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Synthetic Financial Instruments
The price of a reset strike call when the initial value of the stock is a is greater than or equal to a vanilla European strike call and the price of a reset strike put when the initial value of the stock is a is greater than or equal to a vanilla European strike put (as expressed in the cited equation) for all values of S.
The style of an option is, in general and in this case, defined by the date(s) on which the option can be exercised. The European vanilla option may be exercised only at the expiry date of the option; that is, unlike an American style vanilla option, it can only be exercised at a single, a priori-agreed to date. Vanilla options are essentially straightforward options, with few complications in terms of how the payoff is calculated.
As such -- given the simple nature of the ways in which a vanilla option (and especially a European vanilla option) is calculated, the reset strike call or a reset strike put will necessarily in all cases be at least as high as the vanilla call precisely because a reset strike or call will be effected to maintain a price that is no lower than the vanilla European option. No reset option would be effected if the resulting reset price would be lower than the vanilla European option. This is, in a fundamental sense, precisely the reason for the reset option.
2. Referring again to the above equation, there is a point at which the two inequalities shift to being equalities. A reset option pays out at the difference between the vanilla price and the spot price of the stock. Thus, when the vanilla price option and the spot price are the same, then the inequalities will shift to equalities. This occurs when the value of (that is, the date of the reset) co-occurs with this co-occurrence of the vanilla price option and the spot price. (Needless to say, this is a relatively rare circumstance; it is, however, perfectly feasible.)
3. A reset option can be performed for the function at a point in time, an exchange, an exchange complex, a combined compound, or an entire product family. The value of a traditional, non-exotic option is dependent on a single value: The price of the underlying referent on either the day of exercise or the day of expiration (if the two of these are different).
However, the conditions described in this question involve a non-traditional reset option is valued along a path-dependent process that depends either in part or in while on the price pattern that is followed by the underlying in the process of reaching either exercise or expiration. The terminal value (for either of the proposed stock values, that is, 50 or 150) depends on the value of the underlier not only at the point of exercise or expiry but also at all other prior points in time: Thus the terminal value of the option depends upon the path taken by the underlier.
The pricing functions of both reset price strike and reset price put are calculated using the qualifier that this family of derivative instruments is a non-linear one. In fact, the payoff diagram for this scenario is highly non-linear, a quality that arises whenever the derivative either is an option or (as in this case) has a derivative embedded in it.
Thus, in this case, the valuation lines of the two possible values (50 and 150) diverge sharply over time, with the difference between them increasingly difficult to predict as the embedded options in them make exaggerate the difference between the initial value/price over time.
4. Binomial trees are a no-arbitrage calculation that use a lattice-based model (or a discrete-time model) that allow for the calculation and graphing over a period of time of the (varying) price of the underlying financial instrument. One of the important advantages of this form of financial modeling is that it allows for calculations that are not based on closed-form solutions.
To address the question posed here, the following methodology must be used. Any variation of the binomial pricing model assumes that the evolution of the option's key underlying variable(s) must be tracked in discrete time units. (This is both logically and mathematically necessary in terms of reducing the complexity of calculations to a manageable level.) This tracking used the binomial lattice for a set number of time steps (or discrete increments) between the valuation (that is, essentially the first point on what we might conceive of as a time line) and the expiration date (the end node on the time line). Each point (or rather node or intersection) in the lattice can be read as representing a possible price of the underlying at any designated point in time.
Binomial models are iterative ones, which means that they are based on the concept that only possible way in which to extrapolate an accurate model over time is to build the model on the base of feedback loops. The methodology of the model is based on a starting point of each of the final nodes (a final node is one that is reached at the time of expiration). Beginning with a final node, one works backward through the lattice or tree until one reaches the valuation date, which is the site of the first node. At each analogous point in iterative process, the value identified is the value of the option at that point in time.
The option valuation in this process is a three-step process as described above, beginning with the generation of a price tree, followed by the calculation the option value for each of the final nodes, and finally the sequential calculation of the option value for each of the preceding nodes, which vary in number depending on the exact specifications of the model being used.
More specifically, the iterative process is based on the assumption that whatever the underlying instrument may be, it will move either up (u) or down (d) by a specific factor at each of the steps of the lattice or tree (assuming that and ). (A stripped-down sample calculation demonstrates how this works: If the current price is designated as, then the next period will see the price set at either or .)
Both up and down factors for the model are calculated by using the concept of volatility (designated as ?) along with the time (t) in years. This leads us to the following calculations to determine both u and d:
To give proper credit, the above equations are the Cox, Ross, & Rubinstein (CRR) method, the basis for newer models. (For example, there is also the possibility of using a trinomial tree model, in which in addition to up and down paths there is also a stable path).
Key to the assumptions embedded in this model is that the binomial tree is recombinant, that is, that the same value results is an underlying asset moves first down and then up or first moves up and then down. (That is, the two oppositionally directed paths recombine.) This may not be true under the conditions of the real world; however, such assumption creates a model that is mathematically much less time-consuming to calculate. Another way to express this same consequence of the way in which the model is created is that the value at each node of the underlier can be calculated directly -- that is, through the direct application of a formula without the actual creation of the tree itself. If this methodology (or strategy) is pursued, then the node-value will be:
Where Nu is the number of segments that move upward and Nd is the number of segments that move downward.
5. The risk-neutral distribution model makes certain…[continue]
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