Pricing the American Option Using Itô's Formula and Optimal Stopping

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

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

Jonas Bergström

Examensarbete i matematik, 15 hp Handledare och examinator: Erik Ekström

Januari 2014

Department of Mathematics Uppsala University

PRICING THE AMERICAN OPTION USING ITO’Sˆ FORMULA AND OPTIMAL STOPPING THEORY

JONAS BERGSTROM¨

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 Itô’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¨omfor 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 independent whenever u ≤ v ≤ s ≤ t • Ws+t − Ws ∼ N(0, t) W Definition 1.2. The symbol Ft denotes the information generated by the process W = {Ws; s ≥ 0} for s ∈ [0, t]. If Y is a stochastic process s.t W W Yt ∈ Ft , then Y is said to be adapted to the filtration Ft . This simply means that Y can be observed at time t. −rs W For example let Ys := e Ws. Then Yt ∈ Ft . This is because, given the trajectory of W between 0 and t, the value of Y can be determined. −rs W If we define Ys := e Ws+, > 0 then Yt ∈/ Ft since Wt+ exists in the ”future” beyond our information at time t. 1 2 JONAS BERGSTROM¨

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

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 Z t Z t Xt = x + u(Xs) ds + σ(Xs) dWs 0 0

3. stochastic integrals This section is devoted to give a good interpretation of Z t f(s) dWs. 0 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

n−1 Z t X f(s) dWs = f(tk)(Wtk+1 − Wtk ) 0 k=0 where 0 = t0 < ... < tn = t. PRICING THE AMERICAN OPTIONUSING ITO’Sˆ FORMULA AND OPTIMAL STOPPING THEORY3

For a non-simple f one creates a sequence of simple functions, fn with certain properties s.t Z t Z t f(s) dWs = lim fn(s)dWs. 0 n→∞ 0 Proposition 3.1. For any process f, with conditions • E[f 2] < ∞ W • f(τ) is adapted to Fτ then Z t W E[ f(τ)dWτ |Fs ] = 0 s 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 Z t Z t W W W E[E[ f(τ)dWτ |Ft ]|Fs ] = E[ f(τ)dWτ |Fs ] s s Now looking at the left-hand side inside the ﬁrst expectation it follows that

Z t n−1 W X W E[ f(τ)dWτ |Ft ] = E[f(τk)(Wτk+1 − Wτk )|Ft ] s k=0 where s = τ0 < ... < τn = t. Since f(τk) depends on the value of the process W from s to τk (which are W all given by Ft , 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

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

Z t Z t W W W ⇒ E[ f(τ)dWτ |Ft ] = 0 ⇒ E[ f(τ)dWτ |Fs ] = E[0|Fs ] = 0 s s 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 Z t Z t Xt = Xs + u(Xτ ) dτ + σ(Xτ )dWτ , s ≤ t. s s 4 JONAS BERGSTROM¨

4. Ito’sˆ Formula Itô’sformula helps to give a description of the local dynamics of a stochastic process that is a function of an underlying process with a given stochastic differential. Theorem 4.1. (Itô’sFormula) Assume that the stochastic process X = {Xt; t ≥ 0} has the differential dXt = u(Xt)dt + σ(Xt)dWt where W u,σ ∈ Ft . Define a new stochastic process f(t, Xt), where f is assumed to be smooth. Given the multiplication table  2  (dt) = 0 dt · dWt = 0 2  (dWt) = dt the stochastic differential for f becomes ∂f ∂f 1 ∂2f (1) df = dt + dX + (dX )2 ∂t ∂x t 2 ∂x2 t or equivalently ∂f ∂f 1 ∂2f ∂f df = ( + u + σ2 )dt + σ dW ∂t ∂x 2 ∂x2 ∂x t

