MATH3090/7039: Financial mathematics Lecture 6
Introduction
General properties of arbitrage-free prices
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One-period binomial model
Introduction
General properties of arbitrage-free prices
One-period binomial model
MATH3090/7039 – the second half
• MATH 3090/7039 (second half): mathematical foundation of option (derivatives) pricing
Or more briefly, “Option (derivatives) pricing theory”
• MATH4090/7049 (semester 2): Computations in Financial Mathematics
Or more briefly, “Computational option pricing”
• The second half of MATH3090/7039 cover the necessary financial mathematics foundations for MATH4090/7049.
• Option pricing is meant in a broad sense: the pricing (and hedging) of options and other derivatives (at an introductory level)
The main idea
A financial derivative (e.g. an option) is a financial contract whose value is defined in terms of some underlying asset(s) which already exists on the market.
A derivative cannot therefore be priced arbitrarily in relation to the underlying asset prices, if we want to avoid mispricing between the derivative price and the underlying price.
We thus want to price a derivative in a way that is consistent with the underlying asset prices given by the market.
We are not trying to compute the price of the derivative in some “absolute” sense. The idea instead is to determine the price of the derivative in terms of the market prices of the underlying assets.
Frictionless markets
Asset: any financial product whose value is quoted or can, in principle, be measured.
We will specify a set of basic assets, and a set of times.
At any such time, each basic asset has a market price, and we can buy
or sell arbitrary quantities at that time, at that price.
In other words, assume frictionless markets (unless otherwise stated).
In particular,
• No transaction costs: no commissions, no taxes
• No market impact
• Can hold fractional quantities of assets
• Can short: sell what you do not own, and hold negative quantities
Introduction
General properties of arbitrage-free prices
One-period binomial model
• The market lives in a probability space, with a physical probability measure P.
• The market consists of N assets, each with time-t price Xti, i = 1,…,N.
X0 = (X01,…,X0N) is non-random and Xt>0 = (Xt1,…,XtN) is (possibly) random.
• In this section, we do not make any assumptions about which times t ∈ (0, T ) exist in our market. So our general analysis applies to a one-period model (which includes only 0 and T), as well as a continuous-time model (which includes all of t ∈ (0, T )), as well as any intermediate model.
Examples of assets
• A zero-coupon bond or discount bond with maturity T :
Each unit pays at time T a fixed amount, which we take to be 1.
• Non-dividend-paying stock: each unit has random time-t price St ≥ 0.
• A bank account: default risk-free asset – an initial investment of $1 rolled continually at the risk-free interest rate, possibly random, r.
Each unit has non-random time-t price
r(u)du . Note: B solves the differential equation:
dB(t) = r(t)B(t) with B(0) = 1. dt
Static portfolios
A static portfolio is a vector of quantities
Θ ≡ (θ1,…,θN), θi ∈ R, i = 1,…,N,
where each θi denotes the number of units of asset i. Here, θi is a non-random and constant in time.
The time-t value of the portfolio Θ is
Vt =Θ·Xt =θ1 Xt1 +…+θNXtN.
If we are dealing with multiple portfolios, we may give V a superscript to indicate which portfolio.
A static portfolio Θ is a “type 1” arbitrage if its value V satisfies
V0 = 0, and both:
(Zero initial investment, and no chance of loss, some chance of gain.)
A static portfolio Θ is a “type 2” arbitrage if its value V satisfies V0 <0and(VT ≥0)=1
(Initially receive a credit . . . which you will definitely not repay.)
A static portfolio Θ is an arbitrage if it’s either type 1 or type 2 arbitrage.
P(VT ≥0)=1 P(VT >0)>0
No-arbitrage assumption
• Prices which admit arbitrage are, in some sense, incorrect. Existence of arbitrage is a severe form of inconsistency and mispricing that we want to avoid.
• Assume no-arbitrage, unless otherwise stated.
Thus, when we try to price some derivative, we are looking for an arbitrage-free price.
• Some authors define arbitrage without “type 2”.
The distinction between our definition and their definition is
essentially harmless, because:
If there exists an asset whose price is always positive, then type 1 arb exists whenever type 2 arb exists (See Tutorial 7).
(Aside: Want to know more about arbitrage in real life? Read “Flash Boys: A Wall Street Revolt” by )
Absent arbitrage, prices satisfy consistency conditions
Suppose portfolio Θa super-replicates portfolio Θv, which means that P(VTa≥VTb) = 1. Then V0a≥V0b, otherwise arbitrage exists.
If instead V0a < V0b, then construct portfolio Θ = Θa − Θb. (In other words, go long Θa and short Θb.) Itstime-0valueisV0 =V0a−V0b <0.
Its time-T value is VT = VTa − VTb ≥ 0 with probability 1. Hence Θ is an arbitrage.
In this proof, we used a general technique for constructing arbitrage
• Go long what is cheap (undervalued), and short what is rich (overvalued). In other words: buy low, sell high
The law of one price
Likewise, if Θa sub-replicates portfolio Θb, meaning P(VTa ≤ VTb) = 1, then V0a ≤ V0b. By combining the two inequalities, therefore
IfP(VTa =VTb)=1thenV0a =V0b. In other words, if Θa replicates Θb, then V0a = V0b.
