Knock Out Power Options in Foreign Exchange Markets

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U.U.D.M. Project Report 2014:10

Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Maj 2014

Department of Mathematics

Knock Out Power Options in Foreign Exchange Markets

Tomé Eduardo Sicuaio

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UPPSALA UNIVERSITET

Matematiska Institutionen

MASTER’S THESIS

KNOCK OUT POWER OPTIONS IN FOREIGN EXCHANGE MARKETS

COURSE: DEGREE PROJECT E IN MATHEMATICS

Student’s name: Sicuaio, Tom´e Eduardo Supervisor: Prof. Johan Tysk

tomeeduardosicuaio@uem.mz Civic registration number: 790603-P112

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Acknowledgements

I would like to express my sincere thanks to Prof. Johan Tysk for his helpful comments, suggestions and for the continuous support in achieving this work. I thank as well the Mathematics Department of Eduardo Mondlane University for the given opportunity to take my masters studies in one of the best Universities in the world. I am also very grateful to International Science Program team for their hospitality. I would also like to express my sincere thanks to my family, friends and loved ones for their continuous love and encouragement.

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Abstract

In recent years, the pressure of governments in maintaining currency parity has led to the break down of quite a few exchange rate mechanisms and has, thus, strengthened the need for companies, in particular, to make foreign exchange hedging decisions in order to avoid erosion of profit margins. This thesis deals with the pricing of Foreign exchange options. The Knock out options and the Power options will be treated in the sense that their payoffs are computed in closed form. We also combine the previous options in order to yield a new financial product. We find an explicit pricing formula for such financial product. Additionally, a new product of First generation exotic options, called Knock out power options, is described and its payoff is computed and compared with the payoff of the Knock out options.

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List of Figures

1.1 Spot Exchange Rate . . . 6

2.1 Brownian path with reﬂection after ﬁrst passage time . . . 11

3.1 Knock out call option . . . 20

4.1 Asymmetric power call option . . . 27

4.2 Symmetric power call option . . . 27

4.3 Up and out Asymmetric power call options using (Volatility) σ = 0.0853, (Interest rates) rU SD = 0.08, rSEK = 0.12, (Maturity time) T = 1year, (Strike price) K = 7, (Upper barrier) B = 8.4, n = 2. . . . 33

4.4 Discounted payoﬀ of Down and out asymmetric power call and Down and out call (Strike price) K = 7, (Lower barrier) B = 4.8, (Interest rates) rU SD = 0.08, r_SEK = 0.12, (Volatility) σ = 0.0853, (Maturity time) T = 1year, n = 2 . . 43

4.5 Down and out asymmetric power call options with (Volatility) σ = 0.0853, (Interest rates) rU SD = 0.08, rSEK = 0.12, (Maturity time) T = 1year, (Strike price) K = 7, (Lower barrier) B = 4.8, n = 2. . . . 45

4.6 Delta’s comparison and Gamma’s comparison of call options (Volatility) σ = 0.0853, (Interest rates) rU SD = 0.08, rSEK = 0.12, Maturity Time T = 1year, (Strike price) K = 7, (Lower barrier)B = 4.8, (Upper barrier)B = 8.4, n = 2. . . . 46

4.7 Replicated payoff function of Up and out asymmetric power call with (Strike price) K = 7, (Leverage coefficient) A = 10, (Interest rates) rU SD = 0.08, rSEK = 0.12, (Volatility) σ = 0.0853, (Upper barrier) B = 8.4, (Maturity time) T = 1year, n = 2. 48 4.8 Call options with different Maturities (Volatility) σ = 0.0853, (Interest rates) r_{U SD}= 0.08, rSEK = 0.12, Strike price K = 7, (Upper barrier) B = 8.4, n = 2. . . . 52

4.9 Call options with diﬀerent Maturities (Volatility) σ = 0.0853, (Interest rates) r_{U SD} = 0.08, r_SEK = 0.12, (Strike price) K = 7, (Lower barrier) B = 4.8, n = 2. . . . 53

4.10 Call options with diﬀerent Maturities (Volatility) σ = 0.0853, (Interest rates) rU SD= 0.08, r_SEK = 0.12, (Strike price) K = 7, n = 2. . . . 53

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List of Tables

4.1 USD appreciation, (Strike price) K = 7, (Lower barrier) B = 4.8, (interest rates) r_{U SD} = 0.08, r_SEK = 0.12, (Volatility) σ = 0.0853, (Upper barrier) B = 8.4, (Maturity time) T = 1year, n = 2. . . . 50 4.2 USD depreciation, (Strike price) K = 7, (Lower barrier) B = 4.8, (Interest rates) r_{U SD} = 0.08, r_SEK = 0.12, (Volatility) σ = 0.0853, (Upper barrier) B = 8.4, (Maturity time) T = 1year, n = 2. . . . 51 4.3 USD appreciation, (Strike price) K = 7, (Lower barrier) B = 4.8, (Interest rates) r_{U SD}= 0.08, r_SEK = 0.12, (Volatility) σ = 0.0853, (Upper barrier) B = 8.4, (Maturity time) T = 1year, n = 2. . . . 51 4.4 USD appreciations, (Strike price) K = 7, (Lower barrier) B = 5.6, (Interest rates) r_{U SD}=

