What are the basic bulding blocks of derivatives?
 Options
 Forwards and futures
e.g., swaps = hybrid of options + forwards/futures
What is arbitrage?
Taking simultaneous positions in different assets so that one guarantees a riskless profit higher than the riskfree rate
Describe the arbitrage theorem and its relevance.
Given two states with a positive probability of ocurrence,
 If StatePrices > 0 can be found such that the asset prices satisfy S = D StatePrices, then no arbitrage opportunities exist
 Conversely, if no arbitrate opportuinities exist, StatePrices can be found
This provides a general method for pricing derivative assets.
What are some uses of arbitragefree prices?
 Launch of a financial product by derivatives house (new product)
 Measure risks in a portfolio (risk management)
 Markingtomarket assets held in mortfolios to understand MV of iliquid assets (MtM)
 To identify mispriced profit opportunities
How can you price using real world probabilities?
 Modern portfolio theory and utility functions
 Certainty equivalent value under the real random walk:
 Find expected utility under real random walk
 Certaintly equivalent = U^{1}(Expected utility)
 Real Expected PV of Option Payoff +/ k * Std Dev
 Find expected utility under real random walk
 Certaintly equivalent = U^{1}(Expected utility)
What are two arbitragefree methods to price derivatives?
 Equivalent measures / Martingale transformations
 Tools include:

DobbMeyer decomposition into a trend and a martingale piece

Normalization by dividing the martingale by another arbitragefree price (e.g., divide by bond price B_{t})
 Equivalent measures through Girsanov's theorem to derive riskneutral probabilities and remove predictability of realworld
 Arbitrage on financial assets which are martingales
 Implies converting assets to martingale
 Partialdifferential equations (PDE)
 Construct a riskfree portfolio to obtain an BlackScholes PDE by the no arbitrage theorem

e.g., BlackScholes PDE
 Tools include:
 DobbMeyer decomposition into a trend and a martingale piece
 Normalization by dividing the martingale by another arbitragefree price (e.g., divide by bond price B_{t})
 Equivalent measures through Girsanov's theorem to derive riskneutral probabilities and remove predictability of realworld
 Arbitrage on financial assets which are martingales
 Implies converting assets to martingale
 Construct a riskfree portfolio to obtain an BlackScholes PDE by the no arbitrage theorem

e.g., BlackScholes PDE
What are the main steps in riskneutral pricing of deriatives?
 Obtain the underlying's pricing model (e.g., lognormal, skewness/kurtosis,...)
 Calcualte the derivative's payoffs at expiration w.r.t. the underlying
 Obtain riskneutral probabilities
 Calculate expected payoffs
 Discount at r.f.r.
Define the three types of convergence
 Almost sure convergence
 Pr(  lim_{n→inf} X_{n}  X  > d ) = 0
 Meansquared convergence
 lim_{n→inf} E[(X_{n}X)^{2}] = 0
 Weak convergence
 X_{n}→X and P_{n}→P if E^{Pn}[f(X_{n})] → E^{P}[f(X)]
 Pr(  lim_{n→inf} X_{n}  X  > d ) = 0
 lim_{n→inf} E[(X_{n}X)^{2}] = 0
 X_{n}→X and P_{n}→P if E^{Pn}[f(X_{n})] → E^{P}[f(X)]
Define previsible
That the random variable only depends on previous history
Define a filtration
A collection of σalgebras F_{t}, 0 ≤ t ≤ T for a positive fixed number T where:
 Ω is a nonempty sample space
 For each t in [0,T], there a σalgebra F_{t}.
 If s ≤ t, then every set in F_{s} is also in F_{t}.
Define a field and a σalgrebra
A family or collection of subsets of Ω where:
 ∅ ∈ F
 If A ∈ F, then A^{C} ∈ F
 If A_{1}, A_{2}, ... ∈ F, then
 for a field U_{i=1 to n} A_{i} ∈ F
 for a σalgrebra U_{i=1 to inf} A_{i} ∈ F
Define a probability measure P on {Ω,F}
 P({∅}) = 0
 P({Ω}) = 1
 If A_{1}, A_{2}, ... ∈ 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})
Define a realvalued random variable X
A function X: Ω → R such that {ω
Define an adapted stochastic process
Let Ω be a nonempty 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.
Define conditional expectation
E(XG) is any RV that satisfies:
 E(XG) is Gmeasurable
 For every set A in G:
 ∫_{A} E(XG) dP = ∫_{A }X dP
where X is a nonnegative, integral RV and G is a subσalgebra of F.
