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In Finance, We Need . . . How to Make . . . From a Theoretical . . . How to Make . . . What If We Only Have Stochastic . . . What if the Stochastic . . . Approximate Stochastic Additional Reasonable . . . Dominance? The Assumption 0 < .

  1. In Finance, We Need . . . How to Make . . . From a Theoretical . . . How to Make . . . What If We Only Have Stochastic . . . What if the Stochastic . . . Approximate Stochastic Additional Reasonable . . . Dominance? The Assumption 0 < . . . How To Make . . . Vladik Kreinovich 1 , Hung T. Nguyen 2 , 3 , and, Home Page Songsak Sriboonchitta 3 Title Page 1 Department of Computer Science, University of Texas at El Paso ◭◭ ◮◮ El Paso, TX 79968, USA, vladik@utep.edu 2 Department of Mathematical Sciences, New Mexico State University ◭ ◮ Las Cruces, New Mexico 88003, USA, hunguyen@nmsu.edu 3 Faculty of Economics, Chiang Mai University Page 1 of 15 Chiang Mai, Thailand, songsak@econ.chiangmai.ac.th Go Back Full Screen Close Quit

  2. In Finance, We Need . . . How to Make . . . 1. Outline From a Theoretical . . . • In many practical situations, How to Make . . . Stochastic . . . – we need to select one of the two alternatives, and What if the Stochastic . . . – we do not know the exact form of the user’s utility Additional Reasonable . . . function – e.g., we only know that it is increasing. The Assumption 0 < . . . • Stochastic dominance result: if for cdfs, F 1 ( x ) ≤ F 2 ( x ) How To Make . . . for all x , then the 1st alternative is better. Home Page • This criterion works well in many practical situations. Title Page ◭◭ ◮◮ • However, often, F 1 ( x ) ≤ F 2 ( x ) for most x , but not for all x . ◭ ◮ • In this talk, we show that in such situations: Page 2 of 15 – if the set { x : F 1 ( x ) > F 2 ( x ) } is sufficiently small, Go Back – then the 1st alternative is still provably better. Full Screen Close Quit

  3. In Finance, We Need . . . How to Make . . . 2. In Finance, We Need to Make Decisions Under From a Theoretical . . . Uncertainty How to Make . . . • In financial decision making, we need to select one of Stochastic . . . the possible decisions. What if the Stochastic . . . Additional Reasonable . . . • For example, we need to decide whether we sell or buy The Assumption 0 < . . . a given financial instrument (share, option, etc.). How To Make . . . • Ideally, we should select a decision which leaves us with Home Page the largest monetary value x . Title Page • However, in practice, we cannot predict exactly the ◭◭ ◮◮ monetary consequences of each action. ◭ ◮ • Because of the changing external circumstances, the Page 3 of 15 same decision can lead to gains or to losses. Go Back • Thus, we need to make a decision in a situation when we do not know the exact consequences of each action. Full Screen Close Quit

  4. In Finance, We Need . . . How to Make . . . 3. In Finance, We Usually Have Probabilistic Un- From a Theoretical . . . certainty How to Make . . . • Numerous financial transactions are occurring every Stochastic . . . moment. What if the Stochastic . . . Additional Reasonable . . . • For the past transactions, we know the monetary con- The Assumption 0 < . . . sequences of different decisions. How To Make . . . • By analyzing past transactions, we can estimate, for Home Page each decision, the frequencies of different outcomes x . Title Page • Since the sample size is large, the corresponding fre- ◭◭ ◮◮ quencies become very close to the actual probabilities. ◭ ◮ • Thus, in fact, we can estimate the probabilities of dif- Page 4 of 15 ferent values x . Go Back • Comment: this is not true for new, untested financial instruments, but it is true in most cases. Full Screen Close Quit

  5. In Finance, We Need . . . How to Make . . . 4. How to Describe the Corresponding Probabil- From a Theoretical . . . ities How to Make . . . • As usual, the corresponding probabilities can be de- Stochastic . . . scribed: What if the Stochastic . . . Additional Reasonable . . . – either by the probability density function (pdf) f ( x ) The Assumption 0 < . . . – or by the cumulative distribution function (cdf) How To Make . . . def F ( t ) = Prob( x ≤ t ). Home Page • If we know the pdf f ( x ), then we can reconstruct the Title Page � t cdf as F ( t ) = −∞ f ( x ) dx . ◭◭ ◮◮ • Vice versa, if we know the cdf F ( t ), we can reconstruct ◭ ◮ the pdf as its derivative f ( x ) = F ′ ( x ). Page 5 of 15 Go Back Full Screen Close Quit