Proof. To prove that (1) holds for all t we look at the taylor expansion around the ﬁxed point (t, x). ∂f ∂f 1 ∂2f ∂2f ∂2f f(t+h, x+k) = f(t, x)+h (t, x)+k (t, x)+ (h2 +2hk +k2 )+I ∂t ∂x 2 ∂t2 ∂t∂x ∂t2 1 ∂ ∂ I = (h + k )3f(t + sh, x + sk) 3! ∂t ∂x where 0 < s < 1. Now let h → dt and k → dXt to get ∂f ∂f 1 ∂2f ∂2f 1 ∂2f f(t+dt, x+dX ) = f(t, x)+ dt+ dX + (dt)2+ dt·dX + (dX )2+I t ∂t ∂x t 2 ∂t2 ∂t∂x t 2 ∂x2 t PRICING THE AMERICAN OPTIONUSING ITO’Sˆ FORMULA AND OPTIMAL STOPPING THEORY5

2 Since (dt) = 0, dt · dXt = dt(udt + σdWt) = 0 and the fact that ∂ ∂ 3 ( ∂t dt + ∂x dXt) = 0 it follows that ∂f ∂f 1 ∂2f df = f(t + dt, x + dX ) − f(t, x) = dt + dX + (dX )2 t ∂t ∂x t 2 ∂x2 t Now we obtain 2 2 2 2 2 2 (dXt) = u (dt) + 2uσdt · dWt + σ (dWt) = σ dt ∂f ∂f 1 ∂2f ∂f ⇒ df = ( + u + σ2 )dt + σ dW ∂t ∂x 2 ∂x2 ∂x t Theorem 4.2. (Feynman-Kac˘) Assume that F solves the boundary value problem ∂F ∂F 1 2 ∂2F ∂t + u ∂x + 2 σ ∂x2 − rF = 0 F (T, s) = Φ(s). ∂F −rs 2 ∂F −rt Also assume that E[|σ(Xs) ∂x (s, Xs)e | ] < ∞ and σ(Xt) ∂x e is adapted W to Ft . Assume that Xs is the solution to dXs = u(Xs)ds + σ(Xs)dWs Xt = x Then it follows that −r(T −t) F (t, x) = e Et.x[Φ(XT )] where Et,x[ . ] = E[ . |Xt = x].

Proof. To prove this deﬁne a new stochastic process −rs f(s, Xs) = e F (s, Xs) and use Itˆo’sformula. ∂F ∂F 1 ∂2F ∂F df = e−rt(−rF + + u + σ2 )dt + σe−rt dW ∂t ∂x 2 ∂x2 ∂x t where u = u(Xt), σ = σ(Xt),F = F (t, Xt) etc. The solution to this SDE is Z T 2 Z T −rs ∂F ∂F 1 2 ∂ F −rs ∂F f(T,XT ) = f(t, Xt)+ e (−rF + +u + σ 2 )ds+ σe dWs t ∂t ∂x 2 ∂x t ∂x where the integrands are evaluated at (s, Xs). Since F solves the PDE above it follows that Z T Z T −rs ∂F −rT −rt −rs ∂F f(T,XT ) = f(t, Xt)+ σe dWs ⇔ e F (T,XT ) = e F (t, x)+ σe dWs t ∂x t ∂x 6 JONAS BERGSTROM¨

Z T −r(T −t) −r(s−t) ∂F ⇔ F (t, x) = e F (T,XT ) − σe dWs t ∂x Taking expected values yields −r(T −t) F (t, x) = e 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 dB t = rB ⇔ B = B ert dt t t 0 and that the local rate of return is equal to: dB t = r Bt · dt 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: dX dW t = u + σ t Xt · dt dt

Theorem 5.1. The solution to the equation dXt = rXtdt + σXtdW t X0 = x0 1 2 is given by Xt = x0 exp((r − 2 σ )t + σWt) and in addition we have rt E[Xt] = x0e

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 1 2 Use Itˆosformula on f(t, Wt) = x0 exp((r − 2 σ )t + σWt). 1 f(t, x) = x exp((r − σ2)t + σx) 0 2 PRICING THE AMERICAN OPTIONUSING ITO’Sˆ FORMULA AND OPTIMAL STOPPING THEORY7

