MTH6155 Financial Mathematics II Weeks 3, 4, and 5 (part 1)

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MTH6155 Financial Mathematics II Weeks 3, 4, and 5 (part 1)

4 The Central Limit Theorem (CLT) and the Geometric Brownian Motion (GBM)

The binomial model we have discussed in previous sections can be viewed as an approximation of a continuous model according to which the price of an asset (say share) is governed by the GBM.

In this section, we first state an important version of the CLT and then use it to explain how the GBM arises as a natural choice of a model describing the behaviour of the price of an asset.

4.1 A useful version of the CLT

The first part of the following statement of the Central Limit Theorem should be familiar to you from second year probability or statistics courses.

Theorem 4.1. Let Y₁, Y₂, · · · , Y_n be a sequence of i.i.d. random variables, E(Yn) = a, Var(Yn) = σ², |Yn| < C. Then the sequence

Z_n= Pn

j=1Yj− na

√nσ

converges in distribution, as n → ∞, to the standard normal random variable. Namely 1. For all x lim_n→∞P(Zn< x) = Φ(x), where Φ(x) = ^√¹

2π

Rx

−∞e⁻^y2² dy.

2. Let g : R → R be a continuous function satisfying, for some constants c1 > 0, c₂ > 0, the estimate |g(x)| < c₁e^c²^|x|. Then

n→∞lim E(g(Zⁿ)) = E(g(Z))

where Z ∼ N (0, 1) (has the standard normal distribution). In other words, we have

n→∞lim E(g(Zn)) = 1

√2π Z ∞

−∞

g(x)e⁻^x2² dx Remarks 4.2.

1. The two statements in this theorem are equivalent (each of them implies the other one).

3. The proof of the above theorem as well as of the equivalence of its two statements is beyond the scope of this course. However, we are going to use this theorem and you are required to know its statement and to be able to apply it as it is done in the proofs explained below.

4.2 Why does the GBM describe the behaviour of the prices?

Recall that the GBM is a model according to which the price of an asset (say, a share) is given by S(t) = S(0)e^µt+σW^t, where W_t is the standard Wiener process and µ, σ are the drift and the volatility parameters of the Brownian motion Y (t) = µt + σW_t.

Our aim is to show that the random process S(t) can be obtained as a limit of a sequence of discrete processes describing the behaviour of the prices. More precisely, we shall construct a sequence of binomial models Sn(t) which converge to S(t) as n → ∞.

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4.2.1 Construction of the approximating binomial models

The construction of the process approximating our GBM is carried out in two steps.

Step 1. Divide [0, T ] into n equal intervals of length h = ^T_n and define for each time of the form jh the value S_n(jh) as follows:

1. At time t = 0, S_n(0) = S(0), where S(0) is the same as in the GBM.

2. At time t = jh, 1 ≤ j ≤ n, set S_n(jh) = S(0)e^µhj+σ

√

h(Y1+Y2+...Yj) = S(0)e^µt+σ

√ hPj

k=1Yj, (1)

where Y₁, Y₂, ..., Y_n, ... is a sequence of independent identically distributed random variables each taking either value 1 or −1 with probability ¹₂:

P(Yj = 1) = 1

2, P(Yj = −1) = 1 2. Step 2. For t ∈ [0, T ] set

S_n(t) = S_n(jh) where j is such that jh ≤ t < (j + 1)h . (2) Remarks 4.3. 1. For every fixed n, the sequence of prices S_n(jh), j = 0, 1, 2, ..., n, evolves according to a binomial model. Indeed, S_n(jh) = S_n((j − 1)h)e^µh+σ

√

hYj. We can therefore say that

either S_n((j − 1)h)u or S_n(jh) = S_n((j − 1)h)d, (3) where u = e^µh+σ

√

h (which corresponds to Y_j = 1) and d = e^µh−σ

√

h (which corresponds to Y_j = −1). As we know, relations (3) is the one defining the binomial model.

3. The additional feature of this binomial model is that each sequence of prices S_n(jh), j = 0, 1, 2, ..., n occurs with a certain real life probability, namely with probability 2⁻ⁿ(prove this statement!). This fact plays a very important role for the theorems we discuss in this section.

