Option Pricing with Delayed Information
aa r X i v : . [ q -f i n . M F ] J u l OPTION PRICING WITH DELAYED INFORMATION
Tomoyuki Ichiba
University of California, Santa Barbara [email protected]
Seyyed Mostafa Mousavi
University of California, Santa Barbara [email protected]
We propose a model to study the effects of delayed information on option pricing. We first talkabout the absence of arbitrage in our model, and then discuss super replication with delayed informationin a binomial model, notably, we present a closed form formula for the price of convex contingent claims.Also, we address the convergence problem as the time-step and delay length tend to zero and introduceanalogous results in the continuous time framework. Finally, we explore how delayed information exag-gerates the volatility smile.
Keywords: Delayed information, binomial model, continuous-time limit, incomplete market, super replication,volatility smile.
1. Introduction
All participants in financial markets have access only to delayed information. Delay adds more uncertaintyto the market, and it is of great importance to study it. A universal assumption in options pricing literatureis that a trader makes his decisions with full access to the prices of the assets (i.e, no delayed information).However, in practice, there is a lag between when the order is decided and its execution time. In particular,there are two important types of delays in financial markets. First is the delay in order execution, that is,the order would be executed with some delay after the trader places it. For example, if the order is madein the morning, it would be executed in the afternoon. Second is the delay in receiving information, thatis, the trader observes the prices and other important information with some delay, usually because of thetechnological barriers, exacerbated by having long physical distance from the exchange.In the view of traders, these two types of delays act similarly. In both cases, orders are executed withprices which are unknown at the time they are made. In other words, the source of the delayed informationdoes not change the decisions of the trader. For example, let { , , . . . } be a discrete trading horizon. Ifthere exists a delay with length of 1 period, then regardless of what the source of delay is, no trade happensat time 0, and in later times trades happen based on the information available up until the previous period.The reason is that if the delay is only in receiving information, then, at time 0 the trader does not have anyinformation, so he waits till time 1 to get time-0 prices to make a trade and those trades would of coursebe executed with time-1 prices. If the delay is only in order execution, then at time-0 and based on time-0prices, the trader makes an order, but that order would be executed with time-1 prices.In this work, we start with the binomial model proposed by Cox et al. (1979) and consider fixed periods ofdelay in the flow of information. Therefore, agents have an information stream smaller than the information The authors would like to thank Jean-Pierre Fouque, Mike Ludkovski, Yuri Saporito and Andrey Sarantsev for severalhelpful discussions and feedbacks at different stages of the work. The first author is supported in part by NSF grants DMS1313373 and 1615229. N -period binomialmodel with H = N − N -periodbinomial model with H periods of delay in subsection 2.5 using both dynamic programming and directapproaches. A geometrical representation of the strategy is presented in subsection 2.6. In section 3, westudy the asymptotic behavior of the model as the time step and delay length tend to zero. In particular,subsection 3.2 is devoted to the discussion of how delayed information affects the volatility smile.
2. Discrete Time Model
Before introducing delays, let us recall the N -period binomial tree model of Cox et al. (1979) for a financialmarket with a single risky asset and a single risk-free asset (e.g., stock). Given N ∈ N , let us denote by(Ω , F , P ) a probability space for the canonical space Ω := { , } N of N -period binomial tree with theBorel σ -algebra F generated by Ω . For every ω := ( ω , . . . , ω N ) ∈ Ω we define a coordinate map by Z k ( ω ) = ω k for each k = 1 , . . . , N . Let P be the probability measure under which Z k , k = 1 , . . . , N are independent, Bernoulli random variables with P ( Z k = 1) = P ( Z k = 0) = 1 / k = 1 , . . . , N . Definethe filtration F := {F k , k = 0 , . . . , N } , where F k is the σ -field σ ( Z , . . . , Z k ) generated by the first k variables for k = 1 , . . . , N and F is the trivial σ -field, i.e., F = {∅ , Ω } .In the N -period binomial tree model, the risky asset price S k : Ω → R and its discounted price2 S k : Ω → R , discounted by instantaneous rate r >
0, at time k , are defined by S k ( ω ) := S u I k ( ω ) d k − I k ( ω ) , I k ( ω ) := k X l =1 Z l ( ω ) , e S k ( ω ) := e − rk S k ( ω ) , k = 1 , . . . , N , (2.1)where S is a given initial price of risky asset at time 0 , and u (or d ) is a fixed ratio by which the priceprocess goes up (or down) in one period with u > r > d > F . We shall introduce delays in the flow of information in the N -period binomial model. For simplicity, letus consider the situation where an investor sends buy or sell orders to the market at time t , but herorders are not executed until time t + H with H ∈ { , . . . , N − } delay periods . The investor herselfknows that she has H delay periods when she is sending orders. Then we define the delayed filtration G := {G k , k = 0 , , . . . , N } , where G k := F , for k = 0 , . . . , H − G k := F k − H , k = H , . . . , N . (2.2)In other words, G k is the information set of the price process until time min( k − H,
0) , rather than time k . In the following, we shall consider investments based on this delayed information.Let A G be the set of all G -adapted stochastic processes ∆ := { ∆ k , k = 0 , . . . , N − } with ∆ k ≡ k = 0 , . . . , H − ∈ A G represents a strategy for this investor based on the delayedinformation, that is, the positive ∆ k > k < k , giveninformation G k . In other words, the order made at time k − H to buy or sell (∆ k − ∆ k − ) shares of therisky asset, gets executed at time k with price S k (not S k − H ), because of H periods of delay. Thus theinvestor has to deal with the risk of price changes between the time of order submission and execution.For an initial investment of x in the risk free asset and a strategy ∆ ∈ A G , we shall consider theportfolio value process V k ( x , ∆)( ω ) , k = 0 , . . . , N , ω ∈ Ω . The first order ∆ H submitted at time 0 isexecuted at time H , and the portfolio value process is not observed until time H . Thus we define(2.3) V H ( x , ∆)( ω ) := x · e rH + ∆ H · S H ( ω ) , V ( x , ∆)( ω ) := e − rH · V H ( x , ∆)( ω ) = x + ∆ H · e S H ( ω ) , and in general(2.4) V k ( x , ∆)( ω ) := e − r ( H − k ) · V H ( x , ∆) ( ω ) , k = 0 , . . . , H − ,e rk x + k − P l = H S l ( ω ) · (cid:0) ∆ ( l − ∨ H − ∆ l (cid:1) + S k ( ω ) · ∆ ( k − ∨ H , k = H, . . . , N .