∂f 1 ∂f ∂2f = (r − σ2)f(t, x), = σf(t, x), = σ2f(t, x) ∂t 2 ∂x ∂x2 1 1 ⇒ df = f(t, x)((r − σ2)dt + σdW + σ2(dW )2) 2 t 2 t 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 Z t Z t Xt = x0 + rXτ dτ + σXτ dWτ 0 0 Now take expected values to get Z t E[Xt] = x0 + r E[Xτ ] dτ 0 Z t m(s) := E[Xs] ⇒ m(t) = x0 + r m(τ) dτ 0 dm ⇒ = rm(t) ⇒ m(t) = ertm(0) = ertx dt 0

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 X to FT 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} respectively, 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. 8 JONAS BERGSTROM¨

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.

−rt Proof. f(t, X(t)) := e Xt where Xt has the following dynamics when taking expected values dXt =uX ˆ tdt + σXtdWt. In this proof one can assume thatu ˆ > r. By Itô’sformula one arrives at ∂f ∂f 1 ∂2f ∂f df = ( +uX ˆ + σ2X2 )dt + σ dW ∂t t ∂x 2 t ∂x2 ∂x t which has the solution Z t 2 Z t −rt −rτ ∂f ∂f 1 2 2 ∂ f ∂f e Xt = e Xτ + ( +uX ˆ s + σ Xs 2 )ds + σ dWs τ ∂t ∂x 2 ∂x τ ∂x By replacing Xt by the deterministic function x it follows that ∂f ∂f ∂2f f(t, x) := e−rtx, = −re−rtx, = e−rt, = 0. ∂t ∂x ∂x2 Inserting this into the integrands in the expression above yields Z t Z t −rt −rτ −rs −rs e Xt = e Xτ + (û − r)xe ds + σe dWs τ τ and by taking expectations it follows that Z t −r(t−τ) −r(s−τ) (2) E[e Xt|Fτ ] = Xτ + (û − r) xe ds > Xτ .∀τ ≤ t τ The left hand side can be seen as the value of a contract at time τ that gives Xt at 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.eu ˆ = r then (2) becomes −r(t−τ) E[e Xt|Fτ ] = Xτ PRICING THE AMERICAN OPTIONUSING ITO’Sˆ FORMULA AND OPTIMAL STOPPING THEORY9 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 mathematical 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 Itô’sformula and introduces dynamics for portfolios and then uses certain hedging positions (see [1]) and the above assumptions one can derive that

2 ∂V + rX ∂V + 1 σ2X2 ∂ V − rV = 0 (3) ∂t t ∂x 2 t ∂x2 V (T,XT ) = Φ(XT ) which according to Feynman-Ka˘c has the solution −r(T −t) V (t, Xt) = e E[Φ(XT )]. If one replaces the stochastic variables with deterministic ones (3) becomes

∂V ∂V 1 2 2 ∂2V ∂t (t, x) + rx ∂x (t, x) + 2 σ x ∂x2 (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 < ∞. 10 JONAS BERGSTROM¨

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 Xt be 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 [ = {Xt ∈ A}. t=1

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 continue 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 Xs and 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

−r(τ−t) V (t, x) = sup Et,x[e Φ(τ, Xτ )]. t≤τ≤T

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 . Letτ ˆt ∈ [t, T ] be the stopping time that satisﬁes

−r(ˆτt−t) V (t, x) = Et,x[e Φ(Xτ )].

