The volume of up-and-out, or knock-out, puts now being traded makes their pricing and hedging a matter of some importance to writers, buyers and market-makers. (First published Risk Magazine, March 1991, Mike Hudson)
An up-and-out stock index option is a simple put contract between an option writer and holder which is cancelled if at some time during the life of the option the index rises above a specified level, known as the knock-out boundary. Suppose, for example, that at some time when the index equals 100 an investor purchases a one-year at-the-money European up-and-out index put with a knock-out boundary of 105. Assuming that he holds the option until expiration, and that at no point during the life of the option the index rises above 105, his pay-out at expiration will be identical to that of a one-year at-the-money simple European put. In a case where the index does rise above 105, the contract will be cancelled and the investor will not receive a pay-out at expiration, regardless of the index level. From the perspective of the option writer, his contingent liability to the option buyer is cancelled if at any time during the life of the option the index rises above 105. Up-and-outs are also known as "knock-outs", "over-and-out puts", "extinguishing puts", "over-the-top puts", "barrier puts", etc. The process whereby the index rises above the knock-out boundary, cancelling the contract, is known as "going-out", "extinguishing", "vanishing" or "knocking-out".
Although a similar product, the "down-and-out call", has traded over the counter sporadically in the US markets since 1967, the up-and-out put only began to emerge as a significant product in the late 1980s. During that period Nikkei linked bonds with embedded short European up-and-out puts were marketed with great success to Japanese investors, and consequently options based on the Nikkei 225 now dominate the market in OTC up-and-out puts. Because a European up-and-out put has some chance of going out before expiration, it will clearly be worth less than an identical European put without a knock-out boundary.
The price of a European up-an-out put, U, should be that of a simple European put, P, less some quantity Y: U=PÑY
Black and Scholes approached the problem of pricing simple European options by assuming that, over a large number of trials, the distribution of index prices at an option's expiration could be described by a lognormal distribution curve centred at the forward rate of the index at expiration, the "width" of which was determined by the volatility of the index. To extend this approach to an "up-and-out" put it is useful to visualise the development of this lognormal curve as time moves from to to te (where te Ñ to is the time from origination to expiration of the option). At any point, the probability distribution of index prices will be represented by a lognormal curve centred on whatever is the forward rate at tx. As x increases, the distribution curve becomes "wider". Practically, the forward curve of an equity index is always upward-sloping. By making the visualisation that the distribution curve is centred on a rising forward curve, and that the distribution curve is continuously widening as time goes on, two properties of up-and-out puts become apparent. Firstly, up-and-out puts with low knock-out boundaries are more likely to extinguish than those with high knock-out boundaries, and are hence cheaper. Second, the more time the distribution curve has to scale the forward curve the more likely the option is to go out; long-term up-and-out puts tend to be cheaper than otherwise identical options with shorter maturities.
The volatility of an index not only determines how much the distribution curve widens, but also how likely an up-and-out is to go out at any value of t. Knowing that the risk-free rate determines the steepness of the forward curve itself, it can be seen intuitively that for a pay-out protected European up-and-out put: Y = f(H,K,S,r,t,sigma)
where:
H = knock-out boundary
K = strike
S = index spot
r = risk-free rate
t = time to maturity
sigma= volatility
Because the knock-out boundary for an up-and-out put is known in advance, it has been possible to derive a closed-form formula for the valuation of a payout protected European up-and-out put. Cox and Rubinstein have published such a formula:
U = SN(Ñx) + Kr--tN(sigma…t Ð x) - (1) [(ÑS(S / H)-2E N(Ñy) + Kr--t (S / H) -2E+2 N(sigma…t Ð y))]
where :
E = (Iogr)/sigma2 +0.5
x = (Iog(S/K))/sigma…t +Esigma…t
y = (Iog(H2/SK))/sigma…t + Esigma…t
Several of the earliest Nikkei 225 up-and-out puts were priced without reference to this formula. Pricing methods such as Monte-Carlo runs and modified binomial processes were used. Currently most major participants in the OTC stock index option markets are aware of the closed-form solution to the payout protected European up-and-out put. Several participants, realising its shortcomings, have continued to use a modified binomial process.
The original Black-Scholes formula is strictly valid only for the valuation of payout-protected (non-dividend paying) stocks or indices. However, whether a dividend payment is made towards the beginning or end of a European option's life is irrelevant to the pay-out of a European option. Consequently, the problem of valuing dividend-paying securities with the Black-Scholes formula maybe circumvented by replacing S in the formula by (S Ñ present value of dividends), thereby effectively lowering the spot, price to account for future dividend payments. One of the shortcomings of formula (1) is that it is also valid only for payout-protected stocks or indices. In this case, however, the technique of simply dropping the spot price by the discounted value of future dividends is not appropriate. Such a technique effectively increases the gap between the index spot level and the knock-out boundary, reducing the probability that the option will go out and thereby overstating its price. Because the dividend yield on the Nikkei 225 is so small (about 1%) some market practitioners continue to use formula (1). Others continue to use a modified binomial process, which may be adapted to cope with the dividend problem. In the Cox-Ross- Rubinstein binomial model for American puts, a "tree" of stock price movements is constructed, where at each node the stock may branch up or down. At each node the stock price is checked to discover whether or not the option should be exercised. In a modified binomial process for valuing European up-and-out puts, the stock price is checked at each node to ascertain whether or not the option has gone out.
