# Probability Theory on Coin Toss Space

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1 Probability Theory on Coin Toss Space 1 Finite Probability Spaces 2 Random Variables, Distributions, and Expectations 3 Conditional Expectations

2 Probability Theory on Coin Toss Space 1 Finite Probability Spaces 2 Random Variables, Distributions, and Expectations 3 Conditional Expectations

3 Inspiration A finite probability space is used to model the phenomena in which there are only finitely many possible outcomes Let us discuss the binomial model we have studied so far through a very simple example Suppose that we toss a coin 3 times; the set of all possible outcomes can be written as Ω = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT } Assume that the probability of a head is p and the probability of a tail is q = 1 p Assuming that the tosses are independent the probabilities of the elements ω = ω 1 ω 2 ω 3 of Ω are P[HHH] = p 3, P[HHT ] = P[HTH] = P[THH] = p 2 q, P[TTT ] = q 3, P[HTT ] = P[THT ] = P[TTH] = pq 2

4 Inspiration A finite probability space is used to model the phenomena in which there are only finitely many possible outcomes Let us discuss the binomial model we have studied so far through a very simple example Suppose that we toss a coin 3 times; the set of all possible outcomes can be written as Ω = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT } Assume that the probability of a head is p and the probability of a tail is q = 1 p Assuming that the tosses are independent the probabilities of the elements ω = ω 1 ω 2 ω 3 of Ω are P[HHH] = p 3, P[HHT ] = P[HTH] = P[THH] = p 2 q, P[TTT ] = q 3, P[HTT ] = P[THT ] = P[TTH] = pq 2

5 Inspiration A finite probability space is used to model the phenomena in which there are only finitely many possible outcomes Let us discuss the binomial model we have studied so far through a very simple example Suppose that we toss a coin 3 times; the set of all possible outcomes can be written as Ω = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT } Assume that the probability of a head is p and the probability of a tail is q = 1 p Assuming that the tosses are independent the probabilities of the elements ω = ω 1 ω 2 ω 3 of Ω are P[HHH] = p 3, P[HHT ] = P[HTH] = P[THH] = p 2 q, P[TTT ] = q 3, P[HTT ] = P[THT ] = P[TTH] = pq 2

6 Inspiration A finite probability space is used to model the phenomena in which there are only finitely many possible outcomes Let us discuss the binomial model we have studied so far through a very simple example Suppose that we toss a coin 3 times; the set of all possible outcomes can be written as Ω = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT } Assume that the probability of a head is p and the probability of a tail is q = 1 p Assuming that the tosses are independent the probabilities of the elements ω = ω 1 ω 2 ω 3 of Ω are P[HHH] = p 3, P[HHT ] = P[HTH] = P[THH] = p 2 q, P[TTT ] = q 3, P[HTT ] = P[THT ] = P[TTH] = pq 2

7 Inspiration A finite probability space is used to model the phenomena in which there are only finitely many possible outcomes Let us discuss the binomial model we have studied so far through a very simple example Suppose that we toss a coin 3 times; the set of all possible outcomes can be written as Ω = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT } Assume that the probability of a head is p and the probability of a tail is q = 1 p Assuming that the tosses are independent the probabilities of the elements ω = ω 1 ω 2 ω 3 of Ω are P[HHH] = p 3, P[HHT ] = P[HTH] = P[THH] = p 2 q, P[TTT ] = q 3, P[HTT ] = P[THT ] = P[TTH] = pq 2

8 An Example (cont d) The subsets of Ω are called events, e.g., The first toss is a head = {ω Ω : ω 1 = H} = {HHH, HTH, HTT } The probability of an event is then P[ The first toss is a head ] = P[HHH] + P[HTH] + P[HTT ] = p The final answer agrees with our intuition - which is good

9 An Example (cont d) The subsets of Ω are called events, e.g., The first toss is a head = {ω Ω : ω 1 = H} = {HHH, HTH, HTT } The probability of an event is then P[ The first toss is a head ] = P[HHH] + P[HTH] + P[HTT ] = p The final answer agrees with our intuition - which is good

10 An Example (cont d) The subsets of Ω are called events, e.g., The first toss is a head = {ω Ω : ω 1 = H} = {HHH, HTH, HTT } The probability of an event is then P[ The first toss is a head ] = P[HHH] + P[HTH] + P[HTT ] = p The final answer agrees with our intuition - which is good

