Structure of hedging portfolio for American Put and Russian options

Structure of hedging

portfolio for American Put

and Russian options

Master’s Thesis in Financial Mathematics Alexander Stromilo

School of Information Science, Computer and Electrical Engineering Halmstad University

Alexander Stromilo

Halmstad University Project Report IDE0736

Master’s thesis in Financial Mathematics, 15 ECTS credits Supervisor: Prof. Albert N. Shiryaev

Examiner: Prof. Ljudmila A. Bordag External referees: Prof. Vladimir Roubtsov

February 21, 2008

Department of Mathematics, Physics and Electrical engineering School of Information Science, Computer and Electrical Engineering

Halmstad University

A lot of financial problems have several approaches. In this paper we try to consider the underlying problem from the probabilistic point of view. In the quantitative finance there are two general problems, namely estimating of the fair price of derivatives (different kinds of options) and hedging portfolio construction. And the main aim is obtaining concrete formulas in the closed form for the components β^∗ and γ^∗ of the hedging portfolio π^∗ for some American options in special models.

For a well-known Black-Scholes model Paavo Salminen (see [3] and [4]) studied both option types - American Put and Russian ones. For the former case he gives only the pricing technique for finite and infinite horizons omit- ting the hedging strategy construction, which one can find in the Karatzas’

lectures [7]. For the Russian option Salminen showes the main idea of hedg- ing constructing. Also he introduces extended version of the Russian option with the following payoff

Gt= e^−λtS_t^asup

s≤t

S_s^b. (1)

Using some properties of geometrical Brownian motion he reduces the pricing of option with payoff (1) to the pricing of common Russian one. He leaves the hedging strategy searching for readers. And we do it in the present work introducing more general payoff function.

Albert Shiryaev and Goran Peshkir in [1] give very accurate explanation of the price calculating process. They show how to reduce an optimal stop- ping problem to a free-bounadary one. In Chapter 7 one can find examples and calculations connected with pricing of American Put and Russian op- tions under the Black-Scholes conditions. From this book we borrow general notations and solution algorithm.

Thank my supervisor Albert N. Shiryaev for useful comments, the Master Program’s coordinator Ljudmila A. Bordag for many pieces of advice, and all people who were with me during the time I worked on my diploma.

valuations and estimations of the fair price of American options. But the formulas for hedging portfolio are interesting as well and are known for very particular cases only. In our work we study different cases of American Put and Russian options on finite and infinite horizon.

iii

1 Models of the Financial Markets, Option Problems, Basic

Formulas 1

1.1 European type . . . 1

1.2 American type . . . 2

1.3 Hedge . . . 2

1.4 Lines of the research . . . 3

1.5 Solution Algorithm . . . 4

1.6 Mathematical model . . . 5

1.7 Step I. Optimal stopping (OS) and free-boundary (FB) problems 7 1.8 Step II. Wealth process . . . 9

1.9 Step III. Hedging . . . 10

2 Hedging Formulas in Black-Scholes Model 13 2.1 The American Put . . . 14

2.2 The Russian option . . . 17

2.3 Extended Russian Options . . . 19

3 Hedging Formulas in Bachelier model 25 3.1 American Put option . . . 26

3.1.1 Problem . . . 26

3.1.2 Solution of the optimal stopping problem . . . 26

3.1.3 Derivation of the Hedging strategy . . . 27

3.2 Russian option . . . 28

3.2.1 Problem . . . 29

3.2.2 Solution of the optimal stopping problem . . . 29

3.2.3 Derivation of the Hedging strategy . . . 31

4 Sammary and Conclusions 33

Notation 35

v

vi

Models of the Financial

Markets, Option Problems,

Basic Formulas

We consider a (B, S)-market, arbitrage-free and complete. From mathemat- ical point of view market is represented by two sequences

B = (Bt)t≥0 and S = (St)t≥0, where

St is a stock price at time t ∈ [0, T ] (discrete or continuous), which is a random variable (r.v.),

Bt is a bank account or a value of bond at time t ∈ [0, T ], which is a predictable variable,

[0, T ] is the time horizon, where T can be infinite (in the latter case we view the interval [0, ∞) as the time horizon).

1.1 European type

Payoff in European case is FT-measurable r.v. GT and price of option V (for details see [2]) is given by

V = B0E˜G_T BT

, 1

where expectation is taken under a probability measure ˜P (so-called the mar- tingale measure) such that the process

S B =

µSt

Bt

¶

t≥0

is ˜P-martingale, defined on the filtered probability space (Ω, F, (F_t)_t≥0, ˜P).

1.2 American type

In American case we consider a payment process G = (G_t)_t≥0, adapted to the filtration (Ft)t≥0. At each time t until certain deadline T (which can take infinite value) we can either

ˆ stop, and our gain in this case is Gt

ˆ or continue observation.

At pointed out deadline T (if finite) the value of the payment process GT is known. The price of an option V is given by

V = B0 sup

τ ∈M^T0

E˜Gτ

B_τ.

For more details see [2]. Supremumm in the formula above is taken over all stopping times τ taking value in [0, T ], i.e. r.v.’s τ , satisfying two conditions:

1. for every fixed time t events {τ < t}, {τ = t} and {τ > t} are from σ-algebra Ft, i.e. at each time t we know exactly either τ < t, τ = t or τ > t (this explains the term ’stopping time’ above);

2. 0 ≤ τ ≤ T.

1.3 Hedge

We also construct a portfolio π (so-called the hedging portfolio) π = (β, γ),

where

β = (βt)0≤t≤T and γ = (γt)0≤t≤T,

and such a process V^π (so-called the wealth process or capital process) given by

V_t^π = βtBt+ γtSt

that

i) V₀^π = V ii) and

• V_T^π = G_T in European case

• V_t^π ≥ G_t for all 0 ≤ t ≤ T in American case.