4. We thus suppose that at each time moment the change of the price is driven by the product of two factors: one is e^µh and e^±σ

√ h.

The first factor is not random and would usually be pushing the price up: it is natural to expect that µ > 0 (can you explain why?). The second factor is random and can push the price both up and down with the same probability. Note that when h is small, √

h is much larger than h (in the sense that

√ h

h → ∞ as h → 0). So, at each particular step the influence of the second factor is much stronger than that of the first one. However, in a long run, with probability which is close to one, the first factor accumulates a much stronger influence on the price because of the cancelations in the second sum.

4.2.2 Convergence of binomial approximations to the GBM

Theorem 4.4. When n → ∞, the process S_n(t) → S(t) in the following sense: if g(z₁, · · · , z_k) is a ”good enough” function of k variables (say, continuous, and growing no faster than ex- ponentially in each variable) and

0 < t₁ < t₂· · · < t_k≤ T are time moments then

n→∞lim E[g(Sn(t₁), S_n(t₂), · · · S_n(t_k))] = E[g(S(t1), S(t₂), · · · S(t_k))] (4)

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Proof. We prove this theorem only for the case k = 1, and so we can simplify our notation and write t in place of t₁. We thus have to show that if 0 < t ≤ T then, as n → ∞,

E(g(Sn(t))) → E(g(S(t))) = 1

√2π Z ∞

−∞

g(S(0)e^µt+σ

√

tx)e⁻^x2² dx.

Note that according to (2), for every t there is a j such that jh ≤ t < (j + 1)h. To simplify (slightly) the further steps of the proof, suppose that t = jh. Then h = ^t_j and

S_n(t) = S_n(jh) = S(0)e^µjh+σ

√

h(Y1+Y2+...Yj) (∗)= S(0)e^µt+σ

qt j

Pj k=1Yj

, where (∗) is due to the fact that we can replace jh by t and √

h by q

t

j. Now, h → 0 when n → ∞ and therefore also j → ∞.

Since Y_iare independent with E(Yj) = 1×¹₂−1×¹₂ = 0 and Var(Y_j) = E(Yj²)−(EYj)² = 1, we can now make use of Theorem 4.1 in order to be able to control the behaviour of qt

j

Pj

k=1Y_k. Namely, according to Theorem 4.1 r t

j

X

k=1

Y_k =√ t × 1

√j

j

X

k=1

Y_k →√

tZ as j → ∞.

where Z is the standard normal random variable, Z ∼ N (0, 1) and the convergence is under- stood as explained by Theorem 4.1. Namely, for a “good” function G we have

E G 1

√j

j

X

k=1

Y_k

!!

→ E(g(Z)), where Z ∼ N(0, 1).

In particular, as n → ∞, E(g(Sn(t))) = E

g(S(0)e^µt+σ

√t^√¹_jPj k=1Yk

)

→ E

g(S(0)e^µt+σ

√ tZ)

= 1

√2π Z ∞

−∞

g(S(0)e^µt+σ

√tx)e⁻^x2² dx.

5 Convergence of the risk-neutral probabilities

As we have just seen in the previous section, the real life probability on the space of functions Sn(t) (defined by (1) and (3)) converges to a probability corresponding to the Geometric Brownian motion.

What can we say about the convergence of the risk-neutral probabilities defined on the space of sequences (1) and thus also on the space of functions (3)? The answer is given by the following statement.

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Theorem 5.1. Suppose that S(t) = Se^µt+σW^t and let r be the interest rate compounded continuously. Then the risk-neutral probability defined on the space of functions S_n(t) converges to a risk neutral probability on the space of trajectories of the GBM given by

S(t) = Se˜ ^µt+σW^˜ ^t, where ˜µ = r −1 2σ².

Definition 5.2. The process ˜S(t) = Se^µt+σW^˜ ^t is called the risk-neutral Geometric Brow- nian motion.

Proof. Has not been explained in lectures and is not examinable.

