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U.U.D.M. Project Report 2014:12

Examensarbete i matematik, 30 hp

Handledare och examinator: Erik Ekström Maj 2014

Department of Mathematics

Valuation of American put options with exercise restrictions

Domingos Celso Djindja

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UPPSALA UNIVERSITY MATHEMATICS DEPARTMENT

Master Thesis

Valuation of American put options with exercise restrictions

By:

Domingos Celso Djindja

Master student in Mathematics

Advisor:

Erik Ekstrom

Uppsala, May 6, 2014

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Acknowledgements

This work involved the collaboration of several people that I would to express my sincere acknowledgments.

First to Erik Ekstrom, my supervisor, for his patience with me, suggestions and his commit- ment throughout the job. For the Professors of the Mathematics Department, for the teachings and moral support. Moreover, to my family, friends and all who contributed on some way to this work success.

Domingos Celso Djindja

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Abstract

In this work we price an American put with exercise restriction on weekends. The idea is to remove weekends and shrink the interval (useful days) and so we may have jumps. Thus, we hedge and price it both on analytical valuation and numerical one. On this last, we focus essentially on nd the optimal exercise boundary. We describe a version of the nite dierence method given in B. Kim et all [5] and we extend it to nd the boundary of an American put with exercise restriction on weekends.

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Contents

Acknowledgements i

Abstract ii

Introduction 1

1 Pricing an American put numerically. Finite dierence method 3

1.1 The equation for the boundary . . . 4

1.2 Numerical algorithm . . . 8

2 American put options with jumps 10 2.1 Itô formula for diusion with jumps . . . 10

2.2 The stock price for the problem in analysis . . . 11

Case 1: The stock price is traded continuously at any time . . . 11

Case 2: The stock price can not be traded during the weekends but during the useful week days it is continuously traded . . . 13

2.3 The pricing problem . . . 15

2.3.1 Analytical valuation . . . 16

2.3.2 Numerical valuation . . . 21

Conclusion 26 Appendix 27 A. Maple code for the critical stock price of an American put (standard case) . . . 27 B. Maple code for the critical stock price of an American put (for the main problem) 29

Bibliography 31

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Introduction

An option which the holder has the right to sell or buy it at any time along the time life of it is called American option. Thus, pricing an American option consists on nd the optimal time to exercise it (optimal time to stop and exercise) which corresponds to the optimal value. We will focus on American put with nite maurity. Such problems may rely on optimal stoping problems. Many authors have been writing on optimal stoping theory, for example, on G.

Peskir [7], American problems on formulation of stopping and free-boundary problems are treated. We will focus on the problem of pricing an American put by using free boundary problem formulation. The conversion from an optimal stopping to a free-boundary problem for pricing American put is done on G. Peskir [7], for instance. Since we want to optimize our gains, the strategy of pricing an American put consists on wait if the option value is bigger than the pay-o and exercise when on the opposite case. Acting so, there will be a boundary (critical stock price) on which we will have this two dierent situations, either wait or exercise the option. Such boundary is also part of the solution, this becomes a free-boundary problem.

Therefore, pricing an American put also leads to nd the boundary.

It is well known that there is no closed formula for pricing an American put. Thus, numerical methods have been used. Since such methods are approximations it is good to rene them and increase the algorithms eciency. Therefore, many authors have been focussed on such numerical methods to price an American put and nd the corresponding boundary. Mostly they use nite dierence method versions, binomial and Monte Carlo methods, for example on Paul Willmott [9] the rst two methods are explained. They rely on solving the solving numerically the Black-Scholes partial dierential equation (see chapter 7, 21 [1]) or solve numerically the analytic valuation formula presented, for instance on G. Peskir [7], which the American put is represented as sum of European put and a premium for exercise early the option. For example, on J. E. Zhang [10], the boundary and the American put is determined numerically by solving the corresponding integral equation (similar to the one on Peskir [7]) numerically.