We can view ourselves as buyers or sellers:

ˆ Buyer of option needs to know how to compute

– fair price V = B0E˜^G_B^T_T (for European type of option)

– or fair price V and optimal stopping time τ^∗ (for American type of option) such that

V = B0 sup

τ ∈M^T0

E˜G_τ Bτ

= B0E˜G_τ^∗ Bτ^∗

ˆ Seller of option (or writer) needs to know additionally how to con- struct such a self-financing portfolio π = (β, γ) that he/she can satisfy the buyers requirement at each (admissible) time t.

Further we will view ourselves only as option sellers and will be very interested in the structure of the hedging portfolio.

To find the latter is

the main aim of the present work!

1.4 Lines of the research

We will consider two famous models of market:

ˆ Bachelier model,

ˆ Black-Scholes model,

and study two types of options:

1. American Put Option (APO),

2. Russian Option (RO).

1.5 Solution Algorithm

General method to solve these problems consists of three main steps:

I. We compute the fair price of option which in our framework has the following expression

V^∗ = B0 sup

τ ∈M0

EGτ

Bτ

(1.1) by solving the corresponding free boundary problem of the type

LXV = 0 on C (1.2)

V = G on D (1.3)

V^∗ = min{V : V ≥ G} (1.4)

with some special “initial” conditions, making the solution unique.

II. The knowledge about the fair price is particularly helpful for the next step, namely for the calculating wealth process V^π∗. So-called essential supremum plays the main role in doing this. In this context we can define V_t^π∗ via

V_t^π∗ = Btess sup

τ ≥t E µGτ

B_τ

¯¯

¯¯ F^t

¶

(1.5)

III. Derivation of the process γ^∗ = (γ_t^∗)t≥0completes the algorithm. On this step we use the well-known Itˆo formula for the stochastic process V^π∗/B and uniqueness of the Doob decomposition of super-martingales. It can be written in short form as follows

V_t^π∗ Bt

= V^∗+ M_t− C_t, (1.6)

where M is a martingale which is equal to Rt 0

γ_s^∗d (Ss/Bs) and can be derived via Itˆo formula. Due to the uniqueness we obtain an expression for γ_t^∗.

In the next sections we consider all those steps in more details with precise notation. Now it is important to note that we can obtain an explicit form to the objects we are interested in not in any case . And such cases seem to be attractive for scientists.

1.6 Mathematical model

We work on a filtered probability space (Ω, F, (Ft)τ ≥0, P), where Ω = {ω} is the set of all possible events, F is σ-algebra of subsets of Ω, and (Ft)τ ≥0 - family of sub-σ-algebras satisfying t < s ⇒ Ft ⊂ Fs, and P is a probability measure on Ω.

Further we will need a few basic definitions. One can find it in [1].

Definition 1 (Wiener process or Brownian motion). A standard Brow- nian motion (or Wiener process) W = (W_t)_t≥0 is a process defined on a probability space (Ω, F, P) satisfying the following properties:

(a) W0 = 0;

(b) the trajectories of (Wt)t≥0 are continuous functions;

(c) the increments Wtk−W_tk−1, Wt_k−1−W_tk−2, . . . , Wt1−W_t0 are independent (for any 0 = t0 < t1 < . . . < tk, k ≥ 1);

(d) the random variables Wt− Ws, s ≤ t, have the N (0, t − s)-distribution, i.e.

E(Wt− Ws) = 0, D(Wt− Ws) = t − s.

Let X = (Xt)t≥0 be a time-homogeneous diffusion process defined on a filtered probability space (Ω, F, (F_t)_{τ ≥0}, P) solving the stochastic differential equation (SDE)

dX_t= ρ(X_t)dt + σ(X_t)dW_t, (1.7) where X0 = x (initial condition), W is a standard Brownian motion and co- efficients ρ(y) and σ(y) satisfy the local Lipschitz condition and the condition of linear growth:

1. for any n ≥ 1 there exist a constant θn such that

|ρ(y) − ρ(y^′)| ≤ θn|y − y^′| , |σ(y) − σ(y^′)| ≤ θn|y − y^′| for |y| ≤ n and |y^′| ≤ n;

2. and for any n ≥ 1

|ρ(y)| ≤ θ_n(1 + |y|), |σ(y)| ≤ θn(1 + |y|)) for |y| ≤ n.

Then (see [1], Theorem 4.1, pp.73-74) the process X exists and it is unique.

Thus X is a strong Markov process taking value from E ≡ R.

Definition 2 (Infinitesimal generator). The infinitesimal generator associated with the process X is defined on a function F : E → R as follows

LXF (x) := lim

t→0

ExF (Xt) − F (x)

t (1.8)

Infinitesimal generator also can be expressed in the following differential form:

LX = ρ(x) ∂

∂x + σ²(x) 2

∂²

∂x² (1.9)

It follows from the general theory of Markov processes (see [5] for more details). However, to get an intuitive meaning of this statement we suggest the following scheme. But first of all let us consider one of the central results in stochastic calculus, namely Itˆo’s Lemma.

Lemma 1 (Itˆo). Let Xt solve the SDE

dXt= a(t, Xt)dt + b(t, Xt)dWt, (1.10) where W is a standard Brownian motion (see Defenition 1). Which means that X is Itˆo process. And let F (t, x) ∈ C^1,2. Then F (t, Xt) is also Itˆo process, and the following representation holds

F (T, X_T) = F (0, X₀) + Z T

0

µ∂F

∂t + a∂F

∂x +1 2b²∂²F

∂x²

¶

(t, X_t)dt +

Z T 0

µ b∂F

∂x

¶

(t, Xt)dWt.

