1. Stochastic Calculus

Question 1

What are the basic bulding blocks of derivatives?

Answer 1

1. Options
2. Forwards and futures

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

Question 2

What is arbitrage?

Answer 2

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

Question 3

Describe the arbitrage theorem and its relevance.

Answer 3

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.

Question 4

What are some uses of arbitrage-free prices?

Answer 4

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

Question 5

How can you price using real world probabilities?

Answer 5

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

Question 6

What are two arbitrage-free methods to price derivatives?

Answer 6

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 B_t)
     • 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

Question 7

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

Answer 7

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.

Question 8

Define the three types of convergence

Answer 8

1. Almost sure convergence
   
   • Pr( | lim_n→inf X_n - X | > d ) = 0
2. Mean-squared convergence
   
   • lim_n→inf E[(X_n-X)²] = 0
3. Weak convergence
   
   • X_n→X and P_n→P if E^P_n[f(X_n)] → E^P[f(X)]

Question 9

Define previsible

Answer 9

That the random variable only depends on previous history

Question 10

Define a filtration

Answer 10

A collection of σ-algebras F_t, 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 F_t.
• If s ≤ t, then every set in F_s is also in F_t.

Question 11

Define a field and a σ-algrebra

Answer 11

A family or collection of subsets of Ω where:

1. ∅ ∈ F
2. If A ∈ F, then A^C ∈ F
3. If A₁, A₂, … ∈ F, then
   
   1. for a field U_{i=1 to n} A_i ∈ F
   2. for a σ-algrebra U_{i=1 to inf} A_i ∈ F

Question 12

Define a probability measure P on {Ω,F}

Answer 12

1. P({∅}) = 0
2. P({Ω}) = 1
3. If A₁, A₂, … ∈ F and {A_i}_{i=1 to inf} is disjoint such that A_i∩A_j=∅ i<>j, then P(U_{i=1 to inf} A_i) = Σ_{i=1 to inf} P(A_i)

Question 13

Define a real-valued random variable X

Answer 13

A function X: Ω → R such that {ω

Question 14

Define an adapted stochastic process

Answer 14

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

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

Question 15

Define conditional expectation

Answer 15

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 = ∫_A X dP

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

Question 16

What are the properties of conditional expectation?

Answer 16

• Conditional probability
  
  • If 1_A is an indicator RV for an event A, then E(1_A|G) = P(A|G)
• Linearity
  
  • If X₁,…, X_n are integrable RVs and c₁,…, c_n are constants, then E(c₁X₁ + … c_nX_n | G) = c₁E(X₁|G) + … + c_nE(X_n|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)]

Question 17

Define the partial averaging property

Answer 17

∫_AE(X|G)dP = ∫_AXdP

Question 18

What is a martingale?

A submartingale? A supermartingale?

Answer 18

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

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

1. E^P[X_t+s | I_t] = X_t
2. E(|X_t|) < infinity
3. X_t is I_t-adapted

A submartingale will hold E^P[X_t+s | I_t] ≥ X_t

A superartingale will hold E^P[X_t+s | I_t] ≤ X_t

Question 19

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

Answer 19

Ingredients:

1. Process
2. Measure
3. Information set

Properties:

1. St is It-adapted
2. E^P|S_t| < infinity
3. E_t^P(S_T) = S_t for all t < T

Question 20

When is a continuous martingale a “continuous square integrable martingale”?

Answer 20

If Xt has finite second moment E[X_t²] < infinity for all t > 0

Question 21

What is a Markov process? A strong Markov process?

Answer 21

A process {X₁, … X_t} is Markov if

Pr(X_t+s <= x_t+s|x₁,…,x_t) = Pr(X_t+s <= x_t+s|x_t)

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

In other words, the conditional distribution of X_t given F_s only depends on F_s.

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

Question 22

State the key Markov and Martingale properties.

Compare and contrast.

Answer 22

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

Question 23

Define a Wiener process and its properties

Answer 23

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

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

⇒ ​Properties:

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

Question 24

Define Brownian Motion

Answer 24

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

Question 25

Describe six Brownian motion properties

Answer 25

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(t_i) - X(t_i-1) ~ N(0, σ² = t_i - t_i-1)

Question 26

What are the key expectations of Brownian Motion?

