Stochastic Process

Brownian Motion and Martingales

  • W(0) = 0
  • Continuous
  • Independent increments
  • normal increments

Levy's Theorem

Martingales:

  • $$M_t^2 - [M,M]_t$$
  • $$exp(M_t^2 - \frac{1}{2} [M,M]_t)$$

Momentum of a Normal RV: $$E (Y-\mu)^n = \sigma^n (n-1)!!$$, if n is odd, 0 otherwise

  1. derive the probability that B_1 > 0 and B_2 < 0? (use indepedent normal)

Ito's Lemma and Black- Scholes

  1. Prove the Martingales listed above (drift = 0)

    1. valuation

    varaince swap

    swaptoion

    barrier options

    quanto, exchange options

    chooser, cap, forward-start option

  3. What is transition density? Transition density of BM/GBM?

    p (t,x,T,y)

  4. Derive Kolmogorov Backward Equation/ Kolmogorov Forward Equation?

  5. Derive Dupire’s Equation.

    c_k = e^{-rt} P(S_t >K) c_{kk} = e^{-rt} p(0,S_0,T,K)

  6. Martingale? Exponential Martingale? (- Moment Generating Functions) Quadratic Variation?

  7. Derive B-S P.D.E

  8. Delta Hedge with futures/ options/ other instruments?

  9. Self-financing condition?

  10. Hedging Portfolio with Multiple Rates/ Collaterals

  11. Stochastic Vol/ Interest Rate P.D.E

  12. Asian Option/ Corridor Option (path dependent option) P.D.E

  13. Derive B-S formula - Risk Neutral Pricing

    One Stock Option: h(S_T) as payoff ( eg. S_T^2, ln(S_T), 1/ (S_T)

    Two Stock Options: Spread option: max(S1- S2,0)

  14. Futures and Forward

  15. derive Futures price (prove it is a martingale)

  16. derive forward price

  17. Forward and Futures Spread ( negatively correlated underlying and discount process, futures> forward)

  18. Fixed Income - Bond, Short Rate, Swap

  19. Short Rate Models - Hoo- Lee, Vasiek, Hull-White, CIR model - Bond P.D.E - Affine Interest

    Models

  20. value a swap

  21. value a swaption

  22. LIBOR, foward interest rate

  23. interest rate futures, forward interest rate

  24. Forward LIBOR

  25. Derive HJM condition

  26. (explain) Change of Measure- Change of Numeraire- Girsanov Theorem (Change Drift)

  27. Foreign Exchange

  28. Change of Measure - foreign exchange measure, foreign exchange measure

  29. price a Quanto

  30. Standard S.D.E

  31. Solve GBM

  32. Solve O-U process ( Vasicek model)

  33. Browian Motion

  34. Levy’s Theorem

  35. Reflection Principle

  54. Measure-based probability

    1.1 Prob Space- \Omega(sample space), P(prob measure), F(sigma-algebra)

    Sigma Algebra

    infinite (all inclusive) sigma-algebra

    **Kolmogorov Extension Theorem

    Borel Sets

    Measure – Countabily additive functions

    Lebesgue Measure

    Probability Measure

1.2 Expectation

  1. Measurability

  2. Partial Averaging

    Lebesgue Integral

    properties: discrete form, comparison, linearity, Jensen’s Inequality

    Random Variable (F measurable random variables)

    sigma algebra generated by R.V.

    distribution/ distribution measure/ density

1.3 Independence

Probability definition, sigma-algebra definition

expectation definition

1.4 Conditional Expectation

A f-measurable random variable satisfies partial averaging identity

\int_F E(X\f) dp = \int_F Xdx

Conditional Probabilities

A F-measurable random variable, integral upto interaction probability

\int_F P(A_i|\f) = P(A_i \and F)

Randon-Nikodym Theorem

Properties

  1. Linearity

    2 Taking out what is know

    3 iterated conditioning

  2. independence property (like no information) 5. Jensen’s Inequality

1.5 Filtration and adapted process

Filtration generated by X ( family of sigma algebras)

Adapted Process (X(t) is t-measurable)

1.6 Martingale & Markov Process

Martingale

Stopping Times (function that the event is in filtration)