(1.11)

Thus we can write using Lemma 1 LXF (x) = lim

t→0

ExF (Xt) − F (x) t

= lim

t→0

ExhRt 0

¡ρF^′+ ¹₂σ²F^′′¢

(Xs)ds +Rt

0(σF^′) (Xs)dWs

i t

= Exlim

t→0

Rt 0

¡ρF^′+¹₂σ²F^′′¢

(Xs)ds t

= ρ(x)dF

dx(x) + 1

2σ²(x)d²F dx²(x).

In the second line the expectation of the second integral is equal to zero, be- cause it is taken w.r.t. Wiener increments. Finally we see that formula (1.9) holds.

What kind of problem do we want to solve? For the simplicity in the present chapter we consider only a problem of the form

V^∗(x) = sup

τ ExG(X_τ) (1.12)

with the further derivation of the wealth process V^π∗ and the hedging process γ^∗ which correspond to the value function V^∗.

Remark 1. In the sections 1.7-1.9 of this chapter we work with the infinites- imal generator of the form (1.9) just to show the main idea. But in the next chapters we will solve more complicated problems, than (1.12). It requires more general form of an infinitesimal generator. In particular we should maximize the expectation of the discounted payoff process

V^∗(x) = sup

τ Exe^−λτG(Xτ), (1.13) where λ is called “killing” rate. Here the trick is based on changing the underlying process X to the new process eX, in terms of which the problem is reduced to (1.12), where instead of Xτ we put eXτ. Here eX is also Markov process with the infinitesimal generator given by

LXe = LX − λI, (1.14)

where I is the identity operator (for more details see [1, Chapter 3, Subsection 1.3]).

Now we are ready to consider the step-by-step algorithm of the solution.

1.7 Step I. Optimal stopping (OS) and free-

boundary (FB) problems

To reduce the OS-problem to the FB-problem we use well-known theory of harmonic function. Let us start with some definitions and results we need.

Definition 3 (Superharmonic or excessive function). A measurable function F = F (x) is superharmonic (or excessive) for a process X if

ExF (Xτ) ≤ F (x), ∀τ ∈ M0, ∀x ∈ E. (1.15)

The informal explanation is the following: there is no stopping strategy which can give us, in average, grater value of such a function F by shifting initial point x in accordance with the distributional law of the process X.

Now we introduce the continuation set

C = {x ∈ E : V^∗(x) > G(x)} (1.16) and the stopping set

D = {x ∈ E : V^∗(x) = G(x)}. (1.17) If function V is lower semicontinuous (lsc) and G upper semicontinuous (usc) then C is open and D is closed.

Also we introduce the first entry time τD of the process X into the set D τD = inf{t ≥ 0 : Xt∈ D}. (1.18) The random time τD is a stopping (Markov) time w.r.t. filtration (Ft)t≥0

if D is closed set since both the X and (Ft)t≥0 are right-continuous.

Next result plays the most important role for the search of V^∗.

Theorem 1 (Necessary conditions for the existence of an optimal stopping time). Let τ^∗ be an optimal to the problem (1.12), i.e.

V^∗(x) = ExG(Xτ^∗) (1.19) for all x ∈ E. Then the value function V^∗ is the smallest superharmonic function which dominates the gain function G on E.

If we additionally assume that V is lsc and G is usc, then

i) the stopping time τD satisfies τD ≤ τ^∗ Px-a.s. for all x ∈ E and is optimal in (1.12);

ii) the stopped process (V^∗(X_t∧τD))_t≥0 is a right-continuous martingale un- der Px for every x ∈ E.

One can find the proof of the above theorem in [1], pp. 38-39.

From now on we are able to construct the problem which is equivalent to the initial one, namely (1.12), and is expressed in terms of the FB-problem.

It is clear that the family of superharmonic functions V dominating gain function G should solve the following problem:

LXV ≤ 0 (1.20)

V ≥ G (V > G on C & V = G on D by definition). (1.21) And we have to find a minimal of the functions:

V = min{V : (1.20) − (1.21)}.b (1.22)

Remark 2. The fact that we need to find both function V and set C (or D) explains the term “free-boundary problem”.

Remark 3. (see [1, Chapter 4]) If the process X is a diffusion and boundary

∂C is sufficiently regular e.g. Lipschitz, then X after starting at ∂C enters interior of D immediately. Keeping (1.20) it leads to

∂V

∂x

¯¯

¯¯

∂C

= ∂G

∂x

¯¯

¯¯

∂C

(smooth fit). (1.23)

Otherwise the following condition holds:

V |_∂C = G|_∂C (continuous fit). (1.24) When the function V^∗ is known together with the optimal stopping sets D we are able to do the next step.

1.8 Step II. Wealth process

The following important result can be found in [6], p. 57.

Theorem 2 (the Fundamental pricing theorem of perpetual Amer- ican contingent claims). There exist a SF-portfolio π^∗ = (β^∗, γ^∗) and a consumption process C^∗ such that the wealth process is given by

V_t^π∗ = Btess sup

τ ∈Mt

Ex

µG(Xτ) Bτ

¯¯

¯¯ F^t

¶

P − a.s. ∀ 0 ≤ t ≤ ∞, (1.25) where Mt is the set of all stopping times taking value from [t, ∞).

This theorem brings the tool to derive the wealth process via value func- tion V^∗:

V_t^π∗ = Btess sup

τ ∈Mt

Ex

µG(Xτ) Bτ

¯¯

¯¯ F^t

¶

= Btess sup

τ ∈M⁰ Ex

µG(Xt+τ) Bt+τ

¯¯

¯¯ X^t

¶

= ess sup

τ ∈M⁰ EXt

µG(X_τ) Bτ

◦ θ_t

¶

= V^∗(X_t), (1.26)

where θt: Ω → Ω is usual shift operator:

θ_t(ω)(s) = ω(t + s) for ω = (ω(s))_s≥0∈ Ω and t, s ≥ 0. (1.27) It is possible to get the second line from the first line of the equation se- quence (1.26) if the risk-free asset process B has the following (multiplicative) property:

Bt+s = BtBs, (1.28)

which takes place if the process is of the exponential form

Bt = e^µt. (1.29)

Remark 4. Consequence (1.26) would be more complicated if we used more complicated gain process e.g. discounted process, but the explicit dependence on function V^∗(·) holds whenever the killing factor is of the exponential form.