Answer 26

• E₀(W_t²) = V₀(W_t) = t
• E₀(W_t⁴) = 3t² (kurtosis)
• E0(Wt2k) = (2k)! t^k / (2^kk!)
• E₀(W_t^k) = 0, for odd k’s
• E₀[exp(σW_t)] = exp(.5σ²t)

Question 27

What is the distribution of W_i - W_j?

Answer 27

N ( µ = 0, σ² = |i - j| )

Question 28

Summarize the Levy theorem

Answer 28

Any Wiener process relative to I_t is Brownian Motion

Question 29

State properties of the Ito integral

Answer 29

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, (dW_t)² = dt

Question 30

What are the three jump frameworks described in MFD?

Answer 30

1. Simple jump
   
   1. X_t 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
     
     • dS_t / S_t = (μ - λκ)dt + σdW_t + (e^J - 1)dN_t
   • Where a jump dN_t = 1 occurs with probability λdt, and the expected proportional jump size is κ = E(e^J - 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*)

Question 31

Fill in the blanks:

• Normal events​
  
  • r_i =
  • q_i =
  • Limiting size =
  • Limiting prob =
  • Time independence =
• Rare
  
  • r_i =
  • q_i =
  • Limiting size =
  • Limiting prob =
  • Time independence =

Answer 31

Fill in the blanks:

• Normal events
  
  • r_i = 1/2
  • q_i = 0
  • Limiting size = 0
  • Limiting prob = p_i > 0
  • Time independence = probability
• Rare
  
  • r_i = 0
  • q_i = 1
  • Limiting size = w_i > 0
  • Limiting prob = 0
  • Time independence = size

Question 32

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

Answer 32

a₀ + a₁F_t + a₂F_S + a₃F_SS + a₄F_tt + a₅F_St = 0

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

Question 33

What are the SDEs for:

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

Answer 33

1. dS_t = µ dt + σ dW_t
2. dS_t = µ S_t dt + σ S_t dW_t
3. dS_t = µ S_t dt + σ sqrt(S_t) dW_t
4. dS_t = λ (µ - S_t) dt + σ S_t dW_t
5. dS_t = -µS_t dt + σ dW_t

Question 34

State Ito’s lemma

Answer 34

Suppose that:

1. F(S_t,t) is a twice-differentiable function
2. dS_t = a_tdt + σ_tdW_t
3. at and σt are well-behaved drift and diffusion parameters

Then

• dF_t = dF/dS_t dSt + dF/dt dt + 0.5σ_t² d²F/dS_t² dt
• (substitute dS_t above)

Question 35

What are uses of the Ito lemma?

Answer 35

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

Question 36

Define the Ito integral

Answer 36

I_t = ∫_[0,t] f(W_s,s)dW_s = lim_n→infΣ_i=[0,n-1] f(W_ti, t_i) (W_ti+1 - W_ti)

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

and t_i = it/n, 01<…n-1n=t

Question 37

Define Ito’s Isommetry

Answer 37

E( ∫f(W_s,s) dW_s ) = E( ∫ [f(W_s,s)]²ds )

Question 38

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

Answer 38

V(X_t,t) = E[Ψ(X_T) - exp(-∫_[t,T]r(u)du|F_t) ]

Can be used for Monte Carlo simulation

Question 39

What are the properties of Ito’s Integral?

Answer 39

1. Paths of I_t are continuous
2. I_t is F_t-measurable
3. _​I_t is a martingale
4. Quasi-linearity, where:
   
   • If It and Jt are Ito integrals w.r.t. Ws,
   • then c_iI_t + c_jJ_t = ∫ [c_if(W_s,s) + c_jg(W_s,s)] dW_s
5. Ito isommetry
6. Quadratic variation _t = _​∫_{0 to t}f(W_s,s)₂ds

Question 40

What are two types of solutions to SDEs?

Answer 40

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

Question 41

What are the BS assumptions?

