Barrier options

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Abstract

In this thesis we study barrier options. As a background we first consider European options. To price barrier options we also need to know the distribution of a Brownian motion absorbed at a certain value. Such distributions are found using the reflection principle. We then price some standard contracts and find the implicit volatility of some contracts that are currently traded. Finally we study some gen- eral properties of barrier options, in particular the asymptotic behavior of the price as the volatility tends to infinity. These results we have not seen elsewhere.

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Acknowledgments

I would like to take the opportunity to thank my supervisor, Associate Pro- fessor Johan Tysk at the Department of Mathematics at Uppsala University for his great support during the work with this thesis. I will also thank my fianc´ee Linda for our useful discussions.

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1 Introduction

This thesis is written to fulfill the requirement of the Master of Science de- gree in Mathematics at Uppsala University.

A European call (put) option is a contract based, for example on a stock, which gives the owner the right but not the obligation to buy (sell) the underlying stock at a certain time in the future for a predefined price. This kind of options, where the price of the options on the expiry date is independent of the stock price prior to the exercise day, is called simple contingent claims. The main use of this type of options is to reduce the risk in a port- folio, but they are also used as high risk investments with opportunity to really high profits. There exist also contracts which are path dependent i.e.

the payoff at the exercise day depends also on the stock price prior to the exercise day. For example we have barrier options, where the payoff on the exercise day depends on whether or not the underlying stock hits a certain barrier during the lifetime of the option. The advantage of a barrier option is that it is cheaper than the corresponding ordinary option. Thus if the in- vestor is convinced that the stock price is not going to hit the barrier during the life time of the contract, he will have a contract similar to the ordinary option but to a lower price.

2 Pricing of simple contingent claims

2.1 Asset dynamics

In this section we present a description of how to price simple contingent claims in the Black-Scholes market. This market has two securities, one risk- free asset B_t and one risky asset S_t. We assume unrestricted borrowing and lending of money at the same interest rate and that the market is frictionless, which means that there are no transaction costs. Short selling is also allowed and we can trade continuously.

The risk-free security, B_t, represents a savings account with constant interest r. The dynamics of Bt is given by

dBt = rBdt (1)

where B₀ is always assumed to be equal 1. The solution to (1) is easily obtained to be Bt= e^rt.

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The risky security, S_t, is often a stock. Before we say anything about the dynamics of the risky security, we need to know what a Brownian motion is.

Definition 2.1. A standard Brownian motion W is a stochastic process with the following properties:

1. W₀= 0.

2. The increments of W are independent, i.e if t₁ < t₂ ≤ t3 < t₄ then are W_t₂− Wt1 and W_t₄ − Wt3 independent.

3. Wt2 − W^t1is normally distributed with expectation 0 and standard deviation √

t₂− t1.

4. W has continuous trajectories with probability 1.

The dynamics of S_t is,

dS_t= S_tµdt + S_tσdW_t, (2) where µ and σ > 0 are constant and often called the drift respectively the volatility of S. Wt is a Brownian motion which is defined on the probability space (Ω, (Ft)_{t∈[0,T ]}, P) where Ft can be seen as all the information until time t and P is the objective probability measure. It should be noted that (2) is only a shorthand for the integral equation,

S_t = S₀+ Z t

0

S_uµdu + Z t

0

S_uσdW_u, S₀> 0

where the first integral is a ordinary Riemann integral and the second one is a Itô integral, for information about Itô integrals see [Ø]. To solve (2) we need a theorem from Itô calculus.

Theorem 2.1. (Itˆo’s formula) Let X_t be the solution to the stochastic differential equation

dX_t = µdt + σdW_t. Let g = g(t, X_t) ∈ C², then

dg = ∂g

∂tdt +∂g

∂xdX_t+1 2

∂²g

∂x²(dX_t²),

with the following multiplication rules (dt)²= 0, dtdW_t = 0 and (dWt)² = dt.

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Proof. The proof is omitted, but it can be found in [Ø] at page 44.

Now we can solve (2) by first applying Itˆo’s formula to ln (S_t) d(ln(S_t)) = 1

S_tdS_t− 1

2S_t²(dS_t)² = 1

S_t(S_tµdt + S_tσdW_t) − 1

2S_t²S_t²σ²dt

= µdt + σdW_t−1 2σ²dt.

Now integration of both sides from 0 to t gives ln(S_t) = ln(S₀) + (µ − 1

2σ²)t + σW_t, which is the same as

S_t = S₀e^((µ−¹²^σ²^)t+σW^t⁾.

This kind of process is often called Geometric Brownian Motion, abbreviated GBM.

2.2 Trading strategies

A trading strategy is an (Ft)_{t∈[0,T ]} adapted stochastic process φ = (φ¹, φ²) which is RCLL¹ and defined on the probability space (Ω, (Ft)_{t∈[0,T ]}, P). The trading strategy is said to be self financing if the value process

V_t = φ¹_tS_t+ φ²_tB_t, ∀t ∈ [0, T ], satisfies

dV_t(φ) = φ¹_tdS_t+ φ²_tdB_t, ∀t ∈ [0, T ], which is equivalent to

Vt(φ) − V⁰(φ) = Z t

0

φ¹_udSu+ Z t

0

φ²_udBu, ∀t ∈ [0, T ].

