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1

What are the basic bulding blocks of derivatives?

  1. Options
  2. Forwards and futures

e.g., swaps = hybrid of options + forwards/futures

2

What is arbitrage?

Taking simultaneous positions in different assets so that one guarantees a riskless profit higher than the risk-free rate

3

Describe the arbitrage theorem and its relevance.

Given two states with a positive probability of ocurrence,

  1. If StatePrices > 0 can be found such that the asset prices satisfy S = D StatePrices, then no arbitrage opportunities exist
  2. Conversely, if no arbitrate opportuinities exist, StatePrices can be found

 

This provides a general method for pricing derivative assets.

4

What are some uses of arbitrage-free prices?

  1. Launch of a financial product by derivatives house (new product)
  2. Measure risks in a portfolio (risk management)
  3. Marking-to-market assets held in mortfolios to understand MV of iliquid assets (MtM)
  4. To identify mispriced profit opportunities

5

How can you price using real world probabilities?

  1. Modern portfolio theory and utility functions
  2. Certainty equivalent value under the real random walk:
    1. Find expected utility under real random walk
    2. Certaintly equivalent = U-1(Expected utility)
  3. Real Expected PV of Option Payoff +/- k * Std Dev

6

What are two arbitrage-free methods to price derivatives?

  1. Equivalent measures / Martingale transformations
    • Tools include:
      • Dobb-Meyer decomposition into a trend and a martingale piece
      • Normalization by dividing the martingale by another arbitrage-free price (e.g., divide by bond price Bt)
      • Equivalent measures through Girsanov's theorem to derive risk-neutral probabilities and remove predictability of real-world
    • Arbitrage on financial assets which are martingales
    • Implies converting assets to martingale
  2. Partial-differential equations (PDE)
    • Construct a risk-free portfolio to obtain an Black-Scholes PDE by the no arbitrage theorem
    • e.g., Black-Scholes PDE

 

7

What are the main steps in risk-neutral pricing of deriatives?

  1. Obtain the underlying's pricing model (e.g., lognormal, skewness/kurtosis,...)
  2. Calcualte the derivative's payoffs at expiration w.r.t. the underlying
  3. Obtain risk-neutral probabilities
  4. Calculate expected payoffs
  5. Discount at r.f.r.

8

Define the three types of convergence

  1. Almost sure convergence
    • Pr( | limn→inf Xn - X | > d ) = 0
  2. Mean-squared convergence
    • limn→inf E[(Xn-X)2] = 0
  3. Weak convergence
    • Xn→X and Pn→P if EPn[f(Xn)] → EP[f(X)]

9

Define previsible

That the random variable only depends on previous history

10

Define a filtration

A collection of σ-algebras Ft, 0 ≤ t ≤ T for a positive fixed number T where:

  • Ω is a non-empty sample space
  • For each t in [0,T], there a σ-algebra Ft.
  • If s ≤ t, then every set in Fs is also in Ft.

11

Define a field and a σ-algrebra

A family or collection of subsets of Ω where:

  1. ∅ ∈ F
  2. If A ∈ F, then AC ∈ F
  3. If A1, A2, ... ∈ F, then
    1. for a field Ui=1 to n Ai ∈ F
    2. for a σ-algrebra Ui=1 to inf Ai ∈ F

12

Define a probability measure P on {Ω,F}

  1. P({∅}) = 0
  2. P({Ω}) = 1
  3. If A1, A2, ... ∈ F and {Ai}i=1 to inf is disjoint such that Ai∩Aj=∅ i<>j, then P(Ui=1 to inf Ai) = Σi=1 to inf P(Ai)

13

Define a real-valued random variable X

A function X: Ω → R such that {ω

14

Define an adapted stochastic process

Let Ω be a non-empty sample space with filtration Ft, where t is between 0 and T; and let Xt be a collection of random variables indexed by t.

The collection of random variables Xt is an adapted stochastic process if for each t, the random variable Xt is Ft measurable.

15

Define conditional expectation

E(X|G) is any RV that satisfies:

  1. E(X|G) is G-measurable
  2. For every set A in G:
    • A E(X|G) dP = ∫X dP

where X is a non-negative, integral RV and G is a sub-σ-algebra of F.

16

What are the properties of conditional expectation?