0.02, r_SEK = 0.03, (Volatility) σ = 0.20, (Upper barrier) B = 8.4, (Maturity time) T = 1year, n = 2, DAO - Down and out, UAO - Up and out. . . . 54

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Introduction

Foreign exchange options are of great importance because nowadays are basic ingredients to international trading, they are many and each with its strategy for hedging.

Wytup [16] in his work describes the Power options whose payoﬀ is just a vanilla option (call or put) raised to the power of n, that is,[(X_T−K)⁺]ⁿ, [(K−XT)⁺]ⁿ in the symmetric case; and each term of vanilla option(call or put) is raised to the power of n, that is,(X_Tⁿ− Kⁿ)⁺, (Kⁿ− XTⁿ)⁺ in the asymmetric case. Therefore, we encounter two kinds of Power options, the asymmetric and symmetric.

Furthermore, Wytup [16] also realizes that these options (Power options) are always equipped with high payoﬀ compared to vanilla options. This, limits the risk of a short position even as the option premium for the holder.

This is one of the reasons that motivates speculators to invest in Power options once they request a high option premium. Yet, Wytup [16] and Tompkins [15] discuss hedging possibilities for Power options and they mentions that these options could be hedged using a combination of vanilla options with diﬀerent strike prices.

Knock out options are Barrier options options whose payoﬀ a Vanilla option if up to maturity time the underlying foreign exchange rate never hits the barrier agreed previously.

The holder of the Knock out option provides protection against the rising of the foreign exchange rate and according to this protection the holder has to pay a premium.

Lipton [7] discusses Barrier options for Foreign exchange options and give diﬀerent forms of cal- culation of their payoﬀ under risk neutral valuation.

Wytup [16] describes some of advantages and disadvantages of Knock out power options. Ones of the advantages are: Knock out options are cheaper than Vanilla options, Knock out options provide a conditional protection against, for example, stronger USD/weaker SEK, Knock out options give a complete participation, for example, in a weaker USD/stronger SEK. Some of the disadvantages of Knock out options are: The exchange rate may hit the barrier before maturity time and another one is having to pay a premium.

One of the main aims in this thesis is to combine both options, Knock out options and Power options, and forming a new ﬁnancial product in foreign exchange option which we will call it as Knock out power option. And we will compute its pricing function under risk neutral valuation using diﬀerent methods.

The thesis consists of ﬁve chapters, the ﬁrst one describe foreign exchange options, and herein,

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we give an overview about foreign exchange options, the dynamics od foreign exchange rate are derived and is also described the pricing partial diﬀerential equation (known as Black-Scholes partial diﬀerential equation) for foreign exchange options.

In chapter two we describe the joint density of Brownian motion and its maximum and minimum.

For this purpose we discuss the Reﬂection Principle in which we derive the main probability result that will be used in description of the joint density of Brownian motion and its maximum and minimum.

In the third chapter we describe the types of Foreign exchange options and special attention is given to Knock out options and Power options, and we compute the payoﬀ under risk neutral valuation using diﬀerent methods.

In the fourth chapter we describe our new financial product, Knock out power option, and also evaluate the payoff under risk neutral of the Down and out asymmetric power call option and Up and out symmetric power call option, using different methods. We also describe the sensitivity’s analysis and we discuss the static hedging for Up and out power call option.

In the last chapter, the ﬁfth, we describe the conclusions about our new ﬁnancial product, Knock out power option.

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Chapter 1 Foreign Exchange Options

1.1 Foreign Exchange Options Overview

Foreign Exchange Options are of great importance in the business world, allowing international trading over all the world. These options give the holder the right but not the obligation to exchange one currency into another currency at a determined exchange rate on a maturity date previously agreed. Nowadays, trade is just a small part of the domain of foreign exchange exchange market whose leading participants are Commercial Banks, Brokers, The International Monetary Market, Investors, Money Managers, Funds and Central Banks.

The market of foreign exchange options is one of the oldest, by Shamah [12], just because between 9000 to 6000 BCE (Before Common/Current/Christian Era) was seen cattle as well as camels being used as the ﬁrst and the form of money in exchange options.

The principal characteristic of any foreign exchange option for its holder are those of limited risk and, at the same time, proﬁt potential is unlimited.

Foreign exchange options have been developed to protect companies from some randomness of the exchange rate displacement. They are used by companies as contingent cover because their exposure can lead to unnecessary losses agreement on currency transactions.