The discounted value process V_t^∗ = _B^V^t

t = φ¹_tS_t^∗+ φ²_t is often used instead of V_t, here S_t^∗= ^S_B^t

t is the discounted stock price. If φ is self financing V_t^∗(φ) − V0^∗(φ) =

Z t 0

φ¹_udS_u^∗, ∀t ∈ [0, T ], (3)

1RCLL means that almost every sample path is right-continuous and has finite left limits

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is satisfied, as can be seen from the calculations below

dV_t^∗(φ) = d(e^−rtV_t(φ)) = −re^−rtV_t(φ)dt + e^−rtdV_t(φ)

= −re^−rt(φ¹_tS_t+ φ²_tB_t)dt + e^−rt(φ¹_tdS_t+ φ²_tdB_t)

= −rφ¹tS_t^∗dt − rφ²tdt + e^−rtφ¹_tdS_t+ rφ²_tdt

= −rφ¹tS_t^∗dt + e^−rtφ¹_tdS_t

= φ¹_tdS_t^∗. (4)

In the last equality, we use the fact that dS^∗ = _B¹

t(dS_t−rStdt), which follows directly from Itˆo’s formula. Now integration from 0 to t of (4) gives (3). The family of all self-financing trading strategies is too large from our point of view, because it does not exclude the opportunity to make a riskless profit, called arbitrage. We now define the concept of arbitrage more precisely.

Definition 2.2. An arbitrage opportunity is a trading strategy φ such that V₀(φ) = 0, P {VT(φ) ≥ 0} = 1 and P {VT(φ) > 0} > 0.

To exclude those trading strategies which are arbitrage opportunities we need to define what a P^∗ admissible trading strategy is, but first we need to know what a martingale respectively a spot martingale measure is.

Definition 2.3. A martingale measure is a measure Q ∼ P(∼ means that Q and P are equivalent probability measures) on (Ω, (F^t)_{t∈[0,T ]}) such that S^∗ is a local martingale under Q, and P^∗ ∼ P on (Ω, (Ft)_{t∈[0,T ]}) is a spot martingale measure if V^∗ is a local martingale under P^∗.

It can be shown that Q is a martingale measure if and only if it is a spot martingale measure. It is also possible to show that in the Black-Scholes model the martingale measure Q uniquely determined by the Radon-Nikod´ym derivative,

dQ

dP = exp r − µ

σ W_T − 1 2

(r − µ)² σ² T

, P-a.s.

Under Q the discounted stock price S^∗ has the following dynamics,

dS_t^∗= S_t^∗σdW_t^∗. (5)

If we apply Itˆo’s formula on B_tS_t^∗, we get that the dynamics for S_t under Q is,

dSt = Strdt + StσdW_t^∗ (6)

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The same method as we used to solve (2) gives us now the solutions to equation (5) and equation (6),

S_t^∗= S0e⁽⁻¹²^σ²^t+σW^t^∗⁾ respectively St = S0e^((r−¹²^σ²^)t+σW^t^∗⁾.

In the equations above W_t^∗ = Wt−^r−µ_σ t is a standard Brownian motion on the probability space (Ω, (Ft)_{t∈[0,T ]}, Q).

We can now define a P^∗-admissible trading strategy.

Definition 2.4. φ is a P^∗-admissible trading strategy if V_t^∗(φ) is a martingale under P^∗. We let Φ(P^∗) denote the class of all P^∗-admissible self financing trading strategies.

Let M be the Black-Scholes market where only P^∗-admissible trading strategies are allowed, then we have,

Theorem 2.2. M is free from arbitrage opportunities.

Proof. Assume that φ ∈ Φ(P^∗) is such that V0(φ) = 0 and P {V^T(φ) ≥ 0} = 1. Then the martingale property of V_t^∗ implies that

E ^∗ VT(φ) BT

F0

= E ^∗ VT(φ) BT

= V₀(φ) B0

= V₀(φ)

1 = V₀(φ) = 0.

This and the assumptions on φ gives that P (VT(φ) > 0) = 0 which means that there are no arbitrage opportunities

A contingent claim X with exercise date T is a FT adapted stochastic variable. If it only depends on the stock prise at time T i.e X = Ψ(ST) then it is called a simple contingent claim and Ψ is called the contract function. The contingent claim is said to be attainable if it is replicated by a P^∗ admissible trading strategy φ. The unique arbitrage free price π_t(X ) of X for any time t ∈ [0, T ] is then given by Vt(φ) where φ ∈ Φ(P^∗) is such that V_T(φ) = X . Theorem 2.3. (Arbitrage free pricing of contingent claims) Let X be a contingent claim with expiry date T which is attainable in M. Then its arbitrage free price π_t(X ) is given by:

π_t(X ) = BtE ^∗

X B_T

Ft

. This equation is called the risk-neutral valuation formula.

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Proof.

