Mathematical Finance

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FINANCIAL MATHEMATICS

I-Liang Chern

Department of Mathematics

National Taiwan University

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Chapter 1 Introduction

1.1 Financial Markets

A society improves its welfare through investment. The financial market provides a link between saving and investment. Savers can earn high returns from their saving and bor-rowers can execute their investment plans to earn future profits. In financial markets, assets are traded in. There are many kinds of financial markets:

• stock markets, • bond markets,

• currency markets, foreign exchange markets, • commodity markets (oil, wheat, gold), • futures and options markets.

In futures or options, more complex contracts than simple buy/sell trades have been intro-duced. These are called financial derivatives.

1.2 Financial Derivatives

1. Forwards contract: A forward contract is an agreement which allows the holder of the contract to buy or sell a certain asset at or by a certain day at a certain price. Here,

• the certain day—maturity or expiration date, • the certain price—delivery price,

• the person who write the contract (has the asset) is called in short position, • the person who holds the contract is called in long position.

2. Futures (futures contracts): A future contract, like a forward contract, except,

• it is normally traded in an exchange;

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• it has standard features (including contract size, quality, delivery arrangement,

price quotes, daily price movement, position limit, etc.);

• it is a margin trading (certain minimal amount of money should be maintained

in a margin account);

• clearinghouse.

3. Options: There are two kinds of options — call options and put options. A call (put) option is a contract between two parties, in which the holder has the right to buy (sell) and the writer has the obligation to sell (buy) an asset at certain time in the future at a certain price. The price is called the exercise price (or strike price). The holder is called in long position, while the writer is called in short position. The underlying assets of an option can be commodity, stocks, stock indices, foreign currencies, or future contracts.

There are two kinds of exercise features:

• European options : Options can only be exercised at the maturity date.

• American options : Options can be exercised any time up to the maturity date.

1.3 Examples

Notation

• t current time • T maturity date • S current asset price • ST asset price at time T

• E strike price

• c premium, the price of call option • r bank interest rate

1. An investor buys 100 European call options on IBM stock with strike price $140. Suppose

E = 140, St = 138,

T = 2 months,

c = 5 (the price of one call option).

If at time T , ST > E, then he should exercise this option. The payoff is 100 × (ST −

E) = 100 × (146 − 140) = 600, The premium is 5 × 100 = 500. Hence, he earns

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The payoff function for a call option is Λ = max{ST − E, 0}. One needs to pay

premium (ct) to buy the options. Thus the net profit from buying this call is

Λ − cter(T −t).

2. Suppose

today is t = 8/22/95, expiration is T = 4/14/96 , the strike price E = 250

for some stock. If ST = 270 at expiration, which is smaller than the strike price, we

should exercise this call option, then buy the share for 250, and sell it in the market immediately for 270. The payoff Λ = 270 − 250 = 20. If ST = 230, we should

give up our option, and the payoff is 0. Suppose the share take 230 or 270 with equal probability. Then the expected profit is

1 2 × 0 +

1

2× 20 = 10.

Ignoring the interest of bank, then a reasonable price for this call option should be 10. If ST = 270, then the net profit= 20 − 10 = 10. This means that the profits is

100% (He paid 10 for the option). If ST = 230 the loss is 10 for the premium. The

loss is also 100%. On the other hand, if the investor had instead purchased the share for 250 at t, then the corresponding profit or loss at T is ±20. Which is only ±8% of the original investment. Thus, option is of high risk and with high return.

1.4 Payoff functions

At the expiration day, the payoff of a future or an option is the follows. 1. The payoff function of a future is

Λ = ST − E. K S T Λ future (long) K S T Λ future (short)

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Payoff of a future, long position (left) and short position (right) 2. The payoff function of a call option is

Λ = max{ST − E, 0}.

K

S

T

Λ

call option (long):max{S_T−K,0}

K

S

T

Λ

call option (short):−max{S_T−K,0}=min{K−S_T,0}

Payoff of a call, long position (left) and short position (right) 3. The payoff function of a put option is

Λ = max{E − ST, 0}.

K

S_T

Λ

put option (long):max{K−S_T,0}

K

S_T

Λ

put option (short):−max{K−S_T,0}=min{S_T−K,0}

Payoff of a put, long position (left) and short position (right) 4. Below is a portion of a call option copied from the Financial Times.

the current time t = Feb 3 the expiration T = end of Feb,

T − t ≈ 10 days St = 2872

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E = 2650, 2700, 2750, 2800, 2850, 2900, 2950, 3000 c = 233, 183, 135, 89, 50, 24, 9, 3 2600 2650 2700 2750 2800 2850 2900 2950 3000 3050 −50 0 50 100 150 200 250 K c

The FT-SE index call option values versus exercise price.

1.5 Other kinds of options

• Barrier option: The option only exists when the underlying asset price is in some

prescribed value before expiry.

• Asian option: It is a contract giving the holder the right to buy or sell an asset for its

average price over some prescribed period.

• Look-back option: The payoff depends not only on the asset price at expiry but

also its maximum or minimum over some period price to expiry. For example, Λ = max{J − S0, 0}, J = max0≤τ ≤TS(τ ).

1.6 Types of traders

1. Speculators (high risk, high rewards)

2. Hedgers (to make the outcomes more certain)

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1.7 Basic assumption

Arbitrage opportunities cannot last for long. Only small arbitrage opportunities are ob-served in financial markets. Our arguments concerning future prices and option prices will be based on the assumption that “there is no arbitrage opportunities”.

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Chapter 2 Asset Price Model

2.1 Efficient market hypothesis

The asset prices move randomly because of the following efficient market hypothesis: 1. The past history is fully reflected in the present price, which does not hold any future

information. This means the future price of the asset only depends on its current value and does not depends on its value one month ago, or one year ago. If this were not true, technical analysis could make above-average return by interpreting chart of the past history of the asset price. This contradicts to the hypothesis of no arbitrage opportunities. In fact, there is very little evidence that they are able to do so.

2. Market reponds immediately to any new information about an asset.

2.2 The asset price model

We shall introduce a discrete model and a continuous model. We will show that the contin-uous model is the contincontin-uous limit of the discrete model.

2.2.1 The discrete asset price model

The time is discrete in this model. The time sequence is n∆t, n ∈ N. Let us denote the asset price at time step n by Sn. We model the asset price by

Sn+1 Sn = ½ u with probability p d with probability 1 − p. (2.1) Here, 0 < d < 1 < u. The information we are looking for is the following transition probability P (Sn = S|S0), the probability that the asset price is S at time step n with

initial price S0. We shall find this transition probability later.

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2.2.2 The continuous asset price model

Let us denote the asset price at time t by S(t). The meaningful quantity for the change of an asset price is its relative change

dS S ,

which is called the return. The change dS

S can be decomposed into two parts: one is

deterministic, the other is random.

• Deterministic part: This can be modeled by dS

S = µdt.

Here, µ is a measure of the growth rate of the asset. We may think µ is a constant during the life of an option.

• Random part: this part is a random change in response to external effects, such as

unexpected news. It is modeled by a Brownian motion

σdz,

the σ is the order of fluctuations or the variance of the return and is called the

volatil-ity. The quantity dz is sampled from a normal distribution which we shall discuss

below.