What are the properties of conditional expectation?
 Conditional probability
 If 1_{A} is an indicator RV for an event A, then E(1_{A}G) = P(AG)
 Linearity
 If X_{1},..., X_{n} are integrable RVs and c_{1},..., c_{n} are constants, then E(c_{1}X_{1} + ... c_{n}X_{n}  G) = c_{1}E(X_{1}G) + ... + c_{n}E(X_{n}G)
 Positivity
 If X ≥ 0 almost surely, then E(XG) ≥ 0 almost surely.
 Monotonicity
 If X,Y are integrable RVs and X≤Y almost surely, then E(XG) ≤ E(YG)
 Compute expectations
 E[E(XG)] = E(X)
 Take out what is known
 If X, Y are integrable RV and X is Gmeasurable, then E(XYG) = X E(YG)
 Tower property
 If H is a subσalgebra of G, then E[E(XG)H] = E(XH)
 Measurability
 If X is Gmeasurable, then E(XG) = E(X)
 Independence
 If X is independent of G, then E(XG) = E(X)
 Conditional Jensen's inequality
 If f is a convex function, then E[f(X)G] ≥ f[E(XG)]
 If 1_{A} is an indicator RV for an event A, then E(1_{A}G) = P(AG)
 If X_{1},..., X_{n} are integrable RVs and c_{1},..., c_{n} are constants, then E(c_{1}X_{1} + ... c_{n}X_{n}  G) = c_{1}E(X_{1}G) + ... + c_{n}E(X_{n}G)
 If X ≥ 0 almost surely, then E(XG) ≥ 0 almost surely.
 If X,Y are integrable RVs and X≤Y almost surely, then E(XG) ≤ E(YG)
 E[E(XG)] = E(X)
 If X, Y are integrable RV and X is Gmeasurable, then E(XYG) = X E(YG)
 If H is a subσalgebra of G, then E[E(XG)H] = E(XH)
 If X is Gmeasurable, then E(XG) = E(X)
 If X is independent of G, then E(XG) = E(X)
 If f is a convex function, then E[f(X)G] ≥ f[E(XG)]
Define the partial averaging property
∫_{A}E(XG)dP = ∫_{A}XdP
What is a martingale?
A submartingale? A supermartingale?
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
 E^{P}[X_{t+s}  I_{t}] = X_{t}
 E(X_{t}) < infinity
 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}
What are the three martingale ingredients and the three martingale properties?
Ingredients:
 Process
 Measure
 Information set
Properties:
 St is Itadapted
 E^{P}S_{t} < infinity
 E_{t}^{P}(S_{T}) = S_{t} for all t < T
When is a continuous martingale a "continuous square integrable martingale"?
If Xt has finite second moment E[X_{t}^{2}] < infinity for all t > 0
What is a Markov process? A strong Markov process?
A process {X_{1}, ... X_{t}} is Markov if
Pr(X_{t+s} <= x_{t+s}x_{1},...,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}.
State the key Markov and Martingale properties.
Compare and contrast.
 Markov
 The expected value of S_{i} conditional on all past event I_{i1} depends only on its previous value S_{i1}
 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
 The expected value of S_{i} conditional on all past event I_{i1} depends only on its previous value S_{i1}
 The expected future value of Si is equal to its current value, i.e. E_{j}(S_{i}) = S_{j} for all j < i
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:
 W_{t} is a square integrable martingale
 W_{t} is continuous (i.e., no jumps)
 W_{0} = 0
 "Stationary increments" property:
 W_{t+s}  W_{t} ~ N(0,s)
 E[ (W_{t}  W_{s})^{2} ] = t  s
 "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
Define Brownian Motion
 B_{0} = 0
 B_{t} has stationary, independent increments
 B_{t} is continuous in t
 B_{t}  B_{s} ~ N(0, t  s)
Describe six Brownian motion properties
 Finiteness  Brownian motion path is almost surely finite
 Continuity  paths are continuous
 Markov
 Martingale
 Quadratic variation
 [0,t] has quadratic variation that meansquare converges to t
 Normality
 X(t_{i})  X(t_{i1}) ~ N(0, σ^{2} = t_{i}  t_{i1})
 [0,t] has quadratic variation that meansquare converges to t
 X(t_{i})  X(t_{i1}) ~ N(0, σ^{2} = t_{i}  t_{i1})
What are the key expectations of Brownian Motion?
