Choosing the tenor for a hedge can be no easy task. In many cases--for example, balance sheet hedging or long-dated exposures--it is impossible or undesirable to hedge the entire period. Also, other factors such as adverse forward points or unfavorable market levels may make it unattractive to do so. In such cases, is there a systematic way of deciding the tenor of the hedge instrument to use?
The problem is more complex when hedging with options. Here we must deal not only with market direction, but take into consideration how the value of the option is affected by the passage of time and by changes in volatility. Longer-dated options undoubtedly are more expensive, but is the extra expense ever justified, and under what market conditions?
We shall address these points below, using a simple example: the choice between a three-month and six-month USD put/JPY call to hedge a long dollar exposure. We use a spot reference of JPY106.60, and assume flat forward points at first to clarify the other factors. Implied volatility is taken at 15.0% for both options. Using these inputs, the price for a three-month option struck at-the-money-forward (ATMF) is 315 JPY pips, while the six-month ATMF USD put would cost 438 pips, 39% more.
DOES GREATER PREMIUM EQUAL GREATER COST?
The important factor in the true cost of option ownership is time decay (theta), which measures how quickly premium erodes with the passage of each day. At the outset, the mark-to-market P/L is zero (neglecting bid/offer spread). This obviously cannot remain the case, as in three months our shorter-dated option would expire worthless if there were no spot movement. Meanwhile the six-month option would still have three months of time value left, and this remaining value can be realized if the position is unwound.
Time decay erodes premium values slowly at first, a rate which accelerates as they get closer to expiry. Because of this, longer-dated options are hurt less by the passage of time. Imagine a situation where the spot market does not move in the first three months: the advantage of the six-month option is greatest after three months, when the short-term option expires worthless. With three months life still remaining, the six-month option is still worth 315 JPY pips, more than the difference in premium of 123 pips. In these circumstances, the six-month option will lose less due to time decay than the three-month, more than recouping the additional premium cost.
* In a sideways market, the true cost of the shorter-dated option is larger.
HOW DO MARKET MOVEMENTS AFFECT MY POSITION?
The benefits of option ownership can be summed up as being long gamma. This means that an option owner will make money at an accelerating rate when the market moves favorably, and will lose money at a decelerating rate in an adverse move. Gamma is the flip-side of theta: an option with higher time decay cost is assured of more benefit from a higher gamma.
This means that, if the market does move significantly, the shorter-dated option will outperform the longer-dated option. In our example, if the dollar were to move immediately higher, the higher gamma of the three-month USD put allows more of the profit in the underlying to be enjoyed. The low-gamma six-month option, in contrast, will continue to erode the profitability of the underlying position.
A downward move in the market would also favor the three-month hedge. Here the profitability of the USD put rises rapidly to offset losses in the underlying, giving an enhanced cushioning effect. The six-month option will make less in these circumstances, so the losses in the underlying will have a greater impact. This is shown in the graph:
This is a P/L snapshot after three months: the heavy line gives the net P/L of hedging with the shorter-dated option at expiry, while the thin line is the mark-to-market P/L of the six-month position which still has three months remaining (N.B.: the positions shown are a long underlying position hedged with a long USD put). The gray line shows that the three-month option is least attractive if spot is at the strike rate, becoming the better choice if spot has moved substantially in either direction.
* Large market movements in either direction favor the shorter-dated hedge.
IMPACT OF FORWARD POINTS AND IMPLIED VOLATILITY
In this example, the forward points were taken as flat to illustrate the time decay and directional effects more clearly. In reality, however, the cost or benefit of selling forward at a discount or premium is a factor that cannot be ignored. Any convexity of the forward curve can enable options to be struck at better levels; in our example (where the forward curve is negative), if the six-month forward points were more than twice the three-month, a shorter-term hedge would give the opportunity of reducing the forward point cost. However, this is not without risk, as we have full exposure to changes in the three-month forward points until the second position is taken in three months time.
The same is true of implied volatilities: If the volatility curve is positively sloping (as it often is in quiet conditions), the three-month hedge will be disproportionately cheaper. The risk is that three-month points may have risen by the time the first hedge has expired although if this happens because of a large market movement, the short-term hedge will have already paid off as shown above, and forward cover may prove the better option for the remaining period. In the first three months, the shorter-term hedge has lower vega (price sensitivity to implied volatility), and so will be affected less by changes in the volatility curve prior to expiry.
CONCLUSIONS
* In a directionless market, the longer-term option is better,
as the time decay cost is less;
* If the market moves by a large amount, the shorter-dated
option provides a better level of either protection or
participation;
* A rolling short-term hedge can reduce forward point cost,
but leaves the hedger exposed to adverse or favorable
moves in the forward points;
* Short-term premiums can be even lower if the volatility
curve is positive, and the positions are less affected by
movements in implied volatilities.
This week's Learning Curve was written by Chris Attfield, an analyst with strategic risk management advisory at Bank Onein London.