For k = H, . . . , N , the first term in the portfolio value process ( e rk x ) in (2.4) corresponds to the initialinvestment in the risk free asset. The second term ( P k − l = H S l ( ω ) · (∆ ( l − ∨ H − ∆ l )) is due to the cash flowin the risk free asset up until time k , and the third term ( S k ( ω ) · ∆ ( k − ∨ H ) relates to the investment in therisky asset at time k . We call V k ( x , ∆), k = 0 , . . . , N the value process from the strategy ( x , ∆) ∈ ( R , A G ).By construction, the changes in the portfolio value process ( V k ( x , ∆)) in (2.4) starting from its firstrealization at time H , are only due to the variation in asset prices. In other words, no money is added to orwithdrawn from the portfolio.Note that the initial portfolio value V ( x , ∆)( ω ) in (2.3) is a random variable, not a constant. This isbecause it is defined by discounting the time- H portfolio value V H ( x , ∆)( ω ), which is the first time theportfolio value is observed due to the existence of delay.For k = H, . . . , N , ∆ k is G k -measurable, but V k ( x , ∆) is F k -measurable. Thus V k ( x , ∆) is F k ∨ H -measurable for k = 0 , . . . , N . In this sense, the portfolio is constructed based on the delayed information.3 .2. Absence of Arbitrage We shall first introduce the notion of arbitrage in our model. In general, arbitrage means that one cannotreap any benefit for free, that is without taking any risk. In our model with delayed information, as it isshown in (2.3), the initial portfolio value V ( x , ∆) is a random variable, because of the existence of delay.Therefore, we need to adjust the classical notion of arbitrage in the domain of ( R , A G ) strategies, to takethis into account. Definition 2.1 (Arbitrage) . An arbitrage opportunity is the strategy ( x , ∆) ∈ ( R , A G ) such that max ω ∈ Ω { V ( x , ∆) ( ω ) } = 0 , P ( V N ( x , ∆) ≥
0) = 1 , (2.5) P ( V N ( x , ∆) > > . The primary difference with the classical definition of arbitrage is the condition that the maximum oftime-0 portfolio value needs to be zero (max ω ∈ Ω { V ( x , ∆) ( ω ) } = 0). It is obvious that in the case of completeinformation (i.e, H = 0), this definition boils down to the classical definition of arbitrage opportunity.We need to show that there is no arbitrage in our discrete time model with delayed information.Kabanov and Stricker (2006) proves that in a general discrete time model with restricted information, theredoes not exist classical arbitrage, if and only if there exists a probability measure e P equivalent to P such thatthe optional projection under e P of the discounted stock price on the delayed filtration, is a e P -martingale Thesetup of our model is a bit different than that in Kabanov and Stricker (2006), given that our first order tobuy/ sell the risky asset is executed at time H , rather than at time 0 (i.e. ∆ k = 0, k = 0 , . . . , H − V ( x , ∆)) a random variable, rather than always a constant. Theorem 2.1shows that still in our model, there does not exists arbitrage, in the sense of Definition 2.1. Theorem 2.1.
There does not exists any arbitrage opportunity in our discrete time model, in the domainof ( R , A G ) strategies.Proof. According to Definition 2.1, absence of arbitrage means that for any strategy ( x , ∆) ∈ ( R , A G ) suchthat max ω ∈ Ω { V ( x , ∆) ( ω ) } = 0, the condition P ( V N ( x , ∆) ≥
0) = 1 implies that P ( V N ( x , ∆) = 0) = 1.In the domain of ( R , A G ), according to (2.3), the condition max ω ∈ Ω { V ( x , ∆) ( ω ) } = 0 is equivalent tomax ω ∈ Ω { V H ( x , ∆) ( ω ) } = 0 , which means that in all ( N − H )-period binomial models starting from time H , the initial values for the( x , ∆) strategy are non-positive.If we consider all these ( N − H )-period binomial models individually, they lie in the general discrete timemodel framework in Kabanov and Stricker (2006). Therefore, in each of these models, even if we consider theinitial values of the strategy to be zero, the condition P ( V N ( x , ∆) ≥
0) = 1 implies P ( V N ( x , ∆) = 0) = 1,given that we show that there exists a probability measure e P ∼ P such that the e P -optional projection of thediscounted stock price on the delayed filtration, is a e P -martingale, that is E e P (cid:16) e S k +1 |G k (cid:17) = E e P (cid:16) e S k |G k (cid:17) , k = H, . . . , N − . (2.6)Define the probability measure e P such that the coordinate maps Z k , k = 1 , . . . , N are still independentBernoulli random variables, but with parameters e P ( Z k = 1) = ue r − du − d = 1 − e P ( Z k = 0) , k ∈ { , . . . , N } , which are the risk-neutral probabilities in the usual binomial model without any delay.Given that the discounted stock price ( e S k ) is ( F k )-martingale under e P , it follows that condition (2.6)holds, which shows that there is no arbitrage opportunity from time H to N . Consequently, given (2.3), weconclude that there is no arbitrage in the model in the domain of ( R , A G ) strategies.4 emark 2.1. The domain of ( R , A G ) strategies in Theorem (2.1) does not include all F -adapted strategies,but only those which are G -adapted. In other words, we are excluding the case that an agent with fullinformation comes and exploits the advantage over the investors with delayed information in the market. Ifwe include all F -adapted strategies, it is likely to have arbitrage opportunities. Given that there is no arbitrage in the market, it now makes sense to discuss about pricing.
Definition 2.2 (Super-replication price and the value process of super-replicating portfolio) . For any con-tingent claim with payoff function ϕ : Ω → R and expiration time N , its super-replication price ¯ π ( ϕ ) isdefined as the minimal initial value of portfolio which exceeds the value ϕ at time N , i.e., ¯ π ( ϕ ) := inf ( x , ∆) ∈ Γ max ω ∈ Ω n V ( x , ∆)( ω ) = x + ∆ H e S H ( ω ) o , (2.7) where Γ := { ( x , ∆) ∈ R × A G : V N ( x , ∆) ≥ ϕ P − a.s. } . (2.8) If there exists a pair ( x ∗ , ∆ ∗ ) that attains the infimum in (2.7), i.e., ¯ π ( ϕ ) = max ω ∈ Ω V ( x ∗ , ∆ ∗ )( ω ) , thenthe time- k super replicating portfolio value V k ( ω ) is defined as V k ( ω ) := V k ( x ∗ , ∆ ∗ )( ω ) , k = 0 , . . . , N , (2.9) and consequently, ¯ π ( ϕ ) = max ω ∈ Ω V ( ω ) . Remark 2.2.
The super-replication price is the most conservative pricing approach for the seller of theoption, considering the worst-case scenario. In other words, it is straightforward to show that any pricegreater than the super-replication price causes arbitrage in the market.
Remark 2.3.