Remark 5.3. It was mentioned in lectures that the proof of this theorem is similar to the proof of Theorem 4.4. The important difference is that rather than dealing with the real life probability this time we have to consider the risk-neutral probability (RNP for short) on the same space of functions S_n(t) (defined by (1) and (3)). Under the RNP, the random variables Y₁, Y₂, ..., Y_n, ... are independent and take values 1 and -1 as before. Our results for the binomial model imply that in the case of RNP

P(Y^j = 1) = 1 + rh − d

u − d = 1 + rh − exp(µh − σ√ h) exp(µh + σ√

h) − exp(µh − σ√ h)

and P(Yj = −1) = 1 − P(Yj = 1). The result then follows from Theorem 4.1 and the fact that for small values of h (or, equivalently, large values of n)

P(Yj = 1) = 1 2

1 + r − µ − ¹₂σ² σ

√ h

+ O(h). (5)

Exercise. Using the ideas explained in the last remark, write down a detailed proof of Theorem 5.1.

Remark 5.4. The notation O(h) in (5) has the following meaning: O(h) is a function of h such that |O(h)| ≤ A|h| for some constant A > 0.

5.1 Generalization of the B-S Formula

Before considering the first application of Theorem 5.1, let us recall the following definition.

Definition 5.5. A derivative on a share (or any security) is a contract or a product (such as a future, option, or warrant) whose value derives from and is dependent on the value of an underlying asset, such as a share, commodity, currency, security, etc.

Let us also recall the following general principle which allows one to compute the no- arbitrage price of a derivative in terms of the risk-neutral probability. This theorem was discussed in the FMI course and should be known to you. However, since it is a very important principle which we shall use also in the future, we shall prove it here - the more that the proof is very short and simple.

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Theorem 5.6. Suppose that

1. The payoff function of a derivative on a share is R(T ), 2. The payoff time is T .

3. The interest rate compounded continuously is r.

Then the price C of this derivative is

C = e^−rTE[R(T )],˜ (6)

where ˜E is the expectation over the risk-neutral probability (RNP for short).

Proof. If we buy the derivative for C then our return at time T is R(T ) − Ce^rT

(which is simply the difference between the payoff and our debt to the bank). The Arbitrage Theorem states that the expectation of the return over the RNP should be zero, that is

E(R(T ) − Ce˜ ^rT) = 0 or, equivalently, ˜E(R(T )) − Ce^rT = 0.

Hence

C = e^−rTE[R(T )].˜

The following statement is in fact a corollary of Theorems 5.1 and 5.6.

Theorem 5.7. Suppose that

1. The price of a share follows the GBM: S(t) = Se^µt+σW^t

2. The payoff function of a derivative on this share is R(S(t₁), S(t₂), · · · , S(t_k)), where 0 ≤ t₁ < t₂ < ... < t_k ≤ T .

3. The payoff time is T .

4. The interest rate compounded continuously is r.

Then the price C of this derivative is

C = e^−rTE[R(S(t˜ ₁), ˜S(t₂), · · · , ˜S(t_k))], (7) where ˜S(t) = Se^µt+σW^˜ ^t, ˜µ = r −¹₂σ².

Proof. The payoff of our derivative at time T is

R(T ) = R(S(t₁), S(t₂), · · · , S(t_k)).

By Theorem 5.6,

C = e^−rTE [R(S(t˜ 1), S(t₂), · · · , S(t_k))] . (8) where ˜E is the expectation computed over the risk-neutral probability. By Theorem 5.1, the risk-neutral probability on the space of paths of the GBM is that corresponding to ˜S(t). This means that in order to compute the expectation in (8) over the risk-neutral probability we have to replace S(t₁), S(t₂), · · · , S(t_k) by ˜S(t₁), ˜S(t₂), · · · , ˜S(t_k), that is:

E [R(S(t˜ 1), S(t₂), · · · , S(t_k))] = Eh

R( ˜S(t₁), ˜S(t₂), · · · , ˜S(t_k))i . This proves that

C = e^−rTE[R(S(t˜ 1), ˜S(t2), · · · , ˜S(tk))], where ˜S(t) = Se^µt+σW^˜ ^t, ˜µ = r −¹₂σ².

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5.2 Examples

The three example below show how to use Theorem5.7 for the derivation of formulae two of which are known from the FMI course.

