3 Stock under the risk-neutral measure

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1 3 Stock under the risk-neutral measure 3 Adapted processes We have seen that the sampling space Ω = {H, T } N underlies the N-period binomial model for the stock-price process Elementary event ω = ω ω N Ω is interpreted as complete information of the development of the stock over the time N Likewise, ω ω n is seen as partial information about the stock available by time n By time n investor has observed n up or down moves of the stock, encoded into coin-tosses ω ω n The stock price at time n depends only on already known outcomes ω ω n, but not on forthcoming coin-tosses ω n+ ω N, namely S n (ω ω n ) = u #heads(ω ω n) d #tails(ω ω n) S 0, () where the time-0 price S 0 is a given constant A sequence of random variables Y 0, Y,, Y N on sampling space Ω is a random process with discrete time variable n = 0,,, N Definition 3 We say that the process is adapted if Y n only depends on n first coin tosses, ie Y n = Y n (ω ω n ) for every n = 0,,, N In particular, Y 0 is constant (ie not random) Example 32 Let I n = ± depending on whether ω j = H or T The process Y n = I + + I n (with, say Y 0 = 0) is adapted Example 33 The stock process S 0,, S N is adapted The delta process defined by equation (9) in Section 2 is adapted Note that the path S (ω ),, S n (ω ω n ) contains the same information as ω ω n, because S j (ω ω j )/S j (ω ω j ) = u ω j = H (hence S j /S j = d ω j = T ); so if we know S,, S n we can recover ω ω n and vice versa (using ()) In view of this Proposition 34 A process is adapted if and only if the value assumed by Y n can be determined from the path S,, S n, for every n N Example 35 The process X n = S n S n is adapted, while Y n = S N S n not Exercise 36 Conclude from equation (7) in Section 2 that the portfolio-worth process X n is adapted 32 Conditional expectations When a probability (aka probability measure) P on Ω is given, we denote E the associated expectation This P could be a market measure, with given probabilities p, q for H, T and independent coin-tosses, or a more general probability measure For risk-neutral probability P(ω) = p #heads(ω) q #tails(ω) (2)

2 the expectation will be denoted Ẽ For instance, for the random variable X(ω) = #heads(ω) N ẼX = j P(X N ( ) N = j) = j p j q N j = N p, j j=0 which is the expectation of a Binomial(N, p) random variable For random variable X, the expectation EX is a number, obtained by averaging over all possible ω Ω: EX = ω Ω P(ω)X(ω) Very often we need to compute a conditional expectation with fixed ω ω n for some n < N: this can be regarded as an estimate or predicted value of X based on the information available at time n Definition 37 Let (Ω, P) be the coin-tossing space with probability measure P, such that P(ω) > 0 for every ω Ω Let X be a random variable For fixed n, a conditional expectation given the information at time n is a random variable, which depends only on ω ω n, and is obtained as j=0 E n [X](ω ω n ) = X(ω ω n ω n+ ω N ) P(ω ω n ω n+ ω N ), P(A ω n+ ω N {H,T } N n ω ω n ) where A ω ω n = {ω ω N Ω : ω = ω,, ω n = ω n }, so P(A ω ω n ) is the probability that n first coin tosses are some fixed outcomes ω ω n In particular, for the risk-neutral probability (2) Ẽ n [X](ω ω n ) = X(ω ω n ω n+ ω N ) p #heads(ωnωn ) q #tails(ωnωn ) ω n+ ω N {H,T } N n The first part of this definition does not assume that the coin tosses under P are independent The value P(ω ω n ω n+ ω N ) P(A ω ω n ) is the conditional probability of ω ω N given the first n tosses ω ω n (Compare with the definition of the conditional probability P(A B) = P(A B)/P(B)) As is common for random variables, we often write E n [X], Ẽn[X] without indicating explicitly the first n coin tosses ω ω n We have seen that under the RN measure but this is the same as S n (ω ω n ) = r + [ ps n+(ω ω n H) + qs n+ (ω ω n T )], (3) S n = r + Ẽn[S n+ ] (4) 2

