Trees and Martingales Multi-Step Binary Model • Generalize the single-period model. • Still just two assets: the stock and the risk-free bond. • Still no transaction cost, nor limits on amounts of either asset. • But: market is observed at times 0 = t 0 < t 1 < · · · < t N = T . 1
• Over each interval [ t i , t i +1 ], the stock follows the binary model. � � • That is, given S ( t i ), S can take one of only two values. t i +1 • But at time t i , there are up to 2 i possible states, and at time T , up to 2 N . • Assume evenly spaced times with spacing δt = T/N , so t i = iδt , i = 0 , 1 , . . . , N , and write S i = S ( t i ). 2
S 000 3 S 00 2 S 001 3 S 0 1 S 010 3 S 01 2 S 011 3 S 0 S 100 3 S 10 2 S 101 3 S 1 1 S 110 3 S 11 2 S 111 3 3
The Cash Bond • As before, the interest rate over [ t i , t i +1 ] is known in advance, at t = t i . • But it need not be a single constant: – It could depend on t , as r ( t i ); – It could even depend on the starting node, as r ( t i , S i ). – But it is the same along both branches from that node. • For convenience, we continue to assume it is an overall con- stant, r . 4
Backward Induction • How do we price a European option? • As a European option, the payout is a function of S N , so we know the value at every node at t = t N = T . • Now consider any node at t = t N − 1 = T − δt : – We know the value at each of the two branches, and we know S N − 1 at this node. – So we can use the one-step model to find the value at this node, which is also the cost of setting up the replicating portfolio. 5
• Specifically, if S N − 1 = S 0 at this node, and the two values at t = T are S 0 u and S 0 d , then: – the price of the option at this node is ue − rδt − 1 1 − de − rδt � � � � C ( S 0 u ) + C ( S 0 d ); u − d u − d – the number of shares in the replicating portfolio should be = C ( S 0 u ) − C ( S 0 d ) △ φ S 0 u − S 0 d • Note that if ( u + d ) / 2 = e rδt , then both weights are 1 2 e − rδt . • Note also that φ , the hedge ratio , is the gradient of C ( · ). 6
• In this way, we can calculate the value of the option at every node at t = t N − 1 . • Now consider any node at t = t N − 2 : – We know the value at each of the two branches, since they end at t = t N − 1 , and we already calculated these values. – So we can use the one-step model to find the value at this node. • Hence we can calculate the value of the option at every node at t = T N − 2 . 7
• The replicating portfolio that we set up at t = t N − 2 is guar- anteed to give the resources that we need to set up the replicating portfolio at t = t N − 1 . • That is, the strategy is self-financing . • Continue recursively to t = 0, which gives the value of the option, which is also the resources needed to set up the initial replicating portfolio. • Note that the replicating portfolio may need to change at any node: dynamic hedging. 8
Recombining Tree • The nodes of the tree are often chosen to reduce the number of distinct nodes. – For instance, if S 01 = S 10 = S ud 2 , say, the tree has only 3 2 2 nodes at t 2 instead of 4. • The tree can have as few as i + 1 nodes at t = t i . 9
S uuu 3 S uu 2 S u S uud 1 3 S ud S 0 2 S d S udd 1 3 S dd 2 S ddd 3 • Also known as a binomial tree, since there are (1 , 2 , 1) paths to the 3 nodes at t = t 2 , and so on. 10
• Example 2.1.2. European call; S 0 = 100 , K = 100 , r = 0: � � 160 60 140 40 120 1 . 00 � � 120 25 20 S 0 = 100 100 0 . 75 V 0 = 15 10 80 φ 0 = 0 . 50 0 . 50 � � 80 5 0 60 0 . 25 0 0 . 00 � � 40 0 11
Path Probabilities • The pricing exercise provides risk-neutral probabilities along each branch. – These are conditional on the node at which the branch begins. – The product of the branch probabilities along any path is the path probability . • The sum of the probabilities of all paths leading to a given node is the risk-neutral probability of the node. 12
• When the risk-free interest rate is constant, or time-dependent but not state-dependent, the value of the option is the ex- pected value of the payoff, discounted to present value, with respect to the risk-neutral node probabilities. • In general, the value is E Q (path discount factor × path payoff) where Q is the risk-neutral distribution over the paths. – In other words, the value of the option is the expected value of the payoff, discounted to present value, with re- spect to the risk-neutral path probabilities. 13
American Option • An American option can be exercised at maturity or at any earlier time . • Early exercise of an American call on a non-dividend-paying stock is never optimal; consider two portfolios: Portfolio A: One American call plus (at t = 0) Ke − rT cash; Portfolio B: One share. 14
• If the option is exercised at time t < T , then: – Portfolio A is one share plus Ke − r ( T − t ) − K < 0 cash; – Portfolio B is one share, so Portfolio B is worth more. • If the option is exercised only at t = T , if at all, then: – Portfolio A is worth max( S T , K ); – Portfolio B is worth S T , hence no more than Portfolio A. • So early exercise does not add value. 15
• American puts are more complicated: – If the risk-free interest rate is zero, early exercise never adds value; – If the risk-free interest rate is positive, early exercise may be optimal. • See Example 2.2.3 for the backward recursion solution for a particular case. 16
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