5.2.1 The European Call Option and the Black-Scholes Formula

Recall the following notation: we write Call(K, T ) instead of saying European call option with strike price K and expiration time T

The payoff function for the Call(K, T ) is R(S(T )) = (S(T ) − K)⁺. This means that we should use Theorem 5.7 with k = 1 and t1 = T . The price C of this option is therefore given by

C = e^−rTE(S(T ) − K)˜ ⁺. (9)

This formula is thus a very particular case of (7). Yet (9) is the main part of the calculation which leads to the famous Black-Scholes formula:

C = C(S, T, K, σ, r) = SΦ(ω) − Ke^−rTΦ(ω − σ√

T ) , (10)

where

ω = ln_K^S + rT σ√

T + 1

2σ√ T

and Φ(x) is the cumulative distribution function of a standard Normal random variable, that is

Φ(x) = Z x

−∞

√1

2πe^−u²^/2du .

Remark 5.8. In this course, the expression ‘the Black-Scholes Formula’ (BS formula for short) is usually associated with formula (10) (which is the most classical form of this formula).

Also,in financial literature the name BS formula is often referring to more complicated versions of (10).

However, I would like to stress that (10) is an easy corollary of another, and in fact more important form of the BS formula, namely (9). Indeed, once (9) has been established, we can write it as an integral

C = e^−rT Z ∞

−∞

(Se^{µT +σ}^˜

√

T x − K)⁺ 1

√2πe⁻^x2² dx (11)

and the rest is a Calculus 2 exercise. (See appendix where this exercise is solved.) 5.2.2 The European Put Option

As you will know, the European put option Put(K, T ) allows you to sell the underlying share for £K at the pre-negotiated time T (which is its expiry time). To compute the gain, consider 2 cases.

1. If S(T ) ≥ K then your option is useless and you have to return to the bank £Ce^rT which is the value at time T of the £C you have borrowed from the bank at time t = 0 to buy the option. Hence you gain is −Ce^rT.

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2. If S(T ) < K, then

(a) you buy the share for £S(T ) and sell it for £K, (b) you return to the bank £Ce^rT (as in the first case).

Your gain thus is (K − S(T )) − Ce^rT.

In other words, your payoff function is R(S(T )) = (K − S(T ))⁺ (and your gain is (K − S(T ))⁺− Ce^rT). So, by Theorem 5.7, we obtain

C = e^−rTE(K −S(T ))˜ ⁺.

This expression can be transformed into a formula which is similar to the one obtained for the price of a European call option. The calculations are similar to those for the Call(K, T ). How- ever, the Call-Put parity formula provides a much shorter derivation which will be considered in the corresponding exercise sheet.

5.2.3 The derivative with payoff R(T ) = _T¹ RT

0 S(t)dt

Suppose that the price of an asset is driven by a GBM: S(t) = Se^µt+σW^t. Consider a derivative on this asset with a payoff function

R(T ) = 1 T

Z T 0

S(t)dt.

What is the risk-neutral price C of this derivative? Note the particularity of this example: the payoff function depends on all values of S(t), t ∈ [0, T ].

The answer follows from Theorems 5.1 and 5.6. Namely, by Theorem 5.1 and by the definition of R(T ),

C = e^−rTE˜ 1 T

Z T 0

S(t)dt

= e^−rT T

E˜

Z T 0

S(t)dt

. (12)

By Theorem 5.1,

E˜

Z T 0

S(t)dt

= E

Z T 0

S(t)dt˜

.

The remarkable fact is that it is possible to change the order of the two operations:

E

Z T 0

S(t)dt˜

= Z T

0

E ˜S(t)

dt.

In words, rather than first computing the integral and then the expectation, we can first compute the expectation and after that compute the integral. We shall use this fact now and also later in the course but its proof is beyond our means.

The rest is simple because we know (see Notes 1) that E ˜S(t)

= E Se^µt+σW^˜ ^t = Se^µt+^˜ ^σ2² ^t.

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Since ˜µ + ^σ₂² = r, we have

E ˜S(t)

= Se^rt and we obtain

E

Z T 0

S(t)dt˜

= Z T

0

Se^rtdt = S

r(e^rT − 1).

Finally we obtain from (12):

C = e^−rTS

rT (e^rT − 1) = S

rT(1 − e^−rT).