11 Definitions A finite probability space consists of a sample space Ω and a probability measure P. The sample space Ω is a nonempty finite set and the probability measure P is a function which assigns to each element ω in Ω a number in [0, 1] so that P[ω] = 1. ω Ω An event is a subset of Ω. We define the probability of an event A as Note: and if A B = P[A] = ω A P[ω] P[Ω] = 1 P[A B] = P[A] + P[B]

12 Definitions A finite probability space consists of a sample space Ω and a probability measure P. The sample space Ω is a nonempty finite set and the probability measure P is a function which assigns to each element ω in Ω a number in [0, 1] so that P[ω] = 1. ω Ω An event is a subset of Ω. We define the probability of an event A as Note: and if A B = P[A] = ω A P[ω] P[Ω] = 1 P[A B] = P[A] + P[B]

13 Definitions A finite probability space consists of a sample space Ω and a probability measure P. The sample space Ω is a nonempty finite set and the probability measure P is a function which assigns to each element ω in Ω a number in [0, 1] so that P[ω] = 1. ω Ω An event is a subset of Ω. We define the probability of an event A as Note: and if A B = P[A] = ω A P[ω] P[Ω] = 1 P[A B] = P[A] + P[B]

14 Probability Theory on Coin Toss Space 1 Finite Probability Spaces 2 Random Variables, Distributions, and Expectations 3 Conditional Expectations

15 Random variables Definition. A random variable is a real-valued function defined on Ω Example (Stock prices) Let the sample space Ω be the one corresponding to the three coin tosses. We define the stock prices on days 0, 1, 2 as follows: S 0 (ω 1 ω 2 ω 3 ) = 4 for all ω 1 ω 2 ω 3 Ω { 8 for ω 1 = H S 1 (ω 1 ω 2 ω 3 ) = 2 for ω 1 = T 16 for ω 1 = ω 2 = H S 2 (ω 1 ω 2 ω 3 ) = 4 for ω 1 ω 2 1 for ω 1 = ω 2 = H

16 Random variables Definition. A random variable is a real-valued function defined on Ω Example (Stock prices) Let the sample space Ω be the one corresponding to the three coin tosses. We define the stock prices on days 0, 1, 2 as follows: S 0 (ω 1 ω 2 ω 3 ) = 4 for all ω 1 ω 2 ω 3 Ω { 8 for ω 1 = H S 1 (ω 1 ω 2 ω 3 ) = 2 for ω 1 = T 16 for ω 1 = ω 2 = H S 2 (ω 1 ω 2 ω 3 ) = 4 for ω 1 ω 2 1 for ω 1 = ω 2 = H

17 Distributions The distribution of a random variable is a specification of the probabilities that the random variable takes various values. Following up on the previous example, we have P[S 2 = 16] = P{ω Ω : S 2 (ω) = 16} = P{ω = ω 1 ω 2 ω 3 Ω : ω 1 = ω 2 } = P[HHH] + P[HHT ] = p 2 Is is customary to write the distribution of a random variable on a finite probability space as a table of probabilities that the random variable takes various values.

18 Distributions The distribution of a random variable is a specification of the probabilities that the random variable takes various values. Following up on the previous example, we have P[S 2 = 16] = P{ω Ω : S 2 (ω) = 16} = P{ω = ω 1 ω 2 ω 3 Ω : ω 1 = ω 2 } = P[HHH] + P[HHT ] = p 2 Is is customary to write the distribution of a random variable on a finite probability space as a table of probabilities that the random variable takes various values.

19 Distributions The distribution of a random variable is a specification of the probabilities that the random variable takes various values. Following up on the previous example, we have P[S 2 = 16] = P{ω Ω : S 2 (ω) = 16} = P{ω = ω 1 ω 2 ω 3 Ω : ω 1 = ω 2 } = P[HHH] + P[HHT ] = p 2 Is is customary to write the distribution of a random variable on a finite probability space as a table of probabilities that the random variable takes various values.