  6. In Finance, We Need . . . How to Make . . . 5. How to Make Decisions Under Probabilistic Un- From a Theoretical . . . certainty: A Theoretical Recommendation How to Make . . . • Let us assume that we have several possible decisions Stochastic . . . whose outcomes are characterized by the pdfs f i ( x ). What if the Stochastic . . . Additional Reasonable . . . • Decisions of a rational person can be characterized by The Assumption 0 < . . . a function u ( x ) called utility function . How To Make . . . • Namely, a rational person should select a decision with Home Page � the largest f i ( x ) · u ( x ) dx . Title Page • A decision corresponding to pdf f 1 ( x ) is preferable to ◭◭ ◮◮ the decision corresponding to pdf f 2 ( x ) if ◭ ◮ � � f 1 ( x ) · u ( x ) dx > f 2 ( x ) · u ( x ) dx. Page 6 of 15 Go Back � • This condition is equivalent to ∆ f ( x ) · u ( x ) dx > 0 , Full Screen def where ∆ f ( x ) = f 1 ( x ) − f 2 ( x ). Close Quit

  7. In Finance, We Need . . . How to Make . . . 6. From a Theoretical Recommendation to Prac- From a Theoretical . . . tical Decision How to Make . . . • Theoretically, we can determine the utility function of Stochastic . . . the decision maker. What if the Stochastic . . . Additional Reasonable . . . • However, since a determination is very time-consuming. The Assumption 0 < . . . • So, it is rarely done in real financial situations. How To Make . . . Home Page • As a result, in practice, we only have a partial infor- mation about the utility function. Title Page • One thing we know for sure if that: ◭◭ ◮◮ – the larger the monetary gain x , ◭ ◮ – the better the resulting situation. Page 7 of 15 • So, the utility function u ( x ) is increasing. Go Back Full Screen Close Quit

  8. In Finance, We Need . . . How to Make . . . 7. How to Make Decisions When We Only Know From a Theoretical . . . that Utility Function Is Increasing How to Make . . . def � • The first decision is better if I = ∆ f ( x ) · u ( x ) dx ≥ 0, Stochastic . . . where ∆ f ( x ) = f 1 ( x ) − f 2 ( x ). What if the Stochastic . . . Additional Reasonable . . . • When is the integral I non-negative? The Assumption 0 < . . . • In reality, both gains and losses are bounded by some How To Make . . . � T � value T , so ∆ f ( x ) · u ( x ) dx = − T ∆ f ( x ) · u ( x ) dx. Home Page � ∆ F ( x ) · u ′ ( x ), • Integrating by parts, we get I = − Title Page def where ∆ F ( x ) = F 1 ( x ) − F 2 ( x ). ◭◭ ◮◮ • Here, u ′ ( x ) ≥ 0. ◭ ◮ • Thus, if ∆ F ( x ) ≤ 0 for all x , i.e., if F 1 ( x ) ≤ F 2 ( x ) for Page 8 of 15 all x , then I ≥ 0 and so, the 1st decision is better. Go Back • This is the main idea behind stochastic dominance . Full Screen Close Quit

  9. In Finance, We Need . . . How to Make . . . 8. Stochastic Dominance: Discussion From a Theoretical . . . • We showed: the condition F 1 ( x ) ≤ F 2 ( x ) for all x is How to Make . . . sufficient to conclude that the first alternative is better. Stochastic . . . What if the Stochastic . . . • This condition is also necessary: if F 1 ( x 0 ) > F 2 ( x 0 ) for Additional Reasonable . . . some x 0 , then the 2nd decision is better for some u ( x ). The Assumption 0 < . . . • Sometimes, we have additional information about the How To Make . . . utility function. Home Page • E.g., the same amount of extra money h is more valu- Title Page able for a poor person than for the rich person: ◭◭ ◮◮ if x < y then u ( x + h ) − u ( x ) ≥ u ( y + h ) − u ( y ) . ◭ ◮ Page 9 of 15 • This condition is equivalent to convexity of u ( x ). Go Back • For convex u ( x ), we can perform one more integration by parts and get an even more powerful criterion. Full Screen Close Quit

  10. In Finance, We Need . . . How to Make . . . 9. What if the Stochastic Dominance Condition From a Theoretical . . . Is Satisfied “Almost Always” How to Make . . . • Let us return to the simple situation when we only Stochastic . . . know that utility is increasing, i.e., that u ′ ( x ) ≥ 0. What if the Stochastic . . . Additional Reasonable . . . • In this case, if we know that F 1 ( x ) ≤ F 2 ( x ) for all x , The Assumption 0 < . . . then the first alternative is better. How To Make . . . • In many cases, we can use this criterion. Home Page • Sometimes, the inequality F 1 ( x ) ≤ F 2 ( x ) holds for Title Page most values x – except for some small interval. ◭◭ ◮◮ • It would be nice to be able to make decisions even if ◭ ◮ we have approximate stochastic dominance. Page 10 of 15 • We show that, under reasonable assumptions, Go Back – we can make definite decisions Full Screen – even under such approximate stochastic dominance. Close Quit

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