6 The Greeks and the dependence of the BS price on the parameters of the model

6.1 The Greeks

In this section we shall study the dependence of the Black-Scholes price C = C(S, T, K, σ, r) = e^−rTE((S(T ) − K)⁺) on the parameters S, T , K, σ, and r.

More precisely, we shall compute the first order partial derivatives of C with respect to each of the parameters. The value of each such derivative shows the sensitivity of the price to small changes in the parameter. And if we know just that a derivative is positive (negative) then we also know that C is growing (decaying) when the parameter is growing.

But first, let us introduce the names for these derivatives. First of all they have a collective name - they are called the Greeks. The reason for this term is that these derivatives are denoted by Greek letters. Here are the most common of them:

∆ = ∂C

∂S is called delta ν = ∂C

∂σ is called vega (see remark below) ρ = ∂C

∂r is called rho θ = ∂C

∂T is called theta

Remark Vega is not the name of any Greek letter. The letter we actually use is called nu.

Nevertheless, this is how this derivative is called in financial literature - presumably because ν looks similar to the Latin V (vee).

The very fact that these names were invented and are widely used shows how important these derivatives are.

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6.2 The derivatives of C with respect to S, T, K, σ, r

Remark1. The calculations presented below have not been discussed in lectures and you are not required to remember them. I do however strongly suggest that you understand them.

2. If a formula for any of these derivative is require for solving a problem in the exam then it will be stated as a part of the corresponding question.

In what follows, we shall use the Black & Scholes formula for the price of the call option given by

C = C(S, T, K, σ, r) = SΦ(ω) − Ke^−rTΦ(ω − σ√ T ) where

T + 1

2σ√

T and Φ(x) =

Z x

−∞

√1

2πe^−u²^/2du . (13) Observe that the price C of the call option does NOT depend on the drift parameter µ of the geometric Brownian motion which describes the behaviour of the price S(T ) of the underlying share.

We shall show that C = C(S, T, K, σ, r) is a decreasing function of K (while all other parameters remain constant) and it is an increasing function of all other parameters, that is of S, T, σ, r. These properties follow from the following lemma.

Proposition 6.1. The partial derivatives of C = C(S, T, K, σ, r) = e^−rTE((S(T ) − K)⁺)

are: ∂C

∂K = −e^−rTΦ(ω − σ√

T ) (14)

∂C

∂S = Φ(ω) (15)

∂C

∂r = KT e^−rTΦ(ω − σ√

T ) (16)

∂C

∂σ = S√

T Φ⁰(ω) (17)

∂C

∂T = σ 2√

TSΦ⁰(ω) + Kre^−rTΦ(ω − σ√

T ) . (18)

The above formulae show that _∂K^∂C < 0 and all other derivatives are positive, namely

∂C

∂S > 0, ^∂C_∂r > 0, ^∂C_∂σ > 0, ^∂C_∂T > 0. Hence C is indeed a decreasing function in K and increasing function in all other parameters.

The proof of Proposition 6.1 is not examinable. Those who want to know more can find the proof in Appendix A.

7 Hedging within the framework of the MPBM and the GBM model

We have considered hedging in the context of the One-Period Binomial Model. We shall see that the formulae derived there will be instrumental for the analysis of Multi-Period Binomial Model (MPBM).

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7.1 Hedging within the framework of the MPBM

We start with the statement of the problem.

Suppose that the price of a share S(j), 0 ≤ j ≤ n, follows a MPBM with parameters s, u, d, r. As usual, the no-arbitrage condition d < 1 + r < u is supposed to be satisfied.

Recall that there are n + 1 possible values of S(n): sdⁿ, sudⁿ⁻¹, ..., suⁱdⁿ⁻ⁱ, ..., suⁿ. Consider a derivative on this share with the payoff time n and the payoff function V . We shall consider payoff functions which depend only on S(n) - the price of the share at time n, that is

V (suⁱdⁿ⁻ⁱ) = V_n,i, 0 ≤ i ≤ n.

In words: if at time n the price of the share is S(n) = suⁱdⁿ⁻ⁱ then the owner of the derivative is paid £V_n,i.

Denote by C the no arbitrage price of this derivative.

Question 1. What is the price C of this derivative?