It is remarkable to note that call-put parity does not hold anymore. The reason is that thesuper-replication price ¯ π is a coherent risk measure on the space L ∞ (Ω , F , P ) of payoff functions, andtherefore it is subadditive. All of the results in this paper are for European-style contingent claims with convex payoff functions.In section 2.4, we consider first the case H = N − N -period binomial model with H = N − periods of delay We determine the super-replication price and the corresponding strategy for the European contingent claimswhen H = N − H = N − S is observed, but the order ∆ H , sent by the investor at time 0 , would be executed at time H . For example, when N = 2 and H = 1 , the order ∆ sent at time 0 is executed at time 1 with twopossible prices S = S d or S = S u (see Figure 1).Let us observe that in the case of H = N − V N ( x , ∆) in (2.4) is simplified to(2.10) V N ( x , ∆)( ω ) = e rN x + S N ( ω ) · ∆ N − . There are ( N + 1) possible values of S N ( ω ) , ω ∈ Ω in (2.1) and there are only two controls ( x , ∆ N − )in the terminal value. Since there are ( N + 1) constraints and only two controls, the minimization problemin (2.7) has possibly infinitely many solutions. In other words, in an economic sense, the market is notcomplete. To learn more about pricing in incomplete markets, we refer to Staum (2007).5 S dS u S d S udS u time 0 time 1 time 2Figure 1: Asset price process S k in a 2-period binomial model Theorem 2.2.
For a European-style contingent claim with payoff ϕ := Φ( S N ) for some convex function Φ( · ) in the N -period binomial model with H = N − periods of delay, the super-replication price is ¯ π ( ϕ ) = max (cid:0) x ∗ + e − rH ∆ ∗ H · S u H , x ∗ + e − rH ∆ ∗ H · S d H (cid:1) , (2.11) where the corresponding strategy ( x ∗ , ∆ ∗ ) is given by ∆ ∗ j ≡ , j = 0 , , . . . , H − , (2.12) ∆ ∗ H = ∆ ∗ N − = Φ( S u N ) − Φ( S d N ) S · ( u N − d N ) and x ∗ = e − rN · u N Φ( S d N ) − d N Φ( S u N ) u N − d N . Proof.
First, we shall prove that for any ω ∈ Ω , ( x ∗ , ∆ ∗ ) in (2.12) satisfiesinf ( x , ∆) ∈ Γ n V ( x , ∆)( ω ) = x + ∆ H e S H ( ω ) o = V ( x ∗ , ∆ ∗ )( ω ) = x ∗ + ∆ ∗ H · e S H ( ω ) . (2.13)Here the infimum is taken over the set Γ in (2.8), that is, x ∈ R and ∆ ∈ A G must satisfy V N ( x , ∆) ≥ ϕ ( S N ) almost surely. Note that V N ( x , ∆) = ( e rN x + x · ∆ N − ) | x = S N in (2.10) is realized as the value at x = S N of linear function y = e rN x + x · ∆ N − with the slope ∆ H and the y -intercept e rN x in the( x, y ) coordinates. Moreover, since the payoff function Φ( · ) is convex, by Jensen’s inequality, one can verifyΓ = { ( x , ∆) ∈ R × A G : e rN x + S u N · ∆ N − ≥ Φ( S u N ) , e rN x + S d N · ∆ N − ≥ Φ( S d N ) } . (2.14)That is, in order to check whether the inequality V N ( x , ∆) ≥ Φ( S N ) holds with probability one, it sufficesto check it just at the extreme cases, in which the asset price S N at time N is the minimum S d N or themaximum S u N in the binomial tree model. Then it is easy to check that the choice ( x ∗ , ∆ ∗ H ) in (2.12)belongs to the set Γ as we have e rN x ∗ + ∆ ∗ H S u N = ϕ ( S u N ) , e rN x ∗ + ∆ ∗ H S d N = ϕ ( S d N ) . In otherwords, the minimization problem is reduced to a linear programming problemminimize ( x , ∆ H ) ∈ R x + ∆ H · e S H ( ω )subject to e rN x + S u N · ∆ H ≥ Φ( S u N ) , and e rN x + S d N · ∆ H ≥ Φ( S d N ) . Define the Lagrangian as L := x + ∆ H e S H ( ω ) + λ [Φ( S u N ) − (cid:0) e rN x + S u N ∆ H (cid:1) ] + λ [Φ( S d N ) − (cid:0) e rN x + S d N ∆ H (cid:1) ] , where λ and λ are the Lagrangian multipliers. Then, it is easy to check that the quantities x ∗ = e − rN · u N Φ( S d N ) − d N Φ( S u N ) u N − d N , ∆ ∗ H = Φ( S u N ) − Φ( S d N ) S u N − S d N ,λ ∗ = e S H ( ω ) − e − rN S d N S · ( u N − d N ) , λ ∗ = e − rN S u N − e S H ( ω ) S u N − S d N ( x o , ∆) ∈ Γ max ω ∈ Ω V ( x , ∆)( ω ) = max ω ∈ Ω inf ( x o , ∆) ∈ Γ V ( x , ∆)( ω ) . Thus, we get(2.16) ¯ π ( ϕ ) = max ω ∈ Ω inf ( x o , ∆) ∈ Γ V ( x , ∆)( ω ) = max ω ∈ Ω V ( x ∗ , ∆ ∗ )( ω ) . Then, the proof is completed by the following observationmax ω ∈ Ω { V ( x ∗ , ∆ ∗ )( ω ) = x ∗ + ∆ ∗ H e S H ( ω ) } = max (cid:0) x ∗ + e − rH ∆ ∗ H · S u H , x ∗ + e − rH ∆ ∗ H · S d H (cid:1) . (2.17)By using Theorem 2.2, the portfolio value V H ∈ F H in (2.9) at time H of the super-replicating strategycan be calculated as V H = e rH x ∗ + ∆ ∗ H · S H = H X j =0 e − r ( N − H ) E Q j [Φ( S N )] · { S H = S u j d H − j } = H X j =0 e − r ( N − H ) (cid:2) p j Φ( S u N ) + q j Φ( S d N ) (cid:3) · { S H = S u j d H − j } . (2.18)Here { Q j } Hj =0 are probability measures on (Ω , F ) defined by Q j ( I N = N ) := p j = 1 − Q j ( I N = 0) = 1 − q j , p j := u j d H − j e r − d H +1 u H +1 − d H +1 , j = 0 , . . . , H . (2.19) Remark 2.4.