  • Conditional probability
    • If 1A is an indicator RV for an event A, then E(1A|G) = P(A|G)
  • Linearity
    • If X1,..., Xn are integrable RVs and c1,..., cn are constants, then E(c1X1 + ...  cnXn | G) = c1E(X1|G) + ... + cnE(Xn|G)
  • Positivity
    • If X ≥ 0 almost surely, then E(X|G) ≥ 0 almost surely.
  • Monotonicity
    • If X,Y are integrable RVs and X≤Y almost surely, then E(X|G) ≤ E(Y|G)
  • Compute expectations
    • E[E(X|G)] = E(X)
  • Take out what is known
    • If X, Y are integrable RV and X is G-measurable, then E(XY|G) = X E(Y|G)
  • Tower property
    • If H is a sub-σ-algebra of G, then E[E(X|G)|H] = E(X|H)
  • Measurability
    •  If X is G-measurable, then E(X|G) = E(X)
  • Independence
    • If X is independent of G, then E(X|G) = E(X)
  • Conditional Jensen's inequality
    • If f is a convex function, then E[f(X)|G] ≥ f[E(X|G)]

17

Define the partial averaging property

AE(X|G)dP = ∫AXdP

18

What is a martingale?

A submartingale? A supermartingale?

Suppose one has information It at time t, i.e., market information.

A RV Xt for all s > 0 is a martingale w.r.t. P if

  1. EP[Xt+s | It] = Xt
  2. E(|Xt|) < infinity
  3. Xt is It-adapted

 

A submartingale will hold EP[Xt+s | It] ≥ Xt

A superartingale will hold EP[Xt+s | It] ≤ Xt

19

What are the three martingale ingredients and the three martingale properties?

Ingredients:

  1. Process
  2. Measure
  3. Information set

Properties:

  1. St is It-adapted
  2. EP|St| < infinity
  3. EtP(ST) = St for all t < T

20

When is a continuous martingale a "continuous square integrable martingale"?

If Xt has finite second moment E[Xt2] < infinity for all t > 0

21

What is a Markov process? A strong Markov process?

A process {X1, ... Xt} is Markov if

Pr(Xt+s <= xt+s|x1,...,xt) = Pr(Xt+s <= xt+s|xt)

that is, only the last piece of information is relevant.

 

In other words, the conditional distribution of Xt given Fs only depends on Fs.

 

A strong Markov process is such that Xt+s - Xt is independent of Xt.

22

State the key Markov and Martingale properties.

Compare and contrast.

  1. Markov
    • The expected value of Si conditional on all past event Ii-1 depends only on its previous value Si-1
  2. Martingale
    • The expected future value of Si is equal to its current value, i.e. Ej(Si) = Sj for all j < i 

23

Define a Wiener process and its properties

For a probability space (Ω,F,P), a stochastic process {Wt: t≥0} is a standard Wiener process if:

  1. Wt is a square integrable martingale
  2. Wt is continuous (i.e., no jumps)
  3. W0 = 0 
  4. "Stationary increments" property:
    • Wt+s - Wt ~ N(0,s)
    • E[ (Wt - Ws)2 ] = t - s
  5. "Independent increments" property: Wt+s - Wt is independent of Wt

⇒ ​Properties:

  • Wt has uncorrelated increments
  • Wt has zero mean
  • Wt has a variance t
  • The process is continuous so, in infinitesimally small intervals t, movements of Wt are infinitesimal

24

Define Brownian Motion

  1. B0 = 0
  2. Bt has stationary, independent increments
  3. Bt is continuous in t
  4. Bt - Bs ~ N(0, |t - s|)

25

Describe six Brownian motion properties

  1. Finiteness - Brownian motion path is almost surely finite
  2. Continuity - paths are continuous
  3. Markov
  4. Martingale
  5. Quadratic variation
    • [0,t] has quadratic variation that mean-square converges to t
  6. Normality
    • X(ti) - X(ti-1) ~ N(0, σ2 = ti - ti-1)

26

What are the key expectations of Brownian Motion?

  • E0(Wt2) = V0(Wt) = t
  • E0(Wt4) = 3t2 (kurtosis)
  • E0(Wt2k) = (2k)! tk / (2kk!)
  • E0(Wtk) = 0, for odd k's
  • E0[exp(σWt)] = exp(.5σ2t)

27

What is the distribution of Wi - Wj?

N ( µ = 0, σ2 = |i - j| )

28

Summarize the Levy theorem

Any Wiener process relative to It is Brownian Motion

29

State properties of the Ito integral

  1. Existence
    • If f is continuous and nonanticiapting, then an Ito integral exists
  2. Martingale
    • The Ito integral is a Martingale
  3. Additive
  4. Ito isommetry
    • It follows that for a square integrable Xt, (dWt)2 = dt

30

What are the three jump frameworks described in MFD?