The overall asset price model is then given by

dS

S = µdt + σdz. (2.2)

We shall look for the transition probability density function P(S(t) = S|S(0) = S0). Or

equivalently, the integral

Z _b

a

P(S(t) = S|S(0) = S0) dS

is the probability that the asset price S(t) lies in (a, b) at time t and is S0 initially.

2.3 Random walk

To study the discrete asset price model, we study a simple model—the random walk in one dimension—first. Consider a particle moving randomly on a uniformly distributed grid points on the real lines. Suppose the grid points are located at m∆x, m ∈ Z. In each time step, the particle moves to its left adjacent grid point or right adjacent grid point with equal probability. Suppose the particle is located at 0 initially. Let Zndenote the location of this

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located at the m∆x cell at the time n∆t. That is, w(m∆x, n∆t) = P (Zn= m∆x|Z0 = 0). By our rule, Zn+1− Zn = ½ ∆x with probability 1 2 −∆x with probability 1 2 and w(m∆x, (n + 1)∆t) = 1 2w((m − 1)∆, n∆t) + 1 2w((m + 1)∆, n∆t). (2.3) Suppose in n times, the particle moves p times toward right and n − p time toward left. Then

m = p − (n − p) = 2p − n or p = 1

2(n + m).

Notice that m is even(odd), when n is even(odd). There is a one-to-one correspondence between {p | 0 ≤ p ≤ n} and {m | − n ≤ m ≤ n, m + n is even}. Notice also that the number of choices in n steps that the particle moves p times toward right is¡n_p¢:= n!

(n−p)!p!. When p = 1 2(n + m), we have w(m∆x, n∆t) = ½ 0, if m + n is odd, ¡_n p ¢ (1 2)n, if m + n is even.

We may check that w(m∆x, n∆t) is a probability density function. Namely, 1. w(m∆x, n∆t) ≥ 0.

2. P_mw(m∆x, n∆t) = 1.

Given any function f (m), we define its expectation value at n∆t by

< f (m) >:=X

m

f (m)w(m∆x, n∆t).

The moments < mk _{>, k ∈ N are particularly important. The first moment < m > is}

called the mean, while the second moment of the variation from mean < (m− < m >)2 _>

is called the variance. They can be found by computing < pk _{>, which in turn can be}

computed through the help of the following generating function:

G(u) := X p up µ 1 2 ¶_nµ n p ¶ = µ 1 + u 2 ¶_n . Hence = G0(1) =X p p µ 1 2 ¶_nµ n p ¶ = n 2. From m = 2p − n, we have < m >= 2 −n = 0.

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To compute the second moment < m2 _{>, from m = 2p − n, we have}

< m2 >= 4 < p2 > −4n +n2.

With the help of the generating function,

G00_{(1) =} n X p=0 p(p − 1) µ n p ¶ µ 1 2 ¶_n = < p2 _{> − } = < p2 _{> −}n 2 On the other hand, from G(u) = ¡1+u

2 ¢_n , we obtain G00_{(1) =} n(n−1) 4 . Hence, < p2 >= n2 4 + n4 and < m2 _{>= 4 < p}2 _{> −4n +n}2 _{= n.}

The mean of this random walk is < m >= 0, while its variance is < (m− < m >)2 _{>= n.}

Exercise

1. Find the transition probability, mean and variance for the case

Zn+1− Zn =

½

∆x with probability p

−∆x with probability 1 − p

2. One can also find the transition probability w by solving the difference equation (2.3).

2.4 The solution of the discrete asset price model

Let us consider the case

Sn+1 Sn = ½ u with probability 1 2 d with probability 1 2.

for simplicity. In n movements of the asset price, if the price goes up p times, then the price at time step n∆t is Sn = S0updn−p. Since there are

¡_n

p

¢

such choices, we then obtain the transition probability of the asset:

P (Sn= S|S0) = ½ ¡_n p ¢ ¡₁ 2 ¢_n if S = S0updn−p, 0 otherwise. (2.4)

2.5 The Brownian motion

2.5.1 The definition of a Brownian motion

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1. ∀t, z(t) is a random variable.

2. The increment z(t + s) − z(t), z(t) − z(t − u), u > 0, s > 0 are independent. 3. z(t) is continuous in t.

4. ∀s > 0, zt+s − zt is normally distributed with mean zero and variance s, i.e., its

probability density is N (0, s)(i.e., _√1 2πse

−x2

2s ).

2.5.2 The Brownian motion as a limit of random walk

We may realize the Brownian motion as the limit of the random walk in the previous sec-tion. Namely, Zn → z(t) as n → ∞ with m∆x → x, n∆t → t and (∆x)

2

∆t = σ fixed. This

can be proved by the Stirling formula:

n! ≈ √2πnn+1 2e−n.

Recall that the probability

P (Zn= m∆x|Z0 = 0) = µ n 1 2(m + n) ¶ µ 1 2 ¶_n .

Using the Stirling formula, we have for n, p, n − p >> 1, µ n 1 2(m + n) ¶ µ 1 2 ¶_n = (1 2) n n! (1 2(n + m))!(12(n − m))! ≈ (1 2) n √ 2πnn+1 2e−n √ 2π(1 2(n + m)) 1 2(n+m)+ 1 2 √ 2π(1 2(n − m)) 1 2(n−m)+ 1 2 = ( 2 πn) 1 2(1 + m n) −1 2(n+m)−12(1 − m n) −1 2(n−m)−12 ≈ ( 2 πn) 1 2(1 − (m n) 2₎−1₂n = ( 2 πn) 1 2 h (1 − (m n) 2₎(_mn)2i−m22n ≈ ( 2 πn) 1 2 exp(−m 2 2n) As m∆ → x, n∆t → t, (∆x)2_{/∆t = σ fixed, we obtain} P (Zn = m∆x|Z0 = 0)/2∆x ≈ ( 2 πn) 1 2 1 2∆xe −m2_2n = ( 1 2πn∆t) 1 2e− (m∆x)2 n∆tσ2 · √ ∆t 2(∆x) → √ 1 2πσ2_te − x2 2σ2t

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2.5.3 Properties of Brownian motion

By definition P(z(t) = x|z(0) = 0) = √1 2πte −x2_2t_. We can check 1. < z(t) >= 0 2. < z(t)2 _{>= t}

3. Independence of disjoint increments

P(z(t) = x|z(0) = 0) =

Z _∞

−∞

P(z(t) = x|z(s) = y) P(z(s) = y|z(0) = 0) dy.