3 Exercise 38 Give a detailed proof to (4) using (2), (3) and the fact that S 0,, S N is an adapted process The unconditional expectation E may be seen as a special case, E = E 0 The conditional expectation has the property of linearity: E n [ax+by ] = ae n X+bE n Y (for constants a, b), and E n = Other important properties are as follows (i) If X depends on ω ω n only, then E n [XY ] = X E n [Y ] In particular, for such X we have E n X = X (ii) Iterated conditioning or tower property : for 0 n m N E n [E m [X]] = E n X (iii) Under the risk-neutral P, if X is independent of the n first coin tosses, then Ẽ n X = ẼX (conditional expectation is equal to the unconditional expectation) Recall that ω ω n contains the same information as S,, S n This allows us to re-write the conditional expectation E n X as a function of the path S,, S n In these terms, the identity (4) can be also written as Ẽ[S n+ S = s,, S n = s n ] = (r + )s n, where the conditioning is on any admissible values of the first n stock values Example 39 By (i) and by (4) in agreement with pu + qd = r + S2 Ẽ = Ẽ [S 2 ] S S = S (r + )S = r +, Note that by (4) S n+ = r+ẽn+[s n+2 ] Applying Ẽn to both sides and exploiting the tower property so using (4) Ẽ n [S n+ ] = Ẽn r + Ẽn+[S n+2 ] = [Ẽn+ [S n+2 ]] = r + Ẽn r + Ẽn[S n+2 ], S n = (r + ) 2 Ẽn[S n+2 ] Exercise 30 Show a more general formula S n = (r+) k Ẽn[S n+k ], in particular show that S 0 = (r+) N Ẽ[S N] 3

4 33 Martingales Martingales are random processes which have no tendency to decrease or increase Loosely speaking, on the average a martingale stays in the future where it is at present time Example 3 (Random walks: symmetric and biased) Suppose I, I 2, are independent with P(I n = ) = p, P(I n = ) = p = q The sum M n = M 0 + I + + I n is the fortune of a gambler with starting capital M 0 after n games with unit bets on heads, with probability p of winning in each game Consider three cases If p = /2 the game is fair, because EI n = 0 and E n [M n+ ] = E n [M n + I n ] = E n [M n ] = M n After n games, the wealth is M n, and the expected wealth by playing one more game is also M n If p > /2 the game is advantageous, because EI n = p q > 0 and E n [M n+ ] = E n [M n + I n ] = E n [M n ] + p q > M n If p < /2 the game is disadvantageous, because EI n = p q < 0 and E n [M n+ ] = E n [M n + I n ] = E n [M n ] + p q < M n Definition 32 An adapted random process M n (n = 0,, N) is a martingale if E n M n+ = M n, n = 0,, N (5) If instead E n M n+ M n we say that the process is a submartingale, and if E n M n+ M n that a supermartingale Submartingales have a tendency to increase with time, supermartingales to decrease Exercise 33 Consider independent coin-tosses with probabilities p, q for H, T Let X n = #heads(ω ω n ) np, X 0 = 0 Show that X n (n = 0,,, N) is a martingale Exercise 34 Let Y n = #heads(ω ω n ) #tails(ω ω n ), n = 0,, N Is Y n a martingale, sub- or supermartigale? The defining identity (5) can be iterated (using the tower property of conditional expectations) to obtain M n = E n [M m ] for every 0 n m N In particular, EM n = M 0, n = 0,, N the expected value of the martingale is constant over time Theorem 35 Under the risk-neutral measure P the discounted stock price process is a martingale S n ( + r) n, n = 0,, N 4

5 Proof Divide both parts of (4) by ( + r) n and compare with (5) The theorem can be re-formulated as follows: the risk neutral measure P is the probability measure on Ω under which the discounted stock value is a martingale Therefore P is sometimes called the martingale measure Now let ( n, X n ) be a stock+bond portfolio process satisfying the self-financing condition X n+ = n S n+ + (r + )(X n n S n ) (6) Theorem 36 Under the risk-neutral measure, the discounted wealth process is a martingale X n ( + r) n, n = 0,, N Proof Apply Ẽn on both sides of (6) and divide by ( + r) n+ Ẽ n [ X n+ ( + r) n+ ] = Ẽn [ n S n+ ( + r) + X ] n n S n n S n+ = n+ ( + r) n+ Ẽn ( + r) n+ +Ẽn Xn n S n ( + r) n+ We can pull n out of Ẽn because n is adapted (depends on n first coin tosses), and for the same reason cancel Ẽn in the second term S n+ = n Ẽ n + X n n S n ( + r) n+ ( + r) n+ Using the martingale property of the discounted stock = n S n+ ( + r) n+ + X n n S n ( + r) n+ = X n ( + r) n For a given option with payoff V N, let ( n, X n ) (n = 0,, N) be a hedging portfolio Then V n = X n is the no-arbitrage price of the option From Theorem 36 follows Theorem 37 The discounted price of the option is a martingale under P, that is V n ( + r) = Ẽ Vn+ n ( + r) n In particular, the risk-neutral pricing formula holds V 0 = Ẽ V N ( + r) N 5