Lecture 3: Review of mathematical finance and derivative pricing modelswang913/teaching/stat598w2014/LectureNotes/... · Lecture 3: Review of mathematical nance and derivative pricing

Lecture 3: Review of mathematical finance andderivative pricing models

Xiaoguang Wang

STAT 598W

January 21th, 2014

(STAT 598W) Lecture 3 1 / 51

Outline

1 Some model independent definitions and principals

2 European Options PricingBinomial TreesBlack-Scholes Model

3 American Options Pricing


Outline





Self Financing Portfolio

Notations:

N = the number of different types of assets or stocks.

hi (t) = number of shares of type i held during the period [t, t + ∆t).

h(t) = the portfolio [h1(t), · · · , hN(t)] held during period t.

c(t) = the amount of money spent on consumption per unit timeduring the period [t, t + ∆t).

Si (t) = the price of one share of type i during the period [t, t + ∆t).

V (t) = the value of the portfolio h at time t.

Definition

A self-financing portfolio is a portfolio with no exogenous infusion orwithdrawal of money (apart from the consumption term c). It must satisfy

dV (t) = h(t)dS(t)− c(t)dt


Dividends

Definition

We take as given the processes D1(t), ·,DN(t), where Di (t) denotes thecumulative dividends paid to the holder of one unit of asset i during theinterval (0, t]. If Di has the structure

dDi (t) = δi (t)dt

for some process δi , then we say that asset i pays a continuous dividendyield.

Taking into account the dividend payments, a self-financing portfolioshould now satisfy

dV (t) = h(t)dS(t) + h(t)dD(t)− c(t)dt


Free of Arbitrage

Definition

An arbitrage possibility on a financial market is a self-financed portfoliosuch that

V h(0) = 0

P(V h(T ) ≥ 0) = 1

P(V h(T ) > 0) > 0

We say the market is arbitrage free if there are no arbitrage possibilities.

In most cases, we assume that the market of interest is arbitrage free.


Martingale measure

Definition

Consider a market model containing N assets S1, · · · ,SN and fix the assetS1 (in most cases this will be the risk free rate account) as the numeraireasset. We say that a probability measure Q defined on Ω is a martingalemeasure if it satisfies the following conditions:1. Q is equivalent to P, i.e.

Q ∼ P

2. For every i = 1, · · · ,N, the normalized (discounted) asset price process

Z it =

S it

S1t

is a martingale under the measure Q.


First Fundamental Theorem

Theorem

Given a fixed numeraire, the market is free of arbitrage if and only if thereexists a martingale measure Q.

Under the martingale measure Q, every discounted (normalized) priceprocess (either underlying or derivative) is a martingale.


Complete market

Definition

A contingent claim is a stochastic variable X of the form

X = Φ(ST ),

where the contract function Φ is some given real valued function.A given contingent claim X is said to be reachable if there exists aself-financing portfolio h such that

V hT = X ,P − a.s.

with probability 1. In that case we say that the portfolio h is a hedgingportfolio or a replicating portfolio. If all contingent claims can bereplicated we say that the market is (dynamically) complete.


Pricing principal

A reachable claim X must be priced properly such that there is noarbitrage opportunity caused by this pricing.

Theorem

If a claim X is reachable with replicating (self-financing) portfolio h, thenthe only reasonable price process for X is given by

Π(t; X ) = V ht , 0 ≤ t ≤ T


General pricing formula: martingale approach

Theorem

In order to avoid arbitrage, a contingent claim X must be priced accordingto the formula

Π(t; X ) = S0(t)EQ

[X

S0(T )|Ft

]where Q is a martingale measure for [S0, S1, · · · ,SN ], with S0 as thenumeraire.In particular, we can choose the bank account B(t) as the numeraire.Then B has the dynamics

dB(t) = r(t)B(t)dt

where r is the (possibly stochastic) short rate process. In this case thepricing formula above reduces to

Π(t; X ) = EQ[e−

∫ Tt r(s)dsX |Ft

](STAT 598W) Lecture 3 11 / 51

Second Fundamental Theorem

Theorem

Assuming absence of arbitrage, the market model is complete if and only ifthe martingale measure Q is unique.

For incomplete market, different choices of Q will generally give rise todifferent price process for a fixed claim X . However, if X is attainable(reachable) then all choices of Q will produce the same price process,which then is given by

Π(t; X ) = V (t; h) = EQ[e−

∫ Tt r(s)dsX |Ft

]where h is the hedging portfolio. Different choices of hedging portfolio (ifsuch exist) will produce the same price process.