(2.5) In particular, let us define an infinitesimal increment

dz = z(t + dt) − z(t)

We have

1. < dz >= 0 2. < (dz)2 _{>= dt}

In fact we have more, we may think

dz = ²√dt (2.6)

where ² is a random variable with standard Gaussian distribution N (0, 1) (i.e. mean is 0 and variance is 1). And we have

(dz)2 _{= dt with probability 1.} _(2.7)

Exercise

1. Check (2.5).

2.6 Itˆo’s formula

In this section, we shall study differential equations which consist of deterministic part: ˙x = b(x), and stochastic part σ ˙z(t). Here, z(t) is the Brownian motion. We call such an equation a stochastic differential equation and expressed as

dx(t) = b(x(t))dt + σ(x(t))dz(t). (2.8)

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Lemma 2.1 Suppose x(t) satisfies the stochastic differential equation (2.8), and f (x, t) is

a smooth function. Then f (x(t), t) satisfies the following stochastic differential equation: df = µ ft+ bfx+1 2σ 2_f xx ¶ dt + σfxdz (2.9)

Proof. This is not a proof, rather an intuition why (2.9) is true. According to the Taylor expansion, df = ftdt + fxdx + 1 2ftt(dt) 2_{+ f} xtdx dt + 1 2fxx(dx) 2_{+ · · · .}

Plug (2.8) into this equation. We recall that dz = ²√dt, where ² is a random variable with

standard Gaussian distribution N (0, 1). In the Taylor expansion of df (x(t), t), the terms (dt)2_{, dt · dz are relative unimportant as comparing with the dt term and dz term. Using}

(2.8) and noting (dz)2 _{= dt with probability 1, we obtain (2.9).}

A simple application of Itˆo’s lemma is to find the transition probability density function for the s.d.e.

dx = adt + σdz

where a and σ are constants. By letting y = x − at, from Itˆo’s lemma, y satisfies dy = σdz. Thus, the transition probability density function for y is

P(y(t) = y|y(0) = y0) =

1

√

2πσ2_te

−(y−y0)2/2σ2t_.

Or equivalently, the transition probability density function for x is

P(x(t) = x|x(0) = x0) =

1

√

2πσ2_te

−(x−at−x0)2/2σ2t_.

2.7 The solution of the continuous asset price model

In this section, we want to find the transition probability density function for the continuous asset price model:

dS = µS dt + σS dz. (2.10)

with initial data S(0) = S0. We apply Itˆo’s lemma with x = f (S) = log S. Then x satisfies

the s.d.e. dx = µ µ − σ2 2 ¶ dt + σ dz,

and x(0) = x0 := log S0. From the discussion of the previous section, we obtain

P(x(t) = x|x(0) = x0) = 1 √ 2πσ2_te −(x−x0−(µ−σ2₂ )t)2/2σt_. From P(x(t) = x|x(0) = x0)dx = P(x(t) = x|x(0) = x0)dS/S = P(S(t) = S|S(0) = S0)dS,

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we obtain that the transition probability density function for S(t) is P(S(t) = S|S(0) = S0) = √ 1 2πσ2_tS e −(log S S0−(µ− σ2 2 )t)2/2σ2t. (2.11)

This is called the lognormal distribution. Exercise

1. Find the mean and variance of the lognormal distribution.

S S

0

p

2.8 Continuous model as a limit of the discrete model

We want to show that the continuous model (2.10) is the limit of the discrete model (2.1). The parameters in (2.10) are µ and σ. The parameters in (2.1) are u, d and p. We may assume p = 1/2. First, we relate (µ, σ) and (u, d). Both models should have the same mean and variance. For the continuous model, we compute its mean under the condition

S((n − 1)∆t) = Sn−1. Then E(S(n∆t)|Sn−1) = Z SP(S, n∆t|S((n − 1)∆t) = Sn−1)dS = Z S µ 1 √ 2πσ2_∆tSe −(log_Sn−1S −(µ−1₂σ2_)∆t)2_/2σ2_∆t¶ dS

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= Sn−1 Z 1 √ 2πσ2_∆te −(x−(µ−1 2σ2)∆t)2/2σ2∆texdx, = Sn−1eµ∆t Z 1 √ 2πσ2_∆te −(√ x 2σ2∆t− √ 2σ2∆t 2 )2_dx = eµ∆t_S n−1.

Here, we have used the change-of-variable: x = log S

Sn−1. For the second moment for the

continuous model, we have

E(S(n∆t)2_|S n−1) = Z S2_{P(S, ∆t|S} n−1) dS = e(2µ+σ2_)∆t S2 n−1.

On the other hand, the mean and the second moment for the discrete model in one time step

∆t are _µ 1 2u + 1 2d ¶ Sn−1 µ 1 2u 2₊1 2d 2 ¶ S_n−12 .

In order to have the same means and variances in one time step in both models, we should require 1 2u 2₊1 2d 2 _{= e}(2µ+σ2_)∆t 1 2u + 1 2d = e µ∆t_. Or u = eµ∆t_{(1 +}p_eσ2_∆t − 1), (2.12) d = eµ∆t_{(1 −}p_eσ2_∆t − 1). (2.13)

These relate (u, d) and (µ, σ).

Theorem 2.1 Let us fix (µ, σ). Let us choose a ∆t and a ∆x with (∆x)2_{/∆t = σ}2_{. Define}

(u, d) by (2.12) and (2.13). Then

P (S0updn−p|S0)/2∆x −→ P(S(t) = S|S(0) = S0)

as n∆t → t, n → ∞ and S0updn−p → S.

Proof. Let us define x = log S, x0 = log S0. Then

log S0updn−p = x0+ p log u + (n − p) log d.

Thus, what we want to show is equivalent to

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where P(x(t) = x|x(0) = x0) = 1 √ 2πσ2_te −(x−x0−(µ−σ2₂ )t)2/2σ2t_.

To show this, we define m = 2p − n. Then p = 1

2(n + m), n − p = 12(n − m). Hence p log u + (n − p) log d = 1 2(n + m) log u + 1 2(n − m) log d = 1 2n log(ud) + 1 2m log( u d). From (2.12) and (2.13), u · d = e2µ∆t(2 − eσ2∆t) ≈ e2µ∆t· e−σ2∆t, u d = 1 + 2σ √ ∆t + σ2_{∆t ≈ e}2σ√∆t_. Hence 1 2n log ud + 1 2m log( u d) ≈ n(µ − 1 2σ 2_{)∆t + mσ}√_∆t = n(µ −1 2σ 2_{)∆t + m∆x.}

Define ∆x such that (∆x)_∆t2 = σ2_{. Then}

p log u + (n − p) log d = n∆t(µ − 1

2σ

2_{) + m∆x.}

Recall that the probability that the price moves up p times is¡n_p¢(1

2)n. Then the density is

µ n p ¶ µ 1 2 ¶_n /2∆x ≈ ( 2 nπ) 1 2e−m22n −→ √ 1 2πσ2_te −(x−x0−(µ−σ2₂ )t)2/2σ2t

2.9 Simulation of asset price model

Typically, µ = 0.16, σ is 0.20 ∼ 0.40 for a stock. To simulate the model

dS

S = µdt + σdz S(0) = S0

We perform N sample paths ω1, · · · , ωN. In each path, we choose time step ∆t = 0.01, for

instance. We obtain Sk+1 from Sk by discretizing the s.d.e. and sample a number ξ from

the normalized Gaussian distribution N (0, 1):

Sk+1− Sk

Sk

= µ∆t + σξ√∆t

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Here, ξ = 0.5 is the sampled number. Then the transition probability density function Z _b

a

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Chapter 3 Black-Scholes Analysis

3.1 The hypothesis of no-arbitrage-opportunities

The option pricing theory was introduced by Black and Scholes. The fundamental hypoth-esis of their analysis is that ”there is no arbitrage opportunities in financial markets”.

