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John Crosby - possible M.Sc. dissertation project

Possible M.Sc. dissertation project.   Variance swaps are very actively traded especially in the equities markets. Indeed the bid-offer spread on these instruments is much lower than those for at-the-money vanilla options.    Consider a variance swap entered into at time $ t_0 \equiv 0 $ with maturity $ T $. Suppose it has $ N $ monitoring dates $ 0\equiv t_0 \equiv T_0 < T_1 < T_2 < \ldots < T_N \equiv T $. The variance swap is written on an underlying asset whose price at time $ t $ is $ S(t) $. The payoff of the variance swap at time $ T $ is:
\[ \sum_{j=1}^N ( \log( S(t_j) / S(t_{j-1}) ) )^2. \]
We denote the price of the variance swap, at time $ t_0 $, by $ VS(N) $.   Clearly $ VS(N) $ is a function of $ N $. An interesting question is how does $ VS(N) $ depend on $ N $ and, in particular, for large $ N $. In other words, as we increase $ N $ by “filling” in the mesh $ 0\equiv t_0 \equiv T_0 < T_1 < T_2 < \ldots < T_N \equiv T $ with extra points in such a way that $ \sup(t_j-t_{j-1}) \rightarrow 0 $, how does the quantity
\[ \lvert VS(N) – VS(\infty) \rvert. \]
behave?   An existing reference for the convergence rates on variance swaps (under Heston (1993) dynamics or under Merton (1976) dynamics) is Broadie and Jain (2008) “The effect of jumps and discrete sampling on volatility and variance swaps” International Journal of Theoretical and Applied Finance Vol 11 No 8 p761-797 (see Mark Broadie’s website at http://www.columbia.edu/~mnb2/broadie/research.html).   Ongoing research (Crosby, Davis and Raval (2010) – in preparation) shows that
\[ \lvert VS(N) – VS(\infty) \rvert = O(1/N), \mbox{ as } N \rightarrow \infty, \]
where O means the usual big-oh notation. This convergence result is also true if we replace the variance swap by a skewness swap (paying $ \sum_{j=1}^N ( \log( S(t_j) / S(t_{j-1}) ) )^3 $) or a proportional variance swap (paying $ \sum_{j=1}^N ( ( ( S(t_j) - S(t_{j-1}) ) / S(t_{j-1}) ) )^2 $. Crosby, Davis and Rival (2010) show that all these convergence results hold under very general dynamics for $ S(t) $ (and not only under Heston (1993) dynamics or under Merton (1976) dynamics).   So that part we do know now. One unsolved question is what about conditional variance swaps (sometimes called corridor variance swaps) ie swaps paying
\[ \sum_{j=1}^N 1_{ L \leq S(t_{j-1}) \leq U } ( \log( S(t_j) / S(t_{j-1}) ) )^2, \]
where $ L $ and $ U $ are the “corridor levels” and $ 1 $ denote the indicator function (in words, the payoff only accrues when the asset price is inside a “corridor” region”)? It would be possible to examine this empirically for various dynamics of the underlying and see via Monte Carlo simulation or, possibly, other methods, what the convergence rate is empirically. I suspect it is more like $ O(1/N^{\alpha}) $ where $ \alpha &lt; 1 $ and possibly $ \alpha $ is around 0.5.   A general reference for conditional variance swaps is Carr and Lewis (2004) “Corridor variance swaps” Risk magazine February 2004 (see Peter Carr’s website at http://www.math.nyu.edu/research/carrp/research.html).   In general, not much is known about pricing or convergence of conditional variance swaps under jump processes which therefore gives an opportunity to fill in that omission in the extant literature.   In particular, it would be interesting to see empirically if the convergence rate depends on whether there are jumps in the underlying or not (Crosby, Davis and Raval (2010) show that the convergence rate does NOT depend on whether there are jumps in the underlying for standard (ie not conditional) variance swaps). Crosby, Davis and Raval (2010) also shows the circumstances under which the prices of discretely-monitored (i.e. with finite $ N $) variance swaps are greater than, equal to or less than the prices of continuously-monitored (i.e. with $ N = \infty $) variance swaps. It would be interesting to see empirically if there are similar relationships for conditional variance swaps.   If anyone might be interested in doing their M.Sc. dissertation along the lines of what I have just outlined, they can contact me on:   John dot Crosby at ubs dot com   and/or:   johnc2205 at yahoo dot com (in the obvious anti-spam notation and where there are no spaces in the real addresses)   or let Octavia Usher or Christoph Reisinger know.   PS. In the above, "lt;" means "<" - I don't know why the tex conversion is not working.
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Last updated by John Crosby on Wed, 10/02/2010 - 3:09pm.
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