• No results found

Pricing the American Option Using Itô’s Formula and Optimal Stopping Theory

N/A
N/A
Protected

Academic year: 2021

Share "Pricing the American Option Using Itô’s Formula and Optimal Stopping Theory"

Copied!
22
0
0

Loading.... (view fulltext now)

Full text

(1)

U.U.D.M. Project Report 2014:3

Examensarbete i matematik, 15 hp

Handledare och examinator: Erik Ekström Januari 2014

Pricing the American Option Using Itô’s

Formula and Optimal Stopping Theory

Jonas Bergström

Department of Mathematics

Uppsala University

(2)
(3)

USING IT ˆO’S FORMULA AND OPTIMAL STOPPING THEORY

JONAS BERGSTR ¨OM

Abstract. In this thesis the goal is to arrive at results concerning the value of American options and a formula for the perpetual American put option. For the stochastic dynamics of the underlying asset I look at two cases. The first is the standard Black-Scholes model and the second allows for the asset to jump to zero i.e default. To achieve the goals stated above the first couple of sections introduces some basic concepts in probability such as processes and information. Before introducing Itˆo’s formula this paper contains a not too rigorous introduction of stochastic differential equations and stochastic integration. Then the Black-Scholes model is introduced followed by a section about optimal stopping theory in order to arrive at the American option.

Acknowledgements. I would like to thank my supervisor Erik Ek- str¨om for his support. His input has been essential for the progress and structure of this thesis.

1. Introduction

This section contains the definition of one of the most important building blocks in continuous probability namely Wiener process. This is followed by definitions of information and martingales.

Definition 1.1. A Wiener process, W = {Wt; t ≥ 0}, starting from W0 = 0 is a continuous time stochastic process taking values in R s.t

• W has independent increments i.e Wv− Wu and Wt− Ws are inde- pendent whenever u ≤ v ≤ s ≤ t

• Ws+t− Ws∼ N (0, t)

Definition 1.2. The symbol FtW denotes the information generated by the process W = {Ws; s ≥ 0} for s ∈ [0, t]. If Y is a stochastic process s.t Yt ∈ FtW, then Y is said to be adapted to the filtration FtW. This simply means that Y can be observed at time t.

For example let Ys:= e−rsWs. Then Yt∈ FtW. This is because, given the trajectory of W between 0 and t, the value of Y can be determined.

If we define Ys := e−rsWs+,  > 0 then Yt ∈ F/ tW since Wt+ exists in the

”future” beyond our information at time t.

1

(4)

Definition 1.3. (Martingale) The process Xt∈ FtW is called a Martingale if

• E[|Xt|] < ∞

• E[Xt|FsW] = E[Xs], ∀s ≤ t

Xt∈ FtW is called a Submartingale if E[Xt|FsW] ≥ E[Xs], ∀s ≤ t.

Xt∈ FtW is called a Supermartingale if E[Xt|FsW] ≤ E[Xs], ∀s ≤ t.

2. Stochastic Differential Equations

Let Xt be a stochastic process that resembles the value of an asset.

What is a reasonable way to mathematically construct price evolutions in continuous time? That is, what can be said about dXt= Xt+dt− Xt?

One can assume that X should change proportionally with the increment of time, dt. Driven by the assets fundamental values and market expectation, dt, should be amplified by a deterministic function of the asset. Call this function u(Xt). Thus one arrives at

dXt= u(Xt)dt.

To make this model more realistic one also adds a non-deterministic term dWt= Wt+dt− dW ∼ N (0, dt) that is amplified by a deterministic function σ that depends on the same variables as u. The result is what is called a stochastic differential equation (SDE) that describes the local dynamics of a stochastic process in continuous time

 dXt= u(Xt)dt + σ(Xt)dWt X0 = x

were the solution to this system is Xt= x +

Z t 0

u(Xs) ds + Z t

0

σ(Xs) dWs

3. stochastic integrals This section is devoted to give a good interpretation of

Z t 0

f (s) dWs.

If one assumes that f is a simple function over [0, t], meaning that [0, t] can be split in smaller intervals were f is equal to some constant on respective intervals, then one can formulate the stochastic integral as

Z t 0

f (s) dWs=

n−1

X

k=0

f (tk)(Wtk+1− Wtk) where 0 = t0 < ... < tn= t.

(5)

For a non-simple f one creates a sequence of simple functions, fn with certain properties s.t

Z t 0

f (s) dWs= lim

n→∞

Z t 0

fn(s)dWs. Proposition 3.1. For any process f , with conditions

• E[f2] < ∞

• f (τ ) is adapted to FτW then

E[

Z t s

f (τ )dWτ|FsW] = 0

Proof. In this proof one assumes that f is simple because the full proof is outside of the scope of this thesis. From the law of iterated expectations it is true, for s < t, that

E[E[

Z t s

f (τ )dWτ|FtW]|FsW] = E[

Z t s

f (τ )dWτ|FsW]

Now looking at the left-hand side inside the first expectation it follows that E[

Z t s

f (τ )dWτ|FtW] =

n−1

X

k=0

E[f (τk)(Wτk+1− Wτk)|FtW] where s = τ0 < ... < τn= t.