For simplicity, we shall also assume

1. There exists a risk-free investment that gives a guaranteed return with interest rate r. ( e.g. government bond, bank.)

2. Borrowing or lending at such riskless interest rate is always possible. 3. There is no transaction costs.

4. All trading profits are subject to the same tax rate. We will use the following notations:

S current asset price

E exercise price

T expiry time

t current time

µ growth rate of an asset

σ volatility of an asset

ST asset price at T

r risk-free interest rate

c value of European call option

C value of American call option

p value of European put option

P value of American put option Λ the payoff function

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3.2 Basic properties of option prices

3.2.1 The relation between payoff and options

1. Recall that

Λ(t) = max(St− E, 0) for call option

Λ(t) = max(E − St, 0) for put option

2. c(ST, T ) = Λ(T ).

Otherwise, there is a chance of arbitrage. For instance, if c(ST, T ) < Λ, then we can

buy a call on price c, exercise it immediately. If ST > E, then Λ = ST − E > 0

and c < Λ by our assumption. Hence we have an immediate net profit ST − E − c.

This contradicts to our hypothesis. If c(ST, T ) > Λ, we can short a call and earn c.

If the person who buy the call does not claim, then we have net profit c. If he does exercise his call, then we can buy an asset from the market on price ST and sell to

that person with price E. The cost to us is ST − E. By doing so, the net profit we get

is c − (ST − E) > 0. Again, this is a contradiction.

3. Similarly, we have

p(ST, T ) = Λ(T )

C(St, t) = Λ(t)

P (St, t) = Λ(t)

3.2.2 European options

Lemma 3.2 We have the following for European options

max{S − Ee−r(T −t)_{, 0} ≤ c ≤ S} _(3.1)

max{Ee−r(T −t)_{− S, 0} ≤ p ≤ Ee}−r(T −t) _(3.2)

and the put-call parity

p + S = c + Ee−r(T −t) (3.3)

To show these, we need the following definition and lemmae. Definition 2.1 A portfolio is a collection of investments.

For instance, a portfolio I = c − ∆S means that we long a call and short ∆ amount of an asset S.

Lemma 3.3 Suppose I(t) and J(t) are two portfolios containing no American options

such that I(T ) ≤ J(T ). Then under the hypothesis of no-arbitrage-opportunities, we can conclude that I(t) ≤ J(t), ∀t ≤ T .

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Proof. Suppose the conclusion is false, i.e., there exists a time t ≤ T such that I(t) > J(t). An arbitrageur can buy (long) J(t) and short I(t) and immediately gain a profit I(t)−J(t). Since I and J containing no American options, nothing can be exercised before T . At time

T , since I(T ) ≤ J(T ), he can use J(T ) (what he has) to cover I(T ) (what he shorts) and

gains a profit J(T )−I(T ). This contradicts to the hypothesis of no-arbitrage-opportunities. As a corollary, we have

Corollary 2.1 If I(T ) = J(T ), then I(t) = J(t), ∀t ≤ T . Now, we can prove the basic properties of European options 1-5. Proof of Lemma 3.2.

1. Let I = c and J = S. At T , we have

I(T ) = cT = max{ST − E, 0} ≤ max{ST, 0} = ST = J(T ).

Hence, I(t) ≤ J(t) holds for all t ≤ T .

Remark. The equality holds when E = 0. In this case c = S 2. Consider I = c + Ee−r(T −t) _{and J = S. At time T ,}

I(T ) = max{ST − E, 0} + E = max{ST, E} ≥ ST = J(T ).

This implies I(t) ≥ J(t).

3. Let I = p and J = Ee−r(T −t)_{. At time T ,}

I(T ) = max{E − ST, 0} ≤ E = J(T ).

Hence, I(t) ≤ J(t).

4. Consider I = p + S and J = Ee−r(T −t)_{. At time T ,}

I(T ) = max{ST, E} ≥ E = J(T )

. Hence, I(t) ≥ J(t).

5. Consider I = c + Ee−r(T −t) _{and J = p + S. At time T ,}

I(T ) = c + E = max{ST − E, 0} + E = max{ST, E},

J(T ) = p + S = max{E − ST, 0} + ST = max{E, ST}

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3.2.3 Basic properties of American options

Lemma 3.4 For American options, we have

(i) The optimal exercise time for American call option is T and we have C = c.

(ii) The optimal exercise time for American put option is as earlier as possible, i.e. t, and we have P ≥ p.

(iii) The put-call parity for American option:

S − E < C − P < S − Ee−r(T −t) _(3.4)

As a consequence, P ≤ E.

To prove these properties, we need the following lemma.

Lemma 3.5 Let I or J be two portfolios that contain American options. Suppose I(τ ) ≤

J(τ ) at some τ ≤ T . Then I(t) ≤ J(t), for all t ≤ τ .

Proof. Suppose I(t) > J(t) at some t ≤ τ . An arbitrageur can long J(t) and short I(t) at time t to make profit I(t)−J(t) immediately. At later time τ , he can use J(τ ) to cover I(τ ) with additional profit J(τ ) − I(τ ), in case the person who owns I exercises his American option.

Remark. The equality also holds if I(τ ) = J(τ ). Proof of Lemma 3.4.

1. Firstly, we show C ≥ c. If not, then c(τ ) > C(τ ) for some time τ ≤ T , we can buy

C and sell c at time τ to make a profit c(τ ) − C(τ ). The right of C is even more than

that of c. This is an arbitrage opportunity which is a contradiction.

Secondly, we show c ≥ C. Consider two portfolios I = C + Ee−r(T −t) _{and J = S.}

Suppose we exercise C at some time τ ≤ T , then I(τ ) = max{Sτ − E, 0} +

Ee−r(T −τ ) _{and J(τ ) = S}

τ. This implies I(τ ) ≤ J(τ ). By our lemma, I(t) ≤ J(t)

for all t ≤ τ . Since τ ≤ T arbitrary, we conclude I(t) ≤ J(t) for all t ≤ T . Combine this inequality with the inequality of 2) of section 3.2, we conclude c = C. Further, early exercise results C(τ ) + Ee−r(T −τ ) _{< S(τ ). Hence, the optimal exercise time}

for American option is T .

2. Example. Suppose S = 50, E = 40. If C is exercised before expiration, then the investor needs to pay 40 to buy the share. However, he can instead invest $40 into the bank to earn interest and there is a chance that the stock price may go up.

3. Suppose p(t) > P (t). Then we can make an immediate profit by selling p and buying

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Next, we show that if we have a P , we should exercise it immediately. We consider two portfolios I = P + S and J = Ee−r(T −t)_{. If we exercise P at some time τ ,}

t ≤ τ ≤ T , then

I(τ ) = max{E − Sτ, 0} + Sτ = max{E, Sτ} = E.

Putting this money into bank we will receive Eer(T −τ )_{at time T . On the other hand,}

J(τ ) = E−r(T −τ )_{. Hence, I(τ ) ≥ J(τ ). Therefore, I(t) ≥ J(t). Further, we see}

that if we exercise P at t, then I(T ) = Eer(T −t)_{is the maximum. Hence we should}

exercise P as early as possible.

