MATH3090/7039: Financial mathematics Lecture 7
One-period, multiple discrete states
Multi-period, multiple discrete states
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One-period, multiple discrete states
Multi-period, multiple discrete states
Replication in two-state model
Recall: we replicated the option using β = (Cu − Cd)/(Su − Sd) shares.
Match the slope by choosing the appropriate number of shares. Match the level using the appropriate number of bank account units.
Another view: For each asset, write its payoff as a vector of up-state and down-state payoffs. Replication possible because
Cu erT Su
C ∈span erT , S dd
A three-state model
• Times 0 and T. No intermediate trading; all portfolios are static
• Up, middle, down state at time T, each with positive probability
• Bank account: Each unit has time-t value Bt = ert, t = {0, T }
• StockS: LetST takevaluesSu >Sm >Sd inup,mid,down states respectively.
• Option C: Let CT take values Cu, Cm, Cd in up, mid, down states
Replication in three-state model
Example: Letr=0,letS0 =100,Su =130,Sm =100,Sd =80. Consider a 90-call: Cu = 40, Cm = 10, Cd = 0. Can we replicate it?
Can replicate option on the upside by holding 1 share of S.
Can replicate option on the downside by holding 0.5 shares of S. But can’t simultaneously replicate both risks.
Replication and spanning
Another view: Write payoffs as vectors
up-payoff 1 130 Bank acct payoff mid-payoff = 1 . Stock payoff 100
down-payoff 1 80
And the 90-call payoff is
40 1 130
10̸∈span 1,100
Complete markets
Market is said to be complete if every random variable CT can be replicated, meaning there exists Θ such that P(Θ · XT = CT ) = 1, whereΘ=(θ1,…,θN)andXT =(XT1(ω),…,XTN(ω))
• The market of {bank acct, stock} in the two-state model is complete.
• The market of {bank acct, stock} in the three-state model is incomplete, because the call payoff could not be replicated .
Martingale measures and completeness
Any positive pu, pm, pd such that
pu + pm + pd = 1
130pu + 100pm + 80pd = 100 yields a martingale measure. Two examples:
(pu, pm, pd) = (0.20, 0.50, 0.30) (pu, pm, pd) = (0.30, 0.25, 0.45)
Martingale measure exists but is not unique.
The [first] fundamental theorem
The first fundamental theorem still holds in the multiple-state (but one period) setting with an arbitrary number of assets, regardless of completeness.
No arbitrage ⇐⇒ ∃ equivalent martingale measure P
Proof: skipped (covered in MATH4091/7091)
The [second] fundamental theorem
Theorem: An arbitrage-free market is complete iff there exists a unique martingale measure.
(Note: if time permits, we will revisit both fundamental theorems when we introduce different numeraires. For now, the only numeraire is B.)
Proof: skipped (covered in MATH4091/7091)
Discussions
Two approaches to price derivatives • replication
• risk-neutral/martingale measure
What if market is incomplete (e.g. the one period, multiple states example)?
• If a derivative can be replicated perfectly, the price is still unique. All martingale measures give the same price as the replication technique.
• Some derivatives are not replicable
◦ the perfect replication approach breaks down
◦ multiple martingale measures imply martingale approach does not zero in a unique price for derivatives
But we have been using only static portfolio. What if we allow intermediate trading?
One-period, multiple discrete states
Multi-period, multiple discrete states
Now allow intermediate trading
Start with an example. Two periods, so three time instants t = 0, 1, 2. Four outcomes Ω = {UU, UD, DU, DD}. Bank acct with r = 0.
Let S0 = 100.
Let S1(UD) = S1(UU) = 115 and S1(DD) = S1(DU) = 75.
Let S2(UU) = 150, S2(UD) = S2(DU) = 100, S2(DD) = 50.
Replicate a 90-call with expiry T = 2? No way using a static portfolio of bank acct and stock. But suppose we allow trading at time t = 1.
Now allow intermediate trading (cont.)
Filtration
In multi-period models, we need to represent the revelation of information as time passes.
A filtration {Ft : t ≥ 0} represents, for each t, all information revealed at or before time t.
Example: in the previous model
• F1: the information about whether the first step was U or D.
• F2: the information about whether the first two steps were UU, UD, DU, or DD.
More specifically, the sample space is Ω = {UU,UD,DU,DD} and • F1 = the σ-algebra generated by {UU,UD}, {DU,DD}
• F2 = the σ-algebra generated by {UU} {UD}, {DU}, {DD}
Adapted processes
A stochastic process Yt is adapted to Ft if for each t the the value of Yt is determined by the information in Ft.
(More specifically, if Ft is generated by a partition, then the requirement is that Yt be constant on each part of the partition (information set).)
• Construct our models so that asset prices Xt are adapted to Ft Interpretation: At time t the market has revealed the price Xt.
• Define our trading strategies to require that the quantities θt be adapted to Ft.
Interpretation: Allow trading, but determined only by what has been revealed, not by future outcomes.
Trading strategy
A trading strategy on t = 0,1,··· ,T is a sequence Θt adapted to Ft. Let us agree to view Θt as the vector of quantities of the tradeable
assets held after all time-t trading at prices Xt.
Say that the trading strategy is self-financing if for all t > 0, Θt−1 · Xt = Θt · Xt.
This implies that the change in portfolio value from time t to t + 1 is
Vt+1−Vt =Θt+1·Xt+1−Θt·Xt =Θt·Xt+1−Θt·Xt =Θt·(Xt+1−Xt)
So the change in value is fully attributable to gains and losses in asset prices.
Note that we can sum from t = 0 to t = T − 1
VT − V0 = Θt · (Xt+1 − Xt)
This looks like a discrete version of the stochastic integral
Idea: P&L from a self-financing trading strategy is a stochastic integral, namely the integral of quantity with respect to price.
Arbitrage in a multi-period model
Arbitrage is a self-financing trading strategy Θt whose value Vt = Θt · Xt satisfies
V0 = 0, and both:
P(VT ≥0)=1 P(VT >0)>0
V0 <0, andP(VT ≥0)=1
Note that static portfolios are a special case of self-financing trading strategies, so the previous definition is consistent with this one. This definition extends the notion of arbitrage beyond static strategies, to self-financing ones.
Related theorems
Theorem: If Θa and Θb are self-financing strategies such that P(ΘaT ≥ ΘbT ), then Θa0 ≥ Θb0. Otherwise, arbitrage exists.
Theorem: If Θa and Θb are self-financing strategies such that P(ΘaT = ΘbT ), then Θat = Θt0 , for all t ≤ T . Otherwise, arbitrage exists.
Martingales
The time-t conditional expectation of a random variable Y , written EtY or E(Y |Ft)
is the expectation of Y , conditional on the information that has been revealed up to time t.
(Specifically, if Ft is generated by a finite partition, then on each part R of the partition, define E(Y |Ft) to be the expectation of Y , conditional on the occurrence of event R.)
Say that Mt is a martingale with respect to filtration Ft if Mt is adaptedtoFt andforallt