Since f (τk) depends on the value of the process W from s to τk(which are all given by FtW, for ∀k s.t 0 ≤ k ≤ n). Because of independent increments Wτk+1−Wτk does not depend on the interval [s, τk] and hence is independent of f (τk). Then it follows that

E[f (τk)(Wτk+1− Wτk)|FtW] = E[f (τk)|FtW]E[Wτk+1− Wτk|FtW] = 0

⇒ E[

Z t s

f (τ )dWτ|FtW] = 0 ⇒ E[

Z t s

f (τ )dWτ|FsW] = E[0|FsW] = 0

 Theorem 3.2. Assume that

dXt= u(Xt)dt + σ(Xt)dWt. If u = 0 P-a.s ∀t ⇒ Xt is a martingale.

Proof. dXt= u(Xt)dt + σ(Xt)dWt have the solution Xt= Xs+

Z t s

u(Xτ) dτ + Z t

s

σ(Xτ)dWτ, s ≤ t.

(6)

Taking expected value yields E[Xt|FsW] = E[Xs|FsW] + E[

Z t s

u(Xτ) dτ |FsW] + E[

Z t s

σ(Xτ)dWτ|FsW]

= Xs+ E[

Z t s

u(Xτ) dτ |FsW]

= Xs

 4. Itˆo’s Formula

Itˆo’s formula helps to give a description of the local dynamics of a stochas- tic process that is a function of an underlying process with a given stochastic differential.

Theorem 4.1. (Itˆo’s Formula) Assume that the stochastic process X = {Xt; t ≥ 0} has the differential dXt = u(Xt)dt + σ(Xt)dWt where u,σ ∈ FtW. Define a new stochastic process f (t, Xt), where f is assumed to be smooth. Given the multiplication table

(dt)2= 0 dt · dWt= 0 (dWt)2= dt the stochastic differential for f becomes

(1) df = ∂f

∂tdt +∂f

∂xdXt+1 2

2f

∂x2(dXt)2 or equivalently

df = (∂f

∂t + u∂f

∂x+1 2σ22f

∂x2)dt + σ∂f

∂xdWt

Proof. To prove that (1) holds for all t we look at the taylor expansion around the fixed point (t, x).

f (t+h, x+k) = f (t, x)+h∂f

∂t(t, x)+k∂f

∂x(t, x)+1

2(h22f

∂t2+2hk ∂2f

∂t∂x+k22f

∂t2)+I I = 1

3!(h∂

∂t+ k ∂

∂x)3f (t + sh, x + sk) where 0 < s < 1. Now let h → dt and k → dXt to get f (t+dt, x+dXt) = f (t, x)+∂f

∂tdt+∂f

∂xdXt+1 2

2f

∂t2(dt)2+ ∂2f

∂t∂xdt·dXt+1 2

2f

∂x2(dXt)2+I

(7)

Since (dt)2 = 0, dt · dXt= dt(udt + σdWt) = 0 and the fact that (∂tdt +∂x dXt)3 = 0 it follows that

df = f (t + dt, x + dXt) − f (t, x) = ∂f

∂tdt +∂f

∂xdXt+1 2

2f

∂x2(dXt)2 Now we obtain

(dXt)2 = u2(dt)2+ 2uσdt · dWt+ σ2(dWt)2= σ2dt

⇒ df = (∂f

∂t + u∂f

∂x +1 2σ22f

∂x2)dt + σ∂f

∂xdWt

 Theorem 4.2. (Feynman-Ka˘c) Assume that F solves the boundary value problem

 ∂F

∂t + u∂F∂x +12σ2 ∂∂x2F2 − rF = 0 F (T, s) = Φ(s).

Also assume that E[|σ(Xs)∂F∂x(s, Xs)e−rs|2] < ∞ and σ(Xt)∂F∂xe−rtis adapted to FtW.

Assume that Xs is the solution to

 dXs= u(Xs)ds + σ(Xs)dWs Xt= x

Then it follows that

F (t, x) = e−r(T −t)Et.x[Φ(XT)]

where

Et,x[ . ] = E[ . |Xt= x].

Proof. To prove this define a new stochastic process f (s, Xs) = e−rsF (s, Xs) and use Itˆo’s formula.

df = e−rt(−rF + ∂F

∂t + u∂F

∂x +1 2σ22F

∂x2)dt + σe−rt∂F

∂xdWt where

u = u(Xt), σ = σ(Xt), F = F (t, Xt) etc. The solution to this SDE is

f (T, XT) = f (t, Xt)+

Z T t

e−rs(−rF +∂F

∂t+u∂F

∂x+1 2σ22F

∂x2)ds+

Z T t

σe−rs∂F

∂x dWs where the integrands are evaluated at (s, Xs).