Answer 41

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

Question 42

State the BS PDE and the type of PDE it is

Answer 42

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

Linear parabolic PDE

Question 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

Answer 43

• Lookback - floating
  
  • Payoff = (S_T - S_min)₊
• Lookback - fixed
  
  • Payoff = (S_max - 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{S_1T,S_2T} - K)₊
  • Spread call = ( (S_1T - S_2T) - K)₊
  • Portfolio call = ( (θ₁S_1T + θ₂S_2T) - K)₊
  • Dual-strike call = max{0, S_1T - K, S_2T - K}
• Asian option
  
  • Payoff depends on the average price of the underlying asset over the life of the option

Question 44

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

Answer 44

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

Question 45

State the Dobb-Meyer theorem

Answer 45

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

X_t = M_t + A_t, where

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

Question 46

How can it be determined if two measures are equivalent?

Answer 46

Two measures are equivalent if they have

1. the same sample space
2. the same possibilities

(But not necessarily the same probabilities)

Question 47

State the Girsanov Theorem

Answer 47

If

1. ξ_t is a martingale (previsible process) w.r.t. I_t and P
2. The Novikov condition is met, i.e., the non-explosiveness condition where E(exp{ integral( ξ_s²ds)_{from 0 to t} } ) < infinity

Then W_t* defined by

W_t* = W_t - integral(X_udu)_{from 0 to t}

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

Q(A) = E^P(1_AξT)

with A being an event determined by I_T.

Question 48

What is the Radon-Nikodym derivative?

Answer 48

dQ(z_t) / dP(z_t) = ξ(z_t)

Question 49

What are the implications of the Girsanov theorem?

Answer 49

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

Question 50

How is quadratic variation calculated?

Answer 50

Sum{ (S_i - S_i-1)² }

Question 51

Define a convex function

Answer 51

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

f (λx + (1-λ)y) <= λ f (x) + (1-λ) f (y)

for any λ in [0,1].

For example call/put payoffs and x²

Question 52

How do you quantify convexity?

Answer 52

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.

Question 53

State the Jensen’s inequality

Answer 53

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

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

Question 54

What are the requirements to apply risk-neutral pricing?

Answer 54

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

Question 55

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

Answer 55

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

Question 56

What are common confusions between RW and RN?

Answer 56

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))

Question 57

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

Answer 57

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

Question 58

Define a complete market

Answer 58

• 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

Question 59

State models that result in complete markets

Answer 59

1. Lognormal with a constant volatility
2. Binomial

Question 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

Answer 60

1. Trading strategy:
   
   • Π_t = Φ_tS_t + Ψ_tB_t
2. Self-financing trading strategy:
   
   • dΠ_t = Φ_t dS_t + Ψ_t dB_t
   • 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(S_T) is attainable if there exists an admissible trading strategy worth Π_t = Ψ(S_T) 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 N_A = # traded assets and N_R = # of sources of risks, the following holds:
   
   • ​N_A < N_R ⇒ incomplete, no arbitrage
   • ​N_A = N_R ⇒ complete, no arbitrage
   • ​N_A > N_R ⇒ arbitrage

Question 61

What are pricing approaches for incomplete models?

Answer 61

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

Question 62

What are complications of delta hedging?

Answer 62

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

Question 63

Compare and contrast forwards and futures

Answer 63

• 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

Question 64

What is the delta of a forward contract?

A futures contract?

Answer 64

delta_forward = e^{-δ (T-t)}

delta_futures = e^{(r-δ) (T-t)}

Question 65

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

Answer 65

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

Question 66

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

Answer 66

M(λ) = exp( 1/2 t λ²)

Question 67

What is the mgf for Yt ~ N(μt,σ²t) ?

Answer 67

M(λ) = exp( λμt + 1/2 σ²tλ²)

Question 68

dX₁ dX₂ = ?

Answer 68

dX₁ dX₂ = ρdt

Question 69

Given dS_t = rS_tdt + σS_tdW_t, calcualte d(lnS_t)

Answer 69

d(lnS_t) = (r - 1/2 σ²) dt + σ dW_t

Question 70

Given d(lnS_t) = rdt + σ dW_t, calculate dS_t / S_t

Answer 70

dS_t / S_t = (r + 1/2 σ²) dt + σ dW_t

Question 71

d(X_tF_t) = ?

Answer 71

d(X_tF_t) = dX_t F_t + X_t dF_t + dX_t dF_t

Question 72

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

Answer 72

dF = (μ - σ²/2) dt + σ dX