General Pricing formula: PDE approach

Assume the market is complete, under the martingale measure(risk-neutral) Q, any derivative price process Π(t) must have the followingdynamics

dΠ(t) = rΠ(t)dt + σΠ(t)dW (t)

where W is a Q− Wiener process, and σΠ is the same under Q as under P.With Π above, the process Π(t)

B(t) is a Q martingale, i.e. it has a zero driftterm.Assume Π(t) has the form Π(t) = F (t, S(t)), then apply Ito’s formula onF (t,S(t))

B(t) and set the drift term as zero. Then we will come up with a PDFas follows

Ft + rxFx +1

2x2σ2Fxx − rF = 0,

F (T , x) = Φ(x)


Feynman-Kac

Assume that F is a solution to the boundary value problem

∂F

∂t(t, x) + µ(t, x)

∂F

∂x+

1

2σ2(t, x)

∂2F

∂x2(t, x) = 0

F (T , x) = Φ(x)

Assume furthermore that process

σ(s,Xs)∂F

∂x(s,Xs)

is in L2, where X is defined below. Then F has the representation

F (t, x) = Et,x [Φ(XT )]

where X satisfies the SDE

dXs = µ(s,Xs)ds + σ(s,Xs)dWs ,

Xt = x


Greeks

Let P(t, s) denote the pricing function at time t for a portfolio based on asingle underlying asset with price process St . The portfolio can thusconsist of a position in the underlying asset itself, as well as positions invarious options written on the underlying asset. For practical purpose it isoften of vital importance to have a grip on the sensitivity of P withrespect to the following

Price changes of the underlying asset

Changes in the model parameters

Definition

∆ =∂P

∂s, Γ =

∂2P

∂s2, ρ =

∂P

∂r

Θ =∂P

∂t,V =

∂P

∂σ


Incomplete market: market price of risk

Assume that a derivative price F has the following dynamics undermeasure P:

dF = αFFdt + σFFdW

Definition

Assume that the market for derivatives is free of arbitrage. Then thereexists a universal process λ(t) called such that, with probability 1, and forall t, we have

αF (t)− r

σF (t)= λ(t)

regardless of the specific choice of the derivative F . The λ(t) is called”market price of risk”, which can be interpreted as the ”risk premiumper unit of volatility”.


Incomplete Market Pricing formula: PDE approach

Assuming absence of arbitrage, the pricing function F (t, x) of a T -claimΦ(X (T )) solves the following boundary value problem

Ft(t, x) +AF (t, x)− rF (t, x) = 0, (t, x) ∈ (0,T )× R

F (T , x) = Φ(x), x ∈ R

where

AF (t, x) = µ(t, x)− λ(t, x)σ(t, x)Fx(t, x) +1

2σ2(t, x)Fxx(t, x)

and we assume that the asset X (t) has the following dynamics undermeasure P:

dX (t) = µ(t,X (t))dt + σ(t,X (t))dW (t)

Here W is a standard scalar P-Wiener process.


Incomplete market pricing: Martingale approach

Theorem

Assuming absence of arbitrage, the pricing function F (t, x) of the T -claimΦ(X (T )) is given by the formula

F (t, x) = e−r(T−t)EQt,x [Φ(X (T ))]

, where the dynamics of X under the martingale measure Q are given by

dX (t) = µ(t,X (t))− λ(t,X (t))σ(t,X (t))dt + σ(t,X (t))dW (t)

Here W is a Q-Wiener process, and the subscripts t, x indicate as usualthat X (t) = x.


Outline





European Call and Put Options

Definition

A European Call (Put) Option on an underlying asset S with strikeprice K and exercise date T is a contract written at time t = 0 with thefollowing properties:

The holder of the contract has, exactly at time t = T , the right tobuy (sell) one unit of the asset at the strike price K .

The holder of the option has no obligation to buy (sell) the asset.

The pay-off function for a European Call option is

X = Φ(S(T )) = max(S(T )− K , 0)


Model Set Up

Discrete time running from t = 0 to t = T , where T is fixed.

There are two underlying assets, a risk-free bond with price processBt and a stock with price process St .

Assume a constant deterministic short rate of interest R. AndBn+1 = (1 + R)Bn,B0 = 1.

Sn+1 = Sn·Zn, S0 = s.