B_tE ^∗

X B_T

Ft

= B_tE ^∗ V_T(φ) B_T

Ft

= B_tV_t(φ)

B_t = V_t(φ) = π_t(X )

2.3 Black-Scholes formula

The two most commonly traded simple contingent claims are the European call and the European put. A European call(put) option is a contract which gives the holder of the contract the right to buy(sell) the stock S at time T for the predefined amount K, called the exercise price. The contract function is X = max(ST − K, 0) = (ST − K)⁺ respectively X = max(K − ST, 0) = (K − S^T)⁺. The price of a European call option is given by the famous Black-Scholes formula which is given in the next theorem. When the price of the European call C(t, S_t) is known its not hard to find the price P (t, S_t) of the European put, as we shall see later.

Theorem 2.4. (Black-Scholes formula) In M the arbitrage free price at time t of an European call with exercise price K and expiry date T is given by the formula

C(t, St) = StN (d^K₁ (t, St)) − e^{−r(T −t)}KN (d^K₂ (t, St)) where

d^K₁ (t, S_t) = ln ^S_K^t + (r +¹₂σ²)(T − t) σ√

T − t and

d^K₂ (t, S_t) = d^K₁ (t, S_t) − σ√ T − t

Proof. Here we assume that X = (ST− K)⁺is attainable in M (for a proof see [M-R]). The risk-neutral valuation formula gives that

C(t, S_t) = B_tE ^∗ (ST − K)⁺ B_T

Ft

= e^{−r(T −t)}E ^∗ (S_T − K)⁺ Ft

the second equality follows because r is constant. S_T can be written as S_T = S_te^{(r−¹²^σ²)(T −t)+σ(WT^∗−Wt^∗)} where S_t = s hence

C(t, S_t) = e^{−r(T −t)}E ^∗

S_te{^(r−¹²^σ²)(T −t)+σ(WT^∗−Wt^∗)} − K+ Ft

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where S_t is Ft-measurable and W_T^∗− Wt^∗ is independent of Ft. Thus if E ^∗

se{^(r−¹²^σ²)(T −t)+σ(WT^∗−Wt^∗)} − K+

< ∞, (8)

we can use Theorem A.1 in appendix A to rewrite (7) as, e^{−r(T −t)}E ^∗

S_te{^(r−¹²^σ²)(T −t)+σ(WT^∗−Wt^∗)} − K+ Ft

= H(S_t, T − t), where H(s, T − t) is given by

H(s, T − t) = e^{−r(T −t)}E ^∗

se{^(r−¹²^σ²)(T −t)+σ(WT^∗−Wt^∗)} − K+ . Let us now check if (8) holds,

E ^∗

se{^(r−¹²^σ²)(T −t)+σ(WT^∗−Wt^∗)} − K+

≤ E ^∗h

se{^(r−¹²^σ²)(T −t)+σ(WT^∗−Wt^∗)}i + K

= se{^(r−¹²^σ²^{)(T −t)}}E ^∗h

e{^σ(W^T^∗^−W^t^∗⁾}i + K

= se{^(r−¹²^σ²^{)(T −t)}}E ^∗ e

σW_{(T −t)}^∗

+ K

= se{^(r−¹²^σ²^{)(T −t)}}e¹²^σ²^{(T −t)}+ K

= se^{r(T −t)}+ K < ∞.

In the second last equality we have used that the expectation is the moment generating function for a normal distributed random variable, with mean 0 and variance T −t. Since (8) holds it is enough to calculate the unconditional expectation. Define

Y = W√_T^∗− Wt^∗

T − t

then Y ∈ N(0, 1) and ST = se^((r−¹²^σ²^{)(T −t)+σ}^√^{T −tY )}, so e^{−r(T −t)}E ^∗h

(se^((r−¹²^σ²)(T −t)+σ(WT^∗−Wt^∗))− K)⁺i

= e^{−r(T −t)} Z _+∞

−∞

(se^((r−¹²^σ²^{)(T −t)+σ}^√^{T −ty)}− K)⁺ϕ(y)dy, (9)

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where ϕ(y) = √¹

2πe⁻¹²^y². We note that (se^(r−¹²^σ²^{)(T −t)+σ}^√^{T −ty}− K)⁺= 0 if se^(r−¹²^σ²^{)(T −t)+σ}^√^{T −ty} < K which is equivalent to y < ln(K/s)−(r−¹₂σ²)(T −t)

σ√

T −t .

If we define y₀ = ln(K/s)−(r−¹₂σ²)(T −t) σ√

T −t then (9) is equal to e^{−r(T −t)}

Z _+∞

y0

(se^((r−¹²^σ²^{)(T −t)+σ}^√^{T −ty)}− K)ϕ(y)dy

= e^{−r(T −t)} Z _+∞

y0

se^((r−¹²^σ²^{)(T −t)+σ}^√^{T −ty)}ϕ(y)dy

− e^{−r(T −t)} Z _+∞

y0

Kϕ(y)dy = I₁− I2. First look at I1

I₁ = e^{−r(T −t)} Z _+∞

y0

se^((r−¹²^σ²^{)(T −t)+σ}^√^{T −ty)} 1

√2πe⁻¹²^y²dy

= 1

√2πse⁻¹²^σ²^{(T −t)} Z _+∞

yo

e⁻¹²^(y²^−2σ^√^{T −ty)}dy

= 1

√2πse⁻¹²^σ²^{(T −t)} Z _+∞

yo

e⁻¹²^(y−σ^√^{T −t)}²⁺¹²^σ²^{(T −t)}dy [y⁰= y − σ√

T − t, dy⁰ = dy]