On this work, we will focus on determine the critical stock price (exercise boundary) which is an important task to problems of pricing an American put. We will use a version of nite dierence method (FDM) presented in B. Kim et all [5]. Thus, we will present a very close description to this algorithm present on that paper and produce a code in maple for numerical experiments. This is done on the rst chapter. Then, we will study on chapter 2 how would be the critical stock price of an American put when it is not allowed to exercise the option

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during weekends. For simplicity, we will work with Bermudan options, regard that we can only exercise the option on discrete points (days). Therefore, we can remove the weekends and nd the option value only on useful week days. Thus, on a more general case and at a rst glimpse, we may have jumps on the stock price. Thus, questions about completeness of the market, hedging and how to choose a martingale measure may arise. However, this jumps may arise on known dates (weekends). Incidently, we will follow some ideas from authors who treated more general cases, when the jumps arrive on unknown dates, for example R. Cont, P. Tankov [2] and C. R. Gukhal [4]. Mainly, we will study the critical stock price to this problem both analytical and numerical valuation. On the numerical one, we will extend the algorithm presented on B.

J. Kim et all [5] to nd the critical stock price.

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Chapter 1

Pricing an American put numerically.

Finite dierence method

We will determine the optimal exercise boundary for American put options by nite dierence method given in [5], B. J. Kim et all. The method is based on a Lipschitz surface near the free boundary which is designed by Q, see the denition afterwards. The Lipschitz surface avoids the degeneracy of the solution surface near free boundary. This function produces a linearly converging algorithm to locate the free boundary.

We start by considering an nancial market characterized by a risky asset in a risk neutral economy with a constant risk-free interest rate r > 0 and price process {S(t), t ∈ (0, T )}

over the option's life [0, T ] that is specied by constant volatility rate σ > 0. Let the market measure be denoted by P , let {W (t), t ∈ (0, T )} be a P -Brownian motion. The stock dynamics of the Black-Scholes model follows the geometric Brownian motion:

dS = Srdt + σSdW.

Let P (t, S; T ) denote the value of an American put option as a function of the current time t, the current stock price S , and the maturity date T . The critical stock price β(t; T ), t ∈ [0, T ] is dened as the largest price S at which the American put option value P (t, S; T ) equals its exercise value K − S , where K is the strike price. The Black-Scholes partial dierential equation is the following:

Pt+ 1

2σ2S2PSS + SPS − rP = 0 (1.1) when S ∈ (β(t, T ), ∞), t ∈ (0, T )

satisfying

P (T, S, T ) = max{K − S, 0} and β(T, T ) = K.

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and the boundary conditions

lim

S↑∞P (t, S, T ) = 0, lim

S↓β(t,T )P (t, S, T ) = K− β(t, T ), lim

S↓β(t,T )PS(t, S, T ) =−1. (1.2)

Boundary value problems arising in option valuation usually require one terminal condition and two boundary conditions to guarantee an unique solution. Let

e={(t, S) ∈ (0, T ) × (Smin, Smax), S ≤ β(t, T )}

c ={(t, S) ∈ (0, T ) × (Smin, Smax), S > β(t, T )}

be the exercise and continuous region, respectively, (Smin, Smax) the interval of the stock price.

By setting Q =

P − K + S , we see that Q = 0 on the free boundary and Ωe. Also Q has Lipschitz character with no-singular and non-degeneracy property near free boundary. The degeneracy of solution surface causes instability and slow convergence of numerical algorithm.

Therefore, Q is a natural candidate for computation in continuation region.

Since the solution surface in the exercise region is a horizontal plane and in the continuation region it is an inclined surface, we can then nd the boundary. We transform P to Q and we solve the equation for the boundary. In order to nd P , we solve the backward time Black- Scholes equation. To get the boundary, we rst chop the interval [0, T ] in N equal subintervals such that tn = n∆t, n = 0, 1, ..., N, ∆t = NT . We do the same to the stock price, we chop the interval (Smin, Smax) in M equal subintervals, such that Sin= Si(tn) i = 0, 1, 2, ...M, and

∆S = SmaxM−Smin,

Sin = βn+ ρ∆S + i∆S, n = N, N − 1, ..., 0, i = 0, 1, ..., M,

0 < ρ < 1, is a constant parameter and the option price P (tn, Sin, T ) = Pin. Sin is computed after the free boundary βn+1 be computed, and βN = K.

1.1 The equation for the boundary

We will now derive the boundary equation. By using the analyticity of Q in the continuation region, we decompose it as Taylor series (Q(tn, S0n, T )) at βn(t, T ):

Q(tn, S0n, T ) = Q(tn, βn, T ) + QS(tn, βn, T )(S0n− βn) + 1

2QSS(tn, βn, T )(S0n− βn)2+

+O((S0n− βn)3). (1.3)

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We need QS, QSS, thus, dierentiating P = Q2+ K− S, We obtain

Pt= 2QQt, PS = 2QQSS− 1, PSS = 2Q2S + 2QQSS, (1.4) PSSS = 6QSQSS+ 2QQSSS.

