Monday, May 28, 2012

Variance Swap Revisited (Redux)

It's amazing how much I learn every time I re-read the variance swap literatures.

1. The (model-independent) formula for the fair strike of a variance swap depends on the choice of S* (the truncating spot price). Most generally it is

$ rT - \frac {S_0 e^{rT} - S^*} {S^*} - ln \frac {S^*}{S_0} + e^{rT} \int_0^{S^*} \frac {P(K)}{K^2} dK + e^{rT} \int_{S^*}^{\infty} \frac {C(K)}{K^2} dK$

If S* is chosen to be the future stock price then the above expression can be simplified.

To hedge a variance swap, a trader statically holds the continuum of OTM options, plus rolling a future position of (1/F_t - 1/F_0). In a B-S universe the future position would hedge away the delta exposure, except that...

2. Delta of variance swap (one that comes from practicality):
In principal the terminal variance is measured as the INTEGRAL of INSTANTANEOUS variance; in practice, the contract is such that variance is calculated on a daily basis using daily prices. This leads to the need for intra-day Delta hedging.

3. Delta of variance swap (one that comes from smile)
The Coulombe paper show that variance swap Delta is a function of implied volatility skew. This implies that the Delta can be negative.

Reference:
Derman GS paper
de Weert "Exotic Options Trading" Ch. 23
Carr and Madan 2002 "Towards a Theory of Volatility Trading"
Coulombe et al 2008 "An Analytic Formula for the Delta of Variance Swap"

Sunday, May 20, 2012

Short Notes on (Implied) Volatility

  1. At the risk of stating the obvious, implied volatility is nothing but the quoting of price under B-S framework. The use of it for anything else (i.e. hedging) is a recipe for disaster.
  2. In light of this, the purpose of having a stochastic volatility model is not primarily to have better fits to market smiles (though that is important too), but rather:
    • To allow for consistent hedging (especially vega hedging);
    • To prescribe some dynamics to the evolution of the smile^.
  3. In the B-S framework, if one long an option and continuously delta-hedge, the payoff would be 0.5*S*S*Gamma*(realized var. - implied var.)*dt. What about under a stochastic volatility model? Suppose that the variance process is also stochastic and we only delta hedge w.r.t. the stock price. If the variance process is dv = a dt + b dX, then the delta-hedged P&L would have a residual of b*Vv*dX, thanks to the additional source of randomness.
^ Local volatility model is somewhat helpful for the first purpose (hedging). It however produces smile dynamics that are at odd with observation. See the discussion in the original SABR paper.

 Reference: Rebonato, "Volatility and Correlation: The Perfect Hedger and the Fox" Ch. 6
                  Wilmott, "Paul Wilmott on Quantitative Finance" Ch. 12

Monday, May 7, 2012

Time Integral of Brownian Motion

This is a short note on computing the time integral of a Brownian motion:

$ \int_0^t W_s ds$

In fact I should have said the moments of such an integral. The first moment is just

$ E[\int_0^t W_s ds] = \int_0^t E[W_s] ds = 0$


The second moment is

\begin{align*} E[\int_0^t W_s ds \int_0^t W_u du] & =  E[\int_0^t \int_0^t W_s W_u ds du] \\ &= \int_0^t \int_0^t E[W_s W_u] ds du \\ &= \int_0^t \int_0^t min(u,s) ds du \\ &= \int_0^t \int_0^s u du ds + \int_0^t \int_s^t s du ds \\ &=\int_0^t \frac {s^2} {2} ds+\int_0^t (ts-s^2) ds \\ &= \frac {t^3}{3}\end{align*}

The trick is to separate the min function into two 
regimes.