Since F solves the PDE above it follows that f (T, XT) = f (t, Xt)+

Z T t

σe−rs∂F

∂x dWs⇔ e−rTF (T, XT) = e−rtF (t, x)+

Z T t

σe−rs∂F

∂x dWs

(8)

⇔ F (t, x) = e−r(T −t)F (T, XT) − Z T

t

σe−r(s−t)∂F

∂x dWs Taking expected values yields

F (t, x) = e−r(T −t)E[Φ(XT)]

 5. Asset Dynamics in the Black-Scholes model

Within the framework of the Black-Scholes(BS) model there exist two assets: A stock and a bond.

5.1. Bond. Bonds have the following dynamics in the BS-model:

dBt= rBtdt Note that this is equivalent to

dBt

dt = rBt⇔ Bt= B0ert and that the local rate of return is equal to:

dBt

Bt· dt = r

5.2. Stock. Stocks are said to have the following dynamics in the BS-model:

dXt= uXtdt + σXtdWt.

Here the constants u and σ are, respectively, the local mean of return and the volatility of Xt. In contrast to a bond the local rate of return on a stock is stochastic:

dXt

Xt· dt = u + σdWt dt

Theorem 5.1. The solution to the equation

 dXt= rXtdt + σXtdW t X0 = x0

is given by Xt= x0exp((r − 12σ2)t + σWt) and in addition we have E[Xt] = x0ert

Note: Here the drift term of Xt is r rather than u. This is because of so called ”risk neutral valuation” which is explained in theorem 6.2.

proof

Use Itˆos formula on f (t, Wt) = x0exp((r −12σ2)t + σWt).

f (t, x) = x0exp((r −1

2)t + σx)

(9)

∂f

∂t = (r −1

2)f (t, x),∂f

∂x = σf (t, x),∂2f

∂x2 = σ2f (t, x)

⇒ df = f (t, x)((r − 1

2)dt + σdWt+1

2(dWt)2) From the multiplication table in theorem (4) it follows that

df = f (t, x)(rdt + σdWt) ⇔ dXt= rXtdt + σXtdWt To prove the second claim rewrite the expression above as

Xt= x0+ Z t

0

rXτdτ + Z t

0

σXτdWτ Now take expected values to get

E[Xt] = x0+ r Z t

0

E[Xτ] dτ

m(s) := E[Xs] ⇒ m(t) = x0+ r Z t

0

m(τ ) dτ

⇒ dm

dt = rm(t) ⇒ m(t) = ertm(0) = ertx0 6. Options and Option Pricing

An option is a contract derived from some underlying asset which gives the holder of the contract the right (but not the obligation) to buy or sell the underlying asset for a determined amount, called the exercise price. The option is called European if the contract can only be used at a specific time in the future, called maturity. If the option can be exercised at any time between today and maturity then the option is called American. The right to buy is called a call option and the right to sell is called a put option.

Definition 6.1. A contingent claim, χ, is a random variable that is adapted to FTX where T is called the maturity, i.e at time T the payoff of χ can be determined by looking at process X = {Xt; t ≥ 0}.

A claim is said to be ”simple” if χ = Φ(XT), where Φ is called a ”contract- function”. The call option and the put option are two simple contingent claims who has contract functions max{x − K, 0} and max{K − x, 0} respec- tively, where K is the strike price.

To price an option one first assumes that the market where financial assets are traded is complete, meaning that every payoff structure of a contract can be replicated using bonds and stocks.

Another important assumption is the absence of arbitrage opportunities in the market. This assumption leads to the concept of a so called risk free measure when taking expected values of asset prices in the BS-model.

(10)

Theorem 6.2. Let Xt resembles a stock price with dynamics dXt= uXtdt + σXtdWt.

Assume that Xt is traded at a complete market that is free of arbitrage. If one takes expected values, Xt must have the following dynamics

dXt= rXtdt + σXtdWt

in order to make the market free of arbitrage. This is called a risk free measure.

The theorem says that if, while taking an expected measure, the drift term is anything other then r there will exist an arbitrage opportunity.

Proof. f (t, X(t)) := e−rtXtwhere Xt has the following dynamics when tak- ing expected values

dXt= ˆuXtdt + σXtdWt. In this proof one can assume that ˆu > r.

By Itˆo’s formula one arrives at df = (∂f

∂t + ˆuXt

∂f

∂x +1

2Xt22f

∂x2)dt + σ∂f

∂xdWt

which has the solution e−rtXt= e−rτXτ+

Z t τ

(∂f

∂t + ˆuXs

∂f

∂x +1

2Xs22f

∂x2)ds + Z t

τ

σ∂f

∂xdWs

By replacing Xt by the deterministic function x it follows that f (t, x) := e−rtx,∂f

∂t = −re−rtx,∂f

∂x = e−rt,∂2f

∂x2 = 0.

