Fair Price Intervals
In this chapter, we introduce the fair price interval of a European claim and characterize it in various ways for the interval model, giving explicit computational algorithms. We also provide a detailed comparison with the simpler binomial tree model. We
- PDF / 479,804 Bytes
- 19 Pages / 439.36 x 666.15 pts Page_size
- 25 Downloads / 203 Views
Fair Price Intervals
4.1 Fair Price Interval of an Option: The General Discrete-Time Case Consider again the discrete-time setting where time points are indicated by t j , j = 0, 1, 2, . . . . We consider in this section a market with a single underlying asset. There are no conceptual difficulties, however, in extending the analysis to a situation with multiple assets. To simplify formulas, we assume zero interest rates; this assumption is not essential. Our basic framework is nonprobabilistic. Let S denote the asset price path S = {S0 , S1 , S2 , . . . , SN },
(4.1)
where tN represents the time horizon, which will be fixed in the subsequent discussion. A model M is a collection of such sequences of real numbers, M ⊂ (R+ )N+1 ;
(4.2)
no probability structure is imposed at the outset. A European derivative maturing at time tN is specified by a payoff function F(·); the value of the derivative at time tN for a path {S0 , . . . , SN } is F(SN ). We will consider models in which asset prices are always positive, and so we can look at the payoff function as a function from (0, ∞) to R. We note that if such a function is convex, it is also continuous. We consider a portfolio consisting of one option owed (short position) and a quantity γ of the underlying asset held (long position). Positions are closed at the expiry of the derivative. A strategy is a collection of strategy functions {g0 (S0 ), g1 (S0 , S1 ), . . . , gN−1 (S0 , . . . , SN−1 )} that at each time t j determine the quantity of the underlying asset to be held.
P. Bernhard et al., The Interval Market Model in Mathematical Finance, Static & Dynamic Game Theory: Foundations & Applications, DOI 10.1007/978-0-8176-8388-7 4, © Springer Science+Business Media New York 2013
45
46
4 Fair Price Intervals
Note that strategies do not require knowledge of the future, that is, they are nonanticipating by definition. A specific subclass of strategies we will consider are the so-called path-independent strategies. Path-independent strategies take only the current price of the underlying into account and can therefore be characterized by strategy functions g j (S j ), or, stated differently, the trader’s information structure is of the closed-loop perfect state pattern when strategies in general are used, whereas in the case of path-independent strategies a trader just has feedback perfect state information on the price process. As an example we recall from previous sections the following two strategies: • The (left-continuous) stop-loss strategy: g j (S j ) = 0 if S j ≤ X, and g j (S j ) = 1 if S j > X, where X is the strike price, which may be used to hedge, e.g., a short European call option; • The delta strategy [see (3.4), (3.5), where F(.) is the payoff function for a European call option] with parameters F(·), u, and d, which is given by strategy functions Δ j that are defined recursively by F(uSN−1 ) − F(dSN−1 ) , (u − d)SN−1
(4.3)
Δ j (S j ) = λ Δ j+1 (uS j ) + (1 − λ )Δ j+1(dS j ),
(4.4)
ΔN−1 (SN−1 ) =
where λ := uu(1−d) −d , which may be used to co
Data Loading...