Dobb’s Optional Sampling Theorem

Markov Process

Super martingale, Sub martingale

  1. Browian Motion

    Symmetric Random Walk

    Variance, Expectation,

    First Variation (use Mean-Value Theorem when it is differentiable)

    Quadratic Variation (Attention: it is a Sum)

    Brownian Motion

    continuous

    Stationary

    Independent increment,

    normal increments, variation

    Levy’s Theorem (Levy’s Criterion)

  2. Ito Integral

    Martingale Property

    Ito Isometry

    Quadratic Variation

    Ito-Doubelin formula

    Browian filtration

    F(s)< F(t), Browian motion measurable at time t.

    Martingles

    B,

    F(t,W) - \int_o^t (\diff_t f(s,W)- ½ \diff^2 f(s,W)) ds (Ito’s Lemma)

  3. Black-Scholes

  4. Multi-dimensional

Joint Quadratic Variation

Multi-dimensional Ito’s Formula

Multi-dimensional Browian Motion

Multi-Dimensional Levy’s Theorem

One-dimensional, multi-dimensional

  1. Risk-Neutral Pricing

Random-Nikoym Theorem

Girsanov Theorem

Martingale Representation Theorem

Futures and Forward

  1. Relationship with P.D.E

Transition Density

Markov Process / Markov Property

Feynman - Kac Theorem

Kolmogrov Forward Equation

Kolmogrov Backward Equation

Dupire’s Equation - Volatility

Delta Hedge

under different rates

with dividend (rate q

Fixed Income ( Stochastic Interest Rate)

Short Rate Models

Hoo-Lee, Vasicek, Hull-White, CIR

Yield, Yield Curve

Forward Interest Rate - instaneous interest rate

LIBOR

Whole Yield Models

Forward Rate Models

  1. Derive Heath–Jarrow–Morton Condition and Explain

    $$df(t,T) = \alpha dt + \sigma dW_t, B(t,T) = e^{-\int_t^T f(t,u)du},\alpha^(t,T) = \int_t^T \alpha(t,u) du, \sigma^(t,T) = \int_t^T \sigma(t,u) du$$

    $$d(D(t)B(t,T)) = -\sigma^ (t,T) D(t) B(t,T)[dW_t + \frac{\alpha^(t,T) - \frac{1}{2} (\sigma^(t,T))^2}{\sigma^(t,T)} dt]$$

$$\theta(t) = \frac{\alpha^(t,T) - \frac{1}{2} (\sigma^(t,T))^2}{\sigma(t,T)}, dW(t) + \theta(t)dt = d\widetilde{W}(t)$$， no arbitrage: $$\sigma(t,T) \theta(t) = \alpha(t,T) - \sigma^(t,T) \sigma(t,T) $$

$$df(t,T) = \sigma(t,T) \sigma^*(t,T) dt + \sigma(t,T) d\widetilde{W}(t)$$

  4. Change of Measure/ Change of Numeraire

    Multi-dimensionaly Levy’s Theorem

    Girsanov Theorem (Multi-dimensional)

    attention: used on independent Browian Motions

    eg. multi-factor interest models/ volatility models

Change of Numeraire

Forward Measure

numeraire

* Quotient of Martingales is a martingale using the denominator as the numeraire

* The volatility ( vol - R-N Derivative) is the numerators’ vol vector - denominator’s vol vector

———————

做题技巧

  1. Tower Property

  2. For(t,t) = St , f(t,t) = Rt ( the underlying asset price)

  3. Change Measure A-B-C by Zt = *numeraire B/ Numeraire A (with normalization)

  4. 凑 exponential martingale， 凑Black - Scholes Form/ Black’s Form

  5. MGF (构造

  6. 做差

  7. Conditional Expectation- Tower Property

  8. Ito’s Formula

Important Martingales

M^2- [M,M]t

MN - [M,N]

exp (M^2 - [M,M]t/2)

Ito f - \int_0^t ( df/dt +d^2f/dx^2)

Martingale stopped at stopping time

概率：构造partition， patrial averaging

正态： decomposition

martingale ： 作差

results matching ""

    No results matching ""