• This is the law of one price. Any two static portfolios with identical future payouts must have identical current prices.
• “You can summarize the essence of quantitative finance,” according to , as follows
“If you want to know the value of a security, use the price of another security [or portfolio of securities] thats as similar to it as possible.”
Discount bond
Consider a zero-coupon bond Z maturing at T, and a bank account B. If interest rate rt is non-random, then
Z0 = 1/BT .
In particular, if r is constant, then Z0 = e−rT .
A portfolio consisting of 1/BT units of the bank account has time-T value (1/BT ) × BT = 1, which is identical to ZT = 1.
So the time-0 values of the portfolios must be equal: 1/BT = Z0. Intermsofr,wehaveZ0 =exp(−T rtdt).
In particular, if r is constant then Z0 = exp(−rT ).
Forward contract: definition
Consider a random variable ST whose value is revealed at time T .
• A forward contract on ST with maturity / delivery date T and nonrandom delivery price K obligates the holder to, at time T , pay K and receive ST .
• So the forward contract has payoff ST − K. Payoff diagram
• Forward contract is an example of a derivative a security whose payout is contractually related to some underlying variable.
Forward contract: valuation
Consider a forward contract on a non-dividend-paying stock S, with delivery date T and any delivery price K.
Then the time-0 value of the forward contract is S0 − KZ0. Proof.
The portfolio
(1 share, -K bonds) has time-T value ST − K.
The forward contract also has time-T value ST − K.
So the time-0 value of the forward contract must equal the time-0 value of the replicating portfolio, which is 1 × S0 − K × Z0.
Forward price
The forward price F0 which sets at time 0 for delivery at time T is the delivery price K such that the forward contract has zero value at time 0.
• A forward price is not the same thing as the value of a forward contract
• A forward on a non-dividend-paying stock S has time-0 value S0 − K × Z0.
Choice of K that makes value zero is S0/Z0. Thus, F0 = S0/Z0. If r is constant, then F0 = S0erT .
Note: This does not depend on the dynamics of S.
Forward price example
If r = 0.04 and the share price today is S0 = 500, and you and I want to enter costlessly today into a contract for time-1 delivery of S in exchange for a delivery price to be paid at time-1, the only arbitrage-free way to set that delivery price is
500 × e0.04×1 ≈ 520.41
Homework: construct an arbitrage when the delivery price is different from 520.41, say the delivery price is 600.
Affine payoff
More generally, consider the following “affine” contract on a non-dividend-paying stock S. The contract pays, by definition,
where a and b are constants. Then its time-0 value is
because a bonds and b shares perfectly replicate the payoff.
Call option: definition
A European call option with strike K and expiry T on an underlying asset is a contract between two parties, the holder and the writer, according to which:
• the holder has the right, but not the obligation, at time T to buy from the writer the underlying asset at price K,
• the writer is obliged to sell, should the holder decide to buy. Payoff diagram (from the holder’s perspective): (ST − K)+, where
x+ = max(0, x)
Attimet≤T,thecalloptionissaidtobeinthemoneyifSt >K,at themoneyifSt =K,outofthemoneyifSt
Put-call parity
Let C0(K, T ) and P0(K, T ) be time-0 prices of a call and put, with identical (K,T), on a non-dividend-paying stock S. Let Z0(T) be the time-0 price of a T-maturity discount bond. Then
Payoff diagram:
C0(K,T)=P0(K,T)+S0 −KZ0(T)
Payoffs are equal, hence prices at earlier date are equal. How recent is put-call parity?
Put option: bounds wrt underlying, and wrt other puts
The time-0 price of a put on a non-dividend-paying stock S satisfies (KZ0 − S0)+ ≤ P0 ≤ KZ0.
The time-0 put prices P0(K1) and P0(K2), for strikes K1 < K2 (with same expiry, on same underlying) satisfy
0 ≤ P0(K2) − P0(K1) ≤ (K2 − K1)Z0.
Compare payoffs. Or use put-call parity.
Put option: bounds wrt puts, revisited
If K1 < K2 then P0(K1) ≤ P0(K2). Proof by comparing payoffs
Better yet, because
General payoffs
Using static positions in T-expiry bonds, forwards, calls, and puts on S, we can replicate or bound (superreplicate, subreplicate) general functions of ST
• Use bonds to adjust level.
• Use forwards (or perhaps S itself) to adjust slope
• Use calls (or puts) to adjust convexity and concavity
Introduction
General properties of arbitrage-free prices
One-period binomial model
Model specification
• Times 0 and T.
• Time T : only two states, up and down, each with (+) probability
• Stock S: ST takes value Su (resp. Sd) in the up (resp. down) state, where Su > Sd.
• Option C: Let CT take value Cu (resp. Cd) in the up (resp. down) state.