1.2 Exchange Rate Dynamics

Before we go deeper in derivation of the Exchange Rate Dynamics let us have a look in some technical concepts which will be need along the section. These deﬁnitions can also be found at least on Shreve [13], Neftci [11], Khoshnevisan [5] and Shamah [12].

Deﬁnition 1.2.1. An Exchange rate between two currency is the value at which one currency worth to be swapped for another.

Deﬁnition 1.2.2. A probability space consists of a sample space Ω, a set of events F and a probability measure P : F → [0, 1]. The sample Ω is a nonempty set, the collection F satisﬁes the

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following properties 1. ∅ ∈ F ;

2. Whenever A∈ F , its complement A^c∈ F ; 3. whenever A₁, A₂, . . .∈ F , then their union ∪^∞

n=1

A_n ∈ F.

In addition the probability measure P is a function that assigns to each element ω ∈ F a number in [0, 1] so that

1. P(Ω) = 1

2. Whenever A₁, A₂, . . . is a sequence of disjoint events in F, then P( ∪_∞

n=1

A_n )

= ∑^∞

n=1

P(An).

Deﬁnition 1.2.3. Let (Ω,P, F) be a probability space. A real-valued function X deﬁned on Ω with the property that for every Borel subset B of R, the subset of Ω given by {X ∈ B} = {ω ∈ Ω; X(ω)∈ B} is in F, will be called random variable.

Deﬁnition 1.2.4. Let X(t) be a collection of random variables, where t is a parameter that runs over an index set T. The collection of random variables in an indexed time set is called stochastic process.

Deﬁnition 1.2.5. A Wiener process W is a stochastic process which satisﬁes the following properties

1. W (0) = 0.

2. The process W has independent increments, i.e. if r < s ≤ t < u then W (u) − W (t) and W (t)− W (r) are independent stochastic variable.

3. For s < t the stochastic variable W (t)− W (s) has Gaussian distribution with mean zero and standard deviation √

t− s 4. W has continuous trajectories.

Let X(t) be the Spot Exchange Rate at time t quoted as the quotient between the number of units of the domestic currency and the the number of units of the foreign currency. We will use the notation in Wystup [16] for a quote of foreign exchange rate, as follows, FOR-DOM, which mean that one unit of foreign currency worth FOR-DOM units of domestic currency. For example, in the case of USD-SEK with a spot of 6.3800, this means that one unit of USD worth 6.3800SEK.

We regard that B_f and B_d are bank accounts of the foreign currency and domestic currency,

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respectively. We suppose that under objective probability measure, the Spot Exchange Rate X(t) has a dynamics of a Geometric Brownian Motion [1]

dX = αXXdt + σXXd ¯W , (1.1)

where α_X, σ_X, ¯W are constant drift, constant volatility and scalar Wiener Process, respectively.

We consider that the foreign bank account and the domestic bank account have the following dynamics

dB_f = r_fB_fdt, (1.2)

dB_d= r_dB_ddt, (1.3)

where r_f, r_d are interest rates of the foreign account and domestic account, respectively. According to our exposure B_f units of foreign currency worth X· Bf units in the domestic currency. Thus, we regard that

B¯_f = X· Bf, (1.4)

the dynamics of ¯B_f will be obtained applying Ito’s product rule [13] to (1.4), and using the equation (1.1) thus

d ¯B_f = XdB_f + B_fdX + (dX)(dB_f)

= Xr_fB_fdt + XB_fα_Xdt + XB_fσ_Xd ¯W + 0

= B¯_f(r_f + α_X)dt + ¯B_fσ_Xd ¯W .

We know that the interest rate in the domestic account is r_d, so we can write the above equation under martingale measure Q as follows

d ¯Bf = rdB¯fdt + ¯BfσXdW. (1.5) Now, we seek for the dynamics of the spot exchange rate X under martingale measure Q. Using the relation (1.4) we can write X as follows

X = B¯f

B_f. (1.6)

Thus, the Q−dynamics of spot exchange rate X will be obtained applying Ito’s product rule to (1.6) and using relations (1.2) and (1.5) yields

dX = B¯_fdB_f⁻¹+ B_f⁻¹d ¯B_f + (d ¯B_f)(dB_f⁻¹)

= − ¯B_fB_f⁻²dB_f + B_f⁻¹B¯_fr_ddt + B_f⁻¹B¯_fσ_XdW − 0

= − ¯B_fB_f⁻²B_fr_fdt + B_f⁻¹B¯_fr_ddt + B_f⁻¹B¯_fσ_XdW

= −rfXdt + r_dXdt + σ_XXdW

= X(r_d− rf)dt + σ_XXdW.