Inserting this into the integrands in the expression above yields e−rtXt= e−rτXτ+

Z t τ

(ˆu − r)xe−rsds + Z t

τ

σe−rsdWs

and by taking expectations it follows that (2) E[e−r(t−τ )Xt|Fτ] = Xτ + (ˆu − r)

Z t τ

xe−r(s−τ )ds > Xτ.∀τ ≤ t The left hand side can be seen as the value of a contract at time τ that gives Xtat time t. The expression says that this contract is greater than Xτ even though they have the same payoff structure.

Now it exist an arbitrage opportunity in the market. By shortening the contract at time τ , one can immediately buy Xτ while still having capital at ones disposal. At time t one closes the short by selling the stock at value Xt

and pay for the value of the contract at time t wich is Xt hence one makes a risk free profit.

If however we had the risk free measure i.e ˆu = r then (2) becomes E[e−r(t−τ )Xt|Fτ] = Xτ

(11)

which is a correct pricing of such a contract. Hence no arbitrage opportunity exists.

 Note: The last expression in the proof says that standing at time τ the value of a stock is equal to the discounted expected value of the stock given all information available at time τ . This is also in line with the so called efficient-market hypothesis.

7. The black-Scholes equation

To price a contingent claim one first makes a couple of financial and math- ematical assumptions. First we assume that the market is free of arbitrage opportunities and that it is complete. Another financial assumption is that every portfolio consisting of bonds and stocks are so called self financed meaning that every new trades of acquisitions of assets must be financed by selling parts of the portfolio. No exogenous inflow or outflow of capital is allowed.

Here we are also dealing with simple contingent claims meaning that the contract only depends on the value of the asset at maturity, T i.e.

χ = Φ(XT).

Now one defines a stochastic process V (t, St) as the value of the contingent claim χ at time t < T and adds the mathematical assumption that V ∈ C1,2. Now if one uses Itˆo’s formula and introduces dynamics for portfolios and then uses certain hedging positions (see [1]) and the above assumptions one can derive that

(3)

 ∂V

∂t + rXt∂V

∂x +12σ2Xt2∂x2V2 − rV = 0 V (T, XT) = Φ(XT)

which according to Feynman-Ka˘c has the solution V (t, Xt) = e−r(T −t)E[Φ(XT)].

If one replaces the stochastic variables with deterministic ones (3) becomes

 ∂V

∂t(t, x) + rx∂V∂x(t, x) +12σ2x2 ∂∂x2V2(t, x) − rV (t, x) = 0 V (T, x) = Φ(x)

which is called the Black-Scholes equation.

8. Optimal Stopping Theory

This section contains a simplified examination of optimal stopping theory.

For a more rigorous analysis of optimal stopping theory see [3].

Definition 8.1. (Stopping time)

A random variable τ : Ω → [0, ∞] is called a Markov time if

{τ ≤ t} ∈ Ft ∀t ≥ 0. A Markov time is called a stopping time if τ < ∞.

(12)

Note: {τ ≤ t} ∈ Ft means that we can determine, standing at time t if the ”stopping event” has occurred or not.

8.1. Example. Let Xtbe a process in discrete time which is adapted to Ft. Let τ := inf{n ≤ 0; Xn∈ A} i.e τ measures the time when the process takes values in A for the first time.

Here τ is a stopping time because: {τ ≤ n} = {w ∈ Ω; τ (w) ≤ n} for some probability space Ω and

{w ∈ Ω; τ (w) ≤ n} = {w ∈ Ω; Xt(w) ∈ A, for some t ≤ n}

=

n

[

t=1

{Xt∈ A}.

Since Xt is adapted ⇒ {Xt∈ A} ∈ Ft⊆ Fn⇒ {τ ≤ n} ∈ Fn

8.2. Optimal stopping strategy in continuous time. Assume one is holding a contingent claim at time t that expires at a future time T . The value of the contingent claim is assumed to depend on a stock, Xt, with the risk-neutral dynamics

dXt= rXtdt + σXtdWt

Within the time interval [t, T ] one has the option to either exercise or con- tinue to hold the contingent claim. Standing at a fixed point in time, s, the value of exercising the contingent claim at that time is Φ(Xs) were Φ is a contract function. Here, depending on the price levels of Xsand the nature of Φ, s might not be the optimal time to ”stop” i.e exercising the contingent claim.

To analyze this problem further one defines V (t, x) = sup

t≤τ ≤T

Et,x[e−r(τ −t)Φ(τ, Xτ)].

So standing at (t, x), were x = Xt, V (t, x) is by definition the optimal value one can achieve from exercising the contingent claim at some time in the future between t and T .

Let ˆτt∈ [t, T ] be the stopping time that satisfies V (t, x) = Et,x[e−r(ˆτt−t)Φ(Xτ)].

