Stochastic Calculus, MFD

Question 1

Give the practitioner and Ingersoll definition of derivatives

Answer 1

Practitioner - securities that derive their value from cash market instruments
Ingersoll - a financial contract is a derivative security, or a contingent claim, if its value at expiration date T is determined by the market price of the underlying instrument at time T

Question 2

Define swap

Answer 2

The simultaneous selling and purchasing of various financial assets

Question 3

Describe three types of swaps

Answer 3

Cancelabke - gives either party the right to cancel the swap
Callable - payer if the fixed rate has the right to cancel
Puttable - receiver of the fixed rate has the right to cancel

Question 4

Define arbitrage

Answer 4

Taking simultaneous positions in different assets to produce a riskless profit

1. Negative investment with a nonnegative profit
2. Zero investment with a positive profit

Question 5

Define martingale

Answer 5

A random variable X_t s.t. E^P[X_T|I_t] = X_t, t < T
RV should be continuous and non-explosive (E[|X_t|] < inf)

Submartingale change = to >
Supermartingale change = to

Question 6

List uses of arbitrage free prices

Answer 6

1. Pricing a new product
2. Risk management
3. Marking to market
4. Comparing with market

Question 7

State the arbitrage theorem

Answer 7

Vector of Current Prices = State Vector * State Probabilities

Question 8

List important interpretations of the arbitrage theorem

Answer 8

State prices should add up to 1/(1+r)
Risk neutral probability = State probability * (1+r)
Risk neutral probabilities should add up to 1
The state price can be interpreted as the value of $1 today if state i is realized and no money is realized in all other states

Question 9

Briefly describe the two main pricing methods

Answer 9

Method 1: arbitrage on financial assets that are martingales (method of equivalent martingales)
Method 2: construct a risk-free portfolio to obtain a PDE implied by the lack of arbitrage opportunities

Question 10

Define a martingale

Answer 10

A process S_t, t in (0, inf), w.r.t a family of information sets I_t and w.r.t a probability P if for all t > 0:

1. S_t is known given I_t (I_t adapted)
2. E^P[|S_t|] < inf
3. E^P_t[S_T] = S_t for all t < T

Question 11

State the additional requirement for a square integrable martingale

Answer 11

Finite second moment: E[X_t^2] < inf

Question 12

State the Doob-Meyer Theorem

Answer 12

Suppose that:
1. X_t, t in (0, inf), is a right continuous submartingale w.r.t I_t
2. E[X_t] < inf for all t
Then X_t = M_t + A_t:
1. M_t is a right continuous martingale w.r.t. probability P
2. A_t is an increasing process measurable w.r.t. I _t

Question 13

Define an adapted process

Answer 13

If the value of S_t is included in the information set I_t at each t > 0, then it is said that S_t, t in (0, inf), is adapted to I_t, t in (0, inf)

Question 14

Explain why the (dS_t)^2 term cannot be dropped in a stochastic setting

Answer 14

We need to model variance to include the stochastic nature of the process, which is related to the second derivative of the stock process

Question 15

State the distribution of W_t+h - W_t

Answer 15

Normal(0, h)

Question 16

Define a Wiener process

Answer 16

A Wiener process W_t, relative to a family of information sets I_t, is a stochastic process s.t.:

1. W_t is a contiuous, square integrable martingale
2. W_0 = 0
3. E[(W_t - W_s)^2] = t - s, s <= t
4. W_t is continuous over t

Question 17

State properties of a Wiener process

Answer 17

1. W_t has uncorrelated increments
2. W_t has zero mean
3. W_t has variance t
4. The process is continuous and movements are infinitesimal

Question 18

Define Brownian Motion

Answer 18

A random process B_t is a Brownian motion if:

1. B_0 = 0
2. B_t is continuous and has stationary, independent increments
3. B_t is continuous in t
4. B_t - B_s is Normal(0, |t - s|)

Question 19

State the key result of the Levy Theorem

Answer 19

Wiener process and Brownian motion process are the same

Question 20

Differentiate between normal and rare events

Answer 20

Normal events: dependent on size, limited by probability

Rare events: dependent on probability, limited by size

Question 21

State Iso’s Isometry

Answer 21

E[XdW^2] = E[X^2dt]

Question 22

State three properties of the Ito Integral

Answer 22

1. Existence - if f is continuous and nonanticipating, then the Ito Integral exists
2. Martingale - E[Ito Integral] = 0
3. Additive
   Also, the Ito Integral is a random variable.

Question 23

State Ito’s Lemma

Answer 23

Second degree Taylor expansion

Question 24

State the uses of Ito’s Lemma

Answer 24

Provides a tool for obtaining stochastic differentials for functions of random variables
Helps with evaluating certain integrals
1. Guess function
2. Use Ito’s Lemma to obtain SDE
3. Apply the integral operator
4. Rearrange and solve

Question 25

Briefly describe two jump frameworks

Answer 25

Jump-Diffusion Model - adds a jump component to the SDE | Variance-Gamma - perform a stochastic time change

Question 26

Describe the jump-diffusion process

Answer 26

No jumps, assume GBM Jump (N_t = 1) with probability lambda*dt Expected jump size: K = E[e^J - 1] Jump Diffusion SDE: dS/S = (mu - lambda*K)*dt + sigma*dW + (e^J -1)*dN

Question 27

Describe the variance-gamma process

Answer 27

Perform a stochastic time change Stochastic time has a gamma distribution(t; 1, v) Process given stochastic time: b(t, sigma, theta) = theta*t + sigma*dW Unconditional process: X(t; sigma, v, theta) = b(y(t; 1, v), sigma, theta) = theta*y(t; 1, v) + sigma*W(y(t; 1, v))

Question 28

List 5 types of SDEs

Answer 28

1. Linear (ABM): dS = mu*dt + sigma*dW 2. GBM: dS = mu*S*dt + sigma*S*dW 3. Square Root: dS = u*S*dt + sigma*sqrt(S)*dW 4. Mean Reverting: dS = lambda*(mu - S)*dt + sigma*S*dW 5. Ornstein Uhlenbeck: dS = -mu*S*dt + sigma*dW

Question 29

State the two major methods of pricing derivative products

Answer 29

Partial differential equations/risk-free portfolio | Transform underlying processes into martingales

Question 30

State assumptions underlying Black-Scholes

Answer 30

1. Underlying asset is a stock 2. Stock does not pay dividends 3. Derivative asset is a European call 4. Risk free rate is constant 5. No transaction costs

Question 31

State differences between standard and extic options

Answer 31

1. Expiration value may depend on an event 2. Random expiration dates 3. Multiple underlying assets

Question 32

State the two conditions to apply the Girsanov Theorem

Answer 32

1. e_t must be a martingale w.r.t I_t and P | 2. Novikov condition must hold

Question 33

State Girsanov’s Theorem and be able to apply the Markov Condition

Answer 33

Compile results in an iPad note

Question 34

State the MGF of a Normal R.V.

Answer 34

M(lambda) = exp(lambda*mu*t + 0.5*sigma^2*t*lambda^2)

Question 35

State the conditional forecast for GBM

Answer 35

E[S_t|S_u, u