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Therefore, the dynamics of spot exchange rate under martingale measure is given by

dX = X(r_d− rf)dt + σ_XXdW. (1.7)

Now, we compute the solution to the dynamics of spot exchange rate. Let Z = ln X, applying Ito’s formula we get

dZ = 1

XdX− 1 2

1

X²(dX)²

= (r_d− rf)dt + σ_XdW − 1 2σ_X²dt

= (

rd− rf − 1 2σ_X²

)

dt + σXdW, integrating from t to T yields

Z(T )− Z(t) = (

rd− rf − 1 2σ_X²

)

(T − t) + σX(W (T )− W (t)), we know that Z = ln X , so

ln X(T )− ln X(t) = (

r_d− rf − 1 2σ²_X

)

(T − t) + σX(W (T )− W (t)).

Figure 1.1: Spot Exchange Rate Therefore, solution to the dynamics of spot exchange rate is

X(T ) = X(t)· e(^r^d^−r^f⁻¹²^σ²^X)^(T−t)+σX(W (T )−W (t)), (1.8)

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where (W (T )− W (t)) has a normal distribution with zero mean and standard deviation √ T − t.

The graph in the Figure 1.1 shows a simulated path, using Monte Carlo method [2], of exchange rate quoted in (USD-SEK) with spot 6.3800, r_SEK = 12%, r_USD = 8%, σ_X = 0.43, Maturity T = 1year.

1.3 Pricing Foreign Exchange Options

Our aim herein, is to derive the pricing equation for foreign exchange options. Similar results can be found on Bj¨ork [1], Jiang[4] and Kwok [6]. Now, regarding the models given in the previous section deﬁned by (1.1), (1.2) and (1.3) and a simple contingent claim of the form Φ(X(T )), where Φ is a contract function.

Let us denote the relative portfolio invested in foreign exchange and the derivative by uB¯f and u_F, respectively. The dynamics for the value V of the portfolio is as follows

dV = V (

uB¯f

d ¯B_f

B¯_f + u_FdF F

)

, (1.9)

where d ¯Bf = ¯Bf(rf + αX)dt + ¯BfσXd ¯W and F = F (t, X(t)) is the pricing function whose dynamics is, applying Ito’s formula and using (1.1), as follows

dF = ∂F

∂tdt + ∂F

∂XdX + 1 2

∂²F

∂X²(dX)²

= ∂F

∂tdt + (Xα_Xdt + Xσ_Xd ¯W )∂F

∂X + 1

2σ_X² X²∂²F

∂X²dt

= (

∂F

∂t + Xα_X∂F

∂X + 1

2σ_X²X²∂²F

∂X² )

dt + ∂F

∂XXσ_Xd ¯W

= F αFdt + F σFd ¯W , where

αF =

∂F

∂t + Xα_X_∂X^∂F +¹₂σ²_XX^{2 ∂}_∂X²^F2

F and σF =

∂F

∂XXσ_X

F . (1.10)

Taking the dynamics of ¯B_f and F to (1.9) and, collecting dt term and d ¯W term yields dV = V(

uB¯f(r_f + α_X) + u_Fα_F)

dt + V(

uB¯fσ_X + u_Fσ_F) d ¯W . We have to ﬁnd the relative portfolio in a such way that

{ uB¯fσ_X + u_Fσ_F = 0, uB¯f + u_F = 1.

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Solving the above system of linear equations we get the following solution to the system









u_F =− σ_X σ_F − σX

,

uB¯f = σ_F σ_F − σX

.

(1.11)

So the d ¯W term of the portfolio value will vanish and we will remain with dV = V(

uB¯_f(rf + αX) + uFαF

)dt.

The theory of arbitrage free pricing implies that we must have

uB¯f(r_f + α_X) + u_Fα_F = r_d. (1.12) Substituting (1.11) into (1.12) and multiplying the entire equation by σ_F − σX we get

(r_f + α_X)σ_F − σXα_F = r_d(σ_F − σX).

Taking (1.10) into the above equation we yield the following partial diﬀerential equation,

∂F

∂t + (r_d− rf)X∂F

∂X + 1

2σ_X²X²∂²F

∂X² − rdF = 0,

the so called Black-Scholes partial diﬀerential equation or in a short way the Black-Scholes equation.

Proposition 1.3.1. The pricing function F (t, x) of the claim Φ(X_T) solves the boundary value

problem 





∂F

∂t + (r_d− rf)x^∂F_∂x + ¹₂σ²_xx^{2 ∂}_∂x²^F2 − rdF = 0, F (T, x) = Φ(x).

Now, we can apply the Feynman-Kac representation theorem, which may be found on Bj¨ork [1], in the above proposition to give us a risk neutral valuation formula.

Proposition 1.3.2. The pricing function has the representation F (t, x) = e^−r^d^(T^−t)E^Qt,x[Φ(X_T)], where the Q-dynamics of X are given by

dX(t) = (r_d− rf)X(t)dt + σ_XX(t)dW (t).