 E_{0}(W_{t}^{2}) = V_{0}(W_{t}) = t
 E_{0}(W_{t}^{4}) = 3t^{2} (kurtosis)
 E0(Wt2k) = (2k)! t^{k} / (2^{k}k!)
 E_{0}(W_{t}^{k}) = 0, for odd k's
 E_{0}[exp(σW_{t})] = exp(.5σ^{2}t)
What is the distribution of W_{i}  W_{j}?
N ( µ = 0, σ^{2} = i  j )
Summarize the Levy theorem
Any Wiener process relative to I_{t} is Brownian Motion
State properties of the Ito integral
 Existence
 If f is continuous and nonanticiapting, then an Ito integral exists
 Martingale
 The Ito integral is a Martingale
 Additive
 Ito isommetry
 It follows that for a square integrable Xt, (dW_{t})^{2} = dt
 If f is continuous and nonanticiapting, then an Ito integral exists
 The Ito integral is a Martingale
 It follows that for a square integrable Xt, (dW_{t})^{2} = dt
What are the three jump frameworks described in MFD?
 Simple jump
 X_{t} has a value 0 until a certain event occurs under which it assumes a value of 1
 (Merton) Jumpdiffusion 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)
 VarianceGamma 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*)
 X_{t} has a value 0 until a certain event occurs under which it assumes a value of 1
 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)
 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*)
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 =
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
Describe how to classify PDEs into elliptical, parabolic and hyperbolic.
a_{0} + a_{1}F_{t} + a_{2}F_{S} + a_{3}F_{SS} + a_{4}F_{tt} + a_{5}F_{St} = 0
 Elliptical: a_{5}^{2}  4a_{3}a_{4}
 Parabolic: a_{5}^{2}  4a_{3}a_{4} = 0
 Hyperbolic: a_{5}^{2}  4a_{3}a_{4} > 0
What are the SDEs for:
 Linear
 GBM
 Square root process
 Meanreverting
 OrnsteinUhlenbeck
 dS_{t} = µ dt + σ dW_{t}
 dS_{t} = µ S_{t} dt + σ S_{t} dW_{t}
 dS_{t} = µ S_{t} dt + σ sqrt(S_{t}) dW_{t}
 dS_{t} = λ (µ  S_{t}) dt + σ S_{t} dW_{t}
 dS_{t} = µS_{t} dt + σ dW_{t}
State Ito's lemma
Suppose that:
 F(S_{t},t) is a twicedifferentiable function
 dS_{t} = a_{t}dt + σ_{t}dW_{t}
 at and σt are wellbehaved drift and diffusion parameters
Then
 dF_{t} = dF/dS_{t} dSt + dF/dt dt + 0.5σ_{t}^{2} d^{2}F/dS_{t}^{2} dt
 (substitute dS_{t} above)
What are uses of the Ito lemma?
 Tool for obtaining SDEs
 Identify F(S_{t},t) or F(W_{t},t)
 Identify a_{t} and σ_{t} in dSt = a_{t}dt + σ_{t}dW_{t}
 Calcualte partial derivatives of F
 Plug & chug
 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
 Identify F(S_{t},t) or F(W_{t},t)
 Identify a_{t} and σ_{t} in dSt = a_{t}dt + σ_{t}dW_{t}
 Calcualte partial derivatives of F
 Plug & chug
 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
Define the Ito integral
I_{t} = ∫_{[0,t]} f(W_{s},s)dW_{s} = lim_{n→inf}Σ_{i=[0,n1]} 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<...n1n>
Define Ito's Isommetry
E( ∫f(W_{s},s) dW_{s} ) = E( ∫ [f(W_{s},s)]^{2}ds )
What is can the FeynmanFac formula be used for?
V(X_{t},t) = E[ Ψ(X_{T})  exp(∫_{[t,T]}r(u)duF_{t}) ]
Can be used for Monte Carlo simulation
What are the properties of Ito's Integral?
 Paths of I_{t} are continuous
 I_{t} is F_{t}measurable

_{}I_{t} is a martingale
 Quasilinearity, where:
 If It and Jt are Ito integrals w.r.t. Ws,
 then c_{i}I_{t} + c_{j}J_{t} = ∫ [c_{i}f(W_{s},s) + c_{j}g(W_{s},s)] dW_{s}
 Ito isommetry
 Quadratic variation _{t} = _{}∫_{0 to t}f(W_{s},s)_{2}ds
 If It and Jt are Ito integrals w.r.t. Ws,
 then c_{i}I_{t} + c_{j}J_{t} = ∫ [c_{i}f(W_{s},s) + c_{j}g(W_{s},s)] dW_{s}
What are two types of solutions to SDEs?
