(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

Shadow Rate

The original paper by Gorovoi and Linetsky, and a summary of it, is a pretty neat idea to resolve the shortcoming of Gaussian short rate models: that the interest rate can go below zero (but should we be concerned about negative rate, now that CHF Libor has seen negative values?).

The 'shadow rate,' which is a latent unobservable process, is still assumed to be Gaussian. The true short rate process is floored at zero of the latent process. Intuitively, people can always choose to hold cash when rate is below zero so the effective rate should never drop to negative.

They also use an eigenfunction expansion method to construct the solution to the PDE (derivative price).