(a) Complete the proof of the “no arbitrage lemma” for the equality cases. (b) W
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Question
(a) Complete the proof of the “no arbitrage lemma” for the equality cases.
(b) We know for the put-call-parity that an European call is equivalent to an European put plus a future that have the same strike price and maturity assuming the underlying stock pays no dividends. Write down an explicit portfolio to take advantage of the arbitrage opportunity when ct - pt < St - Kexp(r(T-t). Also, what would the put-call parity be if the stock pays dividend with Dt being the present value of all known dividends paid between now (i.e. time t) and the expiration date. In fact, this also tells us the price of a future contract at time t that expires at a later time T on a stock with price St that paying dividends in a continuous way with annual rate D and the riskless annual rate is r.
Explanation / Answer
IN storage demand and price arbitrage a price p1, storage demand is endogenous because storage will affect the equilibrium .8 Buyer price expectations can equivalently be described directly in terms of P 2 (x), if buyers observe the storage level. The equilibrium definition implies that buyers can make the correct inference observing only the period-1 price.
price in period 2. The resulting price in period 2 must, however, be consistent with the initial choice of buyers to store the good in period 1. We begin with arbitrage restrictions on equilibrium prices. Price arbitrage. We know, that buyers are indifferent about buying in advance when p1 = pe 2. Given this, quantity is indeterminate and optimal storage requires only that x lies between 0 and [1 F(p1/)]. Equilibrium, however, implies a tighter characterization of storage behaviour.
Lemma 3. In equilibrium, if x > 0, then p1 = Pe 2 (p1); further, x < x¯ and q 2 (x) > 0 also hold. Intuitively, Lemma 3 states that whenever buyers choose to store the good in equilibrium, prices leave them indifferent. Effectively, the option to store brings the discounted period-2 price into equality with the current price and eliminates any strictly positive arbitrage gains. The proof of Lemma 3 is also instructive as it implies that, in equilibrium, it can never be the case that all period-2 consumption is bought in advance in period 1. In other words, there must be positive sales in period 2. Storage demand. We can now derive aggregate storage and consumption demand for period 1. Formally, for any given price p1, we must find the aggregate storage quantity QS(p1) that is consistent with an optimal storage choice (6) when buyers hold rational expectations about the consequences of storage for period-2 prices. To begin, recall from above (just after Lemma 2) that stronger period-2 demand (larger ) will increase the period-2 price. Intuitively, we can expect that extremely large or small values for will lead to a trivial storage outcome. To rule out such cases, which are easily dealt with as limiting cases of the analysis, assume that c < pC 2 < 1, where pC 2 P 2 (0) is simply the (static) equilibrium Cornet price for period 2 (i.e., no storage and demand of q = [1 F(p/)] in period 2); implicitly, is in an open interval containing 1. Now consider storage demand when p1 > pC 2 . We claim buyers will not store at such a price. The reason is that when x = 0, we will have a relatively low period-2 price. Formally, P 2 (0) = pC 2 = Pe 2 (p1), and we then have p1 > Pe 2 (p1), which means that waiting to purchase in period 2 is optimal. Any x > 0 would only depress the period-2 price and reinforce the decision to wait. Hence, we have QS(p1) = 0 at any such p1. Now suppose p1 < pC 2 . Then, if x = 0, we have P 2 (0) = pC 2 = Pe 2 (p1). Hence, p1 < Pe 2 (p1), and so all buyers with a period-2 valuation above p1/ have a strict incentive to purchase in period 1 and store the good. Consequently, QS(p1) > 0 at any such price. In equilibrium, the extent of storage must be sufficiently large to pull the period-2 price down to p1/. To find this level of storage, recall that by no-arbitrage (Lemma 3), we have p1 = P 2 (x) whenever x > 0. Then, in order for the aggregate storage behaviour of buyers to be consistent with equilibrium expectations for period 2, we must have QS(p1) x = [P 2 ] 1 (p1/). (8) Recall that P 2 (x) is well defined and invertible (as it is strictly decreasing) for all x x¯. If x > x¯ (i.e., when p1 < c), then residual period-2 demand is such that no firm can profitably produce and sell in period 2. As noted after Lemma 1, this market outcome with no sales is consistent with any period-2 price that is sufficiently high. To have QS(p1) well defined for all p1, it is convenient to adopt the convention that if p1 < c, then [P 2 ] 1(p1/) [1 F(p1/)]. Intuitively, if p1 < c, then all period-2 buyers who value the good at or above p1/ will purchase in period 1, knowing that in period 2 there will be no market transactions. We have thus established.