 Strong solution
 S_{t} is I_{t}adapted, and need knowledge of W_{t}
 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
 S_{t} is I_{t}adapted, and need knowledge of W_{t}
 Determined by ~W_{t}, a Wiener process whose distribution is determined simultaneously with ~S_{t}
 Could be or not be I_{t}adapted
What are the BS assumptions?
 Underlying is a stock
 No dividends
 European style
 Riskfree rate is constant
 No transaction costs (commissions, bidask spread) or indivisibilities
State the BS PDE and the type of PDE it is
(dV/dt) + 0.5σ^{2}S^{2}(d^{2}V/dS^{2})^{ }+ rS(dV/dS)  rV = 0
Linear parabolic PDE
Describe the following exotic options:
 Lookback (floating, fixed);
 Ladder;
 Trigger or Knockin, and Knockout;
 Basket;
 Multiassets;
 Multiassets call;
 Spread call;
 Portfolio call;
 Dualstrike call
 Asian options
 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 lockedin
 Trigger or Knockin
 Option to exercise once the spot price falls below/above a barrier, otherwise the option expires with a rebate value
 Knockout
 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
 Multiassets
 Multiasset call = (max{S_{1T},S_{2T}}  K)_{+}
 Spread call = ( (S_{1T}  S_{2T})  K)_{+}
 Portfolio call = ( (θ_{1}S_{1T} + θ_{2}S_{2T})  K)_{+}
 Dualstrike 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
 Payoff = (S_{T}  S_{min})_{+}
 Payoff = (S_{max}  K)_{+}
 Once the underlying asset reaches a threshold, the return of the option is lockedin
 Option to exercise once the spot price falls below/above a barrier, otherwise the option expires with a rebate value
 Option to exercise expires immediately if the spot price falls below/above a barrier and expires at a rebate value
 Underlying is a basket of various financial instruments
 Multiasset call = (max{S_{1T},S_{2T}}  K)_{+}
 Spread call = ( (S_{1T}  S_{2T})  K)_{+}
 Portfolio call = ( (θ_{1}S_{1T} + θ_{2}S_{2T})  K)_{+}
 Dualstrike call = max{0, S_{1T}  K, S_{2T}  K}
 Payoff depends on the average price of the underlying asset over the life of the option
What are three differences between standard BS (vanilla options) and exotic options?
 Expiration value (e.g., average, min, max,...)
 Expiration date (e.g., American)
 Underlying (e.g., basket, multiassets)
State the DobbMeyer theorem
If X_{t} is a rightcontinuous 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 rightcontinuous 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
How can it be determined if two measures are equivalent?
Two measures are equivalent if they have
 the same sample space
 the same possibilities
(But not necessarily the same probabilities)
State the Girsanov Theorem
If
 ξ_{t} is a martingale (previsible process) w.r.t. I_{t} and P
 The Novikov condition is met, i.e., the nonexplosiveness condition where E(exp{ integral( ξ_{s}^{2}ds)_{from 0 to t }} ) < infinity
Then W_{t}* defined by
W_{t}* = W_{t}  integral(X_{u}du)_{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}.
What is the RadonNikodym derivative?
dQ(z_{t}) / dP(z_{t}) = ξ(z_{t})
What are the implications of the Girsanov theorem?
 Every equivalent measure is given by a drift change
 One equivalent riskneutral measure
 Not needed for BlackScholes, but useful for more complicated analysis (e.g., stochastic volatility)
 Important for fixedincome with different maturities
How is quadratic variation calculated?
Sum{ (S_{i}  S_{i1})^{2} }
Define a convex function
A function f is convex on an interval if for every x and y in the interval
f (λx + (1λ)y) <= λ f (x) + (1λ) f (y)
for any λ in [0,1].
For example call/put payoffs and x^{2}
How do you quantify convexity?
1/2 E(ε^{2}) 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.
State the Jensen's inequality
If f is a convex function and x is a random variable, then
E( f (x) ) >= f ( E(x) )
What are the requirements to apply riskneutral pricing?