  1. Simple jump
    1. Xt has a value 0 until a certain event occurs under which it assumes a value of 1
  2. (Merton) Jump-diffusion model
    • Can be based on
      • dSt / St = (μ - λκ)dt + σdWt + (e- 1)dNt 
    • Where a jump dNt = 1 occurs with probability λdt, and the expected proportional jump size is κ = E(eJ - 1)
  3. Variance-Gamma process
    • Stochastic time change adds volatility, where
      • t* = γ(t;1,v) ~ Gamma distribution
      • The unconditional process is X(t;σ,v,θ) = b(t*,σ,θ) = θ t* + σ W(t*)

31

Fill in the blanks:

  • Normal events​
    • ri
    • qi
    • Limiting size = 
    • Limiting prob = 
    • Time independence = 
  • Rare
    • ri
    • qi
    • Limiting size = 
    • Limiting prob = 
    • Time independence = 

Fill in the blanks:

  • Normal events
    • ri = 1/2
    • qi =  0
    • Limiting size = 0
    • Limiting prob = pi > 0
    • Time independence = probability
  • Rare
    • ri = 0
    • qi = 1
    • Limiting size = wi > 0
    • Limiting prob = 0 
    • Time independence = size

32

Describe how to classify PDEs into elliptical, parabolic and hyperbolic.

a0 + a1Ft + a2FS + a3FSS + a4Ftt + a5FSt = 0

  • Elliptical: a52 - 4a3a4 
  • Parabolic: a52 - 4a3a4 = 0 
  • Hyperbolic: a52 - 4a3a4 > 0

33

What are the SDEs for:

  1. Linear
  2. GBM
  3. Square root process
  4. Mean-reverting
  5. Ornstein-Uhlenbeck

  1. dSt = µ dt + σ dWt
  2. dSt = µ St dt + σ St dWt
  3. dSt = µ St dt + σ sqrt(St) dWt
  4. dSt = λ (µ - St) dt + σ St dWt
  5. dSt = -µSt dt + σ dWt

34

State Ito's lemma

Suppose that:

  1. F(St,t) is a twice-differentiable function
  2. dSt = atdt + σtdWt
  3. at and σt are well-behaved drift and diffusion parameters

Then

  • dFt = dF/dSt dSt + dF/dt dt + 0.5σt2 d2F/dSt2 dt
  • (substitute dSt above)

35

What are uses of the Ito lemma?

  1. Tool for obtaining SDEs
    • Identify F(St,t) or F(Wt,t)
    • Identify at and σt in dSt = atdt + σtdWt
    • Calcualte partial derivatives of F
    • Plug & chug
  2. Evaluate stochastic integrals, through the following 4 steps:
    • Guess a function F(Wt,t)
    • Use Ito's lemma to obtain an SDE for F(Wt,t)
    • Apply integral operator on both sides of the equation and simplify into known integrals
    • Rearrange to solve for the desired integral

36

Define the Ito integral

It = ∫[0,t] f(Ws,s)dWs = limn→infΣi=[0,n-1] f(Wti, ti) (Wti+1 - Wti)

where f is a simple process i.e., constant over [ti,ti+1), 

and ti = it/n, 01<...n-1n>

37

Define Ito's Isommetry

E( ∫f(Ws,s) dWs ) = E( ∫ [f(Ws,s)]2ds )

38

What is can the Feynman-Fac formula be used for?

V(Xt,t) = E[ Ψ(XT) - exp(-∫[t,T]r(u)du|Ft) ]

Can be used for Monte Carlo simulation

39

What are the properties of Ito's Integral?

  1. Paths of It are continuous
  2. It is Ft-measurable
  3. It is a martingale
  4. Quasi-linearity, where:
    • If It and Jt are Ito integrals w.r.t. Ws,
    • then ciIt + cjJt = ∫ [cif(Ws,s) + cjg(Ws,s)] dWs
  5. Ito isommetry
  6. Quadratic variation t0 to tf(Ws,s)2ds

40

What are two types of solutions to SDEs?

  1. Strong solution
    • St is It-adapted, and need knowledge of Wt
  2. Weak solution
    • Determined by ~Wt, a Wiener process whose distribution is determined simultaneously with ~St
    • Could be or not be It-adapted

41

What are the BS assumptions?