4. The second inequality follows from the put-call parity (3.3) and the facts that c = C and P ≥ p. To show the first inequality, we consider two portfolios: I = C + E and

J = P + S. Suppose P is exercised at some time τ , t ≤ τ ≤ T . Then we must have E ≥ Sτ (otherwise, we should not exercise our put option). Therefore,

J(τ ) = max{E − Sτ, 0} + Sτ = E

I(τ ) = C(τ ) + Eer(τ −t)

= max{Sτ − E, 0} + Eer(τ −t)

= Eer(τ −t)_.

From lemma, we have I(t) > J(t). Hence C + E > P + S.

Examples.

1. Suppose S(t) = 31, E = 30, r = 10%, T − t = 0.25 year, c = 3, p = 2.25. Consider two portfolios:

I = c + Ee−r(T −t) _{= 3 + 30 × e}−0.1×0.25_{= 32.26,}

J = p + S = 2.25 + 31 = 33.25.

We find J(t) > I(t).

Strategy : long the security in portfolio I and short the security in portfolio J. This results a cashflow: −3 + 2.25 + 31 = 30.25. Put this cash into a bank. We will get 30.25 × e0.1×0.25 _{= 31.02 at time T . Suppose at time T , S}

T > E, we can exercise

c, also we should buy a share for E to close our short position of the stock. Suppose ST < E, the put option will be exercised. This means that we need to buy the share

for E to close our short position. In both cases, we need to buy a share for E to close the short position. Thus, the net profit is

31.02 − 30 = 1.02.

2. Consider the same situation but c = 3 and p = 1. In this case

I = c + Ee−r(T −t) _{= 32.25}

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and we see that J is cheaper.

Strategy: We long J and short I. To long J, we need an initial investment 31 + 1, to short c, we gain 3. Thus, the net investment is 31 + 1 − 3 = 29 initially. We can finance it from the bank, and we need to pay 29 × e0.1×0.25 _{= 29.73 to the bank at}

time T . Now, at T , we must have that either c or p will be exercised. If ST > E,

then c is exercised. We need to sell the share for E to close our short position for c. If ST < E, we exercise p. That is, we sell the share for E. In both cases, we sell the

share for E. Thus, the net profit is 30 − 29.73 = 0.27.

Remark. P − p is called the time value of a put. The maximal time value is E − Ee−r(T −t)_.

3.2.4 Dividend Case

Many stocks pay out dividends. These are payments to shareholders out of the profits made by the company. Since the company’s wealth does not change after paying the dividends, the stock price, the strike prices fall as the dividends being paid. If a company declared a cash dividend, the strike price for options was reduced on the ex-dividend day by the amount of the dividend.

Lemma 3.6 Suppose a dividend D will be paid during the life of an option. Then we have

for European option

S − D − Ee−r(T −t) _{< c ≤ S} _(3.5)

−S + D + Ee−r(T −t) _{< p ≤ Ee}−r(T −t)_. _(3.6)

and the put-call parity:

c + Ee−r(T −t) _{= p + S − D.} _(3.7)

For the American options, we have (i)

S − D − E < C − P < S − Ee−r(T −t) _(3.8)

provided the dividend is paid before exercising the put option, or (ii)

S − E < C − P < S − Ee−r(T −t) _(3.9)

if the put is exercised before the dividend being paid.

Proof. We consider two portfolios:

I = c + D + Ee−r(T −t), J = S.

Then at time T ,

I(T ) = max{ST − E, 0} + D + E = max{ST, E} + D

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Hence I(T ) ≥ J(T ). This yields I(t) ≥ J(t) for all t ≤ T . This proves

c ≥ S − D − Ee−r(T −t)_.

In other word, c is reduced by an amount D. Similarly, we have

p ≥ D + Ee−r(T −t)_{− S.}

That is, p increases by an amount D. For the put-call parity, we consider

I = c + D + Ee−r(T −t) J = S + p.

At time T ,

I = J = max{ST, E} + D.

This yields the put-call parity for all time.

When there is no dividend, we have shown that

C − P < S − Ee−r(T −t).

When there is dividend payment, we know that

CD < C, PD > P

Hence,

CD− PD < C − P < S − Ee−r(T −t).

For the American call option, we should not exercise it early, because the dividend will cause the stock price to jump down, making the option less attractive. We should exercise it immediately prior to an ex-dividend date.

For the American put option, we consider

I = C + D + E, J = P + S.

If we exercise P at τ ≤ T , then Sτ < E and

I(τ ) = D + Eer(τ −t)_,

J(τ ) = E + D.

We have J(τ ) ≤ I(τ ). Hence J(t) ≤ I(t) for all t ≤ τ .

If the put option is exercised before the dividend being paid, then we should consider

I = C + E and J = P + S. At τ ,

I(τ ) = Eer(τ −t)_,

J(τ ) = E.

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3.3 The Black-Scholes Equation

3.3.1 Black-Scholes Equation

The fundamental hypothesis of the Black-Scholes analysis is that there is no arbitrage

opportunities. Besides, we make the following additional assumptions:

(1) The asset price follows the log-normal distribution. (2) There exists a risk-free interest rate r.

(3) No transaction costs. (4) No dividend paid.

(5) Shorting selling is permitted.

Our purpose is to value the price of an option (call or put). Let V (S, t) denotes for the price of an option. The randomness of V (S(t), t) would be fully correlated to that S(t). Thus, we consider a portfolio which contains only S and V , but in opposite position in order to cancel out the randomness. Then this portfolio becomes deterministic. To be more precise, let the portfolio be

Π = V − ∆S. In one time step, the change of the portfolio is

dΠ = dV − ∆dS.

Here ∆ is held fixed during the time step. From Itˆos lemma

dΠ = σS µ ∂V ∂S − ∆ ¶ dz + µ µS∂V ∂S − µ∆S + ∂V ∂t + 1 2σ 2_S2∂2V ∂S2 ¶ dt. (3.10) Now, we can eliminate the randomness by choosing

∆ = ∂V

∂S

at the starting time of each time step. The resulting portfolio

dΠ = µ ∂V ∂t + 1 2σ 2_S2∂2V ∂S2 ¶ dt

is wholly deterministic. From the hypothesis of no arbitrage opportunities, the return, dΠ

Π,

should be the same as Π being invested in a riskless bank with interest rate r, i.e.

dΠ

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Otherwise, there would be either a net loss or an arbitrage opportunity. Hence we must have rΠdt = µ µS∂V ∂S − µ∆S + ∂V ∂t + 1 2σ 2_S2∂2V ∂S2 ¶ dt = µ ∂V ∂t + 1 2σ 2_S2∂2V ∂S2 ¶ dt, or ∂V ∂t + 1 2σ 2_S2∂2V ∂S2 = r µ V − S∂V ∂S ¶ . (3.11)

This is the Black-Scholes partial differential equation (P.D.E.) for option pricing. Its left-hand side is the return from the hedged portfolio, while its right-left-hand side is the return from bank deposit. Note that the equation is independent of µ.

Remark. Notice that the Black-Scholes equation is invariant under the change of variable

S 7→ λS.

3.3.2 Boundary and Final condition for European options

• Final condition:

c(S, T ) = max{S − E, 0} p(S, T ) = max{E − S, 0}.