Z0, · · · ,ZT−1 are assumed to be i.i.d. stochastic variables, takingonly two values u and d with probabilities:P(Zn = u) = pu, P(Zn = d) = pd


Portfolio Strategy

Definition

A portfolio strategy is a stochastic process

ht = (xt , yt); t = 1, · · · ,T

such that ht is a function of S0,S1, · · · ,St−1. For a given portfoliostrategy h we set h0 = h1 by convention. The value process correspondingto the portfolio h is defined by

V ht = xt(1 + R) + ytSt

A self-financing portfolio strategy h will satisfy for all t = 0, · · · ,T − 1,

xt(1 + R) + ytSt = xt+1 + yt+1St


No arbitrage and Completeness

Theorem

The conditiond ≤ (1 + R) ≤ u

is a necessary and sufficient condition for absence of arbitrage.

The binomial model defined above is complete.


Martingale Measure

Theorem

The martingale measure probabilities qu and qd which must satisfy

s =1

1 + REQ [St+1|St = s]

are given by

qu =(1 + R)− d

u − d

qd =u − (1 + R)

u − d


Pricing formulas

Theorem

The arbitrage free price at t = 0 of a T -claim X is given by

Π(0; X ) =1

(1 + R)TEQ [X ]

where Q denotes the martingale measure, or more explicitly

Π(0; X ) =1

(1 + R)T

T∑k=1

(T

k

)qkuqT−k

d Φ(sukdT−k)


Hedging portfolio

A T -claim X = Φ(ST ) can be replicated using a self-financing portfolio. IfVt(k) denotes the value of the portfolio at node (t, k) then Vt(k) can becomputed recursively byVt(k) =

1

1 + RquVt+1(k + 1) + qdVt+1(k)

VT (k) = Φ(sukdT−k)

(1)

(2)

And the portfolio strategy is given byxt(k) =

1

1 + R

uVt(k)− dVt(k + 1)

u − d

yt(k) =1

St−1

Vt(k + 1)− Vt(k)

u − d

(3)

(4)


Model set up

Definition

The Black-Scholes model consists of two assets with dynamics given by

dB(t) = rB(t)dt

dS(t) = αS(t)dt + σS(t)dW (t)

Definition

A contingent claim with date of maturity (exercise date) T , also called aT -claim, is ant stochastic variable X ∈ FS

T . A contingent claim X iscalled a simple claim if it is of the form X = Φ(S(T )).


Model assumptions

The derivative instrument in question can be bought and sold on amarket.

The market is free of arbitrage.

The price process for the derivative asset is of the form

Π(t;X ) = F (t,S(t))

where F is some smooth function.


Black-Scholes formula

Theorem

The price of a European call option with strike price K and time ofmaturity T is given by the formula Π(t) = F (t,S(t)), where

F (t, s) = sN[d1(t, s)]− e−r(T−t)KN[d2(t, s)]

Here N is the cumulative distribution function for the N(0, 1) distributionand

d1(t, s) =1

σ√

T − t

ln( s

K

)+

(r +

1

2σ2

)(T − t)

d2(t, s) = d1(t, s)− σ

√T − t


Other European type of derivatives

X = S(T )− K (Forward contract)

X = max

[1

T

∫ T

0S(t)dt − K , 0

](Asian Option)

X = S(T )− inf0≤t≤T

S(t) (Lookback contract)

X = 1S(T )>K (cash-or-nothing call, exotic option)

Down and Out (Barrier Options):

X =

Φ(S(T )), if S(t) > L for all t ∈ [0,T ]

0, if S(t) > L for some t ∈ [0,T ]


Outline





Definitions

Definition

A American Call (Put) Option on an underlying asset S with strikeprice K and exercise date T is a contract written at time t = 0 with thefollowing properties:

The holder of the contract has, up to time t = T , the right to buy(sell) one unit of the asset at the strike price K .

The holder of the option has no obligation to buy (sell) the asset.

It is well known that for an American call option written on an underlyingstock without dividends, the optimal exercise time τ is given by τ = T .Thus the price of the American option coincides with the price of thecorresponding European option. But for American put options, it is muchharder and there is even no analytical solution.


Optimal stopping problem

For an American Call Option it is not hard to understand that the price attime zero should be of the following form

Π(0) = max0≤τ≤T

EQ [e−rτ maxSτ − K , 0]

where the stock dynamics under the risk neutral measure Q are given by

dSt = rStdt + σtStdWt

Similarly, for American put, we have

Π(0) = max0≤τ≤T

EQ [e−rτ maxK − Sτ , 0]

In order to solve these pricing problems, we need to take use of someOptimal Stopping Theories.


