Stochastic Calculus, MFD Flashcards
Give the practitioner and Ingersoll definition of derivatives
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
Define swap
The simultaneous selling and purchasing of various financial assets
Describe three types of swaps
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
Define arbitrage
Taking simultaneous positions in different assets to produce a riskless profit
- Negative investment with a nonnegative profit
- Zero investment with a positive profit
Define martingale
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
List uses of arbitrage free prices
- Pricing a new product
- Risk management
- Marking to market
- Comparing with market
State the arbitrage theorem
Vector of Current Prices = State Vector * State Probabilities
List important interpretations of the arbitrage theorem
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
Briefly describe the two main pricing methods
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
Define a martingale
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:
- S_t is known given I_t (I_t adapted)
- E^P[|S_t|] < inf
- E^P_t[S_T] = S_t for all t < T
State the additional requirement for a square integrable martingale
Finite second moment: E[X_t^2] < inf
State the Doob-Meyer Theorem
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
Define an adapted process
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)
Explain why the (dS_t)^2 term cannot be dropped in a stochastic setting
We need to model variance to include the stochastic nature of the process, which is related to the second derivative of the stock process
State the distribution of W_t+h - W_t
Normal(0, h)
Define a Wiener process
A Wiener process W_t, relative to a family of information sets I_t, is a stochastic process s.t.:
- W_t is a contiuous, square integrable martingale
- W_0 = 0
- E[(W_t - W_s)^2] = t - s, s <= t
- W_t is continuous over t
State properties of a Wiener process
- W_t has uncorrelated increments
- W_t has zero mean
- W_t has variance t
- The process is continuous and movements are infinitesimal
Define Brownian Motion
A random process B_t is a Brownian motion if:
- B_0 = 0
- B_t is continuous and has stationary, independent increments
- B_t is continuous in t
- B_t - B_s is Normal(0, |t - s|)
State the key result of the Levy Theorem
Wiener process and Brownian motion process are the same
Differentiate between normal and rare events
Normal events: dependent on size, limited by probability
Rare events: dependent on probability, limited by size
State Iso’s Isometry
E[XdW^2] = E[X^2dt]
State three properties of the Ito Integral
- Existence - if f is continuous and nonanticipating, then the Ito Integral exists
- Martingale - E[Ito Integral] = 0
- Additive
Also, the Ito Integral is a random variable.
State Ito’s Lemma
Second degree Taylor expansion
State the uses of Ito’s Lemma
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
Briefly describe two jump frameworks
Jump-Diffusion Model - adds a jump component to the SDE
Variance-Gamma - perform a stochastic time change
Describe the jump-diffusion process
No jumps, assume GBM
Jump (N_t = 1) with probability lambdadt
Expected jump size: K = E[e^J - 1]
Jump Diffusion SDE: dS/S = (mu - lambdaK)dt + sigmadW + (e^J -1)*dN
Describe the variance-gamma process
Perform a stochastic time change
Stochastic time has a gamma distribution(t; 1, v)
Process given stochastic time: b(t, sigma, theta) = thetat + sigmadW
Unconditional process: X(t; sigma, v, theta) = b(y(t; 1, v), sigma, theta) = thetay(t; 1, v) + sigmaW(y(t; 1, v))
List 5 types of SDEs
- Linear (ABM): dS = mudt + sigmadW
- GBM: dS = muSdt + sigmaSdW
- Square Root: dS = uSdt + sigmasqrt(S)dW
- Mean Reverting: dS = lambda(mu - S)dt + sigmaSdW
- Ornstein Uhlenbeck: dS = -muSdt + sigma*dW
State the two major methods of pricing derivative products
Partial differential equations/risk-free portfolio
Transform underlying processes into martingales
State assumptions underlying Black-Scholes
- Underlying asset is a stock
- Stock does not pay dividends
- Derivative asset is a European call
- Risk free rate is constant
- No transaction costs
State differences between standard and extic options
- Expiration value may depend on an event
- Random expiration dates
- Multiple underlying assets
State the two conditions to apply the Girsanov Theorem
- e_t must be a martingale w.r.t I_t and P
2. Novikov condition must hold
State Girsanov’s Theorem and be able to apply the Markov Condition
Compile results in an iPad note
State the MGF of a Normal R.V.
M(lambda) = exp(lambdamut + 0.5sigma^2t*lambda^2)
State the conditional forecast for GBM
E[S_t|S_u, u