In general, the final condition is

V (S, T ) = Λ(S),

where Λ is the payoff function.

• Boundary conditions:

(i) On S = 0:

c(0, τ ) = 0, ∀t ≤ τ ≤ T.

This means that you wouldn’t want to buy a right whose underlying asset costs nothing.

(ii) On S = 0:

p(0, τ ) = Ee−r(T −τ )_.

This follows from the put-call parity and c(0, t) = 0. (iii) For call option, at S = ∞:

c(S, t) ∼ S − Ee−r(T −t)_{, as S → ∞.}

Since S → ∞, the call option must be exercised, and the price of the option must be closed to S − Ee−r(T −t)_.

(iv) For put option, at S = ∞:

p(S, t) → 0, as S → ∞

As S → ∞, the payoff function Λ = max{E − S, 0} is zero. Thus, the put option is unlikely to be exercised. Hence p(S, T ) → 0 as S → ∞.

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3.4 Exact solution for the B-S equation for European

op-tions

3.4.1 Reduction to parabolic equation with constant coefficients

Let us recall the Black-Scholes equation

∂V ∂t + 1 2σ 2_S2∂2V ∂S2 + rS ∂V ∂S − rV = 0. (3.12)

This P.D.E. is a parabolic equation with variable coefficients. Notice that this equation is invariant under S → λS. That is, it is homogeneous in S with degree 0. We therefore make the following change-of-variable:

dx = dS S ,

or equivalently,

x = log S E

The fraction S/E makes x dimensionless. The domain S ∈ (0, ∞) becomes x ∈ (−∞, ∞) and ∂V ∂x = ∂S ∂x ∂V ∂S = S ∂V ∂S, ∂2_V ∂x2 = ∂ ∂x µ S∂V ∂S ¶ = ∂S ∂x ∂V ∂S + S ∂S ∂x ∂2_V ∂S2 = S∂V ∂S + S 2∂2V ∂S2 = ∂V ∂x + S 2∂2V ∂S2.

Next, let us reverse the time by letting

τ = T − t.

Then the Black-Scholes equation becomes

∂V ∂τ = 1 2σ 2∂2V ∂x2 + µ r − 1 2σ 2 ¶ ∂V ∂x − rV.

We can also make V dimensionless by setting v = V /E. Then v satisfies

∂v ∂τ = 1 2σ 2∂2v ∂x2 + µ r − 1 2σ 2 ¶ ∂v ∂x − rv. (3.13)

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The initial and boundary conditions for v become c(x, 0) = max{ex_{− 1, 0}} p(x, 0) = max{1 − ex_{, 0}} c(−∞, τ ) = 0, p(−∞, τ ) = e−rτ_, c(x, τ ) → ex_{− e}−rτ _{as x → ∞} p(x, τ ) → 0 as x → ∞.

Our goal is to solve v for 0 ≤ τ ≤ T .

3.4.2 Further reduction

In investigating the equation (5.3), it is of the following form:

vt+ avx+ bv = vxx (3.14)

The part, vt + avx is call the advection part of (3.14). The term bv is called the source

term, and the tern vxx is called the diffusion term. Here , we have absorbed the diffusion

coefficient 1

2σ2 in to time by setting t = τ /( 1

2σ2). (We somewhat abuse the notation here.

The new t here is different from the t we used before.) The advection part:

vt+ avx = (∂t+ a∂x) v

is a direction derivative along the curve (called characteristic curve)

dx dt = a.

This suggests the following change-of-variable:

y = x − at s = t.

Then the direction derivative become

∂s = ∂t+ a∂x

∂y = ∂x

Hence the equation is reduced to

vs+ bv = vyy.

Next, the equation vs+ bv suggests that v behaves like ebs along the characteristic curves.

Thus, it is natural to make the following change-of-variable

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Then the equation is reduced to

us = uyy.

This is the standard heat equation. Its solution can be expressed as

u(y, s) = Z 1 √ 4πse −(y−z)2_4s _{f (z) dz}

where f is the initial data. A simple derivation of this solution is given in the Appendix of this chapter.

3.4.3 Black-Scholes formula

Lert us return to the Black-Scholes equation (5.3). Let us denote the rescaled payoff func-tion by ¯Λ(x). That is,

¯

Λ(x) = Λ(Eex_)/E.

The change-of-variables above gives

s = τ /(1 2σ 2₎ y = x − as a = 1 − r/(1 2σ 2₎ b = r/(1 2σ 2₎ u = erτv Then v(x, τ ) = e−rτ Z 1 √ 2πσ2_τe −(x−z(r− 12 σ2)τ)2 2σ2τ Λ(z) dz¯ (3.15)

In terms of the original variables, we have the following Black-Scholes formula:

V (S, t) = e−r(T −t) Z 1 p 2πσ2_{(T − t)S}0e −(log( SS0)−(r− 12 σ2)(T −t))2 2σ2(T −t) Λ(S0) dS0 (3.16) We may express it as V (S, t) = e−r(T −t) Z P(S0_{, T, S, t)Λ(S}0_{) dS}0 _(3.17) Here, P(S0_{, T, S, t) :=} _p 1 2πσ2_{(T − t)S}0 e −(log( SS0)−(r− 12 σ2)(T −t))2 2σ2(T −t) . (3.18)

This is the transition probability density of an asset price model with growth rate r and volatility σ. In other words, V is the present value of the expectation of the payoff under an asset price model whose volatility is σ and whose growth rate is r. We shall come back to this point later.

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3.4.4 Special cases

1. European call option. The rescaled payoff function for a European call option is ¯ Λ(z) = max{ez_{− 1, 0}.} Then v(x, τ ) = e−rτ Z _∞ 0 1 √ 2πσ2_τe −(x−z−(1 2σ2−r)τ )2/(2σ2τ )(ez− 1) dz.

This can be integrated. Finally, we get the exact solution for the European call option

c (S, t) = SN (d1) − Ee−r(T −t)N (d2), (3.19) N (y) = √1 2π Z _y −∞ e−z2₂ _dz, _(3.20) d1 = log(S E) + (r + 12σ2)(T − t) σ√T − t , (3.21) d2 = log(S E) + (r − 12σ2)(T − t) σ√T − t . (3.22)

Exercise. Prove the formula (3.19).

2. European put option. Recall the put-call parity

c + Ee−r(T −t) _{= p + S.}

We can obtain the price for p from c:

p(S, t) = Ee−r(T −t)N (−d2) − SN (−d1). (3.23)

Exercise. Show that N (d1) − 1 = N (−d1). Use this to prove (3.23).

3. Forward contract Recall that a forward contract is an agreement between two par-ties to buy or sell an asset at certain time in the future for certain price. The payoff function for such a forward contract is

Λ(S) = S − E.

The value V for this contract also satisfies the B-S equation. Thus, its solution is given by V = Ee−rτ_u, where u(x, τ ) = √ 1 2πσ2_τ Z _∞ −∞ e−(y−z−(r− 12σ2τ2 σ2)τ)2 (ez − 1) dz = ex+rτ _{− 1.}

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Hence,

V (S, t) = S − Ee−r(T −t)_. _(3.24)

This means that the current value of a forward contract is nothing but the difference of S and the discounted E. Notice that this value is independent of the volatility σ of the underlying asset.

