The Gamma-Theta story

• Gamma and theta are usually of opposite signs and of similar magnitude. This comes from the fact that they are on the same side of the pricing PDE, and other terms are usually smaller in magnitude.
• Suppose you are a market-maker selling a call option. The option value decreases with time, does that mean you can just sit there with your premium? No, you have to delta-hedge the short call position. And due to convexity of call option price with respect to asset price (it is actually CONCAVE when you are shorting), you lose a small bit whenever you adjust your delta-hedge discretely. At the end of the day if volatility stays constant at the original implied level, your premium should exactly cover this hedging cost.
• Similar story can be told from the other side (the option buyer), who would get her premium back when she delta-hedges. It is in these senses that theta is considered a 'cost' to purchase gamma, and vice versa depending on the position you hold.
• Of course, if realized volatility turns out to be varying, somebody would get hurt. Delta-hedging of a vanilla option is a way to capture volatility changes, but it is not effective due to path dependence.
  

Reference: Veronesi, Wilmott "Paul Wilmott on Quantitative Finance" Ch. 12

Variance Swap Revisited, Part II

1. Once again, a variance swap is not just the log contract. It's the log contract plus dynamically hedged $1 worth of stock.
2. Greeks of a variance swap (under B-S): Vega decreases linearly with time; Delta zero (but...); dollar Gamma constant.
3. Two disadvantages of straddle (even if delta-hedged), as compared to variance swap, are that 1) volatility exposure rapidly diminishes as soon as asset price moves away from strike; and 2) path dependence of P&L.
4. Due to these subtle differences between variance swap and delta-hedged straddle, the combination of the two is a neat way to trade the variance convexity. However the moving-underlying problem is still present and the delta-hedged straddle leg of this trade has to be periodically re-struck.
5. Taking a step back, the reason why volatility capturing using vanilla option (or a porfolio of them) depends on path is that the Gamma of the option/options changes as underlying moves.
6. Interest rate term structure has a short end (short rate) that is quite stable; on the other hand the short end of the variance term structure can move substantially and abruptly.

Reference: JP Morgan Variance Swap
Further readings: Correlation trading, volatility skew trading using Gamma swap

Variance Swap Revisited, Part I

1. Why is it that Vega Notional = 2K *Variance Notional? Because when calculating P&L, (sigma^2 - K^2)/2K is roughly "d sigma". This approximation, of course, is linear and is good only when close to strike, which brings us to...
2. CONVEXITY. Variance swap payoff is convex with respect to volatility. The further away realized variance is from the strike, the more pronounced the effect of convexity is. This means that 1) the fair strike for a variance swap is always greater than the fair strike of a comparable volatility swap because of the benefit of convexity; and 2) Variance swap price increases with vol-of-vol, but volatility swap price does not.
3. The forward arithmetic of IR is mutiplicative; the forward arithmetic of variance is additive.
4. For capped variance swaps, unwinding by entering into an opposite trade is complicated by the fact that the cap itself would move when one tries to lock in the P&L after some time, causing the offsetting trade to have a different cap level than the original trade.
5. Derman's approximation assumes that implied volatility curve is linear. This approximation is good when vol-of-vol effect is low, e.g. for shorter-term index variance swap (instead of single stock variance swap)
6. Volatility risk premium: the price of variance swap is 'too high' as compared to both historical realized volatility or implied vanilla option volatility. In other words longing variance is in high demand. Not to be confused with the vol-of-vol effect.
7. Variance swap can be replicated by OTM call and put options, hence the fair strike is a weighted average of implied variances across the entire volatility smile, which, given a volatility smile (going up both ways), is greater than sigma_ATM.

Reference: JP Morgan Variance Swaps

Vanna-Volga Approximation

Vanna-Volga approximation is the quick-n-dirty way traders use to tweak B-S price so that smile/skew effects are taken into account. It is used to price exotic products. It consists of 3 components: X_BS, RR (risk reversal) correction and BF (butterfly) correction. Incidentally, unlike in equity market where the smile is simply quoted from the market, the FX market convention is much more convoluted, with only ATM, RR and strangle volatilities directly quotable (note also the RR and strangle and BF volatilites are not really volatility in the strict sense; they are merely traded entities).

• X_BS is the B-S exotic price
• RR stands for risk-reversal. An RR strategy consists of long OTM call and short OTM put.
• BF stands for butterfly. A BF strategy consists of long strangle and short straddle.

Intuitively, RR and BF have opposite parity about the strike (RR increases monotonically, BF increases either way). Hence they together can produce corrections that fit market prices. In fact, RR strategy has large (small) vanna (volga) exposure, while BF strategy has large (small) volga (vanna) exposure (this can be seen by considering the B-S Vega expression, Vega ~ S phi(d_1)). The Vanna-Volga method ignores the cost of hedging Vega.

