Typically the payout of an option at maturity is linear with respect to spot. However, non-linear payouts can offer either enhanced leverage, or a better hedge for a non-linear underlying exposure. Pricing of such structures can be done by: * Monte Carlo simulation
* Analytical closed form solution
* Replication
The solution must be quick to set up and execute, with the pricing tools to hand; it must permit rapid delivery to the client of revaluations and scenario analysis; and, crucially, it must define what trades to book in the risk management system.
Replication
One use of a non-linear payout product is a Power Option, which is an enhanced yield deposit. A typical termsheet is below.
Assume NEUR x (1+Ymin)/(1+Ymoneymarket) is placed in a money market deposit to generate the minimum redemption. A Double One Touch or Range Bet is purchased to cover the extra yield of 2% if the range is broken. The balance of the notional goes to the Power Option premium.
Assume we construct the Power Option with a portfolio of euro calls, all with double knockouts at L and U, and strikes spaced evenly between L and U by a distance dK. The missing link is the notional of each of these options. We know from the termsheet (assuming the range is unbroken):
* Termsheet Payout in dollars = P(S1)= NEUR x
((txYmin+0.50%x(50x(1-S 1/S0))2))-Y min)xS1
* Terminal Delta of Payout in euros = dP/dS1=D(S1)=P/S 1+
NEURxtx0.50%x2x50x50x(-S 1/S0+S 12/S02)
* Terminal Gamma of Payout in euros = dD/dS1=G(S1)=
D/S1-P/S1 2+NEURxtx0.50%x2x50x50x(-1/S 0+2xS1/S 02)
The objective is to build a portfolio of options in which the terminal value and delta match the terminal value and delta of the termsheet at all points. This is done by setting the notional of each option according to:
* Option Notional of euro call with strike L =
D(L)+G(L)/2xdK
* Option Notional of all euro calls with strike K>L =
G(K)xdK
The delta always lags and introduces an error into the replication. By adjusting it by half the gamma at L, the error becomes tiny. In the cases of extreme gamma, a similar adjustment would also be made to G(K).
Why bother calculating the delta and gamma? An alternative is to have the payout at each point and infer the delta and option notionals from there. It is quite common, however, that a non-linear payout will have a constant gamma: in these cases, building the portfolio is trivial as all the notionals are the same. The gamma here is not constant because the deposit and coupon is in euros and we are measuring the payout in dollars which introduces an extra S1 term. We are obliged to do this since the termsheet is expressed in dollars per euros terms.
The terminal value of the portfolio at the starting point L is zero. This must match the termsheet payout at L, and so to complete the replication, a dollar cashflow of P(L) is added.
Note: if the Power Option did not have a barrier condition attached, then it is also necessary to meet a further condition: that the delta outside the range is zero. In practice, this means an extra call with strike U and notional equal and opposite to the sum of all other option notionals. In the more general case where there are no boundary conditions, using replication to price the structure is less trivial as it may require an impractically large number of options to achieve the required level of accuracy; however it is highly unusual and generally impractical to have a non-linear payout with no boundary conditions.
Accuracy & Convergence
One unanswered question is what to choose for dK. Having high accuracy requires closely spaced strikes and many options which make pricing and booking less convenient. In practice, the value of the portfolio quickly converges to its limit with 25-50 options depending on the amount of gamma.
A good way to check the results is to use a portfolio of euro puts instead of calls. Then, starting at strike U and working down to L in the same way and adjusting the value by P(U) instead of P(L), the resultant market value of the portfolio must be the same.
Note that it is important to see which way the convergence occurs. It may be the case that a smaller number of options may be used, with a known error, and this error is simply charged for upfront. Then it is important to know whether the client is buying or selling the option and whether the replication is converging from above or below the limit. It may be that the error should be discounted from the price.
Summary
Using replication to price non-linear payouts this way means that such payouts become, if not trivial, certainly more vanilla to price. With further work it would be possible to do this in a systematic way, simply entering the necessary termsheet formulae and conditions to generate the portfolio. Since the replication uses products which have relatively well-known market values then the replication itself will have a well-known market value (this being the object of the exercise). For instance, the effect of the vol smile can be easily factored in, as can the appropriate bid/offer spreads.
The replication can also be done using digitals instead of options. This is a good check for the price, but digitals are typically themselves constructed from options and thus are not the ideal instrument for replication. This is an important point--deciding which products are fundamental or indivisible and using only those to replicate. This selection is driven by the relative liquidity of each product.
This approach can, and should, also be used to price other exotics, or combinations of exotics. Since it is likely there will be several ways to replicate a combination of exotic products and all those replications must result in the same price, achieving a level of consistency of pricing across products which will allow this is then key to a robust pricing system.
This week's Learning Curve was written by Charlie Brown, head of structuring and solutions for global options at Standard Chartered in London.