Then ˆτt is the optimal stopping time when standing at (t, x).

Now the question is to find certain conditions that V (t, x) must satisfy.

Let s be the smallest element in [t, T ] s.t V (s, Xs) = Φ(Xs). If V (s, Xs) = Φ(Xs) it means that the optimal value one can receive is when exercising

(13)

the contingent claim at time s which is to say that ˆτt= s.

This leads to a better definition of ˆτt namely ˆ

τt= inf{t ≤ s ≤ T ; V (s, Xs) = Φ(Xs)}.

So the optimal strategy is to hold the contingent claim until some time ˆτt when V (ˆτt, Xτˆt) = Φ(Xτˆt). This leads to that V (s, Xs) > Φ(Xs) ∀s < ˆτt

which leads to the inequality

V (τ, Xτ) ≥ Φ(Xτ), ∀τ ∈ [t, T ].

To reach new conclusions on the behavior of V (t, x) we use Itˆo’s formula on the process f e−rtV (t, Xt)

d(e−rtV (t, Xt)) = e−rt(−r + L)V (t, Xt)dt + e−rtσ∂V

∂xdWt

where L = ∂t + rXt∂x +12σ2Xt2∂x22. The solution to this SDE is e−r(t+h)V (t+h, Xt+h) = e−rtV (t, x)+

Z t+h t

e−rs(−r+L)V (s, Xs) ds+

Z t+h t

e−srσ∂V

∂x dWs

where L = ∂t + rXs∂x +12σ2Xs2∂x22.

Multiplying both sides with ert and taking expectations yields (4) e−rhEx,t[V (t + h, Xt+h)] = V (t, x) + Et,x[

Z t+h

t

e−r(s−t)(−r + L)V ds].

Now standing at (t, x), what happens to V if it is not optimal to stop i.e V (t, x) > Φ(t, x). This means that the optimal value is in the future. So instead of exercising the contingent claim at t one holds it till t + h. Here we also let h tend to zero, to minimize loss of information of V between t and t + h. This means that the value of V must be the discounted expected value of this future V .

V (t, x) = e−rhEt,x[V (t + h, Xt+h)]

Inserting this into (4) gives Et,x[

Z t+h t

e−r(s−t)(−r + L)V (s, Xs)ds] = 0.

If one divides this expression with h and continue to let h tend to zero it will result in the integrand evaluated at t. This means that

LV (t, x) = rV (t, x).

Alternatively if we had V (t, x) = Φ(t, x) it would not be optimal to con- tinue to hold the contingent claim till t+h. This leads to the strict inequality

V (t, x) > e−rhEt,x[V (t + h, Xt+h)].

(14)

Inserting this into (4) yields

LV (t, x) < rV (t, x).

Proposition 8.2. Assume that V is enough differentiable and V (t, x) = supt≤τ ≤TEt,x[e−r(τ −t)Φ(τ, Xτ)].

Define the region C := {(t, x); V (t, x) > Φ(t, x)} then the following holds:

V (T, x) = Φ(T, x)

LV (t, x) = rV (t, x) ∀(t, x) ∈ C LV (t, x) < rV (t, x) ∀(t, x) /∈ C where L = ∂t + rx∂x +21σ2x2 ∂∂x22.

The optimal stopping time standing at (t, x) is ˆ

τt= inf{s ≥ t; V (s, Xs) = Φ(s, Xs)}

9. The American Put Option

When the opportunity of exercising an option at any time before maturity exists things get complicated. Here the contract function, in contrast to the European option, can not depend solely on the price of the asset at maturity. When maturity is finite there does not exist an analytic formula for the pricing of an American put option. However, in the case of the call option, it does exist.

Proposition 9.1. The price of an American call option with finite maturity coincides with its European counterpart.

Proof. This is proved by showing that the optimal stopping time for the option equals maturity. The optimal stopping problem is

sup

t≤τ ≤T

Et,x[e−r(τ −t)max(Xτ− K, 0)].

Define a new process

Zs= e−r(s−t)max(Xs− K, 0) = ertmax(e−rsXs− e−rsK, 0) and prove that Zs is a submartingale.

This is done in two steps.

Step 1: Yt:= e−rtXt− e−rtK. When s increases, so does −e−rsK hence, even though it is a deterministic function, −e−rsK is a submartingale.

e−rtXt were Xthas the dynamics under the risk neutral valuation i.e dXt= rXtdt + σXtdWt

is, following from theorem 6.2, a martingale.

So it follows that Yt is a submartingale i.e E[Yt|Fs] ≥ E[Ys], ∀s ≤ t.