= 1

√2πs Z _+∞

y0−σ√ T −t

e⁻¹²^y⁰²dy⁰

= sN (−y0+ σ√ T − t) where N (x) = √¹

2π

R_x

−∞e^y²^/2dy, which is the cumulative distribution function for a standard normal distribution. In the last equality the symmetry of the normal distribution is used. Let d^K₁ (t, s) = −y⁰+ σ√

T − t =

ln(s/K)+(r+¹₂σ²)(T −t) σ√

T −t . For I₂ we note that

I₂ = e^{−r(T −t)}KN (−y0),

where N is defined as above, and the symmetry of the normal distribution is used. Let d^K₂ (t, s) = −y0= ln(s/K)+(r−¹₂σ²)(T −t)

σ√

T −t = d^K₁ (t, s) − σ√ T − t.

We have now shown that

C(t, s) = sN (d^K₁ (t, s)) − e^{−r(T −t)}KN (d^K₂ (t, s))

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To get the price of a European put we can use the put-call parity.

Theorem 2.5. (Put-Call Parity) Let C(t, S_t) be the price of a European call and P (t, S_t) be the price of a European put, both with exercise price K and with the same expiry date T. Then they satisfy

C(t, S_t) − P (t, St) = S_t− Ke^{−r(T −t)} Proof. It is easy to see that

(S_T − K)⁺− (K − ST)⁺= S_T − K.

From this and Theorem 2.3 the required equality follows immediately.

2.4 Dividend paying stocks

To price a option on a dividend paying stock, we need to decide how to model the stock price. We assume that all the dividends that are left in the life time of the option after time t are known and that we know when they are paid out. Let the dividends be κ1, κ2, κ3. . . κm and let the dates when they are paid be t < T₁ < T₂ < T₃ < . . . < T_m < T . Let G_s be the solution to the geometric Brownian motion dG_s= µG_sdt + σG_sdW_s. Now we define Ss, the price of the dividend paying stock to be,

Ss= Gs+

m

X

i=1

κe^−r(Tⁱ^−s)I_{s∈[t,T_j_]}. (10) Let I_s be the present value of all future dividend payments until the expiry day of the option, i.e.

I_s =

m

X

i=1

κe^−r(Tⁱ^−t)I_{T_i_{∈[s,T ]}}, ∀s ∈ [t, T ], (11) from this we see that G_t = S_t− It and G_T = S_T.

Theorem 2.6. (The arbitrage free price of a European call option on a dividend paying stock) Let S_t be as in (10), then the arbitrage free price at time t of an European call, with exercise price K and expiry date T, is given by,

C^κ(t, S_t) = (S_t− It)N (d^K₁ (t, S_t− It)) − e^{−r(T −t)}KN (d^K₂ (t, S_t− It)), where It is given by (11) and d1, d2 is the same as in Theorem 2.4.

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Proof. By the risk-neutral valuation formula we get that, C^κ(t, S_t) = e^rtE^∗ (ST − K)⁺

e^rT Ft

= e^rtE^∗ (GT − K)⁺ e^rT

Ft

, (12) we recognize the right hand side of (12) as the usual Black-Scholes price at time t of a European call option on G with exercise price K, thus,

C^κ(t, S_t) = C(t, G_t)

= G_tN (d^K₁ (t, G_t)) − e^{−r(T −t)}KN (d^K₂ (t, G_t))

= (S_t− It)N (d^K₁ (t, S_t− It)) − e^{−r(T −t)}KN (d^K₂ (t, S_t− It)), here d1 and d2 is the same as d1 and d2 in Theorem 2.4.

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3 Barrier options

3.1 Some results from probability theory

To price barrier options we need to know what a hitting time τ (y) is and find the distribution of the stochastic process that is absorbed in certain value.

The hitting time τ (y) of y for a process X_t with continuous trajectories² is defined as follows.

Definition 3.1. The hitting time τ (y) of y is defined by τ (y) = inf{t ≥ 0|Xt = y}, and we define infimum of the empty set to be infinity.

Let Xt be the solution to,

dX_t = µdt + σdW_t X₀ = α,

where W_t is a standard Brownian motion under P. Then the absorbed process X_{t∧τ (y)} is equal to X_t if X_s is different from y, for every s before time t. X_{t∧τ (y)} equal to y if X_s is equal to y for any s before time t. Or more formally,

Definition 3.2.

X_{t∧τ (y)} =

X_t, if t < τ (y) y, if t ≥ τ(y).

We also define the indicator function which will be used in the proof below.

Let A be a statement. Then the indicator function is defined as, I{A} =

1, when A is true 0, when A is false.

2It is important that Xt has continuous trajectories, otherwise it may “jump” over y.