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Proof. First we have to apply Ito’s formula to the pricing function F (t, X(t)) as follows

dF (t, X) = (

∂F

∂t + (r_d− rf)X∂F

∂X +1

2σ_X²X²∂²F

∂X² )

dt + Xσ_X∂F

∂XdW.

Using Proposition 1.3.1 we can write the above equation in the following way dF (t, X) = r_dF dt + Xσ_X∂F

∂XdW.

Integrating the above equation in t≤ s ≤ T yields

F (s, X(s))− F (t, X(t)) = rd

∫s

t

F (τ, X(τ ))dτ + σ_X

∫s

t

X(τ )∂F (τ, X(τ ))

∂X dW (τ ).

Furthermore, the process X(τ )∂F (τ,X(τ ))

∂X is integrable and we take the expected value, the stochastic integral will be zero and remain the following equation

E_t,x^Q[F (s, X(s))]− Et,x^Q[F (t, X(t))] = r_d

∫s

t

E_t,x^Q[F (τ, X(τ ))]dτ.

Diﬀerentiating the above result with respect to s and integrating from t to T and regarding the initial condition, we get

F (t, x) = e^−r^d^(T^−t)E_t,x^Q[Φ(X(T ))].

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Chapter 2 The Joint Distribution of Brownian

Motion and its Maximum and Minimum

In this section we are going find the joint density of Brownian motion and its maximum or minimum. Similar results, for example, for maximum was discussed in Shreve [13] who has shown the joint density of Brownian motion and its maximum. We also will derive the basic result about Reflection principle which was discussed in Kwok [6] and Shreve [13]. Using reflection principle we are going to derive the basic probability result which will help us to derive the joint density of Brownian motion and its maximum or minimum.

2.1 Reﬂection Principle

Consider a standard Brownian motion W (t), let ¯M (T ) be the maximum of Brownian motion W (t), suppose that the maximum ¯M (T ) is greater than a positive m, fix a time T and regard that the first passage time , τ_m, in which the Brownian motion path reach the level m for the first time, τ_m < T , see Figure 2.1.

Let us deﬁne ¯W (t) as the mirror reﬂection of W (t) at level m between the time interval [τm, T ] and consider ¯W (t) being as follows

W (t) =¯

{ W (t), 0≤ t < τm, 2m− W (t), τm ≤ t < T.

The events {W (T ) < w} and { ¯W (T ) > 2m− w} are equivalents, obviously from τm onward.

For τ_m ≤ t ≤ T, hold the following equality

W (τ¯ _m+ δ)− ¯W (τ_m) = −(W (τm+ δ)− W (τm)), δ > 0. (2.1) The ﬁrst passage time , τ_m, will not aﬀect the Brownian motion at onward time since it depends only on trajectory’s history of Brownian motion, {W (t) : 0 ≤ t ≤ τm}.

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m− w

τ_m T t

w m 2m− w M (t)¯

Figure 2.1: Brownian path with reﬂection after ﬁrst passage time

The Brownian motion increments in (2.1) have the same distribution, it follows from the strong Markov property of Brownian motion.

The maximum of Brownian motion, ¯M (T ) is greater than m if and only if the ﬁrst passage time is lesser than T . Now, suppose that W (T )≤ w, then ¯W (T ) ≥ 2m − w and together with (2.1) we obtain

P ( ¯M (T )≥ m, W (T ) ≤ w) = P (τm ≤ T, W (T ) ≤ w)

= P ( ¯W (T ) ≥ 2m − w)

= P (W (T )≥ 2m − w), for all w ≤ m and m > 0.

Proposition 2.1.1. The probability of standard Brownian motion W (t) being lesser than a ﬁxed level w and the Maximum ¯M (t) being greater than a ﬁxed (2m− w) at time T , as described in Figure 2.1, is given by the formula

P ( ¯M (T )≥ m ∧ W (T ) ≤ w) = P (W (T ) ≥ 2m − w), w ≤ m, m > 0. (2.2)

2.2 Joint density of Brownian motion and its maximum

In this section we will follow the theory described in Shreve [13] and Kwok [6]. Let W (t) be a Brownian motion and let us denote by ¯M (T ) = max₀_≤t≤TW (t) the maximum of Brownian

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motion. We also denote by τ_m the ﬁrst passage time at level m > 0. Furthermore, ¯M (T ) ≥ m if and only if the ﬁrst passage time τ_m is less or equal to time t.