Question 2. How should the trader who sells this derivative invest the £C so that to be able at time n to meet the obligation to payoff £V_n,i whatever the i, 0 ≤ i ≤ n, turns out to be?

Answer to Question 1. The price C of such a derivative is given by

C = (1 + r)⁻ⁿ

n

X

i=0

n i

pⁱqⁿ⁻ⁱVn,i, where p = 1 + r − d

u − d , q = u − 1 − r u − d .

Proof. A sequence of prices S(1), ..., S(n) with S(n) = suⁿ has the risk-neutral probability (r-np for short) pⁱqⁿ⁻ⁱ (the price goes up i times and it goes down n − i times).

Each such sequence is uniquely defined by i time moments j₁, ..., j_i such that S(j_k) = S(j_k − 1)u. The total number of these sequence is equal to ⁿ_i which is the number of possibilities to choose i time-moment out of n.

Therefore, the r-np ˜P(S(n) = suⁱdⁿ⁻ⁱ) = ⁿ_ipⁱqⁿ⁻ⁱ. Hence also ˜P(V = Vn,i) = ⁿ_ipⁱqⁿ⁻ⁱ and ˜E(V ) = Pn

i=0 n

ipⁱqⁿ⁻ⁱV_n,i. By Theorem5.6.

C = (1 + r)⁻ⁿE(V ) = (1 + r)˜ ⁻ⁿ

n

X

i=0

n i

pⁱqⁿ⁻ⁱV_n,i

We shall now answer Question 2 which is the main goal of this section.

At each time moment j ≥ 0, the trader is allowed to buy shares and to deposit money into the bank. It is natural to expect that the number of shares and the total capital the trader must have at time j depends on S(j) = suⁱd^j−i - the value of the price of the underlying share. We thus introduce the following notation: given that S(j) = suⁱd^j−i,

X_j,i denotes the number of shares in the portfolio at time j, V_j,i the total value of the trader’s portfolio at time j.

Then V_j,i− X_j,isuⁱd^j−i is the amount of money that should be deposited in the bank at time j.

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Lemma 7.1. For each j, 0 ≤ j ≤ n − 1, the values (X_j,i, V_j,i, 0 ≤ i ≤ j, are computed as follows.

V_j,i = (1 + r)⁻¹(pV_j+1,i+1+ qV_j+1,i) , X_j,i= Vj+1,i+1− Vj+1,i

S(j)(u − d) (19)

Proof. Recall that V_j+1,i+1and V_j+1,iis the capital the seller of the derivative must have at time j+1 in order to be able to meet the payoff obligations at time n given that S(j+1) = suⁱ⁺¹d^j−i and S(j +1) = suⁱd^j respectively. If S(j) = suⁱd^j−ithen either S(j +1) = S(j)u = suⁱ⁺¹d^j−i or S(j + 1) = S(j)d = suⁱd^j and hence V_j,i must be such that the capital at time j + 1 becomes V_j+1,i+1and V_j+1,irespectively. Hence we are in the setting of the one-period binomial model (OPBM) and formulae (19) are the same as the corresponding formulae for OPBM (we formulae (3) and (5)in Notes 2): we replace V₁, V₂, P in these formulae by V_j+1,i+1, V_j+1,i, V_j,i respectively).

We thus can do the following:

Step 1. Use (19) to compute X_n−1,i and V_n−1,i, 0 ≤ i ≤ n − 1, from known values V_n,i, 0 ≤ i ≤ n.

Step 2. Similarly, compute X_n−1,i and V_n−2,i, 0 ≤ i ≤ n − 2, from V_n−1,i, 0 ≤ i ≤ n − 1 Continue these calculations until, moving from the values found for time j + 1 to values at time j. After n steps, we shall compute X0,0, V0,0.

The hedging strategy now works as follows.

1. At time t = 0, buy X_0,0 shares and deposit V_0,0− X_0,0s (the rest of the initial capital) into the bank.

2. At time t = 1, your total capital is V_0,0S(1) + (V_0,0− X_0,0s)(1 + r) and is either V_1,1 or V_1,0 (depending on whether S(1) = su or S(1) = sd. Use it to buy the corresponding number of shares (which will be either X_1,1 or X_1,0 respectively) and deposit the rest into the bank.