Thus, the Black-Scholes equation (1.1) is transformed to 2QQt+1

2σ2S2(2Q2S+ 2QQSS) + rS(2QQS− 1) − r(Q2+ K− S) = 0.

Since Q → 0 when S ↓ β(t, T ), we have 1

2σ2S2(2Q2S) + rS(−1) − r(K − β(t, T )) = 0.

Thus, when S → β+,

Q2S rK σ2β2(t, T ) and from the boundary condition (1.2) we have

QS = PS+ 1 2

P − K + S > 0 on the continuation region. Therefore,

lim

S→β+QS =

√rK

σβ(t, T ). (1.5)

In order to nd QSS, we dierenciate the Black-Scholes partial dierential equation with respect to S , we have

PtS + σ2SPSS +1

2σ2S2PSSS+ rPS+ rSPSS − rPS = 0.

By simplifying it, we get

PtS+ σ2SPSS+ 1

2σ2S2PSSS + rSPSS = 0. (1.6) Moreover, dierentiating the boundary condition P (t, β(t, T ), T ) = K − β(t, T ), we get

PS(t, β, T )· β+ Pt(t, β, T ) =−β(t, T ).

Since PS(β, t, T ) =−1 on the free boundary, by condition (1.2), we get Pt(t, β, T ) = 0.

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On other hand, we dierentiate PS(t, β, T ) =−1 with respect to t and we obtain PSS(t, β(t, T ), T )· β(t, T ) + PSt(β, t, T ) = 0

and hence

PtS(β, t, T ) =−PSS(t, β, T )· β(t, T ). (1.7) Thus, using this last relation, (1.6) becomes

−PSSβ+ σ2SPSS +1

2σ2S2PSSS + rSPSS = 0.

By using (1.4) when S → β+, we have

PSSS → 6QSQSS, PSS → 2Q2S

and also from (1.5), we have

−β· 2Q2S+ σ2β· 2Q2S+1

2σ2· β2· 6QSQSS + rβ· 2Q2S = 0

⇔ −β· 2rK

σ2β2 + σ2β· 2rK

σ2β2 + 3σ2· β2·

√rK

σβ QSS + rβ· 2rK σ2β2 = 0

⇔ QSS =2 rK

3β3[−β+ (σ2+ r)· β]. (1.8)

By substituting (1.5), (1.8) in (1.3), as S → β(t, T ) we get

Q(tn, S0n, T ) =

√rK

σβn (S0n− βn)

√rK

3βn3[−βn + (σ2+ r)· βn]· (S0n− βn)2+ O((ρ∆S)3) =

=

√rK σ

(S0n βn − 1

)

√rK 3

[

−βn βn

+ σ2+ r ]

· (S0n

βn − 1 )2

+ O((ρ∆)3), (1.9) where βn is considered a dierentiable function of tn. We make the following approximation of βn

βn:

βn

βn = (ln βn) = ln βn+1− ln βn

∆t + O(∆t)

= ln S0n− ln βn− (ln S0n− ln βn+1)

∆t + O(∆t)

= lnSβn0

n − lnβSn+10n

∆t + O(∆t).

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Since lnS0n βn = ln

(

1 + S0n βn − 1

)

S0n

βn − 1 when S0n → βn+, We have then βn

βn = (Sn0

βn − 1)

− lnβSn+1n0

∆t + O(ρ∆S∆t). (1.10)

By replacing (1.10) in (1.9), we get Q(tn, S0n, T ) =

√rK σ

(S0n βn − 1

) +

+

√rK 3 ·

 (Sn0

βn − 1)

− lnβSn+10n

∆t − (σ2+ r)

 ·( S0n βn − 1

)2

+ + O((ρ∆S)2· (ρ∆S + ∆t)).

By setting

ξ = S0n βn − 1, we obtain then

Q(tn, S0n, T ) =

√rK σ ξ +

√rK 3 ·

[

ξ− ln S0n

βn+1∆t− (σ2 + r) ]

· ξ2+ O((ρ∆S)2· (ρ∆S + ∆t)).

After some transformations, we obtain the following equation ξ3

( ln S0n

βn+1 + (σ2+ r)∆t )

ξ2− 3σ2∆tξ− 3∆t

√rK Q(tn, S0n, T ) = O((ρ∆S)2· (ρ∆S + ∆t)).