Thenτ ˆt 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 PRICING THE AMERICAN OPTIONUSING ITO’Sˆ FORMULA AND OPTIMAL STOPPING THEORY11 the contingent claim at time s which is to say thatτ ˆt = s. This leads to a better deﬁnition ofτ ˆt namely

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

So the optimal strategy is to hold the contingent claim until some timeτ ˆt when V (ˆτt,Xτˆt ) = Φ(Xτˆt ). This leads to that V (s, Xs) > Φ(Xs) ∀s < τˆt 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’sformula −rt on the process fe V (t, Xt) ∂V d(e−rtV (t, X )) = e−rt(−r + L)V (t, X )dt + e−rtσ dW t t ∂x t ∂ ∂ 1 2 2 ∂2 where L = ∂t + rXt ∂x + 2 σ Xt ∂x2 .

The solution to this SDE is Z t+h Z t+h −r(t+h) −rt −rs −sr ∂V e V (t+h, Xt+h) = e V (t, x)+ e (−r+L)V (s, Xs) ds+ e σ dWs t t ∂x ∂ ∂ 1 2 2 ∂2 where L = ∂t + rXs ∂x + 2 σ Xs ∂x2 .

Multiplying both sides with ert and taking expectations yields Z t+h −rh −r(s−t) (4) e Ex,t[V (t + h, Xt+h)] = V (t, x) + Et,x[ e (−r + L)V ds]. t 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 . −rh V (t, x) = e Et,x[V (t + h, Xt+h)] Inserting this into (4) gives Z t+h −r(s−t) Et,x[ e (−r + L)V (s, Xs)ds] = 0. t 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 continue to hold the contingent claim till t+h. This leads to the strict inequality −rh V (t, x) > e Et,x[V (t + h, Xt+h)]. 12 JONAS BERGSTROM¨

Inserting this into (4) yields LV (t, x) < rV (t, x). Proposition 8.2. Assume that V is enough diﬀerentiable and −r(τ−t) V (t, x) = supt≤τ≤T Et,x[e 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 ∂ ∂ 1 2 2 ∂2 where L = ∂t + rx ∂x + 2 σ x ∂x2 . The optimal stopping time standing at (t, x) is

τˆt = 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 −r(τ−t) sup Et,x[e max(Xτ − K, 0)]. t≤τ≤T Define a new process −r(s−t) rt −rs −rs Zs = e max(Xs − K, 0) = e max(e Xs − e K, 0) and prove that Zs is a submartingale. This is done in two steps.

−rt −rt −rs Step 1: Yt := e Xt − e K. When s increases, so does −e K hence, even though it is a deterministic function, −e−rsK is a submartingale. −rt e Xt were Xt has 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 PRICING THE AMERICAN OPTIONUSING ITO’Sˆ FORMULA AND OPTIMAL STOPPING THEORY13 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 Xt be 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 1 2 underlying asset looks like Xt = x exp((r − 2 σ )t + σWt) so it is reasonable to demand that b(t) > 0.

−r(τ−t) + Theorem 9.2. Assume that V (t, x) = supt≤τ≤T Et,x[e (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.   ∂ ∂x u(t, x) = −1 x = b 6. ∂ ∂ 1 2 2 ∂2 where L = ∂t + rx ∂x + 2 σ x ∂x2 , then it follows that u(t, x) = V (t, x) −rt Proof. Use Itˆo’sformula on e u(t, Xt). Let τ ∈ [t, T ] be a stopping time. Z τ Z τ −rτ −rt −rs −rs ∂u e u(τ, Xτ ) = e u(t, x)+ e (−r+L)u(s, Xs) ds+ e (s, Xs) dWs. t t ∂x 2. and 3. says that −rτ −rτ + e u(τ, Xτ ) ≥ e (K − Xτ ) Z τ Z τ −rτ + −rt −rs −rs ∂u ⇒ e (K−Xτ ) ≤ e u(t, x)+ e (−r+L)u(s, Xs) ds+ e (s, Xs) dWs. t t ∂x 4. and 5. gives Z τ −rτ + −rt −rs ∂u e (K − Xτ ) ≤ e u(t, x) + e (s, Xs) dWs. t ∂x Now taking expected values on both sides yields: −rτ + −rt −r(τ−t) + Et,x[e (K − Xτ ) ] ≤ e u(t, x) ⇔ E[e (K − Xτ ) ] ≤ u(t, x) −r(τ−t) + ⇒ sup Et,x[e (τ, Xτ ) ] ≤ u(t, x) t≤τ≤T 14 JONAS BERGSTROM¨