20 Expectations Let a random variable X be defined on a finite probability space (Ω, P). The expectation (or expected value) of X is defined as E[X ] = ω Ω X (ω)p[ω] The variance of X is Var[X ] = E[(X E[X ]) 2 ] Note: The expectation is linear, i.e., if X and Y are random variables on the same probability space and c and d are constants, then E[cX + dy ] = ce[x ] + de[y ]

21 Expectations Let a random variable X be defined on a finite probability space (Ω, P). The expectation (or expected value) of X is defined as E[X ] = ω Ω X (ω)p[ω] The variance of X is Var[X ] = E[(X E[X ]) 2 ] Note: The expectation is linear, i.e., if X and Y are random variables on the same probability space and c and d are constants, then E[cX + dy ] = ce[x ] + de[y ]

22 Expectations Let a random variable X be defined on a finite probability space (Ω, P). The expectation (or expected value) of X is defined as E[X ] = ω Ω X (ω)p[ω] The variance of X is Var[X ] = E[(X E[X ]) 2 ] Note: The expectation is linear, i.e., if X and Y are random variables on the same probability space and c and d are constants, then E[cX + dy ] = ce[x ] + de[y ]

23 Probability Theory on Coin Toss Space 1 Finite Probability Spaces 2 Random Variables, Distributions, and Expectations 3 Conditional Expectations

24 We call E n[s n+1 ] the conditional expectation of S n+1 based on the information known at time n Back to the binomial pricing model The risk neutral probabilities were chosen as p = e(r δ)h d, q = 1 p u d Thus, at any time n and for any sequences of coin tosses (i.e., paths of the stock price) ω = ω 1 ω 2... ω n, we have that S n (ω) = e rh [p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T )] In words, the stock price at time n is the discounted weighted average of the two possible stock prices at time n + 1, where p and q are the weights used in averaging Define E n[s n+1 ](ω 1... ω n ) = p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T ) Then, we can write S n = e rh E n[s n+1 ]

25 We call E n[s n+1 ] the conditional expectation of S n+1 based on the information known at time n Back to the binomial pricing model The risk neutral probabilities were chosen as p = e(r δ)h d, q = 1 p u d Thus, at any time n and for any sequences of coin tosses (i.e., paths of the stock price) ω = ω 1 ω 2... ω n, we have that S n (ω) = e rh [p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T )] In words, the stock price at time n is the discounted weighted average of the two possible stock prices at time n + 1, where p and q are the weights used in averaging Define E n[s n+1 ](ω 1... ω n ) = p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T ) Then, we can write S n = e rh E n[s n+1 ]

26 We call E n[s n+1 ] the conditional expectation of S n+1 based on the information known at time n Back to the binomial pricing model The risk neutral probabilities were chosen as p = e(r δ)h d, q = 1 p u d Thus, at any time n and for any sequences of coin tosses (i.e., paths of the stock price) ω = ω 1 ω 2... ω n, we have that S n (ω) = e rh [p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T )] In words, the stock price at time n is the discounted weighted average of the two possible stock prices at time n + 1, where p and q are the weights used in averaging Define E n[s n+1 ](ω 1... ω n ) = p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T ) Then, we can write S n = e rh E n[s n+1 ]

27 We call E n[s n+1 ] the conditional expectation of S n+1 based on the information known at time n Back to the binomial pricing model The risk neutral probabilities were chosen as p = e(r δ)h d, q = 1 p u d Thus, at any time n and for any sequences of coin tosses (i.e., paths of the stock price) ω = ω 1 ω 2... ω n, we have that S n (ω) = e rh [p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T )] In words, the stock price at time n is the discounted weighted average of the two possible stock prices at time n + 1, where p and q are the weights used in averaging Define E n[s n+1 ](ω 1... ω n ) = p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T ) Then, we can write S n = e rh E n[s n+1 ]

28 We call E n[s n+1 ] the conditional expectation of S n+1 based on the information known at time n Back to the binomial pricing model The risk neutral probabilities were chosen as p = e(r δ)h d, q = 1 p u d Thus, at any time n and for any sequences of coin tosses (i.e., paths of the stock price) ω = ω 1 ω 2... ω n, we have that S n (ω) = e rh [p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T )] In words, the stock price at time n is the discounted weighted average of the two possible stock prices at time n + 1, where p and q are the weights used in averaging Define E n[s n+1 ](ω 1... ω n ) = p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T ) Then, we can write S n = e rh E n[s n+1 ]

29 We call E n[s n+1 ] the conditional expectation of S n+1 based on the information known at time n Back to the binomial pricing model The risk neutral probabilities were chosen as p = e(r δ)h d, q = 1 p u d Thus, at any time n and for any sequences of coin tosses (i.e., paths of the stock price) ω = ω 1 ω 2... ω n, we have that S n (ω) = e rh [p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T )] In words, the stock price at time n is the discounted weighted average of the two possible stock prices at time n + 1, where p and q are the weights used in averaging Define E n[s n+1 ](ω 1... ω n ) = p S n+1 (ω 1... ω n H) + q S n+1 (ω 1... ω n T ) Then, we can write S n = e rh E n[s n+1 ]

