Friday, 31 October 2008

On Black Scholes Hedging

Further to yesterday's post about CB arb, a short note on hedging options.

If you sell an option at an implied vol of v, and the Black-Scholes assumptions hold (in particular the underlying is a diffusion with constant delivered volatility w, and you can trade instantaneously at zero spread) then the expected P/L of a delta hedged position depends only on v and w. In fact, as Dupire amongst others pointed out, it depends on v versus the gamma weighted delivered volatility. Thus, on average, if you sell an option at v, and v > w, you will make a profit. If you buy an option at v and v < w, you similarly expect to profit.

Two things can screw you up. Firstly this result only holds if the underlying is a diffusion. Therefore in the real world, with jumps, you can buy a 'cheap' option (i.e. one whose implied is less than realised) and still lose money on hedging. Secondly all the other imperfections (bid/offer spreads, variable interest rates, stock borrow costs etc.) hurt, so in practice you need at least a 2 vol point difference between realised and implied to have a good chance of a profit.

Thus, to return to yesterday, you will judge a CB to be cheap if the implied vol needed to recover the price of the CB is significantly less than your expectation of future delivered vol of the underlying during its life. If you are right, you can make money delta hedging the embedded option. Before 2004 or so CBs were often cheap - the issuers discounted them a bit to be sure of getting the issue off - and so CB arb could be profitable. The problem is that buying a CB to get the option is inherently deleveraged, since the option is only a fraction of the total (the rest being the bond floor). Hence callable convertible asset swaps. But that is a story for another day.

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