Binomial Representation Theorem • Recall the discrete stochastic integral : if { X n } n ≥ 0 is a ( P , {F n } n ≥ 0 )- martingale and { φ n } n ≥ 1 is {F n } n ≥ 0 -previsible, then n − 1 � � � Z n = Z 0 + φ j +1 X j +1 − X j , j =0 where Z 0 is a constant, is also a ( P , {F n } n ≥ 0 )-martingale. • In a binary tree model, the converse is also true. 1
• Suppose that the measure Q is such that the discounted price process { ˜ S n } is a ( Q , {F n } n ≥ 0 )-martingale. If { ˜ V n } is any other ( Q , {F n } n ≥ 0 )-martingale, then there exists a ( Q , {F n } n ≥ 0 )- predictable process { φ n } n ≥ 1 such that n − 1 � � V n = ˜ ˜ S j +1 − ˜ ˜ � V 0 + φ j +1 S j . j =0 • To prove this, we must show that � � V i +1 − ˜ ˜ S i +1 − ˜ ˜ V i = φ i +1 S i where φ i +1 is F i -measurable. 2
� � S i +1 ( u ) , ˜ ˜ • For a given node at t = iδt , write S i +1 ( d ) for the � � two possible values of ˜ V i +1 ( u ) , ˜ ˜ S i +1 , and V i +1 ( d ) for the corresponding values of ˜ V i +1 . • We can solve � � V i +1 ( u ) − ˜ ˜ S i +1 ( u ) − ˜ ˜ V i = φ i +1 + k i +1 S i and � � V i +1 ( d ) − ˜ ˜ S i +1 ( d ) − ˜ ˜ V i = φ i +1 S i + k i +1 to get V i +1 ( u ) − ˜ ˜ V i +1 ( d ) φ i +1 = S i +1 ( d ) . S i +1 ( u ) − ˜ ˜ 3
� � � � S i +1 ( u ) , ˜ ˜ V i +1 ( u ) , ˜ ˜ • Because S i +1 ( d ) and V i +1 ( d ) are known at time t = iδt , this φ i +1 is F i -measurable. • Also � � k i +1 = ˜ V i +1 − ˜ S i +1 − ˜ ˜ V i − φ i +1 S i and because k i +1 is also F i -measurable, � � �� � V i +1 − ˜ ˜ S i +1 − ˜ ˜ k i +1 = E V i − φ i +1 S i � F i � � � � � � � V i +1 − ˜ ˜ S i +1 − ˜ ˜ = E V i � F i − φ i +1 E S i � F i � � = 0 because both { ˜ S n } and { ˜ V n } are ( Q , {F n } n ≥ 0 )-martingales. • That completes the proof. 4
• Note that the tree did not need to be recombining, only binary. • We did, however, tacitly assume that ˜ S i +1 ( u ) � = ˜ S i +1 ( d ) at this node, and hence at all nodes. • Self-financing: we can build a dynamic portfolio of stock and cash with discounted value { ˜ V n } , by holding φ i +1 shares in [ iδt, ( i + 1) δt ) and keeping the balance V i − φ i +1 S i in cash. 5
Continuous Time Limit • Fix a time t > 0, and let δt = t/N ; if N is large, and hence δt is small, the binary tree should approximate a continuous time model. • At a node with stock price s , assume that the successor √ � � nodes are s exp νδt ± σ δt for a drift ν and volatility σ . • Suppose that under the market measure P , these are equally likely. 6
• Then the expected value at the next step, conditionally on being at this node, is √ √ se νδt × 1 ν + 1 � � � � � � δt + e − σ e σ δt 2 σ 2 ≈ s 1 + δt , 2 and the conditional variance is √ √ � 2 1 se νδt � 2 � δt − e − σ ≈ s 2 σ 2 δt. � e σ δt 4 √ whence the conditional standard deviation is sσ δt . 7
• Suppose that by time t = Nδt , the price has moved up X N times, and therefore down N − X N times. • Then √ √ � � S t = S 0 exp Nνδt + X N σ δt − ( N − X N ) σ δt √ � � �� 2 X N − N = S 0 exp νt + σ t √ . N • X N is binomial, so, by the Central Limit Theorem, √ △ = (2 X N − N ) / N is approximately standard normal for Z N large N , so we can write this as √ � � S t = S 0 exp νt + σ tZ N , where Z N is approximately N (0 , 1). 8
• So, under the market measure P , S t is approximately log- normally distributed. • Under the martingale measure Q , the probability of an up- jump is √ e rδt − e νδt − σ δt p = √ √ δt − e νδt − σ e νδt + σ δt 2 σ 2 − r ν + 1 √ ≈ 1 . 1 − δt 2 σ • So X N is also binomial under Q , but with parameter p � = 1 2 . 9
√ • Again using the CLT, under Q , (2 X N − N ) / N is approxi- √ 2 σ 2 − r � � ν + 1 � � mately N − t /σ, 1 . • So now we can write √ r − 1 �� � � 2 σ 2 tZ ∗ S t = S 0 exp t + σ , N √ △ 2 σ 2 − r � ν + 1 � where Z ∗ = Z N + t /σ is, now under Q , ap- N proximately N (0 , 1). • Note that Z ∗ N � = Z N : – Z N is approximately N (0 , 1) under P ; – Z ∗ N is approximately N (0 , 1) under Q . 10
• Option pricing: if a European option with maturity T has payoff C ( S T ), then its arbitrage-free price is E Q � � e − rT C ( S T ) . • We would expect this to be approximately √ r − 1 � � �� � ��� e − rT C 2 σ 2 TZ ∗ E Q S 0 exp T + σ (although this follows from the convergence argument only for bounded, continuous C ( · )). • Here, under Q , Z ∗ is approximately N (0 , 1). 11
• For the special case of a European call with strike K and C ( S ) = ( S − K ) + , we find the classic Black-Scholes price log S 0 log S 0 � r + 1 2 σ 2 � � r − 1 2 σ 2 � K + K + T T − Ke − rT Φ S 0 Φ √ √ σ T σ T • Here Φ( · ) is the cumulative distribution function of the stan- dard normal distribution � z 1 2 πe − x 2 / 2 dx. Φ( z ) = √ −∞ 12
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