(1.11) We just proved the following theorem:

Theorem 1.1.1. Suppose that Q(tn, S0n, T ) is known, then ξ satises the approximate cubic equation (1.11).

After solving the cubic equation (1.11), we get βn from the relation βn = S0n

ξ + 1

and from βn+1, (we do it recursively). Therefore, we need to know wether such ξ exists. The following lemma also in B. Kim et all [5], states the sucient conditions for the existence of ξ . Lemma 1.1.1. Suppose that ln

(

1 + ρ ∆S βn+1

)

<

(4Q2(tn, S0n, T )− (σ2+ r) rK

)

∆t with σ >

0, r > 0. Then, for suciently small ∆S and ∆t, ξ3

( ln S0n

βn+1 + (σ2+ r)∆t )

ξ2− 3σ2∆tξ− 3∆t

√rK Q(tn, S0n, T ) = 0.

has a unique real solution on (0, 1) ⊂ R.

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1.2 Numerical algorithm

By using the time and stock price discretization as before, Q as dened before, the numerical algorithm should then follow the steps:

1. Determine the option price Pin−1, i = 1, 2, ..., M explicitly from Pin. The price Pin of the American put option is a discrete solution to the discrete Black-Scholes equation:

Pin− Pin−1

∆t + 1

2σ2Si2Pi+1n − 2Pin+ Pin−1

∆S2 + rSiPi+1n − Pin

∆S − rPin= 0 for n = N, N − 1, ..., 1, Si = Sin, i = 1, 2, ..., M − 1,

PiN = 0 for i = 0, 1, 2, ..., M.

2. Find P0n−1 by solving:

P0n− P0n−1

∆t + 1

2σ2S02 (Pn

1−P0n

∆S P0nρ∆S−P−1n

∆S+ρ∆S 2

) + rS0

( P1n− P−1n

∆S + ρ∆S )

− rP0n= 0

P−1n = K− βn.

Assuming the computational domain is large, we impose zero boundary condition PMn = 0, n = N, N− 1, ..., 0.

3. Determine βn−1 = S0n−1

1 + ξ, where ξ is the solution for the equation

ξ3 {

lnS0n−1

βn + (σ2+ r)∆t }

ξ2+ 3σ2∆tξ−3σ2∆t

√rK Q(tn−1, S0n−1, T ) = 0, (1.12)

which has a unique real root ξ ∈ (0, 1).

4. Change the values of Sn−1 from old Sin−1 = βn+ ρ∆S + i∆S to new Sin−1 = βn−1 + ρ∆S + i∆S for i = 0, 1, 2, ..., M . Then we update the values of Pin−1 too. If we nish step 4, then we repeat the running step 1 through 4 until t0.

By using the software maple, see the code in appendix A, a numerical experiment with K = 1, r = 0.05, T = 0.5, σ = 0.2, ρ = 0.4, N = M = 100, we get:

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Fig. 1

Remark 1.2.1. For dierent values of the parameter ρ (0 < ρ < 1) we may have slightly dierent boundaries but they tend to be the same as ∆S → 0 and ∆t → 0.

The corresponding plot for the option value at a x time is given below.

Fig. 2 Fig. 3

Remark 1.2.2. It should be smoother for bigger values of M , i.e, when ∆S → 0.

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Chapter 2

American put options with jumps

We start by presenting a very useful formula from Itô for jump-diusion process, given in [2], Cont and Tankov (2004 ).

2.1 Itô formula for diusion with jumps

Let X be a diusion process with jumps dened as the sum of drift term, Brownian stochastic integral and a compound Poisson process:

Xt = X0+

t

0

bsds +

t

0

σsdWs+

N (t)

i=1

∆Xi,

where bt and σt are continuous non-anticipating process with

E

T

0

σt2dt

 < ∞ and the jumps on the stock price are given by ∆Xi.