Optimal Stopping Theory: definitions

Definition

A nonnegative random variable τ is called an (optional) stopping timew.r.t. the filtration F = Ftt≥0 if it satisfies the condition

τ ≤ t ∈ Ft , for allt ≥ 0

In general we refer to an integrable process Z as the ”reward process”which we want to maximize. Then we can formulate our problem as

max0≤τ≤T

E [Zτ ]

We say that a stopping time τ ≤ T is optimal (not necessarily exist) if

E [Zτ ] = sup0≤τ≤T

E [Zτ ]


Some simple results

Theorem

The following hold:

If Z is a submartingale, then late stopping is optimal, i.e. τ = T .

If Z is supermartingale, then it is optimal to stop immediately, i.e.,τ = 0.

If Z is martingale, then all stopping times τ with 0 ≤ τ ≤ T areoptimal.

Assume that the process Z has the dynamics

dZt = µtdt + σtdWt

where µ and σ are adpated process and σ is square integrable. Then thefollowing hold

If µt ≥ 0,P − a.s. for all t, then Z is a submartingale.

If µt ≤ 0,P − a.s. for all t, then Z is a supermartingale.

If µt = 0,P − a.s. for all t, then Z is a martingale.(STAT 598W) Lecture 3 35 / 51

Some simple results

Theorem

If Z is a martingale and g is convex, then g(Zt) is a submartingale.

If Z is submartingale and g is convex and increasing, then g(Zt) is asubmartingale.

If Z is a supermartingale and g is concave and increasing, then g(Zt)is a supermartingale.


Snell Envelope

Theorem

Consider a fixed process Y , we say that a process X dominates the processY if Xt ≥ Yt P − a.s. for all t ≥ 0.Assuming that E [Yt ] <∞, the Snell Envelop S, of the process Y isdefined as the smallest supermartingale dominating Y . More precisely: Sis a supermartingale dominating Y , and if D is another supermartingaledominating Y , then St ≤ Dt ,P − a.s. for all t ≥ 0.

For any integrable semimartingale Y , the Snell Envelope exists.


Optimal value process: discrete case

Assuming discrete time framework, the optimal value process of Z isdefined by

Vn = supn≤τ≤T

E [Zτ |Fn]

A stopping time which realizes the supremum above is said to be optimalat n, and it will be denoted as τn.

Theorem

The optimal value process V is the solution of the following backwardrecursion

Vn = maxZn,E [Vn+1|Fn],

VT = ZT

Furthermore, it is optimal to stop at time n if and only if Vn = Zn. Ifstopping at n is not optimal, then Vn > Zn, and Vn = E [Vn+1|Fn].


Optimal Stopping Rule

Theorem

An optimal stopping rule τ at time t = 0 is given by

τ = minn ≥ 0 : Vn = Zn

For a fixed n an optimal stopping time τn is given by

τ = mink ≥ n : Vk = Zk

For any n we haveVτn = Zτn

Theorem

The optimal value process V is the Snell envelope of the reward process Z .


Continuous case

Definition

For a fixed (t, x) ∈ [0,T ]× R, and each stopping time τ with τ ≥ t, thevalue function J si defined by

J(t, x ; τ) = Et,x [Φ(τ,Xτ )]

The optimal value function V (t, x) is defined by

V (t, x) = supt≤τ≤T

Et,x [Φ(τ,Xτ )]

A stopping time which realizes the supremum for V above is calledoptimal and will be denoted as τtx or τt without confusion.


Assumptions

We assume

There exists an optimal stopping time τt,x for each (t, x).The optimal value function V is ”regular enough”. More precisely weassume that V ∈ C 1,2. All processes interested are ”integrableenough”, in the sense that expected values exist, stochastic integralsare true (rather than local) martingales, etc.