Step 2: γ(y) = max{y, 0} is a convex function of y, hence by Jensens’s inequality

E[γ(Yt)|Fs] ≥ γ(Ys), ∀s ≤ t

(15)

and thus Zt is a submartingale . Hence

Et,x[ZT] ≥ Et,x[Zτ] , ∀τ ≤ T

 So, following the results of proposition 9.1, one focuses on the American put option. Thus, in this context one assumes that the contract function, Φ, has the form Φ(x) = max{K − x, 0} = (K − x)+. With an American put option one hopes for a decrease in the value of the underlying asset so that it is below the strike price K. Let Xtbe the process that resembles the value of the underlying asset with the risk-neutral dynamics: dXt= rXtdt+σXtdWt. One assumes that there is a price level, call it b(t), s.t when X goes below this level it is optimal to stop. For b(t) to be optimal it must be smaller than K i.e be ”in the money”. Also by theorem 5.1 the solution to the SDE of the underlying asset looks like Xt= x exp((r −12σ2)t + σWt) so it is reasonable to demand that b(t) > 0.

Theorem 9.2. Assume that V (t, x) = supt≤τ ≤TEt,x[e−r(τ −t)(K − Xτ)+].

If there exist a b(t) and u(t, x) that satisfies:













u(T, x) = (K − XT)+ 1.

u(t, x) > (K − x)+ x > b 2.

u(t, x) = (K − x)+ x ≤ b 3.

Lu(t, x) = ru(t, x) x > b 4.

Lu(t, x) < ru(t, x) x ≤ b 5.

∂xu(t, x) = −1 x = b 6.

where L = ∂t + rx∂x +21σ2x2 ∂∂x22, then it follows that u(t, x) = V (t, x) Proof. Use Itˆo’s formula on e−rtu(t, Xt). Let τ ∈ [t, T ] be a stopping time.

e−rτu(τ, Xτ) = e−rtu(t, x)+

Z τ t

e−rs(−r+L)u(s, Xs) ds+

Z τ t

e−rs∂u

∂x(s, Xs) dWs. 2. and 3. says that

e−rτu(τ, Xτ) ≥ e−rτ(K − Xτ)+

⇒ e−rτ(K−Xτ)+≤ e−rtu(t, x)+

Z τ

t

e−rs(−r+L)u(s, Xs) ds+

Z τ

t

e−rs∂u

∂x(s, Xs) dWs. 4. and 5. gives

e−rτ(K − Xτ)+≤ e−rtu(t, x) + Z τ

t

e−rs∂u

∂x(s, Xs) dWs. Now taking expected values on both sides yields:

Et,x[e−rτ(K − Xτ)+] ≤ e−rtu(t, x) ⇔ E[e−r(τ −t)(K − Xτ)+] ≤ u(t, x)

⇒ sup

t≤τ ≤T

Et,x[e−r(τ −t)(τ, Xτ)+] ≤ u(t, x)

(16)

To prove the reverse inequality use the same arguments but change τ to τ := T ∧ τb, where τb := inf{t ≤ s ≤ T ; Xs≤ b}. Then it follows that e−rτu(τ, Xτ) = e−rtu(t, x)+

Z τ t

e−rs(−r+L)u(s, Xs) ds+

Z τ t

e−rs∂u

∂x(s, Xs) dWs

Between t and τ Xs will always be above b so from 4. it follows that (−r+L)u(s, Xs) = 0 ⇒ e−rτu(τ, Xτ) = e−rtu(t, x)+

Z τ

t

e−rs∂u

∂x(s, Xs) dWs 1. and 3. ⇒ u(τ, Xτ) = (K − Xτ)+ which leads to

e−rτ(K − Xτ)+= e−rtu(t, x) + Z τ

t

e−rs∂u

∂x(s, Xs) dWs

and multiplying both sides with ert and taking expected values yields:

u(t, x) = Et,x[e−r(τ−t)(K − Xτ)+] ≤ sup

t≤τ ≤T

Et,x[e−r(τ −t)(τ, Xτ)+]

⇒ u(t, x) = V (t, x)

 10. The Perpetual American Put Option

When dealing with an American put option that has a finite maturity, unlike its European counter part, no analytic formula for its value is known.

If however one restricts only to the case when there is no maturity or equiv- alently when it is equal to infinity, an analytic formula can be derived.

With T = ∞ the option is called a perpetual option.

When the option has no expiring date the value of the option does not depend on time but only on the price levels of the underlying asset i.e its derivative with respect to time is zero. The optimal price frontier, b, be- comes a constant.

Now one is looking for b and V (x) that satisfy the conditions of theorem 9.2, and as a consequence of the theorem, the solution to the perpetual American put option has been obtained.

It follows from condition 4. in theorem 9.2 that 1

2x2V00(x) + rxV0(x) − rV (x) = 0 for x > b.

This ODE can be rewritten as an Cauchy Euler equation:

(5) x2V00(x) + αxV0(x) + βV (x) = 0, α = 2r

σ2, β = −α.

(17)

This is solved by seeking a solution of the form V (x) = xm which by inserting into (5) gives V (x) = C1x + C2x−α.