Then, for any C1,2 functions f : [0, T ]×R → R, the process Yt= f (t, Xt)can be represented by

f (t, Xt)− f(0, X0) =

t

0

[∂f

∂s(s, Xs) + bs∂f

∂X(s, Xs) ]

ds + 1 2

t

0

σs22f

∂x2(s, Xs)ds+

+

t

0

∂f

∂X(s, XssdWs+ ∑

i≥1, τi≤t

[f (τi, Xτi+∆Xi)− f(τi, Xτi)]

. (2.1)

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In dierential notation:

dYt = ∂f

∂t(t, Xt)dt + bt∂f

∂x(t, Xt)dt +σt2 2

2f

∂X2(t, Xt)dt+

+∂f

∂X(t, XttdWt+ [f (t, Xt+ ∆Xt)− f(t, Xt)]. (2.2)

2.2 The stock price for the problem in analysis

We regard a problem of pricing an American put when we are not allowed to exercise the option during the weekends. We suppose that have a standard brownian motion [W (t)] under a complete probability space (Ω, F, P ) and (Ft)t≥0 is a ltration which satisfy the usual conditions. Furthermore, we consider a nancial market with a constant risk-free interest rate r > 0 and a stock price S(t) follows a geometric brown motion.

In order to deal with the exception of exercising the American put option during weekends, we suppose the following cases:

1. The stock price is traded continuously at any time;

2. The stock price cannot be traded during the weekends but during the week it is continu- ously traded.

Case 1: The stock price is traded continuously at any time.

Suppose that we the stock price is a geometric brownian motion, it is continuous and follows the dynamic:

dS(t) = µSdt + σSdW (t), t ∈ [0, T ],

where µ is the drift, σ is the volatility, W (t) is a standard brownian motion.

Since we are not allowed to exercise the option during the weekends, we will remove the weekends and consider the stock price during the week. Therefore, we may have jumps from Friday to Monday since the price may change during the weekend. Since S is a geometric brownian motion, with σ, µ constants, by Bjork (Chapter 4, 2003) [1], we have

S(t2) = S(t1) exp{

(µ− σ2/2)(t2− t1) + σ[W (t2)− W (t1)]} , for t1 < t2 and µ, σ constants.

Thus, if we regard τ1 as a Friday and τ2 as the next useful day (Monday), we have S(τ2) = S(τ1)· exp{

(µ− σ2/2)(τ2− τ1) + σ[W (τ2)− W (τ1)]} . Therefore, the jump size of the stock price from τ1 to τ2 is given by

∆S = S(τ2)− S(τ1) = S(τ1)[

exp{(µ − σ2/2)(τ2− τ1) + σ(W (τ2)− W (τ1))} − 1]

=

= S(τ1)(Y − 1),

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where

Y = exp{(µ − σ2/2)(τ2− τ1) + σ[W (τ2)− W (τ1)]}.

We regard now that there are n weekend over the interval [0, T ], and we order them as 1, τ2), ...., (τ2n−1, τ2n).

The jump size at each interval is given by

Yi = exp{(µ − σ2/2)(τ2i− τ2i−1) + σ[W (τ2i)− W (τ2i−1)]}.

By removing the weekends on the stock price, we will have the following dynamic dS(t)

S(t) = µdt + σdW (t) + dJ (t), t ∈ [0, T ] \ {∪ni=12i−1, τ2i)}, where

J (t) =

n(t) i=1

(Yi− 1), n(t) is the number of weekends up to time t and

Yi = exp{(µ − σ2/2)(τ2i− τ2i−1) + σ[W (τ2i−1)− W (τ2i−1)]}.

In other to simplify notations and for calculations proposes, We regard that the stock price has the dynamics:

dS(t)

S(t) = µdt + σdW (t) + [Y (t)− 1]dn(t),

where n(t) is one if we have jump at time t (i.e. if t = τ2j the time just after a weekend), it is zero otherwise and S(t)[Y (t)− 1] represents the jump at time t. We suppose that dW and dn are independent. We will now solve the stochastic dierential equation above by following the standard method for the case without any jump.

Therefore, by changing the variables Z(t) = ln S(t), by Itô we have dZ = 1

SdS +1 2

(

1 S2

) (dS)2

= 1

S (µSdt + σSdW + S(Y − 1)dn(t)) −

1

2S2 (µSdt + σSdW + S(Y − 1)dn(t))2

= µdt + σdW + (Y − 1)dn(t) − σ2 2 dt, since

(dW )2 = dt, dt2 = 0, dtdw = 0, dtdn(t) = 0, [dn(t)]2 = 0, dW dn(t) = 0.