 Complete market
 Enough traded quantities with which to hedge risk
 Continuous hedging
 No transaction costs
 Accurate parameters
 No jumps
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  kσ^{2}) or measure of risk aversion
What are common confusions between RW and RN?
 Forward price = market expected future price
 Forward curve = market's expected value of the spot rates
 A risk premium is built into the forward curve
 Under RN, we replace µ with the riskfree rate
 Only allowed when assumptions allow you to do so
 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(d_{2}*)), which is NOT captured by the BS price delta (i.e., N(d_{1}))
 A risk premium is built into the forward curve
 Only allowed when assumptions allow you to do so
 The probability of ending up in the money depends on the real probabilities and growth rate (i.e., N(d_{2}*)), which is NOT captured by the BS price delta (i.e., N(d_{1}))
What are further arguments for riskneutral pricing in a BS world?
 If hedged correctly in a BS world, all risk is eliminated ⇒ RFR and no compensation for risk
 If we assume dS = µS dt + σS dX, then the µ's cancel in the derivation of the BS equation
 Changing measures converts RV into martingales
 Options can be constructed by putting together vanilla options, and they can be priced through synthesization/replication
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
State models that result in complete markets
 Lognormal with a constant volatility
 Binomial
Define the following:
 Trading strategy
 Selffinancing trading strategy
 Admissible trading strategy
 Attainable contingent claim
 Admissible arbitrage opportunity
 Complete market
 Trading strategy:
 Π_{t} = Φ_{t}S_{t} + Ψ_{t}B_{t}
 Selffinancing 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
 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.
 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)
 Admissible arbitrage opportunity  that where the following holds:
 Π0 = 0
 P(ΠT ≥ 0) = 1
 P(ΠT > 0) > 0
 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
 Π_{t} = Φ_{t}S_{t} + Ψ_{t}B_{t}
 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
 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.
 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)
 Π0 = 0
 P(ΠT ≥ 0) = 1
 P(ΠT > 0) > 0
 N_{A} < N_{R} ⇒ incomplete, no arbitrage
 N_{A} = N_{R} ⇒ complete, no arbitrage
 N_{A} > N_{R} ⇒ arbitrage
What are pricing approaches for incomplete models?
 Actuarial method
 Price in an average sense
 Relies on Central Limit Theorem (CLT)
 e.g., pricing insurance contracts
 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
 Price in an average sense
 Relies on Central Limit Theorem (CLT)
 e.g., pricing insurance contracts
 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
What are complications of delta hedging?
 Transaction costs
 Estimate detla based on a model that may be wrong
 Continuous rebalancing
 Underlying needs to be consistent with model assumptions, usually need to assume Brownian motion with no jumps
Compare and contrast forwards and futures
 MarktoMarket
 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
 Futures: Yes
 Forwards: No
 Futures: Daily
 Forwards: Upon maturity
 Futures: Yes
 Forward: No
 Futures: Exchanges
 Forward: OTC
 Futures: Yes
 Forward: No
 Futures: Less
 Forward: More
 Futures: More
 Forward: Less
 Futures: Less
 Forward: More
What is the delta of a forward contract?
A futures contract?
delta_{forward} = e^{δ (Tt)}
delta_{futures} = e^{(rδ) (Tt)}
Why are the deltas of a futures and forward contracts different?
Because futures are markedtomarket daily, while forward contracts are settled at maturity
What is the mgf for Z_{t}~N(0,t)?
M(λ) = exp( 1/2 t λ^{2})
What is the mgf for Yt ~ N(μt,σ^{2}t) ?
M(λ) = exp( λμt + 1/2 σ^{2}tλ^{2})
dX_{1} dX_{2} = ?
dX_{1} dX_{2} = ρdt
Given dS_{t} = rS_{t}dt + σS_{t}dW_{t}, calcualte d(lnS_{t})
d(lnS_{t}) = (r  1/2 σ^{2}) dt + σ dW_{t}
Given d(lnS_{t}) = rdt + σ dW_{t}, calculate dS_{t} / S_{t}
dS_{t} / S_{t} = (r + 1/2 σ^{2}) dt + σ dW_{t}
d(X_{t}F_{t}) = ?
d(X_{t}F_{t}) = dX_{t} F_{t} + X_{t} dF_{t} + dX_{t} dF_{t}
If dS = μSdt + σSdX, what is the SDE for dF = d(lnS)?
dF = (μ  σ^{2}/2) dt + σ dX