Thus we complete the second step of the solution scheme. The last step is deriving hedging process π^∗ from the wealth process V^π∗. To do this we need two fundamental results from the stochastic calculus. First one, Lemma Itˆo, was presented above in the section 1.6. Second result, namely Doob decomposition, will be presented in the next section.

1.9 Step III. Hedging

Let us first remind of the “martingale” term.

Definition 4 (Martingale and Sub-/Super- martingales). A stochastic process X = (Xt)t≥0 defined on the filtered probability space (Ω, F, (Ft)t≥0, P) is called martingale (submartingale or supermartingale) w.r.t. filtration (Ft)t≥0

if process is adapted to it and

ˆ E|Xt| < +∞ for all t ≥ 0,

ˆ E(Xs− X_t|F_t) = 0 (≥ 0 or ≤ 0 accordingly) for all 0 ≤ t < s.

Then we formulate the Theorem 3 in terms of supermartingales.

Theorem 3 (Doob-Meyer decomposition). Let X be a continuous su- permartingale such that the set {Xτ : τ is a stopping times and τ < ∞} is uniformly integrable. Then it can be expressed as follows:

Xt= X0+ Mt− C_t, t ≥ 0, (1.30) where M is a uniformly integrable martingale and C is an increasing pre- dictable integrable process. The processes (Mt)t≥0 and (Ct)t≥0 are unique P-a.s..

One can find the proof of the Theorem 3 in [8].

We expressed the latter theorem in terms of supermartingales, because it is clear that the process V^π∗/B is supermartingale. We can write (see [2])

V_t^π∗ Bt

= V^∗+ Z t

0

γ_s^∗d µXs

Bs

¶

− C_t^∗. (1.31)

From the other hand we use Itˆo’s lemma to express the process V^π∗/B via finction V^∗:

V_t^π∗ Bt

= V₀^π∗+ Zt

0

σXs

∂³

V^∗ Bs

´

∂x (Xs)dWs+ Zt

0

LX

µV^∗ Bs

¶

(Xs)ds (1.32)

= V^∗+ Mt− C_t^∗.

Comparing equalities (1.31)-(1.32) and keeping uniqueness of the Doob’s decomposition of supermartingale V^π∗/B we obtain the following result:

Mt = Zt

0

σXs

∂(V^∗/Bs)

∂x (Xs)dWs = Z t

0

γsd µXs

Bs

¶

. (1.33)

From this equation we will derive hedging strategy γ^∗ for particular prob- lems. In this chapter we will study already obtained results and develop new ones by using technique presented above.

For a continuous time case one can find calculations connected with Rus- sian and American Put options in the Black-Scholes model in the papers [2]

and [1]. Further we will be guided by it to derive hedging. In the paper [4]

the structure of the hedging of Russian option is described. We will gener- alize a payment process and find a corresponding hedging strategy. For the Bachelier model we will derive both prices of options and hedging strategy by our own calculations.

Hedging Formulas in

Black-Scholes Model

The famous market model assumes that

ˆ the price process X = (Xt)t≥0 solves the SDE

dXt= rXtdt + σXtdWt, (2.1) with X0 = x > 0 under the unique martingale measure Px. Uniqueness follows from the fact that model is arbitrage-free and complete (see [2]);

ˆ there exist a risk-free asset (bonds or bank account) Bt solving the following SDE

dB_t = rB_tdt, where r > 0 is the risk-free interest rate.

The equation (3.1) under Px has a unique solution given by Xt = xe^(r−σ

2

2 )t+σWt, (2.2)

which is easy to check applying Itˆo formula to the process (2.2).

The process X is strong Markov (diffusion) with the infinitesimal gener- ator given by

LX = rx ∂

∂x + σ² 2 x² ∂²

∂x². (2.3)

Use Definition 2 and formula (1.9) to get (2.3).

We will be interested in solving the following problem V (x) = sup

τ Ex(e^−rτGτ), (2.4) where the supremum is taken over all stopping times of the process X and G = (Gt)t≥0 is a gain process.

13

2.1 The American Put

Gain process G is given by

G_t= e^−λt(K − X_t)₊, (2.5) where λ ≥ 0 is the discounting (or killing) factor, K > 0 is the strike price and function x+:= 0 ∨ x = max{0, x} as usual. So far, we consider now the problem of such a type:

V (x) = sup

τ Ex

¡e^−(r+λ)τ(K − Xτ)+

¢. (2.6)

Step I. Derivation of the value function V^∗(x). It is suggested that there exist a point b ∈ (0, K) such that the stopping time

τb = inf{t ≥ 0 : Xt ≤ b} (2.7) is optimal in the problem (2.6). Standard arguments based on the strong Markov property lead to the following free-boundary problem for the unknown value function V and the unknown point b:

LXV = (r + λ)V for x > b, (2.8)

V (x) > (K − x)+ for x > b, (2.9)

V (x) = (K − x)+ for x ≤ b, (2.10)

V^′(x) = −1 for x = b. (smooth fit) (2.11) Equation (2.8) using (2.3) reads

Dx²V^′′+ rxV^′ − (r + λ)V = 0,

where we denote D = σ²/2. Solution of this linear equation is of the form V (x) = x^p.