30 The Definition Let 1 n N and let ω 1,... ω n be given and temporarily fixed. Denote by χ(ω n+1... ω N ) the number of heads in the continuation ω n+1... ω N and by τ(ω n+1... ω N ) the number of tails in the continuation ω n+1... ω N We define E n[x ](ω 1... ω n ) = ω n+1...ω N (p ) χ(ωn+1...ωn ) (q ) τ(ωn+1...ωn ) X (ω 1... ω N ) and call E n[x ] the conditional expectation of X based on the information at time n

31 The Definition Let 1 n N and let ω 1,... ω n be given and temporarily fixed. Denote by χ(ω n+1... ω N ) the number of heads in the continuation ω n+1... ω N and by τ(ω n+1... ω N ) the number of tails in the continuation ω n+1... ω N We define E n[x ](ω 1... ω n ) = ω n+1...ω N (p ) χ(ωn+1...ωn ) (q ) τ(ωn+1...ωn ) X (ω 1... ω N ) and call E n[x ] the conditional expectation of X based on the information at time n

32 Properties E 0[X ] = E [X ], E N[X ] = X Linearity: E n [cx + dy ] = ce n [X ] + de n [Y ] Taking out what is known: If X actually depends on the first n coin tosses only, then E n [XY ] = X E n [Y ] Iterated conditioning: If 0 n m N, then E n [E m [X ]] = E n [X ] Independence: If X depends only on tosses n + 1 through N, then E n [X ] = X

33 Properties E 0[X ] = E [X ], E N[X ] = X Linearity: E n [cx + dy ] = ce n [X ] + de n [Y ] Taking out what is known: If X actually depends on the first n coin tosses only, then E n [XY ] = X E n [Y ] Iterated conditioning: If 0 n m N, then E n [E m [X ]] = E n [X ] Independence: If X depends only on tosses n + 1 through N, then E n [X ] = X

34 Properties E 0[X ] = E [X ], E N[X ] = X Linearity: E n [cx + dy ] = ce n [X ] + de n [Y ] Taking out what is known: If X actually depends on the first n coin tosses only, then E n [XY ] = X E n [Y ] Iterated conditioning: If 0 n m N, then E n [E m [X ]] = E n [X ] Independence: If X depends only on tosses n + 1 through N, then E n [X ] = X

35 Properties E 0[X ] = E [X ], E N[X ] = X Linearity: E n [cx + dy ] = ce n [X ] + de n [Y ] Taking out what is known: If X actually depends on the first n coin tosses only, then E n [XY ] = X E n [Y ] Iterated conditioning: If 0 n m N, then E n [E m [X ]] = E n [X ] Independence: If X depends only on tosses n + 1 through N, then E n [X ] = X

36 Properties E 0[X ] = E [X ], E N[X ] = X Linearity: E n [cx + dy ] = ce n [X ] + de n [Y ] Taking out what is known: If X actually depends on the first n coin tosses only, then E n [XY ] = X E n [Y ] Iterated conditioning: If 0 n m N, then E n [E m [X ]] = E n [X ] Independence: If X depends only on tosses n + 1 through N, then E n [X ] = X

37 An illustration of the independence property In the same example that we have looked at so far, assume that the actual probability that the stock price rises in any given period equals p = 2/3 and consider E 1 [S 2 /S 1 ] (H) = 2 3 S2(HH) S 1 (H) S2(HT ) S 1 (H) = = 3 2 E 1 [S 2 /S 1 ] (T ) = 2 3 S2(TH) S 1 (T ) S2(TT ) S 1 (T ) = = 3 2 We conclude that E 1 [s 2 /S 1 ] does not depend on the first coin toss - it is not random at all

38 An illustration of the independence property In the same example that we have looked at so far, assume that the actual probability that the stock price rises in any given period equals p = 2/3 and consider E 1 [S 2 /S 1 ] (H) = 2 3 S2(HH) S 1 (H) S2(HT ) S 1 (H) = = 3 2 E 1 [S 2 /S 1 ] (T ) = 2 3 S2(TH) S 1 (T ) S2(TT ) S 1 (T ) = = 3 2 We conclude that E 1 [s 2 /S 1 ] does not depend on the first coin toss - it is not random at all

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