By comparing the valuation of a European up-and-out put on a pay-out-protected index with both formula (1) and a binomial process it can be seen, as might be expected, that where the index spot price is some way distant from the knock-out level the binomial answer is similar to that of the dosed-form answer. Unfortunately, as the spot price approaches the knock-out boundary the accuracy of the binomial process (compared to the closed-form solution) becomes significantly lower. To some extent, this inaccuracy can be reduced by increasing the number of binomial iterations. However, the number of iterations required for acceptable accuracy becomes computationally very intensive as the knock-out boundary is approached. Most up-and-out options are traded when the option is some way from going out, and the inaccuracies resulting from the use of a binomial process are relatively insignificant. However, for participants trying to maintain hedged-out puts dose to going out, use of the binomial process may well result in unsatisfactory hedges.
One further obstacle to accurate up-and-out put pricing might be described as "barrier discontinuity". Barrier discontinuity means that an up-and-out put may only be knocked out at certain times. The closed-form solution to the pay-out-protected European up-and-out put pricing problem assumes that the option will go out if the index spot moves above the knock-out boundary at any time. In fact, many up-and-out put contracts stipulate that only daily dosing prices may knock the option out. This problem means that intra-day knockouts are precluded and that closed-form solutions will tend slightly to underprice typical European up-and-out puts.
In addition to the specific pricing difficulties already discussed, OTC up-and-out options also face valuation problems common to all long-term OTC stock index options. Some of these more general problems arise from the necessary relaxation of some of the original Black-Scholes restrictions. For example, the Black-Scholes formula assumes that interest rates will be constant throughout the life of the option. In fact, a changing interest rate has a significant effect on the value of long-term stock options.
Finally, as more and more players become involved in the OTC stock index options markets, and spreads become tighter, so the assessment of creditworthiness will increase. Unlike on-exchange markets, where the credits of all writers are effectively guaranteed by the exchange, options traded OTC are only as creditworthy as their writers. Although not yet the case, it seems likely that in the future the credit rating of a writer will be an explicit determinant of an OTC stock index option's price.
Up-and-out put
An up-and-out put is a simple put which "goes out" (ceases to exist) if at any time during the life of the option the index level reaches the "out" level. For example, an investor holding a one-year European up-and-out put with a strike of 100 and an "out" of 105 would own an option with a pay-out equal to that of a European put if at no time during the life of the option did the index rise above 105. If the index rose above 105 during the life of the option then the value of the asset held by the investor would be zero, regardless of the level of the index at the option's expiration date. An up-and-out put is cheaper than its equivalent simple put.
Up-and-in put
The up-and-in put is the inverse of the up-and-out. The holder of an up-and-in put owns an asset with a payout at expiration of zero unless at some time during the life of the option the index rises above the "in" level. If the index does rise above the "in" level at some time during the life of the option then the holder's asset becomes a simple European put, and pays off accordingly at expiration.
Down-and-out call
In the same way that an up-and-out put goes "out" if the index rises above the "out" level, so a down-and-out call is a simple call which goes "out" (ceases to exist) if the index falls below the "out" level.
Down-and-in call
The inverse of a down-and-out call. A down-and-in call is an asset which becomes a simple call if the index goes below the "in" level at any time during the life of the option. If the index does not at any time in the life of the option go below the "in" level then the option expires worthless.
Barrier discontinuity
Barrier options may be created with specifications which only allow the option to go "out" or "in" on particular days Ñ the last day of each month, for example. Such restrictions make up-and-out and down-and-out options more expensive and up-and-in and down-and-in options cheaper. Whilst barrier options are most frequently created in European style, American and quasi-American formats may also be made available. Quasi-American options allow the option holder to exercise at agreed times other than maturity.
Digital options
Simple European and American options have smooth pay-out profiles : the further a simple option is in the money, the more it pays off. The following option products have fixed pay-outs, and, rather like binary digital circuits, which are either "on" or "off". digital options either pay out this amount or nothing at all.
All-or-nothing
An all-or-nothing put (call) pays out a predetermined amount (the "all") if the index is below (above) the strike price at the option's expiration. How much the index is below (above) the strike is irrelevant; the payout will be "all" or nothing.
One-touch all-or-nothing
A one-touch all-or-nothing put (call) pays out a predetermined amount (the "all") if the index goes below (above) the strike price at any time during the option's life. How far below (above) the strike price the index moves is irrelevant; the pay-out will be the "all" or nothing.
Supershares
In some ways supershares are the most fundamental of option products. The holder of a typical supershare type product would receive a predetermined pay-out if at expiration the index closed exactly at the strike (for all other expiration index values the pay-out would be zero). For practical purposes "strike = 100" might be defined as "pay-out occurs where the index at expiration is above 99.5 and below 100.5".
Digital options can be added together to create assets which exactly mirror investors' anticipated index price movements.