Exercise. Show that the payoff function of a portfolio c − p is S − E. From this and the Black-Scholes formula (3.16), show the formula of the put-call parity.

4. Cash-or-nothing. A contact with cash-or-nothing is just like a bet. If ST > E, then

the reward is B. Otherwise, you get nothing. The payoff function is Λ(S) =

½

B if S > E 0 otherwise.

Using the Black-Scholes formula (3.16), we obtain the value of a cash-or-nothing contract to be

V (S, t) = Be−r(T −t)N (d2). (3.25)

5. Supershare. Supershare is a binary option whose payoff function is defined to be Λ(S) =

½

B if E1 < S < E2

0 otherwise.

One can show that the value for this binary option is

V (S, t) = Be−r(T −t)_{(N (d}

2(E1)) − N (d2(E2)))

where d2(E) is given by (3.22).

6. Deterministic case (σ = 0). In this case, the Black-Scholes equation is reduced to

Vt+ rSVs− rV = 0.

Or in τ, x and u variables:

uτ − rux = 0

with initial data

u(x, 0) = Λ(Eex_),

Thus,

u(x, τ ) = Λ(Sex+rτ ).

Or

V (S, t) = e−r(T −t)_Λ(Ser(T −t)_).

This means that when the process is deterministic, the value of the option is the payoff function evaluated at the future price of S at T (that is Ser(T −t)_{), and then}

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3.5 Risk Neutrality

Notice that the growth rate µ does not appear in the Black-Scholes equation. The option may be valued as if all random walks involved are risk neutral. This means that the drift term (growth rate) µ in the asset pricing model can be replaced by r. The option is then valued by calculating the present value of its expected return at expiry. Recall the lognormal probability density function with growth rate r, volatility σ is

P(S0, T, S, t) := p 1

2πσ2_{(T − t)S}0 e

−(log( SS0)−(r− 12 σ2)(T −t))2

2σ2(T −t) . (3.26)

This is the transition probability density of an asset price model in a risk-neutral world:

dS

S = rdt + σdz. (3.27)

The expected return at time T in this risk-neutral world is Z

P(S0_{, T, s, t)Λ(S}0_)dS0_.

At time t, this value should be discounted by e−r(T −T )_:

V (S, t) = e−r(T −t)

Z

P(S0_{, T, S, t)Λ(S}0_)dS0_.

We may reinvestigate the function N and the parameters di in the Black-Scholes formula.

After some calculation, we find

N (d2) =

Z _∞

E

P(S0, T, S, t)dS0. (3.28)

This is the probability of the event { ˜S ≥ E}, where ˜S obeys the risk-neutral pricing model: d ˜S

˜

S = rdt + σdz.

Similarly, one can show that

N (d1) =

R_∞

E P(S0, T, S, t)S0dS0

Ser(T −t) . (3.29)

is the expectation of ˜S at T when S = 1 at t and under the condition that ˜S ≥ E at T .

3.6 The delta hedging

Hedging is the reduction of sensitivity of a portfolio to the movement of the underlying of asset by taking opposite position in different financial instruments. The Black-Scholes

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analysis is a dynamical strategy. The delta hedge is instantaneously risk free. It requires a continuous rebalancing of the portfolio and the ratio of the holdings in the asset and the derivative product. The delta for a whole portfolio is ∆ = ∂Π

∂S. This is the sensitivity of Π

against the change of S. By taking dΠ − ∆ · dS, the sensitivity of the portfolio to the asset price change is instantaneously zero.

Besides the delta helge, there are more sophisticated trading strategies such as: Gamma: Γ = ∂ 2_Π ∂2_S2, Theta: θ = −∂Π ∂t, Vega: = ∂Π ∂σ, rho: ρ = ∂Π ∂r.

Hedging against any of these dependencies requires the use of another option as well as the asset itself. With a suitable balance of the underlying asset and other derivatives, hedgers can eliminate the short-term dependence of the portfolio on the movement in t, S, σ, r.

For the Delta-hedge for the European call and put options, we have the following propo-sitions.

Proposition 1 For European call options, its ∆ hedge is given by ∆ = N (d1). Proof. By definition, ∂c ∂S = N (d1) + S · N 0_(d 1) · d1S− Ee−r(T −t)N0(d2)d2S. Since d1 = log(S E) + (r + σ2 2 )(T − t) σ√T − t , we have d1S = 1 Sσp(T − t), d2S = 1 Sσp(T − t), N0(di) = 1 √ 2πe −d2 i. Hence, ∂c ∂S = N (d1) + ¡ SN0_(d 1) − Ee−r(T −t)N0(d2) ¢ /(Sσ√T − t) ≡ N (d1) + I/(Sσ √ T − t).

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We claim that I = 0. Or equivalently, S E N0_(d 1) N0_(d₂₎ = e −rτ

This follows from the computation below.

S E N0_(d 1) N0_(d₂₎ = e x_{· e}−(d2 1−d22)/2. From (3.21)(3.22), d2₁− d2₂ = 1 σ2_τ ³ (x + rτ + σ 2τ ) 2_{− (x + rτ −} σ 2τ ) 2´ = 2(x + rτ ) Hence, S E N0_(d 1) N0_(d₂₎ = e x_{· e}−x−rτ _{= e}−rτ_.

Proposition 2 For European put options, its ∆ hedge is given by ∆ = N (−d1).

Proof. From the put-call parity, ∆ = ∂p

∂S = ∂c

∂S − 1 = N (d1) − 1 = −N (−d1),

3.6.1 Time-Dependent r, σ, µ

Suppose r, σ, µ are functions of r, but also deterministic. The Black-Scholes remains the same. We use the change-of-variables:

S = Eex, V = Ev, τ = T − t.

The Black-Scholes equation is converted to

vτ = σ

2_{(τ )}

2 vxx+ (r(τ ) −

σ2_{(τ )}

2 )vx− r(τ )v (3.30)

We look for a new time variable ˆτ such that

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For instance, we can choose ˆ τ = Z _τ 0 σ2(τ ) dτ. Then the equation becomes

vˆτ =

1

2vxx+ a(ˆτ )vx− b(ˆτ )v. (3.31) To eliminate a(ˆτ ), we consider the characteristic equation:

dx

dˆτ = −a(ˆτ )

This can be integrated and yields

x = − Z _ˆ_τ 0 a(τ0_)dτ0_{+ y,} Or equivalently, y = x + Z _τ_ˆ 0 a(τ0)dτ0 ≡ x + A(ˆτ ).

Now, we consider the change-of-variable: µ x ˆ τ ¶ → µ y ˆ τ1 ¶ . Then, ∂ ∂x|ˆτ = ∂y ∂x|τˆ ∂ ∂y = ∂ ∂y, and ∂ ∂ ˆτ1 |y = ∂ ˆτ ∂ ˆτ1 ∂ ∂ˆτ + ∂x ∂ ˆτ1 ∂ ∂x = ∂ ∂ˆτ − a(ˆτ ) ∂ ∂x.