Then we must haveV (t, x) ≥ Φ(t, x)

V (t, x) ≥ Et,x [V (t + h,Xt+h)]

using the Ito operator

Af (t, x) = µ(t, x)∂f

∂x(t, x) +

1

2σ2(t, x)

∂2f

∂x2(t, x)

then we haveV (t, x) ≥ Φ(t, x)(∂

∂t+ A

)V (t, x) ≤ 0


Stopping Rule

It is optimal to stop at (t,x) if and only if

V (t, x) = Φ(t, x)

in which case (∂

∂t+ A

)V (t, x) < 0

It is optimal to continue if and only if

V (t, x) > Φ(t, x),

in which case (∂

∂t+ A

)V (t, x) = 0

Thus we can define the continuation region C by

C = (t, x); V (t, x) > Φ(t, x)


Free Boundary value problem

Theorem

Assuming enough regularity, the optimal value function satisfies thefollowing parabolic equation(

∂

∂t+ A

)V (t, x) = 0, (t, x) ∈ C ,

V (t, x) = Φ(t, x), (t, x) ∈ ∂C

Generally speaking, there is little hope of having an analytical solution of afree boundary value problem, so typically one has to resort to numericalschemes.


Variational inequalities

Theorem

Given enough regularity, the optimal value function is characterized by thefollowing relations

V (T , x) = Φ(T , x)

V (t, x) ≥ Φ(t, x), ∀(t, x)(∂

∂t+ A

)V (t, x) ≤ 0

maxΦ(t, x)− V (t, x),

(∂

∂t+ A

)V (t, x) = 0, ∀(t, x)


Applicable partial results

Theorem

It is never optimal to stop at a point where

∂Φ

∂t(t, x) + µ(t, x)

∂Φ

∂x(t, x) +

1

2σ2(t, x)

∂2Φ

∂x2(t, x) > 0

Expressed otherwise, we have(t, x);

(∂

∂t+ A

)Φ(t, x) > 0

⊆ C


American Call Option

The optimal stopping problem becomes

max0≤τ≤T

EQ [e−rτ maxSτ − K , 0]

where the stock price dynamics under the risk neutral measure Q are givenby


Thus the reward process Z is

Zt = e−rt maxSt − K , 0 = maxe−rtSt − e−rtK , 0

It is not hard to see that Z is a submartingale in this case. Thus theoptimal stopping time should be τ = T .


American Put Option

The optimal stopping problem becomes

max0≤τ≤T

EQ [e−rτ maxK − Sτ , 0]

And the optimal value function is

V (t, x) = supt≤τ≤T

EQt,x [e−r(τ−t) maxK − Sτ , 0]

Under Q we still have



Pricing for American Put

Assume that a sufficiently regular function V (t, s), and an open set C ⊆ R+ × R+,satisfying the following conditions:

C has a continuously differentiable boundary bt , i.e. b ∈ C 1 and (t, bt) ∈ ∂C .

V satisfies the PDE

∂V

∂t+ rs

∂V

∂s+

1

2s2σ2 ∂

2V

∂s2− rV = 0, (t, s) ∈ C

V satisfies the final time boundary condition

V (T , s) = max[K − s, 0], s ∈ R+

V satisfies the inequality

V (t, s) > max[K − s, 0], (t, s) ∈ C

V satisfiesV (t, s) = max[K − s, 0], (t, s) ∈ C c

V satisfiesV (t, s) = max[K − s, 0], (t, s) ∈ C c

V satisfies the smooth fit condition

lims↓b(t)

∂V

∂s(t, s) = −1, 0 ≤ t ≤ T


Pricing for American Put

Then the following hold:

V is the optimal value function

C is the continuation region

The optimal stopping time is given by

τ = inft ≥ 0; St = bt

Unfortunately there is no analytical formulas for the pricing function or theoptimal boundary. For practical use, the following alternatives arefrequently used:

Solve the free boundary value problem numerically.

Solve the variational inequalities numerically.

Approximate the Black-Scholes model by a binomial model andcompute the exact binomial American put price.


Perpetual American Put

A perpetual American Put option is an American put with infinite timehorizon. This option is fairly simple to analyze, since the infinite horizonand the time invariance of the stock price dynamics implies that the optionprice as well as the optimal boundary are constant as functions of runningtime.There exists a constant critical price b such that we exercise the optionwhenever St < b. Since in this case V (t, s) is independent of running timet then we have V (t, s) = V (s), and the free boundary value problemreduces to the ODE

rs∂V

∂s+

1

2s2σ2∂

2V

∂s2− rV = 0, s > b


Perpetual American Put Pricing results

Theorem

For a perpetual American put with strike K , the pricing function V andthe critical price b are given by

b =γK

1 + γ

V (s) =K

1 + γ

(b

s

)γ, s > b

where

γ =2r

σ2


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Lecture 3: Review of mathematical finance and derivative pricing modelswang913/teaching/stat598w2014/LectureNotes/... · Lecture 3: Review of mathematical nance and derivative pricing