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Selling expensive options can produce significant returns and the short volatility player sells options in the hope that the premium he receives more than covers any future price movements. Simply selling naked call options is very risky.

Shorting call options is equivalent to establishing a short position in the underlying shares and if the market price rises, large losses can result. As with the long volatility play it is assumed that the individual has no view on the direction of the underlying and so the classic short volatility portfolio will have short call options perfectly hedged with a long stock position. Let us return to the one-year call option. With the stock price at $99 and the option price at $5.46 the delta will be -0.50. The initial portfolio will com-

prise one short call option and long 50 units of stock. We are at point B in Figure 5.4 which together with Table 5.1 shows the profit and loss to the portfolio, assuming for simplicity, that there is no time decay.

We now consider what happens to the portfolio under two scenarios. If the stock price rises, a loss is suffered on the short call and a profit is enjoyed on the long stock. If the stock price falls, a profit is enjoyed

on the short call and a loss is suffered on the long stock. In Figure 5.4 we have shown the stock profit and loss line displayed as a negative sloping line but of course in reality it is positive. Illustrated this way it is easy to see that the net profit and loss to the hedged portfolio will be the difference between the two value profiles and this is given separately in the lower panel.

Small Stock Price Moves

It the stock price rise is small, the option losses are almost completely cancelled by the profit on the long stock. If the stock price fall is small the option profits are almost completely cancelled by the losses on the long stock. The portfolio is perfectly hedged and remains so with small stock price moves.

Large Stock Price Moves

If the stock price rises significantly, the option component always loses more than is made by the stock component. If the stock price falls significantly. the losses on the stock component always exceed the profits on the option component. The bigger the move. up or down, the bigger the losses. After putting on this position one would hope for little or no stock price movement until expiry and this is why the strategy is called the short volatility trade. The reason the portfolio always produces losses whatever the direction of the underlying is again due to the curvature of the option price. In the short volatility portfolio the curvature of the option is the opposite of that in the long volatility portfolio.

Rehedging the Short Volatility Portfolio

\l point B, the portfolio is perfectly hedged. The exposure of the long " units of stock are completely cancelled by the one short call option ^ 'Hi i delta of -0.5. Say the stock price increases to $105. i.e. point Z i ! li'tirc 5.4. At Z, the portfolio has an unrealised loss of $49 and this ' '-.- \\ ill obviously increase if the underlying price keeps increasing. At ^/ 'hi delta of the option is now —0.66 and so to be strictly market

neutral we should buy an additional 16 shares to bring the total long stock position up to 66. Buying the additional shares re-establishes a delta neutral position. If the underlying price continues to increase there will still be losses but they will be smaller than those suffered if the additional shares were not purchased.

Say that immediately after buying the extra 16 shares at $105, the underlying price begins to fall all the way back to B at $99. At B, the portfolio is unbalanced again with the delta at -0.50 and so we must sell the 16 shares at $99 locking in a loss of 16 X (105 - 99) = $96. If the price continues to fall we must readjust the option by selling more shares at an even lower price. At $93 (point Y) the delta is only -0.34 and so 50 shares is too heavy a hedge; the correct quantity is 34.

Accordingly we would sell 16 shares at $93. If the price turns around again, then to keep market neutral we would have to start buying shares all the way up. At Z the delta is again —0.50 and so one must buy 16 shares at $99, locking in a loss of 16 X (99 - 93) = $96. So being short of volatility and suffering significant price moves always involves selling low and buying high—the exact opposite of being long of volatility. The rehedging process, although always re-establishing a neutral position, locks in losses. Selling low and buying high is a direct result of the fact that the delta of the option position moves in the opposite direction to that of the underlying or put more simply the position has negative gamma. The position is short gamma and this expression is sometimes used instead of short volatility.

So why rehedge? Consider the situation of the stock price moving from B to Z to B to Y to B, etc. as in Chapter 4. Rehedging at Z, B, Y, B ... etc. would involve buying stock at $105 selling at $99, selling more at $93 only to buy back at $99 and so on. Each round trip would essentially lock in a loss of $192. If the stock price always returned to B after a $6 swing to Z or Y and we did not rehedge then no losses would occur. And that is the dilemma of the short volatility player. Once the underlying stock price has moved when does one rehedge? We know that if the price moves all the way back to the starting point we should not, with hindsight, have rehedged. In a rising market it is painful buying stock knowing that you might have to sell it all again if the price falls. But if you do not buy stock on the way up the potential losses are unlimited. The same applies to falling prices. In a falling stock price situation the hedge has to be continually unwound by selling stock. If one decides to wait in the hope that the share price will rise again the potential losses can also be very large.