We can conclude from the form in (2.18) that V H , the value of the super-replicating portfolioat time H , is a function of S and S H . In other words V H ≡ V H ( S , S H ) . Therefore, the value process for the super-replicating portfolio is path dependent , due to the existence of H periods of lag between the times of order submission and execution. Thus, the super-replication price ¯ π ( ϕ ) can be calculated as¯ π ( ϕ ) = max ω ∈ Ω V ( ω ) = max j ∈{ ,...,H } e − rN E Q j [Φ( S N )] = max j ∈{ ,H } e − rN E Q j [Φ( S N )]= max j ∈{ ,H } e − rN (cid:2) p j Φ( S u N ) + q j Φ( S d N ) (cid:3) = e − rN max (cid:0) p u Φ( S u N ) + q u Φ( S d N ) , p d Φ( S u N ) + q d Φ( S d N ) (cid:1) , (2.20)where the third equality follows similarly as in (2.17). Notation:
From now on, we use ( p u , q u ) as ( p H , q H ) and ( p d , q d ) as ( p , q ) , since ( p H , q H ) and ( p , q )correspond to the measures at the extreme points S H = S u H and S H = S d H respectively. N -Period binomial Model with H Periods of Delay
We extend our considerations from section 2.4 and generalize the model to the N -period binomial modelwith H ( ≤ N −
1) periods of delay. We determine the super-replication price and the corresponding strategyfor European style contingent claims with convex payoff functions. Here we shall solve the problem fromboth a dynamic programming (or backward induction) approach and a direct approach.7 .5.1 Dynamic Programming Approach
First, let us define the tree T N (0 ,
0) of length N as the set of nodes ( i, j ) , such that there are i ups and j downs from the node (0 ,
0) with 0 ≤ i + j ≤ N , i.e., T N (0 ,
0) := { ( i, j ) ∈ N : 0 ≤ i + j ≤ N } . Then define its ( H + 1)-period subtree T H +1 ( a, b ) starting from the node ( a, b ) at time a + b by T H +1 ( a, b ) := { ( i, j ) ∈ N : a + b ≤ i + j ≤ a + b + H + 1 , i ≥ a , j ≥ b } , for every ( a, b ) ∈ T N (0 ,
0) such that a + b ≤ N − ( H + 1).We shall identify all N − H subtrees T H +1 ( a, b ) starting from the nodes ( a, b ) at time N − ( H + 1) (i.e, a + b = N − ( H + 1) ). We use the results in section 2.4 and consider the value process of the super-replicatingportfolio at time N − H + 1)-period subtrees starting from thenodes at time N − ( H + 2) . Then, we keep super-replicating backwards in the same manner. Remark 2.5.
Given that in the dynamic programming approach, we are using the results in section (2.4)in each step, and Remark (2.4), we can conclude that the value process at level k ∈ { H, . . . , N } for thesuper-replicating strategy in the general model is also path dependent, that is (2.21) V k ≡ V k ( S k − H , S k ) , k = H, . . . , N .
Therefore, let us define the payoff for the subtree T H +1 ( a, b ) starting from the node ( a, b ) at time a + b at its leaf node ( p, q ) (i.e, p + q = a + b + H + 1 ) by(2.22)Φ T H +1 ( a,b ) ( p, q ) := V p + q (cid:0) S a + b d , S a + b d H +1 (cid:1) if p = a ;max (cid:26) V p + q (cid:0) S a + b d , S a + b u i d H +1 − i (cid:1) , V p + q (cid:0) S a + b u , S a + b u i d H +1 − i (cid:1) (cid:27) if p = a + i ,i = 1 , . . . , H ; V p + q (cid:0) S a + b u , S a + b u H +1 (cid:1) if p = a + H + 1 ;for p + q ≤ N −
1, and Φ T H +1 ( a,b ) ( p, q ) := Φ( S N ) , p + q = N where S N = S u p d q .Intuitively, for the subtree T H +1 ( a, b ) starting at time a + b , there are only two ( H + 1) -period subtrees, T H +1 ( a + 1 , b ) and T H +1 ( a, b + 1), starting at time a + b + 1 that can induce payoff at time p + q . So, weneed to take the maximum of the two possible value process as the new payoff because we always considerworst case scenario in super replication. Note that at the edge points, there exists only one value process. Example 2.1.
In the -period binomial tree model (as in Figure 2) with H = 1 , what new payoff we needto consider on the node S = S u d depends on whether we are considering this node as part of the subtree T (1 , or T (0 , . As part of the subtree T (1 , , the payoff (cid:0) Φ T (1 , (2 , (cid:1) would be the maximum of thecorresponding value processes of the subtrees T (1 , and T (2 , , while as part of the subtree T (0 , , thepayoff (cid:0) Φ T (0 , (2 , (cid:1) would be the corresponding value processes of the subtrees T (1 , . One important ingredient in the dynamic programming approach is that when we start from a convexpayoff function, the payoff in (2.22) for all the intermediary ( H + 1) -period subtrees needs to be convexwith respect to the corresponding risky asset prices, in order to be able to use Theorem (2.2) in each stepand keep super-replicating backwards. Theorem (2.3) formalizes this relation. Theorem 2.3.
For a European-style contingent claim with payoff ϕ := Φ( S N ) for some convex function Φ( · ) in the N -period binomial model with H ≤ N − periods of delay, the payoff function Φ T H +1 ( a,b ) ( ., . ) , a + b = 0 , . . . , N − ( H + 1) in (2.22) for all the intermediary ( H + 1) -period subtrees are convex with respectto the corresponding risky asset prices.Proof. Note that for a + b = N − ( H + 1), the payoff functions Φ T H +1 ( a,b ) ( ., . ) for all ( N − H ) intermediary( H + 1)-period subtrees are convex, since the final payoff function Φ( S N ) is convex.8ow we show that all the payoff functions Φ T H +1 ( a ′ ,b ′ ) ( ., . ), a ′ + b ′ = a + b − T H +1 ( a,b ) ( ., . ), a + b ∈ { , . . . , N − ( H + 1) } are convex. By induction this completes theproof.Given that the payoff function Φ T H +1 ( a,b ) ( ., . ) is convex, by Theorem (2.2), there exists x ∗ and ∆ ∗ suchthat we define h ( t ) := V a + b + H ( S a ′ + b ′ u , S a + b + H ) = e rH x ∗ + ∆ ∗ t, t ∈ { S a ′ + b ′ ud H , . . . , S a ′ + b ′ u H +1 } ; e rH x ∗ + ∆ ∗ t, t = S a ′ + b ′ d H +1 . Similarly, there exists x ∗ and ∆ ∗ such that we define h ( t ) := V a + b + H ( S a ′ + b ′ d , S a + b + H ) = e rH x ∗ + ∆ ∗ t, t ∈ { S a ′ + b ′ d H +1 , . . . , S a ′ + b ′ u H d } ; e rH x ∗ + ∆ ∗ t, t = S a ′ + b ′ u H +1 , we can define h ( t ) := max ( h ( t ) , h ( t )) , t ∈ { S a ′ + b ′ d H +1 , . . . , S a ′ + b ′ u H +1 } ;(2.23)Note that h ( t ) = Φ T H +1 ( a ′ ,b ′ ) ( p, q ) where t := S u p d q , given that Φ T H +1 ( a,b ) ( ., . ) is convex, and (2.22).The discrete function h ( . ) is convex if for any v and w such that S u v d w ∈ { S a ′ + b ′ ud H , S a ′ + b ′ u H d } , wehave h ( t p ) + h ( t n ) ≥ h ( t m ) , (2.24)where t p : = S u v − d w +1 , t n : = S u v +1 d w − and t m : = S u v d w .Depending on the choice of v and w , there are 4 cases:Case 1: h ( t p ) = h ( t p ) and h ( t n ) = h ( t n ). Then, given the form in (2.23), we have h ( t m ) = h ( t m ). Then,it is straightforward to show that (2.24) follows by linearity of the function h ( t m ).Case 2: h ( t p ) = h ( t p ) and h ( t n ) = h ( t n ). This case follows similar to that of case 1.Case 3: h ( t p ) = h ( t p ) and h ( t n ) = h ( t n ). Then, h ( t m ) would equal to either h ( t m ) or h ( t m ). Withoutloss of generality assume that h ( t m ) = h ( t m ). Then given that h ( t n ) = h ( t n ), we conclude by the form in(2.23) that h ( t n ) ≥ h ( t n ). So, we derive h ( t p ) + h ( t n ) = h ( t p ) + h ( t n ) ≥ h ( t p ) + h ( t n ) ≥ h ( t m ) = 2 h ( t m ) , where the last inequality follows by the linearity of the h ( . ) function.Case 4: h ( t p ) = h ( t p ) and h ( t n ) = h ( t n ). This case follows similar to that of case 3.Therefore, given Theorem (2.3), we can apply the dynamic programming approach, and derive theportfolio value V k ( S k − H , S k ) in (2.21) at level k = a + b + H , k ∈ { H, . . . , N − } of the super-replicatingstrategy, using representation (2.18), as V k ( S u a d b , S k ) = H X j =0 e − r (cid:20) p j Φ T H +1 ( a,b ) ( a + H + 1 , b ) + q j Φ T H +1 ( a,b ) ( a, b + H + 1) (cid:21) { S k = S k − H u j d H − j } , (2.25) k = H, . . . , N − , where p j and q j , j = 0 , . . . , H are defined as in (2.19).Plugging in (2.22) for k = H, . . . , N −
2, we obtain the key recursive formula V k ( S k − H , S k ) = H X j =0 e − r (cid:20) p j V k +1 ( S k − H u , S k − H u H +1 ) + q j V k +1 ( S k − H d , S k − H d H +1 ) (cid:21) { S k = S k − H u j d H − j } , (2.26) k = H, . . . , N − . S dS u S d S udS u S d S ud S u dS u S d S ud S u d S ud S u time 0 time 1 time 2 time 3 time 4Figure 2: Asset price process S k in a 4-period binomial model Remark 2.6.