The equation (3.31)is transformed to

vτˆ1 = 1 2vyy − b(ˆτ1)v. Let B(ˆτ1) = R_τ_ˆ₁ 0 b(τ0)dτ0, and u = eB(ˆτ1)v, then uτˆ1 = 1

2uyy. And we can solve this heat

equation explicitly.

3.7 Trading strategy involving options

The options whose payoff are max{ST − E, 0} or max{E − ST, 0} are called vanilla

option. In this section, we shall discuss more general payoff functions. The goal is to design a portfolio involving vanilla option with a designed payoff function.

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3.7.1 Strategies involving a single option and stock

There are four cases:

a. Π = S − c (writing a covered call option). In this strategy, we short a call, long a share to cover c. The payoff of Π is Λ = S − max(S − E, 0) = min{S, E}. In this case, we anticipate the stock price will increase.

b. Π = c − S (reverse of a covered call). In this strategy, we anticipate the stock price will decrease. And Λ = − min{S, E}.

c. Π = p + S (protective put). In this portfolio, we long a p and buy a share to cover p. We anticipate the stock price will increase. The payoff is Λ = S + max{E − S, 0} = max{S, E}.

d. Π = −p − S (reverse of a protective put). We do not anticipate the stock price will increase. The payoff is − max{S, E}.

Below are the payoff functions for the above four cases.

E S Λ E S Λ (a) (b) E S Λ E S Λ

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(c) (d)

3.7.2 Bull spreads

In this strategy, an investor anticipates the stock price will increase. However, he would like to give up some of his right if the price goes beyond certain price, say E2. Indeed, he

does not anticipate the stock price will increase beyond E2. Hence he does want to own a

right beyond E2. Such a portfolio can be designed as

Π = CE1 − CE2, E1 < E2,

where CEi is a European call option with exercise price Ei and CE1, CE2 have the same

expiry. The payoff

Λ = max{ST − E1, 0} − max{ST − E2, 0} =    0 if ST < E ST − E1 if E1 < ST < E2 E2 − E1 if ST > E2 E 1 E2 S Λ E 2−E1

Since E1 < E2, we have CE1 > CE2. A bull spread, when created from CE1 − CE2,

requires an initial investment. We can describe the strategy by saying that the investor has a call option with a strike price E1and has chosen to give up some upside potential by selling

a call option with strike price E2 > E1. In return, the investor gets E2− E1 if the price

goes up beyond E2.

Example: CE1 = 3, CE2 = 1 and E1 = 30, E2 = 35. The cost of the strategy is 2. The

payoff _   0 if ST ≤ 30 ST − 30 if 30 < ST < 35 5 if ST ≥ 35

The bull spread can also be created by using put options Π = PE1 − PE2, E1 < E2.

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3.7.3 Bear spreads

An investor entering into a bull spread is hoping that the stock price will increase. By contrast, an investor entering into a bear spread is expecting the stock price will go down. The bear spread is

Π = CE2 − CE1, E1 < E2.

There is cash flow entered (CE2 − CE1). The payoff is

3.7.4 Butterfly spread

If an investor anticipate the stock price will stay in certain region, say, E1 < ST < E3, he

or she can have a butterfly spread such that the payoff function is positive in that region and he or she gives up the return outside that region.

1. Butterfly spread using calls: Define the portfolio:

Π = CE1 − 2CE2 + CE3, with E1 < E2 < E3.

where E3 = E2+ (E2− E1). Its payoff function is a piecewise linear function and

is determined by Λ(E1) = Λ(E3) = 0, Λ(E2) = E2− E1. Below is the graph of its

payoff function. E 1 S−E 1 E 2−E1 E 2 E3 E 2−S S Λ

Example: Suppose a certain stock is currently worth 61. A investor who feels that it is unlikely that there will be significant price move in the next 6 month. Suppose the market of 6 month calls are

E C

55 10 60 7 65 5 The investor creates a butterfly spread by

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The cost is 10 + 5 − 2 × 7 = 1. The payoff is

55

5

60 65 S

Λ

2. Butterfly spread using puts.

PE1 + PE3 − 2PE2, E1 < E3, E2 = E1+ E3 2 . ϕE2 =    linear ∆E if S = E2 0 if S < E2− ∆E, or S > E2− ∆E

Remark 1. Suppose European options were available for every possible strike price E, then any payoff function could be created theoretically:

Λ(S) = X Λi ∆EϕEi

where Ei = i∆E, Λi is constant. Then Λ(Ei) = Λi and Λ is linear on every interval

(Ei, Ei+1) and Λ is continuous. As ∆E → 0, we can approximate any payoff function by

using butterfly spreads.

Remark 2. One can also use cash-or-nothing to create any payoff function: Λ(S) =X Λi

∆EψS − Ei, where

ψ(S) := H(S) − H(S − ∆E).

The value for such a portfolio is

V = e−r(T −t) Z

P(S0, T, S, t)Λ(S0)dS0,

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Chapter 4 Variations on Black-Scholes models

4.1 Options on dividend-paying assets

Dividends are payments to the shareholders out of the profits made by the company. We will consider two “deterministic” models for dividend. One has constant dividend yield. The other has discrete dividend payments.

4.1.1 Constant dividend yield

Suppose that in a short time dt, the underlying asset pays out a dividend D0Sdt, where D0

is a constant, called the dividend yield. This continuous dividend structure is a good model for index options and for short-dated currency options. In the latter case, D0 = rf, the

foreign interest rate.

As the dividend is paid, the return dS

S must fall by the amount of the dividend payment

D0dt. It follows the s.d.e. for the asset price is

dS

S = (µ − D0)dt + σdz.

For a portfolio : Π = V − ∆S, we choose ∆ = ∂V

∂S in order to eliminate the randomness of

dΠ. In one time step, the change of portfolio is

dΠ = dV − ∆dS − ∆D0Sdt,

the last term −∆D0Sdt is the dividend our assets received. Thus

dΠ = dV − ∆(dS + D0Sdt) = σS µ ∂V ∂S − ∆ ¶ dz + µ (µ − D0)S ∂V ∂S + 1 2σ 2_S2∂2V ∂S2 + Vt− (µ − D0)∆S − ∆D0S ¶ dt = µ Vt− D0S ∂V ∂S + 1 2σ 2_S2∂2V ∂S2 ¶ dt, 41

Mathematical Finance

FINANCIAL MATHEMATICS

I-Liang Chern

Department of Mathematics

National Taiwan University

Contents

Chapter 1

Introduction

1.1

Financial Markets

1.2

Financial Derivatives

1.3

Examples

1.4

Payoff functions

1.5

Other kinds of options

1.6 Types of traders

1.7

Basic assumption

Chapter 2

Asset Price Model

2.1

Efficient market hypothesis

2.2

The asset price model

2.2.1

The discrete asset price model

2.2.2

The continuous asset price model

2.3

Random walk

2.4

The solution of the discrete asset price model

2.5

The Brownian motion

2.5.1

The definition of a Brownian motion

2.5.2

The Brownian motion as a limit of random walk

2.5.3

Properties of Brownian motion

2.6

Itˆo’s formula

2.7

The solution of the continuous asset price model

2.8

Continuous model as a limit of the discrete model

2.9

Simulation of asset price model

Chapter 3

Black-Scholes Analysis