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Theorem 3.1. The density function f_{t∧τ (y)}(x, t, α) of the absorbed process X_{t∧τ (y)} is given by

f_{t∧τ (y)}(x; t, α) = ϕ(x; µt + α, σ√

t) − e^2µ(y−α)^σ2 ϕ(x; µt − α + 2y, σ√ t).

If α > y then f_{t∧τ (y)}(x, t, α) has support in (y, +∞) and if α < y it has support in (−∞, y). The function ϕ(x, µ, σ) is the density function for a normal distribution with mean µ and standard deviation σ, i.e.

ϕ(x; µ, σ) = 1 σ√

2πe⁻^(x−µ)2^2σ2 .

Proof. Define M_t^X = max_s∈[0,t]Xs and m^X_t = min_s∈[0,t]Xs. Let α < y and x < y then,

P{X_{t∧τ (y)} ≤ x} = P{Xt ≤ x, Mt^X < y}

= P{X^t ≤ x} − P{X^t ≤ x, Mt^X ≥ y}

= P µt

σ + W_t ≤ x − α σ

− P µt

σ + Wt ≤ x − α σ , M

µt σ+Wt

t ≥ y − α

σ

= I₁− I2.

We know that W_t is normal distributed with mean 0 and standard deviation

√t so I₁ is easy to compute,

I₁ = P

W_t ≤ x − α − µt σ

= 1

√2πt

Z ^x−α−µt

σ

−∞

e⁻^z2² dz.

In the computation of I₂ we will use the Girsanov Theorem (see Theorem A.2 in appendix).

I2 = P µt

σ + Wt ≤ x − α σ , M

µt σ+Wt

t ≥ y − α

σ

= E

I µt

σ + W_t ≤ x − α σ , M

µt σ+Wt

t ≥ y − α

σ

. If we define a probability measure by the Radon-Nikod´ym derivative,

dˆP dP = exp

−µ

σW_T −µ² σ²

T 2

, P-a.s, (13)

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then, by the Girsanov Theorem, ˆW_t = ^µt_σ + W_t follows a standard Brownian motion under ˆP. We see that equation (13) is equivalent to,

dP

dˆP = exp µ

σW_T +µ² σ²

T 2

, ˆP-a.s and that ^µ_σWT +^µ_σ²2T

2 = ^µ_σWˆT −^µ_σ²²^T₂. Hence we get that, I₂ = E_ˆ

"

e

µ

σWˆT−^µ2_σ2^T2

I

Wˆ_t≤ x − α

σ , M_t^W^ˆ^t ≥ y − α σ

# .

To evaluate this we will use the so-called reflection principle. Assume that τ (^y−α_σ ) < t, then by symmetry of the Brownian motion with respect to ^y−α_σ , the probability that ˆW_t ≤ ^x−α_σ is equal to the probability that ˆW_t ≥ ^2y−x−α_σ . (See figure(1)). This implies that,

Pˆ

Wˆ_t ≤ x − α

σ , M_t^W^ˆ^t ≥ y − α σ

= ˆP

Wˆt ≥ 2y − x − α

σ , M_t^W^ˆ^t ≥ y − α σ

= ˆP

Wˆt ≥ 2y − x − α σ

, the second equality follows because x < y. Hence

I₂ = E_ˆ

"

e

µ

σ(^2y−2α_σ − ˆWT)−^µ2_σ2^T₂

I

Wˆ_t ≥ 2y − x − α σ

#

= e

2µ(y−α) σ2 E_ˆ

"

e⁻

µ σWˆT+^µ2

σ2 T 2

I

Wˆ_t ≥ 2y − x − α σ

# . Define another probability measure by the Radon-Nikod´ym derivative,

d¯P dˆP = exp

−µ

σWˆ_T − µ² σ²

T 2

, P-a.sˆ

and again by the Girsanov Theorem ¯W_t = ^µt_σ + ˆW_t follows a standard Brow- nian motion under ¯Pso,

I2 = e

2µ(y−α) σ2 E¯

I

W¯t−µt

σ ≥ 2y − x − α σ

= e

2µ(y−α) σ2 P¯

W¯_t− µt

σ ≥ 2y − x − α σ

= e

2µ(y−α) σ2 P¯

W¯_t ≥ 2y − x − α + µt σ

= e

2µ(y−α)

σ2 1

√2πt Z _+∞

2y−x−α+µt σ

e⁻^z2² dz.

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Differentiation of P{X_{t∧τ (y)} ≤ x} with respect to x give us the density of X_{t∧τ (y)}. In the case when α > y and x > y we have that,

P{X_{t∧τ (y)} ≤ x} = 1 − P{X_{t∧τ (y)} ≥ x} = 1 − P{Xt ≥ x, m^Xt > y}

= 1 − P{X^t ≥ x} + P{X^t≥ x, m^Xt ≤ y}

= P{Xt ≤ x} + P{Xt ≥ x, m^Xt ≤ y}.

The first term on the left hand side is equal to I₁ above and the second term is calculated in the same way as I₂.

(y−α)/σ (2y−x−α)/σ

(x−α)/σ

Figure 1: A Brownian motion reflected in ^y−α_σ

3.2 Pricing of common barrier options

There are many different types of barrier options. In this section we are going to focus on the four most common namely, up-and-out, down-and- out, up-and-in and down-and-in barrier options. We start with the up-and- out contract. Let X = Ψ(ST) be an ordinary contingent claim. Then the up-and-out contract is defined as follows.