  • Underlying is a stock
  • No dividends
  • European style
  • Risk-free rate is constant
  • No transaction costs (commissions, bid-ask spread) or indivisibilities

42

State the BS PDE and the type of PDE it is

(dV/dt) + 0.5σ2S2(d2V/dS2) + rS(dV/dS) - rV = 0

Linear parabolic PDE

43

Describe the following exotic options:

  • Lookback (floating, fixed);
  • Ladder;
  • Trigger or Knock-in, and Knock-out;
  • Basket;
  • Multi-assets;
    • Multi-assets call;
    • Spread call;
    • Portfolio call;
    • Dual-strike call
  • Asian options

  • Lookback - floating
    • Payoff = (ST - Smin)+
  • Lookback - fixed
    • Payoff = (Smax - K)+
  • Ladder
    • Once the underlying asset reaches a threshold, the return of the option is locked-in
  • Trigger or Knock-in
    • Option to exercise once the spot price falls below/above a barrier, otherwise the option expires with a rebate value
  • Knock-out
    • Option to exercise expires immediately if the spot price falls below/above a barrier and expires at a rebate value
  • Basket
    • Underlying is a basket of various financial instruments
  • Multi-assets
    • Multi-asset call = (max{S1T,S2T} - K)+
    • Spread call = ( (S1T - S2T) - K)+
    • Portfolio call = ( (θ1S1T + θ2S2T) - K)+
    • Dual-strike call = max{0, S1T - K, S2T - K}
  • Asian option
    • Payoff depends on the average price of the underlying asset over the life of the option

44

What are three differences between standard B-S (vanilla options) and exotic options?

  1. Expiration value (e.g., average, min, max,...)
  2. Expiration date (e.g., American)
  3. Underlying (e.g., basket, multi-assets)

45

State the Dobb-Meyer theorem

If Xt is a right-continuous submartingale w.r.t. {It} and E(Xt) < infinity for all t > 0, then Xt admits the decomposition

Xt = Mt + At, where

  • Mt is a right-continuous martingale w.r.t. probability P, known as the randomness component
  • At is an increasing process measurable w.r.t. It, known as the trend or drift component

46

How can it be determined if two measures are equivalent?

Two measures are equivalent if they have

  1. the same sample space
  2. the same possibilities

(But not necessarily the same probabilities)

47

State the Girsanov Theorem

If 

  1. ξt is a martingale (previsible process) w.r.t. It and P
  2. The Novikov condition is met, i.e., the non-explosiveness condition where E(exp{ integral( ξs2ds)from 0 to t } ) < infinity

Then Wt* defined by

Wt* = Wt - integral(Xudu)from 0 to t

is a Wiener process w.r.t. It and Q, where

Q(A) = EP(1AξT)

with A being an event determined by IT.

48

What is the Radon-Nikodym derivative?

dQ(zt) / dP(zt) = ξ(zt)

49

What are the implications of the Girsanov theorem?

  1. Every equivalent measure is given by a drift change
  2. One equivalent risk-neutral measure
  3. Not needed for Black-Scholes, but useful for more complicated analysis (e.g., stochastic volatility)
  4. Important for fixed-income with different maturities

50

How is quadratic variation calculated?

Sum{ (Si - Si-1)2 }

51

Define a convex function

A function f is convex on an interval if for every x and y in the interval

x + (1-λ)y) <= λ (x) + (1-λ) (y)

for any λ in [0,1].

 

For example call/put payoffs and x2

52

How do you quantify convexity?

1/2 E(ε2) f''(S)

i.e., half times the expected randomness and function convexity

Must be accounted for in a contract's pricing if an underlying variable/parameter is random and convex.

53

State the Jensen's inequality

If f is a convex function and x is a random variable, then

E( (x) ) >= ( E(x) )

54

What are the requirements to apply risk-neutral pricing?

  • Complete market
  • Enough traded quantities with which to hedge risk
  • Continuous hedging
  • No transaction costs
  • Accurate parameters
  • No jumps

55

What are the problems/concerns of real world probability pricing?

  • Need to measure real probabilities, i.e., μ 
  • Need to decide on an utility function (U = μ - kσ2) or measure of risk aversion

56

What are common confusions between RW and RN?

  1. Forward price  = market expected future price
  2. Forward curve = market's expected value of the spot rates
    • A risk premium is built into the forward curve
  3. Under RN, we replace µ with the risk-free rate
    • Only allowed when assumptions allow you to do so
  4. The delta of an option is the probability of it ending up in the money
    • The probability of ending up in the money depends on the real probabilities and growth rate (i.e., N(d2*)), which is NOT captured by the BS price delta (i.e., N(d1))

57

What are further arguments for risk-neutral pricing in a BS world?