We can conclude that, when we are super-replicating backwards, the value process V k ( S k − H , S k ) in (2.21) is only required at the two extreme points S k = S k − H u H and S k = S k − H d H , because of the formon the right hand side of the recursive formula (2.26). In other words, we just use ( p u , q u ) = ( p H , q H ) and ( p d , q d ) = ( p , q ) . Therefore, similar to (2.20), the super-replication price ¯ π ( ϕ ) can be finally calculated as¯ π ( ϕ ) = e − rH max (cid:0) V H ( S , S u H ) , V H ( S , S d H ) (cid:1) . (2.27) In this section, we solve the recursive equation (2.26) and obtain the value process V k ( S k − H , S k ) for thesuper-replicating strategy explicitly. As Remark (2.6) suggests, when we super-replicate backwards, wejust need the value process at the extreme points, that is V k ( S k − H , S k − H u H ) and V k ( S k − H , S k − H d H ), k = H, . . . , N − k , F k , Q k ) for k = H, . . . , N − k = { , } e N + H , the Borel σ -algebra F k on Ω k , and e N = N − k . For every ω k = ( ω k, , . . . , ω k, e N + H ) ∈ Ω k , we define a coordinate map by Z k,m ( ω k ) = ω k,m for each m ∈ { , . . . , e N + H } .Let Q k be the probability measure under which Z k,m , m = 1 , . . . , e N + H with initial position Z k, is aMarkov chain, and for l = 1 , . . . , e N −
1, it has transition matrix Q = (cid:18) q d p d q u p u (cid:19) on { , } . (2.28)Besides, for l = e N, . . . , e N + H , Q k (cid:16) Z k, e N + H = · · · = Z k, e N = 1 | Z k, e N − = 1 (cid:17) = p u , Q k (cid:16) Z k, e N + H = · · · = Z k, e N = − | Z k, e N − = 1 (cid:17) = q u , Q k (cid:16) Z k, e N + H = · · · = Z k, e N = 1 | Z k, e N − = 0 (cid:17) = p d , Q k (cid:16) Z k, e N + H = · · · = Z k, e N = − | Z k, e N − = 0 (cid:17) = q d . (2.29)The risky asset price S k − H + m satisfies S k − H + m := S k − H u I k,m d m − I k,m , I k,m = m X l =1 Z k,l , m = 1 , . . . , e N + H . (2.30)
Remark 2.7.
Under measures Q k , k = H, . . . , N − , p u is the probability of an upward move precededwith an upward move, q u is the probability of a downward move preceded with an upward move, p d is the robability of an upward move preceded with a downward move, and q d is the probability of a downward movepreceded with a downward move. Besides, equations (2.29) are to ensure that the last H + 1 moves are alleither upward or downward. Remark 2.8.
Under measures Q k , k = H, . . . , N − , probability of a downward move preceded by a downwardmove ( q d ) is higher than the probability of a downward move preceded by an upward move ( q d ). Similar isalso true for upward moves. So, the variance of the risky asset price is higher under these measures thanthe initial measure P . Remark 2.9.
If we put H = 0 , the transition matrix (2.28) would have duplicate rows (i.e. p u = p d and q u = q d ). Therefore, in this case, the model boils down to the binomial tree model of Cox et al. (1979), andall the equations get significantly simplified accordingly. Theorem (2.4) expresses V k ( S k − H , S k − H u H ) and V k ( S k − H , S k − H d H ), k = H, . . . , N − Q k . Theorem 2.4.
For a European-style contingent claim with payoff ϕ := Φ( S N ) for some convex function Φ( S N ) ∈ L ∞ (Ω k , F k , Q k ) , k = H, . . . , N − , the value process V k ( S k − H , S k − H u H ) and V k ( S k − H , S k − H d H ) , k = H, . . . , N − for the super-replicating strategy, in an N -period binomial model with H periods of delay,can be calculated as V k ( S k − H , S k − H u H ) = e − r e N E Q k (Φ ( S N ) | Z k, = 1) , (2.31) V k ( S k − H , S k − H d H ) = e − r e N E Q k (Φ ( S N ) | Z k, = 0) . (2.32) Proof.