Thus, in order to ﬁnd the joint density of ( ¯M (T ), W (T )) we are going to apply the result from Proposition 2.1.1 derived in previous section about reﬂection principle , so

P ( ¯M (T ) ≥ m ∧ W (T ) ≤ w) = P (W (T ) ≥ 2m − w), w ≤ m, m > 0. (2.3) Regarding the reﬂection principle equality (2.3) and supposing that fM ,W¯ (m, w) is the joint density of ( ¯M (T ), W (T )) for T > 0 we have

∫∞

m

∫w

−∞

fM ,W¯ (x, y)dydx =

∫∞

2m−w

e⁻^2T¹ ^z²

√2πTdz. (2.4)

Now, we may diﬀerentiate (2.4) in order to m to get

−

∫w

−∞

fM ,W¯ (m, y)dy = 2

√2πTe⁻^(2m^2T^−w)2. (2.5)

Herein, we diﬀerentiate (2.5) with respect to w to obtain fM ,W¯ (m, w) = 2(2m− w)

T√

2πT e⁻^(2m−w)2^2T . The following two theorem below were discussed on Shreve [13].

Theorem 2.2.1. Let ¯M (T ) be the maximum of Brownian motion W (t). The joint density of ( ¯M (T ), W (T )) under measure Q, for T > 0 is

fM ,W¯ (m, w) = 2(2m− w) T√

2πT e⁻^(2m^2T^−w)2, on D ={(m, w) : w ≤ m, m ≥ 0}

and zero on complement of D.

Theorem 2.2.2. Let W (t) be a Brownian motion with drift α and let ¯M (T ) be the maximum of W (t). The joint density of ( ¯M (T ), W (T )) under measure ¯Q, for T > 0 is

fM ,W¯ (m, w) = 2(2m− w) T√

2πT e^αw⁻¹²^α²^T⁻^(2m^2T^−w)2, on D ={(m, w) : w ≤ m, m ≥ 0}

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2.3 Joint density of Brownian motion and its minimum

Consider W (t) as a Brownian motion and denote by M (T ) = min₀_≤t≤TW (t) the minimum of Brownian motion W (t). We also denote by τ_m the first passage time at level m < 0, in this case, M (T ) ≤ m if and only if the first passage time is less or equal to time T. Therefore, in order to find the joint density of (M (T ), W (T )) we are going to apply the reflection principle, thus

P (M (t) ≤ m ∧ W (t) ≥ w) = P (W (t) ≤ 2m − w), w ≥ m, m < 0. (2.6) According to the reﬂection principle (2.6) and supposing that f_{M ,W}(m, w) is the joint density of (M (T ), W (T )) for T > 0 we have

∫m

−∞

∫∞

w

f_{M ,W}(x, y)dxdy =

2m−w∫

−∞

√1

2πTe⁻^2T¹ ^z²dz. (2.7) Diﬀerentiating (2.7) with respect to m yields

∫∞

w

f_{M ,W}(m, y)dy = 1

√2πTe⁻^(2m^2T^−w)2. (2.8)

Now, diﬀerentiating (2.8) with respect to w yields

−fM ,W(m, w) = 2(2m− w) T√

2πT e⁻^(2m−w)2^2T .

Theorem 2.3.1. Let M (T ) be the minimum of Brownian motion W (T ). The joint density of (M (T ), W (T )) under measure Q, for T > 0 is given by the formula

f_{M ,W}(m, w) = 2(w− 2m) T√

2πT e⁻^(2m^2T^−w)2, on D ={(m, w) : w ≥ m, m ≤ 0}

Theorem 2.3.2. Let W (T ) be a Brownian motion with drift α and let M (T ) be the minimum of W (T ). The joint density of (M (t), W (t)) under measure ¯Q, for T > 0 is

f_{M ,W}(m, w) = 2(w− 2m) T√

2πT e^αw−¹²^α²^T⁻^(2m−w)2^2T , on D ={(m, w) : w ≥ m, m ≤ 0}

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Chapter 3 Types of Foreign Exchange Options

There exist many types of Foreign exchange options, Shamah [12] and Wystup [16] subdivide For- eign exchange options in two types, the First generation exotic options and the Second generation exotic options.

In the First generation exotic options we can ﬁnd at least options such as Barrier options, Digital options, Touch options, Rebates options, Asian options, Lookback options, Forward Start options, Ratchet options, Cliquet options, Power options and Quanto options.

In the Second generation exotic options we can ﬁnd at least options such as Corridors options, Faders options, Exotic Barrier options, Step up and Step down options, Forward on the harmonic average options, Variance options and Volatility swaps.

Of the above options we have just mentioned we will have a deep look into options, the Knock out options which belong to the class of Barrier options and Power options which is First generation exotic option.

The payoff’s sum of a Down and Out option(call/put) and Down and In option (call/put) is always equal to the payoff of regular vanilla option (call/put) whereas the payoff’s sum of an Up and Out option(call/put) and Up and In option(call/put) is also equal to the payoff of regular vanilla option(call/put). Therefore, it is obvious that Barrier options are cheaper than Vanilla options and that is why most dealers traded these kind of options in many cases.

3.1 Power Options

Power options are those whose payoff function is entirely or by parts raised to the power of order n. There are two types of Power Options, the Asymmetric Power Options and Symmetric Power Options [16]. The difference between these options, Asymmetric and Symmetric reside in the way as are their payoffs computed for the payoff of Asymmetric, this is entirely raised to the power of n while the payoff of Symmetric each parcel is raised to the power of n.