Definition 3.3. The up-and-out contract is defined as, X^bo=

X if S_t < b ∀t ∈ [0, T ] 0 if S_t ≥ b for any t ∈ [0, T ] where b > S0 is a fixed constant, called the barrier.

To price this contract we need Ψ^b(S_T) which is defined as.

Ψ^b(x) =

Ψ(x) if x < b

0 if x ≥ b

The arbitrage free price π^bo(t, s, Ψ) is now given by the following theorem.

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Theorem 3.2. (The arbitrage free price of an up-and-out contract) Let X = Ψ(ST) be a contingent claim, X^bo defined as above and S_t < b.

Then, the arbitrage free price of X^bo is π^bo(t, S_t, Ψ) = π(t, S_t, Ψ^b) − b

S_t

^2¯^r

σ2 π

t,b²

S_t, Ψ^b

, where ¯r = r −¹₂σ².

Proof. We will not lose any generality by letting t = 0, assume also that S₀= s. By the risk-neutral valuation formula we get

π^bo(0, s, Ψ) = B₀E ^∗ X^bo B_T

F0

= e^−rTE ^∗

"

Ψ(S_T)I (

sup

t∈[0,T ]

S_t< b )#

= e^−rTE ^∗h

Ψ^b(S_{T ∧τ (b)})i

= e^−rT Z b

−∞

Ψ^b(x)f (x)dx,

where f (x) is the density function for S_{T ∧τ (b)}. Under the risk-neutral probability P^∗ we have S_T = se^((r−¹²^σ²^{)T +σW}^T^∗⁾= e^{(ln s+(r−}¹²^σ²^{)T +σW}^T^∗⁾. Let X_T = ln s + (r −¹₂σ²)T + σW_T^∗, then X_T is a solution to

dX_t = (r − ¹₂σ²)dt + σdW_t^∗ X₀ = ln s.

From the previous section we know that the density function g(x) for XT ∧τ (ln b) is

g(x) = ϕ(x; ¯rT + ln s, σ√ T )

−e^2¯r(ln b−ln s)

σ2 ϕ(x; ¯rT − ln s + 2 ln b, σ√ T ), where ¯r = r −¹₂σ². If we write S_{T ∧τ (b)} = e^X^{T ∧τ(ln b)}, we then have

e^−rT Z _b

−∞

Ψ^b(x)f (x)dx = e^−rT Z _{ln b}

−∞

Ψ^b(e^x)g(x)dx

= e^−rT Z _{ln b}

−∞

Ψ^b(e^x)ϕ(x; ¯rT + ln s, σ√ T )dx

− e^−rT b s

^2¯^r

σ2 Z ln b

−∞

Ψ^b(e^x)ϕ

x; ¯rT + ln b² s

, σ√

T

dx,

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if x > ln b then Ψ^b(e^x) = 0, so it will not make any difference if we integrate to +∞ instead of ln b. We now get

π^bo(0, s, Ψ) = e^−rT Z _+∞

−∞

Ψ^b(e^x)ϕ(x; ¯rT + ln s, σ√ T )dx

− e^−rT b s

^2¯^r

σ2 Z _+∞

−∞

Ψ^b(e^x)ϕ

x; ¯rT + ln b² s

, σ√

T

dx.

In the first integral we recognize ϕ(x; ¯rT +ln s, σ√

T ) as the density function for XT with X0 = ln s and in the second integral

ϕ

x; ¯rT + ln

b² s

, σ√

T

as the density function for X_T with X₀ = ln^b_s². Let

S_T¹ = e^X^T if X₀ = ln s S_T² = e^X^T if X0 = ln^b_s² This leads to

π^bo(0, s, Ψ) = e^−rTE ^∗h

Ψ^b(S_T¹)i

− b s

^2¯^r

σ2 e^−rTE ^∗h

Ψ^b(S_T²)i

= π(0, s, Ψ^b) − b s

^2¯^r

σ2 π

0,b²

s, Ψ^b

The down-and-out contract is defined in a similar way as the up-and-out contract.

Definition 3.4. The down-and-out contract is defined as, Xbo=

X if S_t > b ∀t ∈ [0, T ] 0 if S_t ≤ b for any t ∈ [0, T ] where b < S₀ is a fixed constant, called the barrier.

To price this contract we need a function Ψ_b(S_T) defined as Ψ_b(x) =

Ψ(x) if x > b

0 if x ≤ b (14)

The price of the down-and-out contract is given by the theorem below, but the proof is omitted because it is almost identical to the proof of Theorem 3.2.

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Theorem 3.3. (The arbitrage free price of a down-and-out contract) Let X = Ψ(ST) be a contingent claim, Xbo defined as above and S_t > b then, the arbitrage free price of Xbo is

π_bo(t, S_t, Ψ) = π(t, S_t, Ψ_b) − b St

^2¯^r

σ2 π

t,b²

St

, Ψ_b

. where ¯r = r −¹₂σ².

