Durations

Duration: parallel yield curve shift
D = -(1/P)(dP/dr)

Modified duration:
MD = -(1/P)(dP/dy)

Macaulay duration: under the mindset of "duration<->time to maturity"
MacD = MD*(1+y/2)

Effective duration:
For securities with embedded options

Spot rate duration:
Spot rate is changes (not shift of the entire yield curve). Usually found numerically.

Factor duration:
Price change with respect to some factors other than rate

Also, since D = (-1/P) (dP/dy) and convexity C =  (1/P) (d^2P/dy^2), it can be shown that

dD/dy = D^2 - C

Number of Paths from (0,0,0) to (n,n,n)

Q: How many paths are there to go from (0,0,0) to (n,n,n)? i.e. walk in a certain direction by one step each time.

A: (3n)C(n)*(2n)C(n)*(n)C(n). In general, for k-dimensional space, the answer would be a product of k terms. Things to ponder:

Why combination, not permutation? -> There are 3n steps to walk in total, and there are 3 dimensions x, y and z. We see the steps, not the dimensions, as the space to pick from. For example, for n = 2, there are 6 steps to make, 1, 2, 3, 4, 5 and 6. We pick two from these to assign to the x-direction, and that's why the order is not important (assigning x-direction to positions 1,5 is the same as assigning x-direction to positions 5,1).

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Merton's Jump Diffusion Model: market completeness etc.

Recall:
Existence of martingale measure <=> no-arbitrage
Uniqueness of martingale measure <=> market completeness

When there are jumps, the market is no longer complete because the jump process 'creates' many more states so that the number of asset becomes too few. Hence we are left with the unknown "market price of jump risk". Merton proposes that this price of risk should be zero because the jump in a stock is non-systematic, i.e. diversifiable.

Note that empirical study suggests that Merton's assumption is quite wrong.

Correlation and Dependence

http://en.wikipedia.org/wiki/Correlation_and_dependence
http://mathforum.org/library/drmath/view/64808.html

- There are many flavors of correlation measures
- Non-zero correlation => Dependence
- Independence => Zero correlation

BUT

- Zero correlation =X=> Independence

So Freaking Many Volatilities

There are a number of volatilities defined differently under the LIBOR Market Model:

1. Forward volatility (of a cap): v(T_j)_cap
a single volatility for each cap that makes the cap value (which is a sum of caplet values) agree with the market price. Aka "flat volatility."

2. Forward forward volatility (of a caplet): v(T_j)_caplet
a set of volatilites, different for each caplet, that makes the cap value agree with the market price. Can be bootstrapped from the forward volatility.

3. Instantaneous volatility (of a forward LIBOR dynamic): sigma
the 'sigma' that appears in the SDE of a certain LIBOR. Can be used to parametrize the forward forward volatility.

4. Average volatility (between two points in time): V(T_j,T_k)
a volatility of which forward forward volatility is a special case. V(0,tau)=v(tau)_caplet

Hitting time

The following applies to both binomial (drunken man) and Wiener process:

Suppose we are at x = 0 and there are two absorbing barriers, a and -b (a,b>0). Then

p(absorbed at a) = b/(a+b)
p(absorbed at -b) = a/(a+b)
E[time until absorption] = ab

For a proof, refer to Zhou. Outline of the proof:

Let S_n be the Wiener process. Both S_n and (S_n)^2-N are martingales (N being the hitting time). Hence we have a set of 2 eqns with 2 unknowns p_a and E[N]:
E[S_n]=p_a*a+(1-p_a)*b=0
E[(S_n)^2-N]=p_a*(a^2)+(1-p_a)*(b^2)-E[N]=0

Combinations of H and T

One of the very popular brain teasers:
What's the mean number of coin toss to get HH/HT/HHH...ect?

Approach 1: Considering absorption state of a Markov chain -> solving a system of simultaneous equation

Approach 2: Recursive formulae. For example, let H = expected number of tosses to get a head, HH = expected number of tosses to get two consecutive heads and so on. Then
H = 0 + 0.5*1 + 0.5*(1+H); T = 0 + 0.5*1 + 0.5*(1+T)
HH = H + 0.5*1 + 0.5*(1+HH)

HT = H + 0.5*1 + 0.5*(1+T)

HTH = HT + 0.5*1 + 0.5*(1+HTH)

HHT = HH + 0.5*1 + 0.5*(1+T)

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A few things about CDO

CDO contains many types of risks (not an exhaustive list). Considering a specific tranche,

- Delta risk: If the value of the underlying credit changes by $1, what is the change of value of the tranche? (first-order/linear approximation)

- Convexity risk: Since Delta is a first-order approximation, it fails to capture all the risk when the change in underlying value is large.

- Correlation: Remember, equity tranche is long correlation, while senior tranche is short correlation (why so?).

LIBOR Market Model vs. Swap Market Model

The two models (or the two families of models) are NOT compatible with one another. The recommended practice is to assume an LMM and then seek the swaption prices under such a model.

Correlation across rates of various maturity is much more important for swaption than for cap/floor, because the swaption payoff cannot be separated into individual expectation terms (in other words, when swaption expires we decide on whether to exercise based NOT on the sum of "swap-lets" - there is no such thing, but on the swap as a whole. cf. cap/floor payoff, which are nothing but sum of payoffs of individual caplet/floorlet).

Now back to LMM. We pick a set of forward LIBOR to fit. Each forward rate F(t,T1,T2) is a martingale under its 'natural' probability measure using P(T2) as the numeraire. However if we pick one single P() as the numeraire for all forward rates, most (except for one) rates will NOT be martingales. Thus we also need a formula for the dynamics of F(t,T1,T2) under some other measures. With this formula we can use MC pricing.

Bottom line: The rates "look like" tradable assets

Variance Swap vs. Volatility Swap

- Theoretical exact hedging recipe: Var swap can be hedged using a log contract, which itself can be replicated with a continuum of OTM calls and puts, this result is model independent; Vol swap hedging is model dependent.

- Risk: From a sellers' perspective, Var swap has higher risk because convexity means the payoff can be very huge in extreme volatility spike events; Vol swap is relatively "safer".

- Hedging in practice: Vol swap is easier to hedge in practice than Var swap. First, the payoff of Vol swap is monotonic in S (<=> less convexity); Second, under high volatility scenario, hedging a Var swap requires many options to be hedged, but usually under these circumstances the option market is not liquid enough.