We need to show that (2.31) and (2.32) satisfy the recursive equation (2.26) for k = H, . . . , N − k = N −
1. For k = N −
1, it is already shown in (2.18), and for k = H, . . . , N − Z k, , (2.31) satisfies V k ( S k − H , S k − H u H ) = e − r e N E Q k (Φ ( S N ) | Z k, = 1) , = e − r e N " E Q k (Φ ( S N ) | Z k, = 1 , Z k, = 0) Q k ( Z k, = 0 | Z k, = 1)+ E Q k (Φ ( S N ) | Z k, = 1 , Z k, = 1) Q k ( Z k, = 1 | Z k, = 1) . Note that by the way the spaces (Ω k , F k , Q k ) and (Ω k +1 , F k +1 , Q k +1 ) are constructed, E Q k (Φ ( S N ) | Z k, = 1 , Z k, = 0) = e − r E Q k +1 (Φ ( S N ) | Z k +1 , = 0) , E Q k (Φ ( S N ) | Z k, = 1 , Z k, = 1) = e − r E Q k +1 (Φ ( S N ) | Z k +1 , = 1) . Also, Q k ( Z k, = 0 | Z k, = 1) = q u and Q k ( Z k, = 1 | Z k, = 1) = p u . Therefore, V k ( S k − H , S k − H u H ) = e − r e N (cid:2) p u E Q k +1 ( ϕ ( S N ) | Z k +1 , = 1) + q u E Q k +1 ( ϕ ( S N ) | Z k +1 , = 0) (cid:3) , = e − r (cid:20) p u V k +1 ( S k − H +1 = S k − H u, S k +1 = S k − H u H +1 )+ q u V k +1 ( S k − H +1 = S k − H d, S k +1 = S k − H d H +1 ) (cid:21) , which completes the proof. Similarly, it can also be shown for (2.32). Remark 2.10.
If we are interested just to find out the time- super-replication price ¯ π ( ϕ ) , we only need theprobability space (Ω H , F H , Q H ) where Ω H = { , } N . Then, we would have ¯ π ( ϕ ) = e − rN max (cid:8) E Q H (Φ ( S N ) | Z H, = 1) , E Q H (Φ ( S N ) | Z H, = 0) (cid:9) . (2.33) 11emma 2.1, whose proof can be found in the appendix, calculates E Q k (Φ ( S N ) | Z k, = 1), k = H, . . . , N −
1. For H + 1 ≤ i ≤ e N + H − , ≤ j ≤ min( i − H, e N + H − i ). Define f ( i, j ) := (cid:18) e N + H − i − j − (cid:19)(cid:18) i − Hj (cid:19) q ( j ) u q ( e N + H − i − j ) d p ( i − j − H ) u p ( j ) d . (2.34)Also for 0 ≤ i ≤ e N − , ≤ j ≤ min( i + 1 , e N − i − h ( i, j ) := (cid:18) e N − i − j − (cid:19)(cid:18) ij − (cid:19) q ( j ) u q ( e N − i − j ) d p ( i − j +1) u p ( j − d + (cid:18) e N − i − j − (cid:19) (cid:20)(cid:18) i + 1 j (cid:19) − (cid:18) ij − (cid:19)(cid:21) q ( j +1) u q ( e N − i − j − d p ( i − j ) u p ( j ) d . (2.35) Lemma 2.1.
For a function Φ( S N ) ∈ L ∞ (Ω k , F k , Q k ) , k = H, . . . , N − , the conditional expectation E Q k (Φ ( S N ) | Z k, = 1) can be explicitly calculated as E Q k (Φ ( S N ) | Z k, = 1) = e N + H X i =0 Q k (cid:16) S N = S k − H u i d e N + H − i | Z k, = 1 (cid:17) Φ( S k − H u i d e N + H − i ) , (2.36) where Q k (cid:16) S N = S k − H u i d e N + H − i | Z k, = 1 (cid:17) is given by min( i +1 , e N − i − P j =1 h ( i, j ) 0 ≤ i ≤ H ; min( i +1 , e N − i − P j =1 h ( i, j ) + min( i − H, e N + H − i ) P j =1 f ( i, j ) H + 1 ≤ i ≤ e N − p ( e N − u q u + min( e N − H − ,H +1) P j =1 f ( i, j ) i = e N − min( i − H, e N + H − i ) P j =1 f ( i, j ) e N ≤ i ≤ e N + H − p ( e N ) u i = e N + H. (2.37)Similarly, Lemma 2.2 calculates E Q k (Φ ( S N ) | Z k, = 0), k = H, . . . , N −
1. Also for H +2 ≤ i ≤ e N + H, ≤ j ≤ min( i − H − , e N + H − i + 1), define e f ( i, j ) := (cid:18) i − H − j − (cid:19)(cid:18) e N + H − ij − (cid:19) q ( j − u q ( e N + H − i − j +1) d p ( i − j − H ) u p ( j ) d + (cid:18) i − H − j − (cid:19) "(cid:18) e N + H − i + 1 j (cid:19) − (cid:18) e N + H − ij − (cid:19) q ( j ) u q ( e N + H − i − j ) d p ( i − j − H − u p ( j +1) d . (2.38)For 1 ≤ i ≤ e N − , ≤ j ≤ min( i, e N − i ), define e h ( i, j ) := (cid:18) i − j − (cid:19)(cid:18) e N − ij (cid:19) q ( j ) u q ( e N − i − j ) d p ( i − j ) u p ( j ) d . (2.39) Lemma 2.2.
For a function Φ( S N ) ∈ L ∞ (Ω k , F k , Q k ) , k = H, . . . , N − , the conditional expectation E Q k (Φ ( S N ) | Z k, = 1) can be explicitly calculated as E Q k (Φ ( S N ) | Z k, = 1) = e N + H X i =0 Q k (cid:16) S N = S k − H u i d e N + H − i | Z k, = 1 (cid:17) Φ( S k − H u i d e N + H − i ) , (2.40) 12 Φ( S ) S d S d S S ud S u S u x ∗ Φ Optimal Line V ( S , S = S u ) V ( S , S = S d )Figure 3: Super-replicating Strategy in a 2-period binomial Model with a 1-period Delay. The optimal linecharacterizes the super-replicating strategy. The slope of it is ∆ ∗ and its intercept is x ∗ . The super-replicationprice is ¯ π ( ϕ ) = max {V ( S , S = S d ) , V ( S , S = S u ) } . where Q k (cid:16) S N = S k − H u i d e N + H − i | Z k, = 1 (cid:17) is given by q ( e N ) d i = 0 min( i, e N − i ) P j =1 e h ( i, j ) 1 ≤ i ≤ H ; q ( e N − d p d + min( H +1 , e N − H − P j =1 e h ( i, j ) i = H + 1; min( i, e N − i ) P j =1 e h ( i, j ) + min( i − H − , e N + H − i +1) P j =1 e f ( i, j ) H + 2 ≤ i ≤ e N − min( i − H − , e N + H − i +1) P j =1 e f ( i, j ) e N ≤ i ≤ e N + H. (2.41) Proof.
