Quant Finance

• The Black-Scholes model is an option pricing formula that uses the lognormal distribution to value European options.
• What are affine linear models?

  Affine linear models are models that are linear functions of hidden states
• What is the state variable in the Vasicek model?

  Interest rate
• Why does a large K push rates up or down relative to the mean in the Vasicek model?

  • K is the mean reversion parameter if $\theta-r$is positive that means the rate lies below the mean, so K will push it up by multiplying the positive quantity by a scalar. Conversely, if $\theta-r$ is negative that means the rate lies above the mean, so K will push it down by multiplying the negative quantity by a scalar
• What does a large $\kappa$mean for yield curve shapes in the Vasicek model?


  The yield curve is flatter
• What does a small kappa mean for yield curve shapes in the vasicek model?

  The yield curve is steeper
• What is Jensen's Inequality?

  $E[f(x)] \geq f(E[x])$
• How does Jensen's Inequality relate to the convexity of bonds?

  If volatility is higher the price/yield relationship becomes more convex, assume that $\tilde{P}_{t} =$$e^{-E^{Q}[\int_{t}^{T} r_{s} ds]}$ and that $P_{t} =$$E^{Q}[e^{-\int_{t}^{T} r_{s} ds}]$ the value of convexity

  $P_{t} - \tilde{P}_{t}$

  as captured by Jensen's ineqality
• What is a Numierare?

  A Numierare is a strictly positive value, often representing a tradeable asset, that serves as the unit of reference for a given probability measure. The numeirare fully defines the drift of all assets in that measure
• 1. What does it mean for two probability measures to be equivalent?

  They have the same null space
• When is a market complete?

  A market is considered complete if for every contingent claim, there exists a portfolio of traded assets that replicates the claim's payoff in every state of the world. Mathematically, a market is complete if the number of independent sources of uncertainty is equal to the number of traded assets, allowing for the construction of a unique replicating portfolio for any claim.
• What is a frictionless market?

  A market without any transaction costs or liquidity constraints
• What is cross variation?

  Cross variation refers to the joint variability between two variables. It measures how the variables move together or in opposite directions. $[f,g]_{T} =$$\lim_{||\pi|| \rightarrow 0} \sum_{i=1}^{n} ( f(t_{i}) - f(t_{i-1}))(g(t_{i}) - g(t_{i-1}))$
• What is the Quadratic Variation of B(t)?

  The quadratic variation of B(t) is t.
• What happens to Quadratic Variation under non-stochastic volatility?

  For a process with constant volatility, the quadratic variation is known in advance and grows linearly with time. The intuition behind this is that with constant volatility, there are no surprises in the "amount of vibration" or variability of the process, and thus the quadratic variation is simply the product of the constant volatility squared and the elapsed time.
• Is Quadratic Variation path dependent??

  Yes
• Conceptually explain Girsanov's Theorem

  Girsanov's Theorem adjusts the mean of the process by changing the Brownian motion. In the new Brownian motion we add a drift parameter to the old Brownian motion at least under P. However, under Q this new Brownian is driftless. We can make this driftless by expressing the process in a new "unit" the unit is the numeraire. To find the appropriate numierare (and change the measure) we evoke Girsanov's Theorem.
• Why does RNPM weight bad states of the world as more likely than good states?

  Risk-Neutral Probability Measure (RNPM) weights bad states of the world as more likely than good states because individuals are risk-averse and assign higher probabilities to negative outcomes to protect themselves from potential losses. This can be seen via a change in brownian motion. By removing the drift, you're essentially making "less drifty" paths more plausible. These "less drifty" paths invariably are bad states of the world. Thus you end up weighting them more heavily.
• What is a cap in interest rate derivatives?

  A cap is like a call option on an interest rate forward. it is composed of caplets
• What is a floor in interest rate derivatives?

  A floor is like a put option on an interest rate forward. It is composed of floorlets
• What is the T-forward measure?

  The measure whose numeirare is the zero coupon bond maturing at time T
• What is a forward rate agreement?

  In a FRA transaction, counterparty A agrees to pay counterparty B a floating rate

  settling t years from now applied to a specified notional amount (say, $100 mm).

  In exchange, counterparty B pays counterparty A a pre-agreed fixed rate (say,

  3:05%) applied to the same notional.
• What is a reverse repo?

  A reverse repo is a financial transaction in which a party sells securities to another party with an agreement to repurchase them at a later date.
• How is a swap decomposable into bonds?

  A swap can be seen as a series of FRA 's (at least conceptually). An FRA is basically the same as going short a zero coupon bond today with maturity time S and going long a zero coupon bond today with maturity T. Where T > S
• What is the formula for the instantenous forward rate?

  $f(t) =$$\frac{\ln(\partial P(t,T))}{\partial T}$
• A bond is a derivative on?

  Rates
• In order to hedge a bond you must use?

  Bonds
• When modeling a non-tradeable quantity we always end up with more unknowns than equations
• Why does modeling an non-tradeable quantity always result in an under determined system of equations?


  Because there are more sources of risk (some unobservable) than there are observable market instruments. Seen from the reverse-angle, imagine a yield curve. There are infinitely many possible rate paths/curves that could result in the same vector of discrete yields seen today. The system is underdetermined, there are more unknown paths and risks, these include the equations for the assets themselves,including than there are instruments to calibrate them.
• Why does modeling a non-tradeable quantity always result in an under-determined system of equations?

  Since the quantity is not traded, we can't use data from market instruments to calibrate to its value. i.e. there's no curve to fit for it. As such, we have an underdetermined system of equations. If we have one non-tradeable quantity we're trying to model say an interest rate we're left with N equations (for N market instruments) and N+1 unknowns where the extra unknown is the uncalibrateable rate itself!
• Why is the introduction of the Market Price of Risk necessary when modeling untradeable quantities?

  Our system of equations is underdetermined. i.e. there are more unknowns than equations so there is no way to arrive at a solution! We use the market price of risk as just that a "price" for the (unknowable) quantity of "risk". So for a short-rate for example we introduce a market price of risk for that unobservable quantity i.e. what is the expected "price" for "owning" this interest rate risk? By introducing the market price of risk we make the system of equations solvable.
• Under the risk neutral measure, there is no market price of risk.
• Under the risk neutral measure. All assets evolve at the risk-free rate
• The market price of risk is the excess return for bearing a risk, by subtracting the market price of risk of all risks (observable and unobservable) we revert the drift back to the "risk-free" level.
• How does the numeirare affect the market price of risk?

  THe numeirare effectively changes the risk-free rate so it affects the market price of risk's "reference" point
• The market price of risk can be seen as a system of equations that represents the excess return for bearing a certain risk both unknowable and knowable.
• What is the market price of risk?

  The market price of risk is the excess return for bearing some risk over the risk free rate per unit of volatility
• What is calibration in yield curves?

  Fitting a yield curve to market instruments
• What is "fitting" in yield curves?

  Fitting a yield curve to market data
• Why do you need infinite degrees of freedom to fit a yield curve?

  Because you're fitting an infinite number of points!