*Volga = (d/d sigma) Vega, Vanna = (d/dS) Vega

Reference:
Bossens 2010 - Vanna-Volga method
Reiswich 2010 - Constructing a (quadratic) smile using the market quoted volatilities

Negative duration of FRN

Floating Rate Note (FRN) usually has duration very close to zero. An FRN is reset to par on every reset date when the floating coupon is paid out. However it could have negative duration when it is traded at substantial discounts. Since, as we have already mentioned, the coupon of an FRN is floating, the discount is most probably due to credit instead of interest rate. Suppose then that an FRN is traded at a discount because of credit concern. Then we can write

FRN = FRN' + X

where FRN' is a note at a discount, FRN is an otherwise identical par note and X is some instrument that can be constructed so that the above expression holds. The point is that X has positive duration (as can be shown easily if we assume that the cash flow of FRN' is that of FRN with an extra spread S). Since the duration of FRN is close to zero, and the duration of X is positive, we must have that the duration of FRN' is negative.

(Possible) Quant Interview Questions

1. Maths
What is i^i? Give your answer to 2 decimal places for both the real and the imaginary parts.

2. Black-Scholes
Assume Black-Scholes and no dividend. P and C are, respectively, European vanilla put and call that are otherwise identical. What is the strike price that would make
i) their price; and
ii) their delta
the same?

3. Fixed income
The current yield curve is upward sloping. You speculate that it will get steeper and want to take advantage of it by long-shorting zero coupon bonds with different maturities (5 yrs and 10 yrs).
i) What is the duration-neutral strategy?
ii) What is the impact to your duration-neutral portfolio if there is a small parallel shift in the yield curve? A large (>>1bps) upward shift? A large (>>1bp) downward shift?

See here for more interview questions/brainteasers

Analogies have their limitations

CDS and IR swap are very similar in many aspects. However one must also be aware of where the analogy ends. As an example, one can lock in the profit of a pay-fixed swap (when swap rate goes up) by entering into a receive-fixed swap, IF THE SWAPS ARE RISK-FREE. Alternatively she could unwind the position (selling the swap) and collect the present value of the cash flow.

For CDS things are more complicated. Unwinding the position would still let the investor cash-out immediately. Entering into an opposite swap, however, does not guarantee the cash flow anymore because there is the possibility of default, which would terminate the cash flows from both positions (the original protection buyer and the opposite seller).

Short note on ways to avoid negative short rate

- BK, BLT or other exponential-Gaussian models
- Imposing absorbing or reflecting BC (Goldstein and Keirstead)
- Kim and Singleton reviewed a few possibilities:

• Affine models like CIR, under which zero is not accessible
• Quadratic Gaussian models
• Black's shadow rate (i.e. treating bond yield as a call option to a latent process that CAN go negative)

MC Simulation for Hull-White Model

Recipe 1
Fit kappa and sigma by calibrating to swaptions/caps (swaption and cap prices can be expressed as Hull-White bond options, which in turn can be expressed as function of H-W bond price). Then calculate theta(t) by taking partial derivative of instantaneous forward rate f(0,t). Then do Euler MC simulation:

$r(t)=r(s)+(\theta (t) - \kappa r)dt + \sigma dW$

Recipe 2
This alternative method doesn't require taking partial on f(0,t). After fitting kappa and sigma,
then do the following MC simulation (it is by integrating the SDE):

$r(t)=r(s)e^{-\kappa (t-s)} + \alpha(t) - \alpha(s)e^{-\kappa(t-s)}+B$

where B is a Gaussian distributed white noise with variance

 $ \frac {\sigma^2}{2 \kappa} [1-e^{-2 \kappa (t-s)}]$

and

$\alpha(t) = f(0,t) + \frac {\sigma^2}{2\kappa^2}(1-e^{-\kappa t})^2$

Bottom line is, theta-fitting is about calibrating to the spot curve. But Recipe 2 does just that without invoking theta, because the instantaneous forward rate contains the same information.

Ref: Brigo pp.73

Also:
One can simulate the stochastic short rate process under the T-forward measure instead of the risk-neutral measure. The drift of the SDE will be different from that above, and would depend on T.

Advantage:
Since under the T-forward measure we discount the payoff with zero coupon bond, and the zero coupon bond price is completely determined by the short rate at a certain moment, we don't have to simulate too many points on a path, but we do need to do that for risk-neutral measure pricing in order to approximate the money-market account well.

Disadvantage:
P(t,T) bond is the natural numeraire to use for discounting, but what if we are pricing an instrument with multiple cash flows?

Short note on Vasicek CDO pricing model

- It is a PRICING model, not a credit default rate model (i.e. neither structural nor reduced-form, it takes prob. of default as input)
- It assumes (in its most simple form) large, homogeneous pool
- It assumes Gaussian copula (can be relaxed)
- Just like B-S option pricing has implied volatility, Vasicek CDO pricing has implied correlation
- In its simplest form, the correlation matrix is assumed to be time-independent and all pairwise correlations are identical
- Industry people use Vasicek implied correlation as a convenient way to quote tranche price (cf. Black volatilities for cap/floor/swaption)
- Not surprisingly, the implied correlation is not constant across tranches - correlation smile
- Base correlation is smoother than compound correlation

Ref: This article by Elizalde