Friday, November 16, 2007

Risk in the Times of Reversal

Standard Black Scholes pricing assumes a constant vol. The underlying implication being that the logarithm of the returns is normally distributed – and thus contained in it, a constant standard deviation (the constant volatility). In the market, there are other factors are play – such as supply/demand, risk-premia etc., -- all that contribute to, what Keynes memorably called “animal spirits” in the option pricing market. Typically, if the market expects a greater likelihood of the underlying exchange rate to go past the strike, the calls on the currency tends to get priced more expensively than the puts.

An option on call USD–CAD put refers to the call on the USD and the put on the CAD. So, the holder of the option has the right to buy the USD (convert the CAD notional at a prespecified rate). Equivalently, the holder of the option has the right to sell the CAD at a prespecified rate.

A spot price of 0.97, i.e., one USD can be exchanged for 0.97 CAD; with a strike of 1.01 on a call USD-CAD put refers to the right to buy one USD in exchange for 1.01 CAD. Tersely, the spot is 0.97 and the strike is 1.01 with a CAD-put. On the expiry date if the spot prices are 1.00, then the buyer of the call (with say 101 CAD in his account) would not exercise his option to buy a USD at 1.01 CAD when he can easily buy the same USD at 1.00 CAD. In this example, the USD is anticipated to appreciate. So, the call option on the USD-CAD is evidently ‘worth more” than a corresponding put option. i.e., if an appreciation is anticipated the corresponding call is priced at a higher level. This supply-demand forces are not a part of the Black Scholes derivation. Since, most parameters are fixed – the only “tweak-able” parameter is the vol – or the implied vol.

A 25-delta call refers to a call option where the strike above the spot (thus an out of the money option). So, in the above – it is clear that a 25-delta call has different implied vol than a 25-delta put. The “25” in the above refers to the fact if the underlying exchange rate increases by 1, the corresponding the call option value rises by 0.25. So, to arrive at a delta-hedge a corresponding position has to be taken in the underlying. The market convention of 25-delta is agreed upon – as one that is sufficient to capture the expectations regarding changing underlying prices. Of course, you can have 10-delta, 50-delta and so on.

A risk reversal is thus, the difference in implied vols between, ceteris paribus, out of the money calls and out of the money puts. Quoted thus, a rise in the risk-reversals means that as the currency appreciates, the volatilities are likely to rise. Instead, if risk-reversal are quoted as put – call. Then a rise in risk-reversal refers to the fact that as currency depreciates the vols are likely to rise.

A useful example of the trade flow is as follows: (courtesy

Assume an appreciation of the USD against the CAD over the next 3 month period (mean-reversion??). 3-month 25 delta USD-CAD risk reversal of 0.15 -.28% at a vol of 8.5% means:

1. Buy the 25 delta USD call/CAD put at 8.65% and sell the USD put/CAD call at 8.5%. The trader shells out 0.15%. i.e., he is paying a skew-premium of 0.15% in anticipation of a USD rise.

2. Sell the 25 delta USD call/CAD put at 8.78% and buy the USD call/CAD put at 8.5%. The trader earns the .28% spread.

On option desks, rules of thumb Rule! So, to extract the implied skewness, it is pretty standard to (a) calculate the risk-reversal (b) calculate risk-reversal per-unit of ATM vol. Risk reversals.

The big challenge is what to do when appreciations have different vols than depreciations. If you know how to deal with that – then there is some money to be made and a heck-of-career to be had!

No comments: