18.175: Lecture 15 .1in Random walks - Mathematicsmath.mit.edu/~sheffield/2016175/Lecture15.pdf ·...

18.175: Lecture 15

Random walks

Scott Sheffield

18.175 Lecture 15

Outline

Risk neutral probability

Random walks

Stopping times

Arcsin law, other SRW stories

18.175 Lecture 15

Outline

Random walks

Stopping times

18.175 Lecture 15

I “Risk neutral probability” is a fancy term for “priceprobability”. (The term “price probability” is arguably moredescriptive.)

I That is, it is a probability measure that you can deduce bylooking at prices.

I For example, suppose somebody is about to shoot a freethrow in basketball. What is the price in the sports bettingworld of a contract that pays one dollar if the shot is made?

I If the answer is .75 dollars, then we say that the risk neutralprobability that the shot will be made is .75.

I Risk neutral probability is the probability determined by themarket betting odds.

18.175 Lecture 15

Risk neutral probability of outcomes known at fixed time T

I Risk neutral probability of event A: PRN(A) denotes

PriceContract paying 1 dollar at time T if A occurs PriceContract paying 1 dollar at time T no matter what

I If risk-free interest rate is constant and equal to r(compounded continuously), then denominator is e−rT .

I Assuming no arbitrage (i.e., no risk free profit with zeroupfront investment), PRN satisfies axioms of probability. Thatis, 0 ≤ PRN(A) ≤ 1, and PRN(S) = 1, and if events Aj aredisjoint then PRN(A1 ∪ A2 ∪ . . .) = PRN(A1) + PRN(A2) + . . .

I Arbitrage example: if A and B are disjoint andPRN(A ∪ B) < P(A) + P(B) then we sell contracts paying 1 ifA occurs and 1 if B occurs, buy contract paying 1 if A ∪ Boccurs, pocket difference.

18.175 Lecture 15

Risk neutral probability differ vs. “ordinary probability”

I At first sight, one might think that PRN(A) describes themarket’s best guess at the probability that A will occur.

I But suppose A is the event that the government is dissolvedand all dollars become worthless. What is PRN(A)?

I Should be 0. Even if people think A is likely, a contractpaying a dollar when A occurs is worthless.

I Now, suppose there are only 2 outcomes: A is event thateconomy booms and everyone prospers and B is event thateconomy sags and everyone is needy. Suppose purchasingpower of dollar is the same in both scenarios. If people thinkA has a .5 chance to occur, do we expect PRN(A) > .5 orPRN(A) < .5?

I Answer: PRN(A) < .5. People are risk averse. In secondscenario they need the money more.

18.175 Lecture 15

Non-systemic event

I Suppose that A is the event that the Boston Red Sox win theWorld Series. Would we expect PRN(A) to represent (themarket’s best assessment of) the probability that the Red Soxwill win?

I Arguably yes. The amount that people in general need orvalue dollars does not depend much on whether A occurs(even though the financial needs of specific individuals maydepend on heavily on A).

I Even if some people bet based on loyalty, emotion, insuranceagainst personal financial exposure to team’s prospects, etc.,there will arguably be enough in-it-for-the-money statisticalarbitrageurs to keep price near a reasonable guess of whatwell-informed informed experts would consider the trueprobability.

18.175 Lecture 15

Non-systemic event

18.175 Lecture 15

Non-systemic event

18.175 Lecture 15

Extensions of risk neutral probability

I Definition of risk neutral probability depends on choice ofcurrency (the so-called numeraire).

I Risk neutral probability can be defined for variable times andvariable interest rates — e.g., one can take the numeraire tobe amount one dollar in a variable-interest-rate money marketaccount has grown to when outcome is known. Can definePRN(A) to be price of contract paying this amount if andwhen A occurs.

I Or, for simplicity, can focus on fixed time T , fixed interestrate r .

18.175 Lecture 15

Prices as expectations

I By assumption, the price of a contract that pays one dollar attime T if A occurs is PRN(A)e−rT .

I If A and B are disjoint, what is the price of a contract thatpays 2 dollars if A occurs, 3 if B occurs, 0 otherwise?

I Answer: (2PRN(A) + 3PRN(B))e−rT .

I Generally, in absence of arbitrage, price of contract that paysX at time T should be ERN(X )e−rT where ERN denotesexpectation with respect to the risk neutral probability.

I Example: if a non-divided paying stock will be worth X attime T , then its price today should be ERN(X )e−rT .

I Aside: So-called fundamental theorem of asset pricingstates that (assuming no arbitrage) interest-discounted assetprices are martingales with respect to risk neutral probability.Current price of stock being ERN(X )e−rT follows from this.

18.175 Lecture 15

Outline

Random walks

Stopping times

18.175 Lecture 15

Outline

Random walks

Stopping times

18.175 Lecture 15

Exchangeable events

I Start with measure space (S ,S, µ). LetΩ = (ω1, ω2, . . .) : ωi ∈ S, let F be product σ-algebra andP the product probability measure.

I Finite permutation of N is one-to-one map from N to itselfthat fixes all but finitely many points.

I Event A ∈ F is permutable if it is invariant under any finitepermutation of the ωi .

I Let E be the σ-field of permutable events.

I This is related to the tail σ-algebra we introduced earlier inthe course. Bigger or smaller?

18.175 Lecture 15

Exchangeable events

18.175 Lecture 15

Exchangeable events

18.175 Lecture 15

Exchangeable events

18.175 Lecture 15

Exchangeable events

18.175 Lecture 15

Hewitt-Savage 0-1 law

I If X1,X2, . . . are i.i.d. and A ∈ A then P(A) ∈ 0, 1.

I Idea of proof: Try to show A is independent of itself, i.e.,that P(A) = P(A ∩ A) = P(A)P(A). Start with measuretheoretic fact that we can approximate A by a set An inσ-algebra generated by X1, . . .Xn, so that symmetricdifference of A and An has very small probability. Note thatAn is independent of event A′n that An holds when X1, . . . ,Xn

and Xn1 , . . . ,X2n are swapped. Symmetric difference betweenA and A′n is also small, so A is independent of itself up to thissmall error. Then make error arbitrarily small.

18.175 Lecture 15

Hewitt-Savage 0-1 law

I If X1,X2, . . . are i.i.d. and A ∈ A then P(A) ∈ 0, 1.I Idea of proof: Try to show A is independent of itself, i.e.,

that P(A) = P(A ∩ A) = P(A)P(A). Start with measuretheoretic fact that we can approximate A by a set An inσ-algebra generated by X1, . . .Xn, so that symmetricdifference of A and An has very small probability. Note thatAn is independent of event A′n that An holds when X1, . . . ,Xn

and Xn1 , . . . ,X2n are swapped. Symmetric difference betweenA and A′n is also small, so A is independent of itself up to thissmall error. Then make error arbitrarily small.

18.175 Lecture 15

Application of Hewitt-Savage:

I If Xi are i.i.d. in Rn then Sn =∑n

i=1 Xi is a random walk onRn.

I Theorem: if Sn is a random walk on R then one of thefollowing occurs with probability one:

I Sn = 0 for all nI Sn →∞I Sn → −∞I −∞ = lim inf Sn < lim supSn =∞

I Idea of proof: Hewitt-Savage implies the lim supSn andlim inf Sn are almost sure constants in [−∞,∞]. Note that ifX1 is not a.s. constant, then both values would depend on X1

if they were not in ±∞

18.175 Lecture 15

I Sn = 0 for all n

I Sn →∞I Sn → −∞I −∞ = lim inf Sn < lim supSn =∞

18.175 Lecture 15

I Sn = 0 for all nI Sn →∞

I Sn → −∞I −∞ = lim inf Sn < lim supSn =∞

18.175 Lecture 15

I Sn = 0 for all nI Sn →∞I Sn → −∞

I −∞ = lim inf Sn < lim supSn =∞I Idea of proof: Hewitt-Savage implies the lim supSn and

lim inf Sn are almost sure constants in [−∞,∞]. Note that ifX1 is not a.s. constant, then both values would depend on X1

18.175 Lecture 15

Outline

Random walks

Stopping times

18.175 Lecture 15

Outline

Random walks

Stopping times

18.175 Lecture 15

Stopping time definition

I Say that T is a stopping time if the event that T = n is inFn for i ≤ n.