Lemma 4. In equilibrium, QS(p1) is non increasing and differentiable everywhere (except at p1 = pC 2 ) for all p1 0. Further, QS(p1) = 0 if p1 > pC 2 [P 2 ] 1(p1/) if p1 pC 2 . (9) Aggregate demand in period 1, which is the sum of consumption and storage demand, is then Q1(p1) = 1 F(p1) if p1 > pC 2 1 F(p1)+[P 2 ] 1(p1/) if p1 pC 2 . (10) Since F and [P 2 ] 1 are each differentiable (except at p1 = pC 2 ), the inverse aggregate demand function, P1(q1), is differentiable everywhere except at the point qK 1 F(pC 2 ). Moreover, since P 2 (x) is strictly decreasing, we see that inverse aggregate period-1 demand decreases everywhere and has a kink at the point qK . Intuitively, at prices above pC 2 , buyers do not seek to buy in advance because the period-2 price will be sufficiently low (at pC 2 ). In this case, aggregate period-1 demand consists only of consumption demand for period 1. At prices below pC 2 , however, some buyers purchase their period-2 consumption in advance and store the good. Consequently, at any price below pC 2 , the aggregate period-1 demand lies above the level of period-1 consumption demand. Graphically, this corresponds to a kink (see Figure 1 above) at the price pC 2 and quantity qK where demand rotates outward, becoming “flatter” in the sense that the slope is less negative, once buyers start augmenting their consumption demands with storage demand. Storage by buyers thus provides an equilibrium rationale for a kinked-demand function in oligopoly markets. The economic logic for the kink, however, extends beyond the oligopoly setting. As we will see in Section 5, the period-2 pricing behaviour of a monopoly supplier also generates a kink in period-1 demand, albeit at a higher price. In both of the oligopoly and monopoly settings, period-2 prices are determined via the residual-demand structure given prior storage. Thus, the kink in demand arises when current and future prices are related via the arbitrage condition and the future price is negatively related to the extent of storage. The contrast in storage demand between our model and that in HMN is helpful for understanding the economic forces created by the underlying source of buyer heterogeneity. In our model, period-2 prices fall with the extent of storage (recall Lemma 2) because buyers have different valuations for the good. Thus, whenever prices induce storage, it is necessarily the case that buyers with valuations that are high relative to current prices are the ones who store the good. Buyers from the lower end of the valuation distribution necessarily remain in the market and seek to purchase for consumption in period 2. In contrast, in HMN, heterogeneity takes the form of captives versus shoppers. There, storage implies that shoppers are removed from future consumption demand. Storage then leads to higher prices (a higher mean price in the equilibrium dispersion) in the future period, since each firm has a stronger incentive to price high and extract surplus from their captive buyers. Two further points on heterogeneity now follow directly. First, with valuation heterogeneity, the extent of storage can vary smoothly and it is part of the equilibrium determination. With a common valuation, all buyers tend to boon the same side of the storage decision. Thus, the storage level in HMN coincides with the number of shoppers when storage occurs, and it is only the price dispersion that adjusts in equilibrium. A common property of both models, however, is the presence of an externality among buyers: storage at the aggregate level, as discussed by HMN, influences future prices. Second, valuation heterogeneity leads to a tight arbitrage linkage in prices across periods when storage occurs (recall Lemma 3), for the same reason that the extent of storage demand varies as buyers with different valuations become active.. Suppose F is uniform on [0, 1]. Then (3) reduces to P2(q2+x) = (q2 + x). The corresponding period-2 price is easily verified to be P 2 (x) = (1/3)[ x + 2c]. With p1 = P 2 (x) from Lemma 3, we have, for 0 < x < ( c), p1 = P 2 (x) = 3 [ x + 2c]. (11) Upon rearranging, we have storage demand of QS(p1) x = + 2c 3p1/. (12) Aggregate demand in period 1 is then given by Q1(p1) = 1 p1 if p1 > ( + 2c)/3 1 + + 2c (1 + 3/)p1 if ( + 2c)/3 p1 c 1 + (1 + 1/)p1 if c > p1. (13) The period-1 demand panel in Figure 1 exhibits the main qualitative features of this example.
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