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Integrating both sides we get:

t

0

dZ =

t

0

[µ− σ2/2]ds +

t

0

σdW s +

n(t) i=1

(Yi− 1)

ln S(t) = ln S(0) + (µ− σ2/2)t + σW (t) +

n(t) i=1

(Yi− 1)

S(t) = S(0)∗ exp

(µ − σ2/2)t + σW (t) +

n(t) i=1

(Yi− 1)

 . (2.3)

In order to avoid arbitrage, We need the model to be a martingale. Without removing the weekends we have the standard Black-Scholes model which is complete and with a unique martingale measure. By removing the weekends and shrinking the time interval we may have a dierent scenario, i.e, some jumps on the stock price may appear at known dates. Since we know that e−rtS(t) is a martingale (in this case µ = r on (2.3)), it should be natural to it keep so even with the referred possible jumps since the stock price is continuously traded. Then, the jumps must be a martingale. So,

E[S(τi)Yi(t)|Fs] = S(τi), s < t, i = 1, ..., n(T ), {Ft}t≥0 is the information ow up to time t. The last formula is equivalent to

E[Yi(t)|Fs] = 1. (2.4)

Thus,

e−rtE[S(t)|Fs] = e−rt· S(0) · ert· e[∑n(t)i=1(E[Yi|Fs]−1)

]

= S(0).

From condition (2.4), follows that the interest rate should be zero along the weekend. There- fore,

Yi = exp{−σ2/2(τ2i− τ2i−1) + σ[W (τ2i−1)− W (τ2i−1)]}.

Remark 2.2.1. The market is still complete since the stock is traded continuously and the adjustments that we do only are made in order to compute the option price which cannot be exercised during weekends.

Case 2: The stock price cannot be traded during the weekends but during the useful week days it is continuously traded.

As before, we suppose that the stock price is a geometric brownian motion and has the following dynamic:

dS(t)

S(t) = µdt + σdW (t), t ∈ [0, T ] \ {∪ni=12i−1, τ2i)},

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where (τ2i−1, τ2i) i = 1, 2, ..., n(T ) are weekends.

Since it is not traded during the weekends, after a weekend we may have a dierent value from the last one we end up with on the week before. There are many reasons that may be in the origin of this change, for instance, a political decision, a natural catastrophe, a terrorist attack, of course, depending on the asset on trading. Therefore, it is more convenient to consider that there will be a jump from Friday to Monday. Then, we have more random sources than traded assets. So, the market is incomplete and we may have then many martingale measures.

However, if we suppose that the jumps are given by a stochastic variable which has log-normal distribution since the stock price has log normal distribution during the useful week days, we would have a bit similar case with the previous one. Thus, the value of the stock price just after a weekend is

S(τ2i+) = S(τ2i−1)ea+b·Z(t),

where a, b are constants and Z(t) ∼ N(0, 1), i.e, Z(t) has standard normal distribution.

Therefore, a similar argument as on the previous case to avoid arbitrage, we must have the jumps to be martingale

E[S(τ2i−1)ea+b·Z(t)|Fs] = S(τ2i−1), s < t which implies that

E[ea+bZ(t)|Fs] = 1.

Consequently, we have a = −b2/2. Since the jumps should reect the stock price behavior dur- ing the weekend if it is traded along this time, then the natural value for b is the corresponding coecient of a standard brownian motion that we have along the week which is

b = [W (ti+1)− W (ti)] · σ, ti < ti+1.

By introducing these possible jumps under martingale measure in S(t), we have

S(t) = S(0)· exp

(r − σ2/2)t + σW (t) +

n(t) i=1

(e−b2/2+bZ(τ2i+)− 1)

 ,

where

b = σ· [W (τ2i)− W (τ2i−1)], i = 1, 2, .., n(T ).

Remark 2.2.2. Both cases have similar (equal) formulas. However they are dierent, on the rst one the market is complete and the second one not. Nevertheless, by choosing the martingale measure as we did on both cases, we have the same price process. Therefore, from now on we will treat them as a unique case, i.e, the stock price is given by

S(t) = S(0)· exp

(r − σ2/2)t + σW (t) +

n(t) i=1

(Yi− 1)

 .

where

Yi = exp{(−σ2/2)(τ2i− τ2i−1) + σ[W (τ2i)− W (τ2i−1)]}, i = 1, 2, ..., n(T ).

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2.3 The pricing problem

We will now price an American option when the stock price is given by the last two cases and suppose that the strike price is K . We regard that P (t, S) is the option price. By applying Itô formula (2.2), we have:

dP = (∂P

∂t + µS∂P

∂S + 1

2σ2S22P

∂S2 )

dt + σS∂P

∂SdW (t) + [P (t, S(t))− P (t, S(t))]dn(t).