The proof follows very similarly as that of Lemma 2.1 with only this difference that since Z k, = 0,we look for upward groups instead of downward groups. In this subsection, we first discuss Theorem 2.2 from a geometrical perspective Then, we represent thedynamic programming approach in subsection 2.5.1 geometrically For convenience, assume that interest rate r = 0, and H = 1 in this subsection.In Theorem 2.2, we discussed that in an N -period binomial model with H = N − ϕ := Φ( S N ) ∈ L ∞ (Ω , F , P ), there exist ∆ ∗ H and x ∗ such that V H ( S , S H ) = x ∗ + ∆ ∗ H S H . (2.42)This suggests that there exists a line with slope ∆ ∗ H and intercept x ∗ such that the super-replicating valuefunction V H ( S , S H ) lie on that line. Figure 3 shows this optimal line, the super-replication price, and thesuper-replicating value functions in a 2-period binomial model with 1 period of delay.It is more intuitive to demonstrate the dynamic programming approach in subsection 2.5.1 geometrically.Figure 2 shows a 4-period binomial model with 1-period delay. Figure 4 shows how to geometrically find13 Φ( S ) S d S d S ud S ud S u d S u d S u d S u S u S d S d S ud S u S u S Φ m k n l o m j h i g f d V ( S , S = S u ) V ( S , S = S d )Figure 4: Geometrical Representation of the Super replicating Strategy in a 4-period binomial Model with1-period Delay using a Dynamic Programming Approachthe super-replication price for a contingent claim with convex payoff function (Φ( . )). For convenience and toavoid a clutter of points on the x -axis, suppose ud = 1, so some of the points in the model lie on each other.Now in order to find the super-replication prices at time 3, it is necessary to consider the three 2-periodbinomial models with 1-period delay T (2 , T (1 ,
1) and T (0 , om ), ( nl ) and( mk ) show the optimal super-replication lines for each of these models respectively. As it can be seen, thereare two payoffs at either of the nodes S = S u d and S = S ud depending on which subtree is used forpricing (i.e. depending on what S is). Now, we go one period further back to find out the payoffs at time2. We need to consider two 2-period models T (1 ,
0) and T (0 , S = S u d (out of two payoffs Φ T (1 , (2 ,
1) and Φ T (0 , (2 ,
1) ) and S = S ud (out of two payoffs Φ T (1 , (1 ,
2) and Φ T (1 , (1 ,
2) ) needs to be chosen. As Theorem (2.3)suggests, the payoff functions for both of these models are convex. The lines ( jh ) and ( ig ) demonstrate theoptimal lines for these models. Similarly, to calculate the payoff at time 1, the 2-period model T (0 ,
0) needsto be used and the line ( f d ) shows the optimal line for this model. Finally, we have the super-replicationprice ¯ π ( ϕ ) = max {V ( S , S = S d ) , V ( S , S = S u ) } .
3. Continuous Time Model
In this section, we discuss the asymptotic behavior of the model. We define the probability spaces (Ω n , F n , Q n ), n ∈ N such that Ω n = { , } n , and F n is the Borel σ -algebra on Ω n . For every ω n = ( ω n , . . . , ω nn ) ∈ Ω n , wedefine a coordinate map by Z nℓ ( ω n ) = ω nℓ for each ℓ ∈ { , . . . , n } . Define the filtration {F nℓ , ℓ = 0 , . . . , n } ,where F nℓ is the σ -field σ ( Z n , . . . , Z nℓ ) generated by the first ℓ variables for ℓ = 1 , . . . , n and F is the trivial σ -field.Let µ, σ, r ∈ [0 , ∞ ), H ∈ N , T > µ n = µT δ n , σ n = σ √ T δ n , u n = exp ( µ n + σ n ) ,d n = exp ( µ n − σ n ) , r n = rT δ n , H n = HT δ n , (3.1)where the order δ n = / √ n , as in Donsker’s theorem. Remark 3.1. H characterizes the number of periods we have delayed information, which is constant inthe asymptotic analysis. However, H n is the amount of time we have delayed information, which shouldvanish in the limit. Otherwise, the super-replication price would explode and converge to the maximum ofthe contingent claim payoff function. .1. Price Process Asymptotic Define the probability measures Q n , similar to (2.28) and (2.29), such that Z nℓ , ℓ = 1 , . . . , n with initialposition Z n is a Markov chain, and for ℓ = 1 , . . . , n − H −
1, it has transition matrix Q n = (cid:18) q n,d p n,d q n,u p n,u (cid:19) on { , } . (3.2)Besides, for m = n − H, . . . , n , Q n (cid:0) Z nn = · · · = Z nn − H = 1 | Z nn − H − = 1 (cid:1) = p n,u , Q n (cid:0) Z nn = · · · = Z nn − H = − | Z nn − H − = 1 (cid:1) = q n,u , Q n (cid:0) Z nn = · · · = Z nn − H = 1 | Z nn − H − = 0 (cid:1) = p n,d , Q n (cid:0) Z nn = · · · = Z nn − H = − | Z nn − H − = 0 (cid:1) = q n,d , (3.3)where p n,u , q n,u , p n,d and q n,d are defined, similar to (2.19) with j = 0 , H , as p n,d := d Hn e r n − d H +1 n u H +1 n − d H +1 n = 1 − q n,d , p n,u := u Hn e r n − d H +1 n u H +1 n − d H +1 n = 1 − q n,u . (3.4)Then, the risky asset price S nℓ , similar to (2.1), satisfies S nℓ = S exp " ℓµ n + σ n ℓ X i =1 X ni , ℓ = 0 , . . . , n, (3.5)where X ni = 2 Z ni −
1. The following Lemma 3.1 provides asymptotic for p n,u and p n,d . Lemma 3.1.
We have p n,u = 2 H + 12( H + 1) − (cid:18) µ − r H + 1) σ + 2 H + 14( H + 1) σ (cid:19) √ T δ n + O (cid:0) δ n (cid:1) , (3.6) p n,d = 12( H + 1) − (cid:18) µ − r H + 1) σ + 2 H + 14( H + 1) σ (cid:19) √ T δ n + O (cid:0) δ n (cid:1) . (3.7) Proof.
The proof simply follows by applying Taylor’s expansion to u n , d n and r n , and plugging them in(3.4).Discretize the time interval by setting t nℓ := T ℓ/n . By interpolating over the intervals [ t nℓ − , t nℓ ) in apiecewise constant manner with ( S nℓ , ℓ = 0 , . . . , n ), we get the risky asset price process S ( n ) = ( S ( n ) t ) ≤ t ≤ T S ( n ) t := S n ⌊ nt ⌋ /T , ≤ t ≤ T, (3.8)where ⌊ . ⌋ is the floor function.The process S ( n ) has trajectories which are right continuous with left limits. Note that in particular S ( n ) t nℓ = S nℓ , ℓ = 0 , . . . , n. Here, S ( n ) under measure Q n is distributed according to a probability measure ρ n on the Skorokhod space D [0 , T ] of right continuous functions with left limits. Theorem 3.1 provides a weak convergence for thesequence ( ρ n ) n ∈ N . Theorem 3.1.