To price, for example a up-and-out European call option, we need to know that the pricing function is linear.

Lemma 3.1. The pricing function π is linear in the third argument i.e.

π(t, S_t, αΨ + γΥ) = απ(t, S_t, Ψ) + γπ(t, S_t, Υ) for some real constants α, γ and any contract functions Ψ and Υ.

Proof. The risk-neutral valuation formula gives π(t, S_t, αΨ + γΥ) = B_tE ^∗ αΨ + γΥ

BT

Ft

= αBtE ^∗

Ψ B_T F^t

+ γBtE ^∗

Υ B_T F^t

= απ(t, S_t, Ψ) + γπ(t, S_t, Υ).

In the second equality we used the linearity of the expectation operator.

We also need a contingent claim h(S_T, b) which is defined as h(S_T, b) =

1 if S_T > b 0 if ST ≤ b .

The arbitrage free price of h(S_T, b) is given by the following lemma.

Lemma 3.2. The arbitrage free price of h(S_T, b) is

π(t, S_t, h(S_T, b)) = e^{−r(T −t)}N

"

ln(^S_b^t) + (r − ¹₂σ²)(T − t) σ√

T − t

#

. (15)

Proof. The risk-neutral valuation formula gives that π(t, S_t, h(S_T, b)) = B_tE ^∗ h(ST, b)

BT

Ft

.

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By the same argument as in the proof of Theorem 2.4 it is sufficient to consider the unconditional expected value, so if we use the same notation as in Theorem 2.4 we get

π(t, s, h(S_T, b)) = B_tE ^∗ h(ST, b) B_T

= e^{−r(T −t)}E ^∗[I{ST > b}]

= e^{−r(T −t)}P^∗n

se^(r−¹²^σ²^{)(T −t)+σ}^√^{T −tY} > bo

= e^{−r(T −t)}P^∗ (

Y > ln(_s^b) − (r −¹₂σ²)(T − t) σ√

T − t

)

= e^{−r(T −t)}N

"

ln(^s_b) + (r − ¹₂σ²)(T − t) σ√

T − t

# .

In the last equality we used the symmetry of the normal distribution.

To get a shorter notation we define C(t, St, x) to be the arbitrage free price of a European call, with exercise price x. We also define H(t, S_t, y) to be the arbitrage free price of h(S_t, y) given by equation (15).

Now we get the arbitrage free price of the up-and-out European call option from the Lemma below.

Lemma 3.3. (The arbitrage free price of an up-and-out call) Let Ψ = (S_T − K)⁺ and S₀ < b then if b ≤ K the price is

π^bo(t, s, Ψ) = 0, and when b > K we get

π^bo(t, s, Ψ)

= C(t, S_t, K) − (b − K)H(t, St, b) − C(t, St, b)

− b S_t

^2¯^r

σ2 C(t,b²

S_t, K) − (b − K)H(t, b²

S_t, b) − C(t, b² S_t, b)

.

Proof. If b ≤ K it is easy to see that Ψ^b= 0, so Theorem 3.2 gives that π^bo(t, s, Ψ) = 0.

If b > K then we see from figure(2) that

Ψ^b(ST) = (ST − K)⁺− (b − K)h(S^T, b) − (S^T − b)⁺,

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so by Theorem 3.2 we now get π^bo(t, S_t, Ψ)

= π t, S_t, (S_T − K)⁺− (b − K)h(ST, b) − (ST − b)⁺

− b St

^2¯^r

σ2 π

t,b²

St

, (S_T − K)⁺− (b − K)h(ST, b) − (ST − b)⁺

.(16)

By Lemma 3.1 equation (16) is equal to

C(t, S_t, K) − (b − K)H(t, St, b) − C(t, St, b)

− b S_t

^2¯^r

σ2 C(t,b²

S_t, K) − (b − K)H(t, b²

S_t, b) − C(t, b² S_t, b)

K b

b−K

Figure 2: The up-and-out-call when b > K

We will now define up-and-in and down-and-in contracts. The in-contracts are in some sense the opposite to the out-contracts because the in contracts is worthless if the underlying stock does not hit the barrier while the out contracts is worthless if the underlying stock hits the barrier. If X = Ψ(S^T) is an ordinary contingent claim it is now clear that the up-and-in and down- and-in contracts are defined as below.

Definition 3.5. The up-and-in contract is defined as, X^bi=

X if S_t ≥ b for any t ∈ [0, T ] 0 if St < b ∀t ∈ [0, T ] where b > S₀ is a fixed constant called the barrier.

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Definition 3.6. The down-and-in contract is defined as, Xbi=

X if S_t ≤ b for any t ∈ [0, T ] 0 if St > b ∀t ∈ [0, T ] where b < S₀ is a fixed constant called the barrier.

To get the arbitrage free prices π^bi(t, s, Ψ) respectively π_bi(t, s, Ψ) we will use the the following lemma.