Let us now make a ∆-hedged portfolio and we regard δ = ∂P

∂S: Π(t) = P (t)− δ · S(t).

We have thus,

dΠ(t) = dP − δdS = (∂P

∂t + µS∂P

∂S +1

2σ2S22P

∂S2 )

dt + σS∂P

∂SdW (t)+

+[P (t, S(t))− P (t, S(t))]dn(t)−∂P

∂SS(µdt + σdW (t) + (Y − 1)dn(t))

= (∂P

∂t + 1

2σ2S22P

∂S2 )

dt + [∆P (t, S(t))− δ · ∆S] dn(t).

In order to avoid arbitrage, the expected return of the hedged portfolio must be equal to the value of the portfolio invested at risk-free interest rate r. Therefore,

∂P

∂t +1

2σ2S22P

∂S2 + E[∆P (t, S)− δ∆S(t)|Fs]· I2i−12i)(t) = r(P − δS),

s < t, I(a,b)(t) is the indicator function of the interval (a, b). Since each Yi− 1 is a martingale, we have

E[δ∆S(t)|Fs] = E[δS(τ2i−1)(Yi− 1)|Fs] = δS(τ2i−1)E[Yi− 1|Fs] = 0.

Thus, for the e−rtP (t, S) be a martingale, we impose the condition E[∆P (t, S(t)· I2i−12i)|Fs] = 0, i = 1, 2, ..., n(T ), which is equivalent to

E[P (τ2i−1, S(τ2i−1)Yi)|Fs] = P (τ2i−1, S(τ2i−1)), i = 1, 2, ..., n(T ).

Therefore, the pricing problem becomes













∂P

∂t +1

2σ2S22P

∂S2 + rS∂P

∂S − rP = 0, t ∈ (0, T )

E[P (τ2i−1, SYi)|Fs] = P (τ2i−1, S(τ2i−1)), i = 1, 2, ..., n(T ), P (T, S) = max{K − S, 0}.

(2.5)

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Remark 2.3.1. For the smooth pasting condition, we may use the result of a more general case in a jump-diusion model. In H. Pham [8] for example, it was proved that for a jump-diusion model with positive volatility (σ > 0) and nite jump intensity the value of an American put P (t, S) is continuously dierentiable with respect to the underlying asset on [0, T ]×[0, ∞) and in particular the derivative is continuous across the exercise boundary:

∂P

∂S(t, S)→ −1, when (t, S)→ (t, S(t)),

for every t in [0, T ], where S is the critical stock price (exercise boundary).

We will have both analytical and numerical valuation for this problem.

2.3.1 Analytical valuation

We will now derive a formula for the option price under the stock price dened on the previous section. Consider an American put on the corresponding asset with strike price K and maturity time T . We consider the value of the American put at time t = T − t as PA(t, S), which is taken on the space D = {(t, S) : S ∈ (0, ∞), t ∈ [0, T ]}. There is a critical stock price S (exercise boundary) at each time t ∈ [0, T ] such that it is optimal to exercise the option when S ≤ S and it should continue otherwise. Thus, the American put can be written as

PA(t, S) =



K− S(t), if S(t) ≤ S(t) PA(t, S) > K− S(t), otherwise , where t = T − t.

We rewrite the pricing problem as

∂P

∂t +1

2σ2S22P

∂S2 + rS∂P

∂S − rP = 0, t ∈ (0, T ) (2.6) satisfying

E[P (τ2i−1, SYi)|Fs] = P (τ2i−1, S(τ2i−1)), i = 1, 2, ..., n(T ), (2.7) with the terminal and boundary conditions (also known as smooth-pasting conditions)

























P (T, S) = max{K − S, 0}

lim

S→∞P (t, S) = 0

Slim→SP (t, S) = K− S

S→Slim

∂P

∂S(t, S) =−1.

(2.8)

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Let F (t, S(t)) = e−rtP (t, S(t)), where P is the value of the American put option dened on the space D. By using the martingale measure for the stock price S and Itô formula (2.2), we have

dF = ∂F

∂tdt + ∂F

∂S(rSdt + σSdW (t)) + σ2 2

2F

∂S2dt + [F [(S(t) + ∆S)]− F (S(t))]

⇐⇒ dF = e−rt (

−rP +∂P

∂t + rS∂P

∂S + σ2 2

2P

∂S2 )

dt+

+e−rtσS∂P

∂SdW (t) + e−rt[P (S(t) + ∆S)− P (S(t)].