Since V (x) is bounded by K it follows that C1 = 0. Condition 3. and 6.

in theorem 9.2 says respectively that V (b) = (K − b)+ and V0(b) = −1.

V (b) = C2b−α = (K − b)+= K − b

⇒ C2 = bα(K − b) and V (x) = (b

x)α(K − b).

V0(x) = −αbαx−(1+α)(K − b) ⇒ V0(b) = −αbαb−(1+α)(K − b) = −1 which is equivalent to

b = αK

1 + α

= 2rK

σ2 σ2 2r + σ2

= 2rK

2r + σ2.

So one arrives at the following formula for an American perpetual put option V (x) =

 b

x

α(K − b) x > b K − x x ≤ b where

b = 2rK

2r + σ2 and α = 2r σ2.

11. jump to default models

Under the BS-model stock prices, which have a positive initial value, can never drop to zero (this follows from theorem 5.1). History suggest that the probability of a so called ”default” is non-zero. In this context default means that the financial entity in question fails to meet its financial obligations, such as interest payments on their loans. If this happens the value of this entity will drop to zero with no chance of a recovery. Here we do not assume that countries can default.

To incorporate default one can use poisson processes.

Let N (t) = the number of defaults between 0 and t.

N (s + t) − N (s) ∼ P o(λ · t) where λ is the default intensity.

Empirical evidence shows a positive correlation between corporate bond yields and credit default swap (CDS) spreads (see [4]). A CDS spread is the

(18)

fee that the buyer of the CDS pays to the issuer. If the chances of default increases so does the fee, for having the insurance.

So in this context one expands the BS-model by saying that there exist a stock and two bonds. A corporate bond and a sovereign bond.

In the normal BS-model λ = 0, that is to say both bonds must have the same rate of return namely r. If however λ > 0 then it follows that if investors hold a corporate bond, they need to be compensated for taking a bigger risk in comparison if they held a government bond. This leads to assume that the rate of return of a corporate bond is r + λ. When using a risk neutral valuation the dynamics of a stock price becomes

dXt= (r + λ)Xt+ σXtdWt− XtdNt

were dNt∼ po(λ · dt)

It is also assumed that λ is a decreasing function of the underlying asset, Xt. Using the above assumptions one can arrive at an ODE for a perpetual American put option with a non zero default intensity, λ, namely

(6) 1

2x2V00+ (r + λ(x))xV0− (r + λ(x))V = −λ(x)K Theorem 11.1. Assume that V1 solves the homogeneous differential equation

V00(x) + p(x) · V0(x) + q(x) · V (x) = 0.

Then another linear independent solution, V2 has the form:

V2= V1· u(x), u(x) = Z x 1

V12 exp(−

Z y

p(z) dz) dy.

Theorem 11.2. Assume that V1 and V2 are two linearly independent solu- tions to the homogeneous differential equation

V00(x) + p(x) · V0(x) + q(x) · V (x) = 0.

The particular solution to the differential equation

V00(x) + p(x) · V0(x) + q(x) · V (x) = β(x) has the form

Vp = u1(x) · V1(x) + u2(x) · V2(x) where

 u01· V1+ u02· V2 = 0 u01· V10+ u02· V20= β(x) This system has a solution if

det(

 V1 V2

V10 V20

 ) 6= 0

(19)

Theorem 11.3. (Cramer’s Rule) AX = B, A =

a11 ... a1n

... ...

an1 ... ann

, X =

 x1

...

xn

, B =

 b1

...

bn

⇒ xi = det(Ai) det(A), Ai =

a11 ... a1i−1 b1 a1i+1 ... a1n

... ...

an1 ... ani−1 bn ani+1 ... ann

 i.e the i:th column of A is replaced by B.

For proofs of all three of the above theorems see [5].

To make calculations easier we write (6) as V00+ α1

xV0− α 1 x2V = β α = 2(r + λ(x))

σ2 , β = −2Kλ(x) σ2x2 . One solution to the homogenous differential equation,

V00+ α1

xV0− α 1 x2V = 0 is V1 = x.

Theorem 11.1 gives the second linearly independent solution, namely V2= V1· u(x), where u(x) =Rx 1

V12exp(−Ry α

z dz) dy.

To find the particular solution, Vp, to (6) one uses theorem 11.2.

Vp = V1· u1(x) + V2· u2(x)

 u10V1+ u20V2 = 0 u10V10 + u20V20 = β which can also be written in matrix form:

 V1 V2 V10 V20

  u10 u20



=

 0 β



V10 = 1 V20 = (V1u)0 = V10u + V1u0

= u + V1

d dx(

Z x 1

V12exp(−

Z y α

z dz) dy)

= u + V1( 1

V12 exp(−

Z x α z dz))

= u + 1 V1

exp(−

Z x α z dz).

(20)

We now apply theorem 11.3.