  1. If hedged correctly in a BS world, all risk is eliminated ⇒ RFR and no compensation for risk
  2. If we assume dS = µS dt + σS dX, then the µ's cancel in the derivation of the BS equation
  3. Changing measures converts RV into martingales
  4. Options can be constructed by putting together vanilla options, and they can be priced through synthesization/replication

58

Define a complete market

  • One where a derivative product can be artificially made from more basic instruments
  • One where there exists the same number of linearly independent securities are there are future states of the world

 

Either way, there exists an unique martingale measure

59

State models that result in complete markets

  1. Lognormal with a constant volatility
  2. Binomial

60

Define the following:

  1. Trading strategy
  2. Self-financing trading strategy
  3. Admissible trading strategy
  4. Attainable contingent claim
  5. Admissible arbitrage opportunity
  6. Complete market

  1. Trading strategy:  
    • Πt = ΦtSt + ΨtBt
  2. Self-financing trading strategy: 
    • t = ΦdSt + ΨdBt
    • i.e., change in portfolio value is solely due to changes in the market and not due to injection/extractiokn fo funds
  3. Admissible trading strategy:
    • A trading strategy with a lower bound profit
    • There exists some α > 0 such that for all t in [0,T], the portfolio value will be greater than -α almost surely.
  4. Attainable contingent claim:
    • Ψ(ST) is attainable if there exists an admissible trading strategy worth Πt = Ψ(ST) at time T
    • A contingent claim is a collection of simple contingent claims (state claims that pay off in only one state)
  5. Admissible arbitrage opportunity - that where the following holds:
    • Π0 = 0 
    • P(ΠT ≥ 0) = 1
    • P(ΠT > 0) > 0
  6. Complete market - given NA = # traded assets and NR = # of sources of risks, the following holds:
    • ​NA < NR ⇒ incomplete, no arbitrage
    • ​NA = NR ⇒ complete, no arbitrage
    • ​NA > NR ⇒ arbitrage

61

What are pricing approaches for incomplete models?

  1. Actuarial method
    • Price in an average sense
    • Relies on Central Limit Theorem (CLT)
    • e.g., pricing insurance contracts
  2. Consistent pricing
    • Make options consistent with each other
    • Common when having stochastic volatility models
    • a.k.a.
      • Pricing with a model that explicitly contains the market price of a risk parameter
      • Pricing options in terms of prices of other options

62

What are complications of delta hedging?

  1. Transaction costs
  2. Estimate detla based on a model that may be wrong
  3. Continuous rebalancing
  4. Underlying needs to be consistent with model assumptions, usually need to assume Brownian motion with no jumps

63

Compare and contrast forwards and futures

  • Mark-to-Market
    • Futures: Yes
    • Forwards: No
  • Settlement
    • Futures: Daily
    • Forwards: Upon maturity
  • Margin system
    • Futures: Yes
    • Forward: No
  • Trading
    • Futures: Exchanges
    • Forward: OTC
  • Standardization
    • Futures: Yes
    • Forward: No
  • Contract flexibility
    • Futures: Less
    • Forward: More
  • Liquidity
    • Futures: More
    • Forward: Less
  • Counterparty risk
    • Futures: Less
    • Forward: More

64

What is the delta of a forward contract?

A futures contract?

deltaforward = e-δ (T-t)

deltafutures = e(r-δ) (T-t)

65

Why are the deltas of a futures and forward contracts different?

Because futures are marked-to-market daily, while forward contracts are settled at maturity

66

What is the mgf for Zt~N(0,t)?

M(λ) = exp( 1/2 t λ2)

67

What is the mgf for Yt ~ N(μt,σ2t) ?

M(λ) = exp( λμt + 1/2 σ22)

68

dX1 dX2 = ?

dX1 dX2 = ρdt

69

Given dSt = rStdt + σStdWt, calcualte d(lnSt)

d(lnSt) = (r - 1/2 σ2) dt + σ dWt

70

Given d(lnSt) = rdt + σ dWt, calculate dSt / St

dSt / St = (r + 1/2 σ2) dt + σ dWt

71

d(XtFt) = ?

d(XtFt) = dXt Ft + Xt dFt + dXt dFt

72

If dS = μSdt + σSdX, what is the SDE for dF = d(lnS)?

dF = (μ - σ2/2) dt +  σ dX