After substitution and division through Dx^p we get quadratic equation for the parameter p:

p²−³ 1 − r

D

´

p − r + λ D = 0.

The latter equation has two roots, p1,2 = 1

2

³1 − r D

´± r1

4

³1 − r D

´2

+ r + λ D .

Thus the general solution of (2.8) can be written as V (x) = C1x^p1 + C2x^p2,

where C1 and C2 are undetermined constants. Let us denote p1 and p2 so that p1 < 0 < p2. Hence, C2 must be zero due to the fact that

V (x) ≤ K for all x > 0.

Equations (2.10) and (2.11) form a system of two equations in two unknowns C1 and b. Solving this system we obtain

C1 = −1 p1

µ Kp1

p1 − 1

¶1−p1

, (2.12)

b = Kp1

p1− 1. (2.13)

Let us recall

D = σ² 2 , p1 = 1

2

³ 1 − r

D

´− r1

4

³ 1 − r

D

´2

+r + λ D , b = Kp1

p1− 1.

Inserting obtained constants completes the following result Theorem 4 (Fair Price of American Put Option). The fair

(arbitrage-free) price V^∗ from (2.6) has the following explicit expression

V^∗(x) =





−

1

¡

b

¢

, x > b,

K − x, x ≤ b.

And the stopping time

τb = inf{t ≥ 0 : Xt≤ b}

is optimal.

For proof see [1, Chapter 7, Section 1, Theorem 1.1].

Step II. Derivation of the wealth process V = (V_t )t≥0. As we pointed out above wealth process V^π∗ can be obtained by taking essential supremum of the ’rest’ gain:

V_t^π∗ = ess sup

τ ≥0 Ex(e^−rτGτ +t|Ft)

= e^−λtess sup

τ ≥0 Ex(e^−rτGτ◦ θt|Xt),

= e^−λtess sup

τ ≥0 EXt(e^−rτGτ ◦ θt), where θt is the usual transition operator:

θt(ω(s)) = ω(t + s) for all ω ∈ Ω, s ≥ 0, t ≥ −s.

From now on we can write

V_t^π∗ = e^−λtV^∗(Xt). (2.15) Step III. Derivation of the hedging portfolio π^∗ = (β^∗, γ^∗). Due to (NA+SF) the process Y = V^π∗/B admits the following representation:

Yt= Y0+ Z t

0

γ_s^∗d µXs

Bs

¶

− Ct, (2.16)

where Y0 = V₀^π∗/B0 = V^∗(x), γ_s^∗ is the risk component of the portfolio π^∗ and C is the nonnegative ’consumption’ process. At the same time we know exact formula for the process Y :

Yt= e^−rtV_t^π∗ = e^−(r+λ)tV^∗(Xt). (2.17) Using Itˆo formula in (2.17) we get

Yt= Y0+ Z t

0

e^−(r+λ)sV^∗′(Xs)σXsdWs+ Z t

0 (LX − (r + λ)I)V^∗(Xs)ds.

Noting that Doob’s decomposition for a supermartingales Y_t = Y₀+ M_t− C_t

is unique we shall compare representations (inserting d³

Xt

Bt

´= e^−rtσXtdWt

into (2.16)) Yt = Y0 +

Rt 0

e^−(r+λ)sV^∗′(Xs)σXsdWs − Rt 0

((r + λ)I − LX)V^∗(Xs)ds Yt = Y0 +

Rt 0

e^−rsγ_s^∗σXsdWs − Ct

Thus we have proved the following

Theorem 5 (Components of the Hedging Portfolio for American Put Option). Under the Black-Scholes assumptions about the market components of the portfolio which hedges perpetual American Put option with the payoff (2.5) have the following explicit representation

γ

= e

V

(X

), β

= e

(V

− γ

X

)

or

γ_t^∗ = −e^−λt³¡_Xt

b

¢p1−1

1_{Xt≥b}− 1{Xt>b}

´,

β_t^∗ = e^−(r+λ)t³ Xt

¡_Xt

b

¢p1−1³

1 −_p¹₁´

1_{Xt≥b}+ K1{Xt>b}

´.

(2.19)

2.2 The Russian option

Gain process G is given by

G_t= e^−λtS_t, (2.20)

where λ ≥ 0 is the discounting (or killing) factor, St= sup_u≤tXu ∨ s is the maximum process started at s. It is known that the pair (Xt, St) is a strong Markov process started at the point (x, s) under the martingale measure Px,s. So far, we consider now a problem of such a type:

V (x, s) = sup

τ Ex,s

¡e^−(r+λ)τS_τ¢

. (2.21)

Step I. Derivation of the value function V^∗(x, s).

Theorem 6 (Fair Price of Russian Option). Solution to the problem (2.21) is given by function

V^∗(x, s) =

(

bx

, s ≤ bx,

s, s ≥ bx,

where

b =

µp1(p2− 1) p2(p1− 1)

¶1/(p1−p2)

, K(y) = p1y^p2 − p2y^p1,

p1,2 = 1 2

³ 1 + r

D

´± r1

4

³ 1 + r

D

´2

+ λ D, D = σ²

2 and x ≤ s

and the optimal stopping time

τb = inf{t ≥ 0 : St≥ bXt}.

It is obtained by the analogous technique as in Section 2.1. For references see [1, Chapter 7, Section 2, Theorem 1.1] or [4, Theorem 3].

Step II-III. Derivation of the process V^π∗ and portfolio π^∗. As we pointed out above wealth process V^π∗ can be obtained by taking essential supremum of the ’rest’ gain:

V_t^π∗ = ess sup

τ ≥0 Ex,s(e^−rτGτ +t|Ft)

= e^−λtess sup

τ ≥0 Ex,s(e^−(r+λ)τSτ +t|X_t, St),

= e^−λtess sup

τ ≥0 Ex,s(e^−(r+λ)τ(Sτ ∨ S_t) ◦ θt|X_t, St),

= e^−λtess sup

τ ≥0 EXt,St(e^−rτGτ◦ θ_t),

V_t^π∗ = e^−λtV^∗(Xt, St). (2.23) We shall discuss only difficulties of this special (aftereffect option) case.