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 Z τ ∗ Z τ ∗ −rτ ∗ ∗ −rt −rs −rs ∂u e u(τ ,Xτ ∗ ) = e u(t, x)+ e (−r+L)u(s, Xs) ds+ e (s, Xs) dWs t t ∂x ∗ Between t and τ Xs will always be above b so from 4. it follows that Z τ ∗ −rτ ∗ ∗ −rt −rs ∂u (−r+L)u(s, Xs) = 0 ⇒ e u(τ ,Xτ ∗ ) = e u(t, x)+ e (s, Xs) dWs t ∂x ∗ + 1. and 3. ⇒ u(τ ,Xτ ∗ ) = (K − Xτ ∗ ) which leads to Z τ ∗ −rτ ∗ + −rt −rs ∂u e (K − Xτ ∗ ) = e u(t, x) + e (s, Xs) dWs t ∂x and multiplying both sides with ert and taking expected values yields: −r(τ ∗−t) + −r(τ−t) + u(t, x) = Et,x[e (K − Xτ ∗ ) ] ≤ sup Et,x[e (τ, Xτ ) ] t≤τ≤T ⇒ 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 equivalently 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, becomes 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 00 0 σ2x2V (x) + rxV (x) − rV (x) = 0 2 for x > b. This ODE can be rewritten as an Cauchy Euler equation:

00 0 2r (5) x2V (x) + αxV (x) + βV (x) = 0, α = , β = −α. σ2 PRICING THE AMERICAN OPTIONUSING ITO’Sˆ FORMULA AND OPTIMAL STOPPING THEORY15

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

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 V 0 (b) = −1.

−α + V (b) = C2b = (K − b) = K − b b ⇒ C = bα(K − b) and V (x) = ( )α(K − b). 2 x 0 0 V (x) = −αbαx−(1+α)(K − b) ⇒ V (b) = −αbαb−(1+α)(K − b) = −1 which is equivalent to αK b = 1 + α 2rK σ2 = σ2 2r + σ2 2rK = . 2r + σ2

So one arrives at the following formula for an American perpetual put option α b (K − b) x > b V (x) = x K − x x ≤ b where 2rK 2r b = and α = . 2r + σ2 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 16 JONAS BERGSTROM¨ 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

1 00 0 (6) σ2x2V + (r + λ(x))xV − (r + λ(x))V = −λ(x)K 2

Theorem 11.1. Assume that V1 solves the homogeneous diﬀerential equation V 00(x) + p(x) · V 0(x) + q(x) · V (x) = 0.

Then another linear independent solution, V2 has the form: Z x 1 Z y V2 = V1 · u(x), u(x) = 2 exp(− p(z) dz) dy. V1

Theorem 11.2. Assume that V1 and V2 are two linearly independent solu- tions to the homogeneous diﬀerential equation V 00(x) + p(x) · V 0(x) + q(x) · V (x) = 0. The particular solution to the diﬀerential equation V 00(x) + p(x) · V 0(x) + q(x) · V (x) = β(x) has the form

Vp = u1(x) · V1(x) + u2(x) · V2(x) where 0 0 u1 · V1 + u2 · V2 = 0 0 0 0 0 u1 · V1 + u2 · V2 = β(x) This system has a solution if V1 V2 det( 0 0 ) 6= 0 V1 V2 PRICING THE AMERICAN OPTIONUSING ITO’Sˆ FORMULA AND OPTIMAL STOPPING THEORY17

Theorem 11.3. (Cramer’s Rule)       a11 ... a1n x1 b1 AX = B,A =  ......  ,X =  ...  ,B =  ...  an1 ... ann xn bn

 a ... a b a ... a  det(A ) 11 1i−1 1 1i+1 1n ⇒ x = i ,A = ...... i det(A) i   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