• Bank account: BT = erT in both states
• S0, Su, Sd, Cu, Cd, K, r: given constants
Example: call option, Cu = max{Su − K, 0}, Cd = max{Sd − K, 0}.
Option pricing via replication
Given S0, Su, Sd, Cu, Cd, K, r, find time-0 option price C0. Solution: construct portfolio (α,β) of (bank account, stock) that
replicates the option.
Want P(time-T portfolio value = time-T option value) = 1.
αerT +βSu =Cu, αerT +βSd =Cd.
Solve for α and β:
β = Cu − Cd ,
α=e−rT(Cd −βSd)=e−rT(Cu −βSu).
By no-arbitrage, option value today (time-0) = portfolio value today. Conclude:
C0 =αB0 +βS0 =α+βS0
Rewrite as an expectation
C0 =α+βS0 =e−rTCd −βSd +βS0erT=e−rTCd +β(S0erT −Sd) =e−rTCd+Cu−Cd(S0erT −Sd)
Su − Sd
=e−rTp C +(1−p )C uuud
pu = S0erT −Sd. Su − Sd
(Note, 0 < pu < 1, otherwise arbitrage exists.) Thus,
C0 = e−rT E[CT ],
where E is the expectation wrt the probability measure P which assigns probability pu to up-move and 1 − pu to down-move.
Two probability measures
• P is called a risk-neutral or risk-adjusted or pricing or martingale measure. Important in derivatives pricing.
• P is called the actual or real-world or statistical or physical probability measure. It has no direct relevance here.
It’s indirectly relevant because it affects the prices of the basic asset (S).
Given the specification of this model, we do not care about the value of P(up)/(down) for the purpose of pricing.
• Irrelevance of physical probabilities ?!
The [First] Fundamental Theorem of Asset Pricing
there exists a probability measure P, equivalent to P. No arbitrage ⇐⇒ such that the discounted prices of all tradable assets
are martingales wrt P.
Definition:
(P equivalent to P means that, ∀ event A, P(A) = 0 ⇔ P(A) = 0.)
In this one-period model, Mt is a martingale means that M0 = EMT . (Today’s level equals to today’s expectation of tomorrow level.)
Thus, to say that the discounted price Xt/Bt is a martingale here means that X0/B0 = E(XT /BT ); equivalently X0 = e−rT EXT .
Proof of Fundamental Theorem
We prove in the case of the one-period binomial model, with an arbitrary number of assets, including a stock and a bank account. (True much more generally, but need technical assumptions)
(See MATH4091/7091: Financial Calculus for more details.)
No arb. ⇒ ∃P:
We already proved it. The measure P can be explicitly calculated as P(up) = pu and P(down) = 1 − pu, where pu was given above. Also we need to, and can, verify that no-arb implies 0 < pu < 1.
Proof of Fundamental Theorem
∃P ⇒ No arb:
We will show that type-1 arb does not exists, no need to concern with type-2 arb (See Tutorial.)
Consider any static portfolio Θ of assets X. Each individual asset price satisfies the martingale property; hence the portfolio value does also (by linearity of E):
Θ·X0 =e−rTE(Θ·XT)
If V0 ̸= 0, then the portfolio is not an arb (done!)
IfV0 =0thenE(VT)=0.
If P(VT ≥ 0) < 1, then the portfolio is not an arb (done!)
If P(VT ≥ 0) = 1, then P(VT = 0) = 1 because a nonnegative, zero-expectation, random variable must be identically zero. (IfP(VT >0)>0thenP(VT >ε)>0forsomeε>0,hence
EVT =E[VTIVT>ε +VTIVT≤ε]≥E[εIVT>ε]=εP(VT >ε) Thus P(VT > 0) = 0, so Θ is not an type-1 arbitrage.
Proof of Fundamental Theorem (cont)
In the above steps, we have showed that Θ is not an type-1 arbitrage using P.
However, arbitrage is defined in terms of P.
How could we conclude from the arguments that I showed you that P⇒ Noarb?
Option pricing via the Fundamental Theorem
Suppose P assigns prob. pu to up- and (1 − pu) to down-move. Infer risk-neutral probabilities via S:
S0 = e−rT EP[ST ] = e−rT [puSu + (1 − pu)Sd], Solve for pu:
pu = S0erT −Sd. Su − Sd
(Note, 0 < pu < 1, otherwise arbitrage exists!) Now use pu to price the option:
C =e−rTEP[C ]=e−rTp C +(1−p )C . 0Tuuud
Sanity check:
α+βS =e−rTC −βS +βS erT=...=e−rTp C +(1−p )C ,
replication
which is what we expected.
risk-neutral
• Frictionless market
• No arbitrage (our guiding principle)
• Bounds forcall/put options (without any modelling assumptions) • One-period binomial model assumption
◦ physical measure v.s risk-neutral measure
◦ 1st theorem: no-arbitrage ⇔ existence of an equivalent
martingale measure
◦ pricing by replication and the fundamental theorem
To be covered next
• One-period, multiple discrete states • Multi-period, multiple discrete states
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