In [1] authors make a transformation from the two dimensional Markov process (X_t, S_t) to the one dimensional process Z_t= S_t/X_t. In the paper [4]

another way of derivation is suggested, namely via Martin boundary theory.

In that work the processes Xt and St are studied under their own framework.

The main problem of the hedging derivation is to understand how to

calculate total derivative of the function V^∗(x, s) along the x-space, because we need to note that processes X and S are not fully independent. However the changing of the second variable s can happen only when s = x, but (see [1, Chapter 7, Section 2]) on the diagonal of the space E = R × R we have the instantaneous reflection condition for the function V^∗:

∂V^∗

∂x (x, x) = 0. (2.24)

It follows that total derivative along the process Xt is simply partial derivative along the state variable x. And we obtain

Theorem 7 (Components of the Hedging Portfolio for Russian Option). Under the Black-Scholes assumptions about the market

components of the portfolio which hedges perpetual Russian option with the payoff (2.20) have the following explicit representation

γ

= e

∂V

∂x (X

, S

), β

= e

(V

− γ

X

).

or

γ_t^∗ = −e^−λtb/K(b) (K(Z_t) − p₁p₂((Z_t)^p2− (Z_t)^p1)) 1_{Zt≤b}, β_t^∗= e^−(r+λ)tX_t¡

bp₁p₂/K(b) ((Z_t)^p2 − (Zt)^p1) 1_{Zt≤b}+ Z_t1_{Zt>b}¢ ,

(2.26)

where Zt= St/Xt.

2.3 Extended Russian Options

P. Salminen in [4] consider an American Contingent Claim (ACC) with the payment process

t 7→ e^−λtX_t^psup

u≤tX_u^q, (2.27)

where p and q 6= 0 are arbitrary real numbers. In the present paper it is suggested to consider more general payoff G^[p,q] which is presented by

G^[p,q]_t := e^−λtX_t^pS_t^[q], (2.28) where S^[q] denotes a process (S_t^[q])_t≥0 given by

S_t^[q] := (sup

u≤t

X_t^q) ∨ s^q. (2.29)

Remark 5. By the notation above we produce an improvement of the paper [4] where author does not separate the cases q > 0 and q < 0. Note that in general

S_t^[q] 6= S_t^q, because a priori q can be negative and thus

sup

u≤t

X_u^q =







 (sup

u≤t

Xu)^q, q > 0,

(infu≤tX_u)^q, q < 0.

(2.30)

Then we will remark when sign of q is important. Moreover, we will improve conclusions of [4] if they are wrong.

Furhter we follow reduction technique suggested by Salminen for introduced extended option. Using multiplicative property of GBM one can analyze this extended case. Note that if X is a GBM(µ, σ²) then X^a is a

GBM(µa, a²σ²) with

µa := aµ − σ²

2 a(1 − a).

1. Firstly we consider the case p = 0. Write

Ex,s(e^−(r+λ)tS_t^[q]) = Ex^qq,s^q(e^−(r+λ)tSt)

= E^qx^q,s^q(e^−rqte^{−(λ+r−rq)t}St), (2.31) where the corresponding measure P^q governs a GBM(rq, q²σ²). The associated optimal stopping problem can be solved using the scheme above and formula 2.22 by changing

ˆ a starting point of the coordinate process : x 7→ x^q, s 7→ s^q,

ˆ the risk-free rate: r 7→ r_q := qr − ^σ₂²q(1 − q),

ˆ the volatility: σ 7→ |q|σ

ˆ and the discount parameter: λ 7→ λq:= λ + r − rq, if and only if λq > 0.

2. If p = 1 then

Ex,s(e^−(r+λ)tXtS_t^[q]) = xeEx,s(e^−λtS_t^[q])

= xeE^qx^q,s^q(e^−λtSt)

= xeE^qx^q,s^q(e^−˜rqte^{−(λ−˜rq)t}S_t), (2.32) where eP is the dual martingale measure and eP^q governs a

GBM(˜rq, q²σ²) with

˜

rq := q(r + σ²) − σ²

2 q(1 − q).

In this case we change parameters analogously:

ˆ x 7→ x^q, s 7→ s^q,

ˆ r 7→ ˜rq := q(r + σ²) −^σ₂²q(1 − q),

ˆ σ 7→ |q|σ,

ˆ λ 7→ ˜λq := λ − ˜rq.

The optimal stopping problem can be solved if and only if ˜λq > 0.

3. And now consider the most arbitrary case p /∈ {0, 1}. Hence, we have

Ex,s(e^−(r+λ)tX_t^pS_t^[q]) = Ex^pp,s^p(e^−(r+λ)tX_tS_t^[q/p])

= x^pEbx^p,s^p(e^{−(r−rp+λ)t}S_t^[q/p])

= x^pEb^q/px^q,s^q(e^{−(r−rp+λ)t}St)

= x^pEb^q/px^q,s^q(e^−ˆrq/pte^{−(r−rp−ˆrq/p+λ)t}St), (2.33) where bP is dual to the martingale measure P^p and governs a

GBM(ˆr, p²σ²) with

ˆ

r := rp+ p²σ² and bP^q/p governs a GBM(ˆrq/p, q²σ²) with

ˆ

rq/p:= q

pr −ˆ (pσ)² 2

q p

µ 1 −q

p

¶ . Parameters are changed by the following way:

ˆ x 7→ x^q, s 7→ s^q,

ˆ r 7→ ˆrq/p:= qr − ^σ₂²q(1 − 2p − q),

ˆ σ 7→ |q|σ,

ˆ λ 7→ λp+q := λ + r − rp+q.

