratic variation of a weiner process

Answer 27

dt

Question 28

Is quadratic variation a martingale

Answer 28

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.

Question 29

Taylors expansion

Answer 29

df=df/dtdt+1/2d2f/dt^2*dt^2

Question 30

ito integral in terms of sums

Answer 30

Its the limit of a riemann sum when always taking the left point of the intervals

Question 31

What is an adapted process

Answer 31

One that only relies on historical data, doesn’t peek into the future

Question 32

What is the ito isometry concept

Answer 32

Process that is integral of change in t adapted to change in Brownian motion has a normal distribution at all times

Question 33

What are the rules of preservation of local martingales

Answer 33

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.

Question 34

Martingales squared or differences

Answer 34

Difference of two martingales is also a martingale
Square is not

Question 35

Whats a OU process

Answer 35

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

Question 36

Ways to solve an SDE which doesnt have a closed form

Answer 36

Eulers approximations
Finite difference method
Monte carlo simulation
Tree method

Question 37

Product rule

Answer 37

d(XtYt)=YtdXt +XtdYt+d(XtYt)

Question 38

d[Pt,Xt] where pt is differentiable

Answer 38

0 because P has finite variation and is continuously differentiable