By integrating along the interval [0, T ], we have

F (T, S) = F (0, S) +

T

0

e−rt (

−rP + ∂P

∂t + rS∂P

∂S +σ2 2

2P

∂S2 )

dt +

T

0

e−rtσS∂P

∂SdW (t)+

+

n(T )

i=1

e−rτ2i−1[P (YiS(τ2i−1))− P (S(τ2i−1))].

By substituting F we get:

e−rTP (T, S) = P (0, S) +

T

0

e−rt (∂P

∂t +σ2 2

2P

∂S2 + rS∂P

∂S − rP )

dt +

T

0

e−rtσS∂P

∂SdW (t)+

+

n(T )

i=1

e−rτ2i−1[P (YiS(τ2i−1))− P (S(τ2i−1))].

By using the critical stock price, we can rewrite P (t, S) as follows

P (t, S) = I{S>S}P (t, S) + I{S≤S}(K− S(t)),

where I(a,b) is the indicator function of the interval (a, b).

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Using the above formula and boundary conditions, we get

e−rT(K− S)+= P (0, S) +

T

0

e−rt· I{S>S} (∂P

∂t +σ2 2

2P

∂S2 + rS∂P

∂S − rP )

dt−

−Kr

T

0

e−rtI{S≤S}dt +

T

0

e−rtσS∂P

∂SdW (t)+

+

n(T )

i=1

e−rτ2i−1· I{S>S}[P (YiS(τ2i−1))− P (S(τ2i−1))]+

+

n(T )

i=1

e−rτ2i−1 · I{S≤S}[P (YiS(τ2i−1))− P (S(τ2i−1))].

Since on the continuation region the American put satises the Black-Scholes partial dier- ential equation, then the rst integral is zero. Thus, we get

e−rT(K− S)+= P (0, S)− Kr

T

0

e−rtI{S≤S}dt +

T

0

e−rtσS∂P

∂SdW (t)+

+

n(T )

i=1

e−rτ2i−1· I{S>S}[P (YiS(τ2i−1))− P (S(τ2i−1))]+

+

n(T )

i=1

e−rτ2i−1 · I{S≤S}[P (YiS(τ2i−1))− P (S(τ2i−1))].

Taking expectations both sides and considering that

E

T

0

e−rtσS∂P

∂SdW (t)

 = 0

by the proposition 4.4 ( in Björk [1]), we thus obtain

PE = PA− Kr

T

0

e−rtE(I{S<S})dt +

n(T )

i=1

e−rτ2i−1· E[I{S>S}[P (YiS(τ2i−1))− P (S(τ2i−1))]]+

+

n(T )

i=1

e−rτ2i−1 · E[I{S≤S}[P (YiS(τ2i−1))− P (S(τ2i−1))]],

(25)

where PE = PE(S, T ), PA = PA(S, 0) are the European and the America put options, respec- tively.

Since we want the American put, we have

PA= PE+ Kr

T

0

e−rtE(I{S≤S})dt−

n(T )

i=1

e−rτ2i−1 · E[I{S>S}[P (YiS(τ2i−1))− P (S(τ2i−1))]]

n(T )

i=1

e−rτ2i−1· E[I{S≤S}[P (YiS(τ2i−1))− P (S(τ2i−1))]].

This we can write as

PA= PE + Kr

T

0

e−rtE(I{S≤S})dt−

n(T )

i=1

e−rτ2i−1· E[I{S>S,Y S>S}[P (YiS(τ2i−1))− P (S(τ2i−1))]]

n(T )

i=1

e−rτ2i−1· E[I{S>S,Y S≤S}[P (YiS(τ2i−1))− P (S(τ2i−1))]]

n(T )

i=1

e−rτ2i−1· E[I{S≤S, Y S≤S}[P (YiS(τ2i−1))− P (S(τ2i−1))]]

n(T )

i=1

e−rτ2i−1· E[I{S≤S, Y S>S}[P (YiS(τ2i−1))− P (S(τ2i−1))]]

= PE + Kr

T

0

e−rtE(I{S≤S})dt−

n(T )

i=1

e−rτ2i−1· E[I{S>S,Y S≤S}[P (YiS(τ2i−1))− P (S(τ2i−1))]]

n(T )

i=1

e−rτ2i−1 · E[I{S≤S, Y S>S}[P (YiS(τ2i−1))− P (S(τ2i−1))]].

References

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