I In finance applications, T might be the time one sells a stock.Then this states that the decision to sell at time n dependsonly on prices up to time n, not on (as yet unknown) futureprices.

18.175 Lecture 15

Stopping time definition

I Say that T is a stopping time if the event that T = n is inFn for i ≤ n.

I In finance applications, T might be the time one sells a stock.Then this states that the decision to sell at time n dependsonly on prices up to time n, not on (as yet unknown) futureprices.

18.175 Lecture 15

Stopping time examples

I Let A1, . . . be i.i.d. random variables equal to −1 withprobability .5 and 1 with probability .5 and let X0 = 0 andXn =

∑ni=1 Ai for n ≥ 0.

I Which of the following is a stopping time?

1. The smallest T for which |XT | = 502. The smallest T for which XT ∈ −10, 1003. The smallest T for which XT = 0.4. The T at which the Xn sequence achieves the value 17 for the

9th time.5. The value of T ∈ 0, 1, 2, . . . , 100 for which XT is largest.6. The largest T ∈ 0, 1, 2, . . . , 100 for which XT = 0.

I Answer: first four, not last two.

18.175 Lecture 15

Stopping time theorems

I Theorem: Let X1,X2, . . . be i.i.d. and N a stopping time withN <∞.

I Conditioned on stopping time N <∞, conditional law ofXN+n, n ≥ 1 is independent of Fn and has same law asoriginal sequence.

I Wald’s equation: Let Xi be i.i.d. with E |Xi | <∞. If N is astopping time with EN <∞ then ESN = EX1EN.

I Wald’s second equation: Let Xi be i.i.d. with E |Xi | = 0 andEX 2

i = σ2 <∞. If N is a stopping time with EN <∞ thenESN = σ2EN.

18.175 Lecture 15

Wald applications to SRW

I S0 = a ∈ Z and at each time step Sj independently changesby ±1 according to a fair coin toss. Fix A ∈ Z and letN = infk : Sk ∈ 0,A. What is ESN?

I What is EN?

18.175 Lecture 15

Wald applications to SRW

I S0 = a ∈ Z and at each time step Sj independently changesby ±1 according to a fair coin toss. Fix A ∈ Z and letN = infk : Sk ∈ 0,A. What is ESN?

I What is EN?

18.175 Lecture 15

Outline

Random walks

Stopping times

18.175 Lecture 15

Outline

Random walks

Stopping times

18.175 Lecture 15

Reflection principle

I How many walks from (0, x) to (n, y) that don’t cross thehorizontal axis?

I Try counting walks that do cross by giving bijection to walksfrom (0,−x) to (n, y).

18.175 Lecture 15

Reflection principle

I How many walks from (0, x) to (n, y) that don’t cross thehorizontal axis?

I Try counting walks that do cross by giving bijection to walksfrom (0,−x) to (n, y).

18.175 Lecture 15

Ballot Theorem

I Suppose that in election candidate A gets α votes and B getsβ < α votes. What’s probability that A is a head throughoutthe counting?

I Answer: (α− β)/(α + β). Can be proved using reflectionprinciple.

18.175 Lecture 15

Ballot Theorem

I Suppose that in election candidate A gets α votes and B getsβ < α votes. What’s probability that A is a head throughoutthe counting?

I Answer: (α− β)/(α + β). Can be proved using reflectionprinciple.

18.175 Lecture 15

Arcsin theorem

I Theorem for last hitting time.

I Theorem for amount of positive positive time.

18.175 Lecture 15

Arcsin theorem

I Theorem for last hitting time.

I Theorem for amount of positive positive time.

18.175 Lecture 15

18.175: Lecture 15 .1in Random walks - Mathematicsmath.mit.edu/~sheffield/2016175/Lecture15.pdf ·...

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