00 1 0 1 V + α V − α V = β x x2 2(r + λ(x)) −2Kλ(x) α = , β = . σ2 σ2x2 One solution to the homogenous diﬀerential equation,

00 1 0 1 V + α V − α V = 0 x x2 is V1 = x. Theorem 11.1 gives the second linearly independent solution, namely R x 1 R y α V2 = V1 · u(x), where u(x) = 2 exp(− dz) dy. V1 z 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:

V V u 0 0 1 2 1 = V10 V20 u20 β

0 V1 = 1

0 0 0 0 V2 = (V1u) = V1 u + V1u d Z x 1 Z y α = u + V1 ( 2 exp(− dz) dy) dx V1 z 1 Z x α = u + V1( 2 exp(− dz)) V1 z 1 Z x α = u + exp(− dz). V1 z 18 JONAS BERGSTROM¨

We now apply theorem 11.3. V1 V2 0 V2 V1 0 A = ,A1 = ,A2 = V10 V20 β V20 V10 β

det(A1) = −βV2, det(A2) = βV1,

0 0 det(A) = V1V2 − V1 V2 1 Z x α = V1(u + exp(− dz)) − V2 V1 z Z x α = V + exp(− dz) − V 2 z 2 Z x α = exp(− dz). z

Theorem 11.3 gives

x x 0 Z α 0 Z α u = −βV exp( dz), u = βV exp( dz) 1 2 z 2 1 z

Z x Z y α Z x Z y α ⇒ u = − βV exp( dz) dy, u = βV exp( dz) dy. 1 2 z 2 1 z

Z x Z y α Z x Z y α ⇒ V = −V βV exp( dz) dy + V βV exp( dz) dy p 1 2 z 2 1 z

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

V1 = x,

Z x 1 Z y 1 V = x exp(−α dz) dy 2 y2 z Z x 1 = x exp(−αlog(y)) dy y2 Z x 1 = x dy yα+2 −1 = (α + 1)xα PRICING THE AMERICAN OPTIONUSING ITO’Sˆ FORMULA AND OPTIMAL STOPPING THEORY19

γ −2Kλ Set β = x2 , γ = σ2 and insert V1 and V2 into Vp yeilds

xγ Z x 1 γ Z x V = dy − yα−1 dy p α + 1 y2 (α + 1)xα −γ γ = − α + 1 α(α + 1) −γ = α 2Kλ σ2 = σ2 2(r + λ) Kλ = (r + λ)

So the solution to (6) where λ is constant is Kλ V = C x + C x−α + . 1 2 (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)

Kλ (i) ⇒ C = 0, (ii) ⇒ V (b) = C b−α + = K − b. 1 2 (r + λ) Kλ This leads to that C = bα(K − b − ). 2 (r + λ)

b Kλ Kλ ⇒ V (x) = ( )α(K − b − ) + x (r + λ) (r + λ)

∂V Kλ (iii) ⇒ = −αbαx−(α+1)(K − b − ) = −1 ∂x x=b (r + λ) x=b

−α Kλ α Kλ which is equivalent to (K−b− ) = −1 ⇔ b = (K− ) b (r + λ) α + 1 (r + λ)

2(r + λ) α 2(r + λ) σ2 α = ⇒ = σ2 α + 1 σ2 2(r + λ) + σ2 2(r + λ) = 2(r + λ) + σ2 20 JONAS BERGSTROM¨

Kλ rK K − = (r + λ) r + λ

2(r + λ) rK 2rK ⇒ b = · = . 2(r + λ) + σ2 r + λ 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

( b )α(K − b − Kλ ) + Kλ if x > b V (x) = x (r+λ) (r+λ) K − x, if x ≤ b 2rK 2(r+λ) where b = 2(r+λ)+σ2 and α = σ2 . References [1] Bjork,¨ 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. BirkhäuserVerlag, 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.