At maturity time T, Wystup [16] suggests, for Symmetric power call options and Symmetric

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power put options, the following payoﬀ functions

V (T ) = (max(X_T − K, 0))ⁿ, (3.1)

V (T ) = (max(K − XT, 0))ⁿ, (3.2) respectively. Here, X_T is the underlying at maturity time T and K the strike price.

The payoﬀ functions for Asymmetric power call options and Asymmetric power put options are given by

V (T ) = max(X_Tⁿ− Kⁿ, 0), (3.3) V (T ) = max(Kⁿ− XTⁿ, 0), (3.4) respectively.

All kind of Power options in foreign exchange options satisfy the same Black-Scholes partial diﬀerential equation

 .





∂F

∂t + (rd− rf)x^∂F_∂x + ¹₂σ_X²x^{2 ∂}_∂x²^F2 − rdF = 0, F (T, x) = Φ(x).

Wystup [16], in his work, only computes the discounted payoﬀ for an Asymmetric power call option. In this project we will present discounted payoﬀ for all Power options.

3.1.1 Discounted Payoﬀ for Power Options

We assume that the dynamics model of the underlying foreign exchange rate is a geometric Brow- nian motion

dX(t) = X(t)(r_d− rf)dt + X(t)σ_XdW (t)

under risk neutral measure Q. The spot of the underlying at maturity time T is given by X_T = X₀e(^r^d^−r^f⁻¹²^σ²^X)^{T +σ}X

√T Z.

• Risk neutral valuation for Asymmetric power call options CAP O = e^−r^d^TE^Q0,x[max(X_Tⁿ− Kⁿ, 0)] = e^−r^d^TE^Q0,x

[(X_Tⁿ− Kⁿ)1_{X_T_>K}]

= e^−r^d^TE^Q0,x

[X_Tⁿ1_{X_T_>K_}]

− e^−r^d^TKⁿE^Q0,x

[1_{X_T_>K_}]

= e^−r^d^T

∫∞

−∞

X_Tⁿ1_{X_T_>K}e⁻¹²^z²

√2πdz− e^−r^d^TKⁿ

∫∞

−∞

1_{X_T_>K}e⁻¹²^z²

√2πdz

= e^−r^d^TX₀ⁿ

∫∞

−z0

eⁿ(^rd−rf−¹₂σ_X²)^{T +nσ}X

√T z· e⁻¹²^z²

√2πdz− e^−r^d^TKⁿ

∫∞

−z0

e⁻¹²^z²

√2πdz

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= e^−r^d^TX₀ⁿeⁿ(^rd−rf+ⁿ⁻¹₂ σ²_X)^T

∫∞

−z0

·e⁻¹²^(z^−nσ^X^√^{T )}²

√2π dz− e^−r^d^TKⁿN (z0)

= e^−r^d^TX₀ⁿeⁿ(^r^d−rf+ⁿ⁻¹₂ σ²_X)^TN(

z₀+ nσ_X√ T)

− e^−r^d^TKⁿN (z₀) where

z₀ = ln^X_K⁰ +(

r_d− rf −¹₂σ_X² ) T σ_X√

T and N (x) =

∫x

−∞

e⁻¹²^z²

√2πdz.

Proposition 3.1.1. The price of an Asymmetric power call option, with exercise price K and maturity time T is given by the formula

CAP O(t, x, T, B) = e^−r^d^(T^−t)X₀ⁿeⁿ(^rd−rf+ⁿ⁻¹₂ σ²_X)^(T−t)N(

z0+ nσX

√T − t)

− e^−r^d^(T^−t)KⁿN (z0) (3.5) where N is the cumulative distribution function for the N [0, 1] and

z₀ = ln^X_K⁰ +(

r_d− rf − ¹₂σ²_X)

(T − t) σ_X√

T − t .

This result from Proposition 3.1.1 can also be found on Wystup[16] and Tompkins[15].

• Risk neutral valuation for Asymmetric power put options P_{AP O} = e^−r^d^TE^Q0,x[max(Kⁿ− XTⁿ, 0)] = e^−r^d^TE^Q0,x

[(Kⁿ− XTⁿ)1_{X_T_<K_}]

= e^−r^d^TKⁿE^Q0,x

[1_{X_T_<K_}]

− e^−r^d^TE^Q0,x

[X_Tⁿ1_{X_T_<K_}]