The sequence of processes ( S ( n ) ) n ∈ N converges in distribution to the process ( S t ) ≤ t ≤ T withdynamics dS t = rS t dt + e σS t dW t , ≤ t ≤ T, (3.9) where ( W t ) ≤ t ≤ T is a Brownian motion, and we have the enlarged volatility e σ = √ H + 1 σ. (3.10) 15 roof. First, note that Q n (cid:0) X nℓ = 1 | X nℓ − (cid:1) = 12 (cid:2) p n,u + p n,d + X nℓ − ( p n,u − p n,d ) (cid:3) , ℓ = 1 , . . . , n − H − , According to Lemma 3.1, we conclude that Q n (cid:0) X nℓ = 1 |F nℓ − (cid:1) = p ℓ (cid:0) ℓ, X nℓ − (cid:1) , ℓ = 1 , . . . , n − H − , where in the notation of Gruber and Schweizer (2006) p n ( ℓ, x ) = 12 [1 + φδ n + λ n x ] + O ( δ n ) , φ = − (cid:20) µ − r H + 1) σ + 2 H + 14 ( H + 1) σ (cid:21) √ T , λ n = HH + 1 + O ( δ n ) . (3.11)Now we apply a functional central limit theorem for generalized, correlated random walks in Gruber and Schweizer(2006) with a n ( t, y ) := λ n and b n ( t, y ) := φ for their Theorem 1 and Remark 3. It follows from the contin-uous mapping theorem that S ( n ) , regardless of the initial distribution of X n , converges in distribution to( S t ) ≤ t ≤ T in (3.9). In particular, since lim n →∞ λ n = H/ ( H + 1), we see the volatility e σ = s n →∞ a n ( t, Y t )1 − lim n →∞ a n ( t, Y t ) · σ = √ H + 1 σ, is constant, and larger than σ , where Y t = log S t . Remark 3.2.
The enlarged volatility in the limit is due to the gap λ n = p n,u − p n,d in (3.11) which is causedbecause of the delay in the flow of information (it would be zero when the number of delayed periods H = 0 ).In fact, this is the main source making the price process under the pricing measure more volatile. In this subsection, we discuss the volatility smile of the model, and how it evolves with the number of periods( n ). Volatility smile is the graph of Black-Scholes implied volatility with respect to the strike price. Impliedvolatility is the value of the volatility in the Black-Scholes pricing model which generates a price equal tothat of our model. Several market features, such as crashphobia, have been attributed as the culprits of themarket smile. The volatility smile has been one of the central topics in option pricing literature, and manymodels have been developed to capture it. We refer to Gatheral (2011) for more discussion in this regard.Our model with delayed information shows that delayed information exaggerates the smile. Figure 5plots the volatility smiles for call and put options in the model with and without delayed information when n = 100. In the model with delayed information ( H n = year ≈ .
52 days), we observe volatility smile,on the contrary with the model without delayed information where we get an almost flat smile, which isexpected according to the Remark 2.9. Note that in the model with delayed information, we have differentsmiles for call and put option, and that is because there is not any call-put parity, as discussed in Remark2.3.Figure 5 plots the volatility smiles for call and put options when the number of periods is very big( n = 250 , H n = , year ≈
30 seconds). We observe almostthe same flat volatility smiles for both call and put options, which can be also calculated by the theoreticalresults in 3.10.These volatility smiles in Figures 5 and 6 confirm the intuition of traders that delayed information wouldexaggerate the volatility smile, but it is not its culprit. This is because in the continuous limit, volatilityis constant and there is no smile, but in the discrete model, we can observe volatility smile. Therefore, itconveys that the smile observed in the market might have been exaggerated by the way we interact withdelayed information, and the smile might not be caused all by the market itself.16igure 5: Volatility smile for the Call and Put options in the binomial model with and without delayedinformation ( H n = year ≈ .
52 days and 0 day respectively). The parameters are σ = 0 . T = 1, r = 0, S = 40, and n = 100Figure 6: Volatility smile for the Call and Put options in the binomial model with delayed information( H n = , year ≈
30 seconds). The parameters are σ = 0 . T = 1, r = 0, S = 40, and n = 250 , . Proof of Lemma 2.1 Proof.
Note that Q k (cid:16) S N = S k − H u i d e N + H − i | Z k, = 1 (cid:17) , i = 0 , . . . , e N + H is the sum of several products of e N elements chosen out of { p u , p d , q u , q d } , and each product term corresponds to a path in the tree startingfrom the node S k − H , and ending in the node S N = S k − H u i d e N + H − i .Given equations (2.29), the last H + 1 moves need to be either upward or downward, and they contributeto as just one single move. Since it is conditioned on Z k, = 1, according to Remark (2.7), the first elementin all of the product terms is either q u or p u . For H + 1 ≤ i ≤ e N −
2, the last ( H + 1)-period move to S N = S k − H u i d e N + H − i can be both downward and upward.In the case that it is upward, we need to consider all the paths starting from S k − H to S N − = S k − H u i − d e N + H − i which consist of i − e N + H − i downward ones. There are (cid:0) e N + H − − (cid:1) of such paths, but these paths are not all equivalent and result in different product terms of e N elementschosen out of { p u , p d , q u , q d } , based on the location of the e N + H − i downward moves in the path.Note that all paths which have the same number of downward groups result in the same product terms,where a downward group is any number of consecutive downward moves preceded (if any) by an upwardmove and also succeeded (if any) by an upward move. For example, both of the sequences րցցցրց and ցցրրցց have two groups of ց moves. The reason for studying downward groups is that the startingelement in all of them is q u .In this notation, j corresponds to the number of groups which starts from 1 (assuming that there existsat least one downward move) and can reach to min( i − H, e N + H − i ). Notice that there are (cid:0) e N + H − i − j − (cid:1)(cid:0) i − Hj (cid:1) paths which have exactly j groups. Therefore, along those path the power of both q u and p d is j andconsequently, the powers of q d and p u are respectively e N + H − i − j and i − j − H . Here f ( i, j ) in equation(2.34) corresponds to these paths.The second case is that the last ( H + 1)-period move is downward. Then, we need to consider all thepaths starting from the node S k − H to S N − = S k − H u i d e N + H − i − which consist of i upward moves and e N + H − i − N − N − (cid:0) e N − i − j − (cid:1)(cid:0) ij − (cid:1) paths which have exactly j groups such that the last 1-period movefrom N − N − q ( j ) u q ( e N − i − j ) d p ( i − j +1) u p ( j − d , andthere are (cid:0) e N − i − j − (cid:1) [ (cid:0) i +1 j (cid:1) − (cid:0) ij − (cid:1) ] paths which have exactly j groups such that the last 1-period move from N − N − h ( i, j ) in equation (2.35) takes all these paths into account.For H + 1 ≤ i ≤ e N −
2, it is necessary to use both f ( i, j ) and h ( i, j ) to take into account that the last( H + 1)-period move can be both upward and downward. The same reasoning works for 0 ≤ i ≤ H and e N ≤ i ≤ e N + H −
1, but here the last ( H + 1)-period move can only be downward for 0 ≤ i ≤ H and upwardfor e N ≤ i ≤ e N + H −
1. For i = e N − H + 1)-period move is downward and i = e N + H , thefunctions f ( i, j ) and h ( i, j ) cannot be used because in all of the paths from S k − H to S N − = S k − H u e N + H − ,there is not any downward move at all to make a downward group (i.e., j = 0). References
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