The optimal stopping problem can be solved if and only if λp+q > 0.

Now one sees that [1] and [2] above are particular cases of [3]. To generalize studied objects we use the following notation

X : (x, u) 7→ x^u, R : (r, σ, u) 7→ ur − σ²

2 u(1 − u), R : (r, σ, p, q) 7→ qr −ˆ σ²

2 q(1 − 2p − q), Σ : (σ, q) 7→ |q|σ,

Λ : (λ, r, σ, u) 7→ λ + r − R(r, σ, u).

Denoting vector of parameters as ξ = (x, r, σ, λ) we produce new vector ξ = (ˆˆ x, ˆr, ˆσ, ˆλ) := (X(x, q), ˆR(r, σ, p, q), Σ(σ, q), Λ(λ, r, σ, p + q)).

Hence, for the general payment process (2.28) we have proved the following Theorem 8. For λ, r and σ in R+ and arbitrary p and q 6= 0 the optimal stopping problem to the extended Russian option

V^[p,q](x, s) = sup

τ Ex,s(e^−rτG^[p,q]_τ ) (2.34) has a solution if and only if ˆλ > 0 and an optimal time in this case is

τ_q^∗ =

½ inf{t ≥ 0 : St≥ ˆb^1/qXt}, q > 0;

inf{t ≥ 0 : I_t≤ ˆb^1/qX_t}, q < 0, , where I_t= (inf

u≤tX_u) ∧ s. Moreover, τ_b^∗ < ∞ P-a.s. and V^[p,q](x, s) = x^pVˆ^∗(ˆx, ˆs)

where ˆV^∗ satisfies the formula (2.22) but with ’hatted’ parameters instead.

Remark 6. In the paper [4] the cases p = 0, p = 1 and p /∈ {0, 1} are not unified and the optimal stopping time is suggested to be only of the form inf{t ≥ 0 : St≥ ˆcXt} for some optimal constant c.

Proof. We use the next idea (e.g. for the simplest case a = 0):

Assume we have a solution of the problem (2.31). And the optimal

stopping time is given (see [4], Theorem 3 or [1], p.398, Theorem 26.1) by τ^∗ = inf{t : S_t≥ ˆbX_t}

So far, we can write the following representation for the optimal stopping time of the initial problem (2.34)

[keeping (Xt, P^q) = (X_t^q, P) and (St, P^q) = (S_t^[q], P), sup

u≤t

X_u^q =

½ (sup_u≤tXu)^q, q > 0, (inf_u≤tX_u)^q, q < 0.

and

a^q₁ ∧ a^q₂ =

½ (a1∧ a₂)^q, q > 0, (a1∨ a₂)^q, q < 0. ]

ˆ q > 0

(τ^∗, P^qx^q,s^q) = (inf{t : St≥ ¯bX_t}, P^qx^q,s^q)

= (inf{t : S_t^[q] ≥ ¯bX_t^q}, Px,s)

= (inf{t : S_t^q ≥ ¯bX_t^q}, Px,s, Px,s)

= (inf{t : St≥ ¯b^1/qXt}, Px,s)

ˆ q < 0

(τ^∗, P^qx^q,s^q) = (inf{t : St≥ ˆbX_t}, P^qx^q,s^q)

= (inf{t : S_t^[q]≥ ˆbX_t^q}, Px,s)

= (inf{t : I_t^q ≥ ˆbX_t^q}, Px,s)

= (inf{t : It ≤ ˆb^1/qXt}, Px,s)

Now we are prepared to make calculations connected to the hedging portfolio. Let us recall that under fundamental pricing theorem of perpetual ACC’s there exist a portfolio process π^∗ and a consumption process C^∗ such that if the initial capital is v0 = V^∗ then the corresponding wealth process V^π∗ is given by

V_t^π∗ = ess sup

τ ≥0 Ex,s(e^−rτGt+τ|Ft) (2.35) where we put process G^[p,q] instead of G. Let τ ≥ 0 and consider

Ex,s(e^−rτG^[p,q]_t+τ|F_t)

= e^−λtEx,s(e^−(r+λ)τX_t+τ^p S_t+τ^[q] |Xt, St)

= e^−λtEx,s(e^−(r+λ)τX_τ^p(S_t^[q]∨ S_τ^[q]) ◦ θ_t|X_t, S_t)

= e^−λtEXt,St(e^−(r+λ)τX_τ^pS_τ^[q]◦ θ_t|X_t, St)

After taking ess sup over all stopping times of the process X it holds V_t^π∗ = e^−λtV^[p,q](X_t, S_t) = e^−λtX_t^pVˆ^∗(X_t^q, S_t^[q])

The number of stocks at time t in the hedging portfolio can be obtained from the coefficient of the martingale part of the wealth process. Hence, by Itˆo formula we get an authentic

Theorem 9 (Components of the Hedging Portfolio for Extended Russian Option). Under the Black-Scholes assumptions about the market components of the portfolio which hedges perpetual Extended Russian option with the payoff (2.28) have the following explicit representation

γ

= e

∂V

∂x (X

, S

), β

= e

(V

− γ

X

).

γ_t^∗= e^−λt(b/K(b))X_t^p+q¡

(p + q)K(Z_t^q) − qp1p2¡

(Z_t^q)^p2− (Z_t^q)^p1¢¢

1_{Zq t≤ˆb}

+pe^−λtX_t^p−1S_t^[q]1_{Zq t>ˆb}, β^∗_t = e^−(r+λ)t(bK(b))X_t^p+q¡

qp1p2¡

(Z_t^q)^p2− (Z_t^q)^p1¢

− (p + q − 1)K(Z_t^q)¢ 1_{Zq

t≤ˆb}

+e^−(λ+r)t(1 − p)X_t^pS_t^[q]1_{Z^q

t>b}, where Zt = St/Xt.