Lemma 3.4. (In-out-parity of barrier options) The price of a up-and- in contract is,

π^bi(t, s, Ψ) = π(t, s, Ψ) − π^bo(t, s, Ψ), (17) and for the down-and-in contract we have,

π_bi(t, s, Ψ) = π(t, s, Ψ) − πbo(t, s, Ψ). (18) Proof. If we buy one up-and-in and one up-and-out contract of X , both with the same barrier, we will receive exactly X at the exercise day. We then have,

π(t, s, Ψ) = π^bi(t, s, Ψ) + π^bo(t, s, Ψ),

and from this equation (17) follows immediately. In the same way we get equation (18).

The arbitrage free price of the two contracts are given by the following two theorems.

Theorem 3.4. (The arbitrage free price of an up-and-in contract) Let X = Ψ(ST) be a contingent claim, X^bidefined as above and S_t< b then, the arbitrage free price of X^bi is

π^bi(t, S_t, Ψ) = π(t, S_t, Ψ_b) + b S_t

^2¯^r

σ2 π

t,b²

S_t, Ψ^b

, where ¯r = r − ¹₂σ².

Proof. By Lemma 3.4 it follows that

π^bi(t, s, Ψ) = π(t, s, Ψ) − π^bo(t, s, Ψ), apply Theorem 3.2 and we get

π^bi(t, s, Ψ) = π(t, S_t, Ψ) − π(t, St, Ψ^b) + b St

^2¯^r

σ2 π

t,b²

St

, Ψ^b

.

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From the definitions of Ψ^b and Ψ_bwe see that Ψ = Ψ^b+Ψ_b. Use the linearity of the price function and we get,

π^bi(t, s, Ψ) = π(t, S_t, Ψ^b) + π(t, S_t, Ψ_b) − π(t, St, Ψ^b) + b

S_t

^2¯^r

σ2 π

t,b²

S_t, Ψ^b

= π(t, S_t, Ψ_b) + b S_t

^2¯^r

σ2 π

t, b²

S_t, Ψ^b

.

Theorem 3.5. (The arbitrage free price of a down-and-in contract) Let X = Ψ(ST) be a contingent claim, Xbidefined as above and S_t > b then, the arbitrage free price of Xbi is

π_bi(t, S_t, Ψ) = π(t, S_t, Ψ^b) + b S_t

^2¯^r

σ2 π

t,b²

S_t, Ψ_b

,

where ¯r = r −¹₂σ².

Proof. The proof is almost identical to the proof of Theorem 3.4 so it is omitted.

3.3 A reversed convertible

In this section we are going to find the price of a reversed convertible, which is a contract on a stock. This contract will give the owner a different amount of money, depending on the stock price on the maturity day and whether or not the stock price stays above a barrier during the life time of the contract.

For example, if the stock stays above 0.8 · (stock price at the start day of the contract) during the whole life time of the contract, or if the stock price at the maturity day is greater or equal to the initial price of the stock, then the owner will receive the invested money plus a rate that is specified at the start day of the contract. If the stock price is less or equal to the barrier any time during the life time of the contract and the stock price at the maturity day is less than the initial stock price, then the owner of the contract will receive the same rate as in the other case plus the invested money times the stock price at the maturity day divided by the start price of the stock. If

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we denote the reversed convertible with Ψ, this can be written as,

Ψ =







1 + ˆr if S_t > ⁴₅S_T₀ ∀t ∈ [T0, T ]

min{S^S_T0^T , 1} + ˆr if St ≤ ⁴5ST0 for any t ∈ [T⁰, T ],

here T0 is the start day of the contract, T is the maturity day of the contract and ˆr is a interest rate, specified on the start day of the contract .

Theorem 3.6. (The arbitrage free price of the reversed convertible)

If Ψ is as defined above, then the arbitrage free price is

π(t, S_t, Ψ) = 1 5H

t, S_t,4

5S_T₀

−1 5

4 5

S_T₀ S_t

^2¯^r

σ2 H t, 4 5

2S_T²₀ S_t ,4

5S_T₀

!

+ S_t

S_T₀ + ˆre^{−r(T −t)}− 1 S_T₀C

t, St,4

5ST0

+ 4 5

ST0

S_t

^2¯^r

σ2 1

S_T₀C t, 4 5

2 S_T²₀ S_t ,4

5ST0

!

− 4 5

ST0

S_t

^2¯^r

σ2 1

S_T₀C t, 4 5

2 S_T²₀ S_t , S_T₀

!

, ∀t ∈ [T0, T ],

where ¯r = r − ¹₂σ².

Proof. We note that Ψ is equal to a down-and-out bond with face value 1 + ˆr and some kind of down-and-in contract. Let BO = 1 + ˆr and MI = min{_S^S_T0^T , 1} + ˆr.

π(t, S_t, Ψ) = π_bo(t, S_t, BO) + πbi(t, S_t, MI). (19) Theorem 3.3 gives that,

π_bo(t, S_t, BO) = π(t, St, BOb) − b S_t

^2¯^r

σ2 π

t, b²

S_t, BOb

,

where BOb(x) is defined in the same way as Ψ_b(x) in equation (14) and b = 0.8S_T₀. It is easy to see that BOb(x) = (1 + ˆr)h(x, b) so by the linearity of the pricing function we get,

Barrier options

Contents

1 Introduction

2 Pricing of simple contingent claims

3 Barrier options