= e^−r^d^TKⁿ

∫∞

−∞

1_{X_T_<K_}e⁻¹²^z²

√2πdz− e^−r^d^T

∫∞

−∞

X_Tⁿ1_{X_T_<K_}e⁻¹²^z²

√2πdz

= e^−r^d^TKⁿ

z0

∫

−∞

e⁻¹²^z²

√2πdz− e^−r^d^TX₀ⁿ

z0

∫

−∞

eⁿ(^rd−rf−¹₂σ_X²)^{T +nσ}X

√T z· e⁻¹²^z²

√2πdz

= e^−r^d^TKⁿN (z₀)− e^−r^d^TX₀ⁿeⁿ(^rd−rf+ⁿ⁻¹₂ σ_X²)^T

z0

∫

−∞

·e⁻¹²^(z^−nσ^X^√^{T )}²

√2π dz

= e^−r^d^TKⁿN (z₀)− e^−r^d^TX₀ⁿeⁿ(^rd−rf+ⁿ⁻¹₂ σ_X²)^TN(

z₀− nσX

√T)

in this case

z₀ =−ln^X_K⁰ +(

r_d− rf − ¹₂σ²_X) T σ_X√

T .

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Proposition 3.1.2. The price of an Asymmetric power put option, with exercise price K and maturity time T is given by the formula

P_{AP O}(t, x, T, K) = e^−r^d^(T^−t)KⁿN (z₀)−e^−r^d^(T^−t)X₀ⁿeⁿ(^rd−rf+ⁿ⁻¹₂ σ_X²)^(T−t)N(

z₀−nσX

√(T − t)) , (3.6) where N is the cumulative distribution function for the N [0, 1] and

z₀ =−ln^X_K⁰ +(

r_d− rf − ¹₂σ²_X)

(T − t) σ_X√

T − t .

Remark : This result from Proposition 3.1.2 is not in the literature as far as we know.

Proposition 3.1.3. Consider an Asymmetric European power call and an Asymmetric Eu- ropean power put both with strike price K and exercise time T. Denoting the corresponding pricing functions by C_{AP O}(t, x, T, K) and P_{AP O}(t, x, T, K) we have the following relation,

C_{AP O}(t, x, T, K)− PAP O(t, x, T, K) = e^−r^d^(T^−t)xⁿeⁿ(^rd−rf+ⁿ⁻¹₂ σ_X²)^(T−t) − e^−r^d^(T^−t)Kⁿ. (3.7)

• Risk neutral valuation for Symmetric power call options C_{SP O} = e^−r^d^TE^Q0,x[(max(X_T − K, 0))ⁿ] = e^−r^d^TE^Q0,x

[((X_T − K)1{XT>K})ⁿ]

= e^−r^d^TE^Q0,x

[ _n

∑

j=0

(n j

)

(−K)^jX_Tⁿ^−j1_{X_T>K}

]

= e^−r^d^T

∑n j=0

(n j

)

(−K)^jE[

X_Tⁿ^−j1_{X_T>K}]

= e^−r^d^T

∑n j=0

(n j

) (−K)^j

∫∞

−∞

X_Tⁿ^−j1_{X_T_>K_}e⁻¹²^z²

√2π dz

= e^−r^d^T

∑n j=0

(n j

)

(−K)^jX₀ⁿ^−j

∫∞

−z0

e⁽ⁿ^−j)(^rd−rf−¹2σ_X²)^{T +(n}−j)σX

√T z· e⁻¹²^z²

√2πdz

= e^−r^d^T

∑n j=0

(n j

)

(−K)^jX₀ⁿ^−je⁽ⁿ^−j)(^r^d−rf+ⁿ^−j−1₂ σ²_X)^T

∫∞

−z0

·e⁻¹²^(z^−(n−j)σ^X^√^{T )}²

√2π dz

= e^−r^d^T

∑n j=0

(n j

)

(−K)^jX₀ⁿ^−je⁽ⁿ^−j)(^r^d−rf+ⁿ^−j−1₂ σ²_X)^TN(

z₀+ (n− j)σX

√T)

where

z₀ = ln^X_K⁰ +(

r_d− rf − ¹₂σ²_X) T σX

√T .

Knock Out Power Options in Foreign Exchange Markets

U.U.D.M. Project Report 2014:10

Department of Mathematics

Knock Out Power Options in Foreign Exchange Markets

Tomé Eduardo Sicuaio

UPPSALA UNIVERSITET

Matematiska Institutionen

MASTER’S THESIS

KNOCK OUT POWER OPTIONS IN FOREIGN EXCHANGE MARKETS

Acknowledgements

Abstract

Contents

List of Figures

List of Tables

Introduction

Chapter 1

Foreign Exchange Options

1.1 Foreign Exchange Options Overview

1.2 Exchange Rate Dynamics

1.3 Pricing Foreign Exchange Options

Chapter 2

The Joint Distribution of Brownian

Motion and its Maximum and Minimum

2.1 Reﬂection Principle

2.2 Joint density of Brownian motion and its maximum

2.3 Joint density of Brownian motion and its minimum

Chapter 3

Types of Foreign Exchange Options

3.1 Power Options

3.1.1 Discounted Payoﬀ for Power Options