Hedging Formulas in Bachelier

model

One of the interesting models of market is the Bachelier model which assumes that the price of an underlying stock is a Brownian motion (maybe with a drift). Girsanov’s theorem gives us a possibility to find so-called martingale measure P such that the underlying process is a Brownian motion without drift, i.e. martingale.

Thus let X = (Xt)t≥0 be a time-homogeneous diffusion process defined by the stochastic differential equation

dXt= σdWt (3.1)

with initial condition X0 = x under Px:=Law(X| P, X0 = x) for x ∈ R, where W = Wt: t ≥ 0 is a standard Brownian motion (SBM) and σ > 0 is a diffusion coefficient. All stochastic objects are adapted to the filtered probability space (Ω, F, (Ft)t≥0, P). The infinitesimal generator associated with the process X is given by

LX = 1 2σ² ∂

∂x. (3.2)

Remark 7. The infinitesimal generator of the killed process bX_t= e^−λtX_t is found by

LXb = LX − λI, where I is the identity operator.

25

3.1 American Put option

3.1.1 Problem

The main aim of the present Subsection is to calculate value function V^∗(x) = sup

τ Ex(e^−λtG(Xτ)), (3.3) where the supremum is taken over all stopping times τ of X, the killing parameter λ > 0 and measurable function

G(x) = (K − x)+, (3.4)

where K > 0 is a so-called strike price of option.

3.1.2 Solution of the optimal stopping problem

It is assumed (for details see [1]) that there exist a number b∗ such that the stopping time

τb_∗ = inf{t : Xt≤ b∗} (3.5) is optimal to the problem (3.3). Thus we have V^∗(x) = (K − x)+ for all x ≤ b∗.

To compute the value function V^∗(x) for x > b∗, and to find the optimal stopping boundary b∗, we formulate the following system:

(LXV )(x) = λV (x) f or x > b, (3.6) V (x) > (K − x)+ f or x > b, (3.7)

V (x) = (K − x)+ f or x ≤ b, (3.8)

V^′(x) = −1 f or x = b (smooth f it) (3.9) V (x) → 0 while x → +∞ (boundary condition) (3.10) with LX as in (3.2).

Equation (3.6) can be rewritten as

V_xx− A²V = 0, where A² = 2λ/σ². Solution to this is given by

V (x, s) = C1e^−Ax+ C2e^Ax, (3.11) where C₁ and C₂ are unknown constants we need to define.

It is easy to see that (3.10) leads to C2 = 0.

From (3.9) we get

V^′(b) = −AC1e^−Ab= −1.

And we can express C1 via b as

C1 = 1

Ae^Ab. (3.12)

Substituting C1 and viewing (3.8) we obtain V (b) = 1

A = K − b.

Now expressions for C1 and b can be obtained C1 = 1

Ae^AK−1 b = K − 1

A. (3.13)

Thus we get the formula for the value function V^∗:

V^∗(x) =



 1

A

e

, x > b

K − x, x ≤ b.

Theorem 10 (Fair Price of American Put Option). The fair (arbitrage-free) price V^∗ of American Put Option with payoff (3.4) and infinite horizon under the Bachelier assumtions about the market model is given by (3.14).

3.1.3 Derivation of the Hedging strategy

Hedging strategy can be found as the coefficient of martingale part of the wealth process. Let us consider a process Y = (Yt)t≥0 defined as

Yt= V_t^π∗ := ess sup

τ ≥t Ex(e^−λτ(K − Xτ)+|F_t) (3.15)

= e^−λtess sup

τ ≥t Ex(e^{−λ(τ −t)}(K − Xτ)+|Xt)

= e^−λtess sup

τ ≥t EXt(e^{−λ(τ −t)}(K − Xτ)+)

= e^−λtV^∗(Xt)

Using Ito’s formula for the process Y we get the following Doob’s decomposition

Yt= Y0+ Mt− Ct,

where

Y0 = V^∗(x), Mt =

Z t 0

e^−λsσV_∗^′(Xs)dWs, Ct =

Z t 0

e^−λsλ(K − Xs)1{Xs<b}ds.

Since

Mt= Z t

0

γsdXs = Z t

0

σγsdWs

we obtain the formula of the hedging strategy for Russian option:

Theorem 11 (Components of the Hedging Portfolio for American Put Option). Under the Bachelier assumptions about the market

components of the portfolio which hedges perpetual American Put option with the payoff (3.4) have the following explicit representation

γ_t^∗ = e^−λtV_∗^′(Xt), β_t^∗ = V_t^π∗− γ_t^∗Xt (3.16) or

γ_t^∗ = −e^−λt¡

e^A(b−Xt)1_{Xt>b}+ 1{Xt≤b}

¢,

β_t^∗ = e^−λt¡

e^A(b−Xt)¡₁

A+ Xt

¢1_{Xt>b}+ K1_{Xt≤b}¢ ,

(3.17)

where A =p

2λ/σ² and b is defined by formula (3.13).

3.2 Russian option

Let us recall that we consider a time-homogeneous diffusion process X = (Xt)t≥0 defined by the stochastic differential equation

dXt= σdWt

with initial condition X0 = x under Px :=Law(X| P, X0 = x) for x ∈ R, where W = Wt: t ≥ 0 is standard Brownian motion (SBM) and σ > 0 is a diffusion coefficient. All stochastic objects are adapted to the filtered probability space (Ω, F, (Ft)t≥0, P). The infinitesimal generator associated with the process X is given by

LX = 1 2σ² ∂

∂x.