A =

 V1 V2

V10 V20

 , A1 =

 0 V2

β V20

 , A2 =

 V1 0 V10 β



det(A1) = −βV2, det(A2) = βV1,

det(A) = V1V20− V10V2

= V1(u + 1

V1 exp(−

Z x α

z dz)) − V2

= V2+ exp(−

Z xα

z dz) − V2

= exp(−

Z xα z dz).

Theorem 11.3 gives u01 = −βV2exp(

Z xα

z dz), u02 = βV1exp(

Z xα z dz)

⇒ u1= − Z x

βV2exp(

Z y α

z dz) dy, u2 = Z x

βV1exp(

Z y α

z dz) dy.

⇒ Vp= −V1

Z x

βV2exp(

Z y α

z dz) dy + V2

Z x

βV1exp(

Z y α z dz) dy

If λ is assumed to be constant the values of V1 and V2 are as follows.

V1 = x,

V2 = x Z x 1

y2exp(−α Z y 1

zdz) dy

= x Z x 1

y2exp(−αlog(y)) dy

= x

Z x 1 yα+2dy

= −1

(α + 1)xα

(21)

Set β = xγ2, γ = −2Kλσ2 and insert V1 and V2 into Vp yeilds

Vp = xγ α + 1

Z x 1

y2 dy − γ (α + 1)xα

Z x

yα−1dy

= −γ

α + 1− γ α(α + 1)

= −γ α

= 2Kλ

σ2 σ2 2(r + λ)

= Kλ

(r + λ)

So the solution to (6) where λ is constant is V = C1x + C2x−α+ Kλ

(r + λ).

Now we would like V to satisfy the conditions in theorem 9.2.

V (x) ≤ K (i) V (b) = K − b (ii)

∂V

∂x(b) = −1 (iii)

(i) ⇒ C1 = 0, (ii) ⇒ V (b) = C2b−α+ Kλ

(r + λ) = K − b.

This leads to that C2= bα(K − b − Kλ (r + λ)).

⇒ V (x) = (b

x)α(K − b − Kλ

(r + λ)) + Kλ (r + λ)

(iii) ⇒ ∂V

∂x

x=b = −αbαx−(α+1)(K − b − Kλ (r + λ))

x=b= −1

which is equivalent to −α

b (K−b− Kλ

(r + λ)) = −1 ⇔ b = α

α + 1(K− Kλ (r + λ))

α = 2(r + λ)

σ2 ⇒ α

α + 1 = 2(r + λ) σ2

σ2 2(r + λ) + σ2

= 2(r + λ) 2(r + λ) + σ2

(22)

K − Kλ

(r + λ) = rK r + λ

⇒ b = 2(r + λ)

2(r + λ) + σ2 · rK

r + λ = 2rK 2(r + λ) + σ2.

Now one has arrived at the following formula for an perpetual American put option on a stock with default intensity λ namely

V (x) =

 (xb)α(K − b −(r+λ) ) +(r+λ) if x > b K − x, if x ≤ b where b = 2(r+λ)+σ2rK 2 and α = 2(r+λ)σ2 .

References

[1] Bj¨ork, T. Arbitrage theory in continuous time. Third edition. Oxford university press. 2009.

[2] Stefanica, D. A primer for the mathmatics of financial engineering. Sec- ond edition. FE Press, New York. 2011.

[3] Peskir, G. and Shiryaev, A. Optimal stopping and free-boundary prob- lems. Birkh¨auser Verlag, Basel-Boston-Berlin. 2006.

[4] Carr, P. and Wu, L. A simple robust link between american puts and credit protection.

[5] Simmons ,G. and Krantz, S. Differential equations: Theory, technique and practice. McGraw-Hill companies. International edition. 2007.

References

Related documents

In this paper we were given a MATLAB script pricing European call options using the Fokker-Planck equation and the A¨ıt-Sahalia method to circumvent the singular nature of the

In  the  Black  and  Scholes  model  five  values  are  imputed  to  calculate  the  option  price.  The 

Keywords: IFRS 9, accounting choice, equity investments not held for trade, FVOCI option, irrevocable, recycling, changes in fair value, salient volatility, leverage,

column represents the F-statistic of the joint hypothesis test with the null that the constant (intercept) is equal to zero and the slope coefficient is equal to one. The coefficient

The project is taken from Volvo Powertrain AB and we use the valuation model Real Options Analysis (ROA), and more specifically, the option to defer, which

It will be shown how a financial derivative priced with the binomial model satisfies Black-Scholes equation, and how the price of the underlying stock in the binomial model converge

To see how portfolio risk using Value at Risk (results for Expected Shortfall is provided in Appendix A) is affected by pricing errors in the Monte Carlo method, an arbitrary set of

Här nämndes att för de som bara vill ta sig igenom skolan går det bra att sova mindre och ta mer tid till att göra andra saker, men inte om du vill ha bättre betyg.. Därmed ser