3. At each next time moment, say j you act as follows. If the price S(j) = suⁱd^j−i then the total value of your portfolio is V_j,i. Use it to buy X_j,i shares and deposit V_j,i− X_j,iS(j) into the bank.

Remark Note that C = V_0,0. Indeed, we have two investments.

Investment 1: buy the derivative for £C.

Investment 2: Invest £V_0,0 in the way prescribed by the hedging strategy.

Both investments produce the same result at time n. Hence, by the Law of One Price, their cost should be the same.

7.2 Hedging within the framework of the GBM model

Throughout this section we suppose that the price of a share follows the Geometric Brownian Motion (GBM) S(t) = Se^{µt+σW (t)}, t ≥ 0,, where S, µ, σ are the parameters of the GBM, W (t) is the standard Brownian motion. The interest rate compounded continuously is r.

We consider a derivative on this option with a payoff function G(S(T )) and the payoff time T . We know (Theorem 5.7) that the price C of such a derivative is given by

C = e^−rTE[G(S(T ))] = e˜ ^−rT Z ∞

−∞

G

S(0)e^{µT +σ}^˜

√ T x

f (x)dx, (20)

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where f (x) = ^√¹_2πe⁻^x2² is the p.d.f. of the standard norma random variable.

Question. Can the seller of the derivative invest £C so that to be able to meet the payoff obligation at time T . The seller is allowed to buy the underlying shares and to deposit the money into the bank.

The answer is: Yes, by continually reinvesting the portfolio.

To make this answer precise, we have to answer two more questions.

Question (a). What should be the total value of the portfolio at time t, 0 ≤ t ≤ T ? Question (b). How many shares should be in the portfolio at time t, 0 ≤ t ≤ T ?

Answer to Question (a). The total value of the portfolio should be equal to the price C(S(t), t) of the derivative at time t, given that the price of the share is S(t).

We thus have introduced a a new notation, C(S(t), t) – the price of the derivative which is a function of two variables, S(t) and t. The important fact is that we can compute this price.

Namely, since the conditions of the market allow the investor to buy the derivative at any time t, 0 ≤ t ≤ T , C(S(t), t) can be computed in the same way as C in (20). The only difference is that the starting price of the share now is S(t) and the payoff takes place in T − t units of time. Thus

C(S(t), t) = e^{−r(T −t)} Z ∞

−∞

G

S(t)e^{µ(T −t)+σ}

√T −tx

f (x)dx. (21)

Remark 7.2. We don’t prove this statement. However, we note that if at time t = 0 the £C can be used for hedging, then it should also be possible to use £C(S(t), t) given by (21) for hedging which starts at time t with the price of the share being S(t).

Answer to Question (b). The number of shares in the portfolio at time t should be

∆(t) = ∂C(S, t)

∂S , where S = S(t). (22)

The hedging strategy is: at each time t, have ∆(t) shares and keep £[C(S(t), t)−∆(t)S(t)]

in the bank.

This can be achieved by re-investing the total capital in our portfolio at times 0, h, 2h, ..., T , where h as a short period of time as possible.

At time 0, we own £C(S(0)) and use them to buy ∆(0) shares and deposit the rest of the capital in the bank.

At time h, we change the number of shares in the portfolio to ∆(h) and deposit the rest of the capital in the bank. We can do that because we know the price S(h) (from our observation of the development in the market of shares) and we can use S(h) to carry out the necessary calculations.

The same operation is then repeated ta times 2h, 3h, ..., T .

If h is very small then this process looks like a continual reinvestment of the portfolio.

This strategy is designed so that to make sure that at time T the value of our portfolio is R(S(T )) which means we can meet our financial obligations.

Remark. (22) explains why the term Delta-hedging is used in finance.

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Example. If, in the above formulae, G(x) = (x − K)⁺ then this is case of the call option Call(K, T ). In this case we know the explicit expressions for C(S, t) and ∆(t). Namely, using (10), we obtain that the value of the portfolio at time t should be

C(S(t), t) = S(t)Φ(ω(t)) − Ke^{−r(T −t)}Φ(ω(t) − σ√

T − t) , (23)

where

ω(t) = ln^S(t)_K + r(T − t) σ√

T − t +1 2σ√

T − t

Φ(x) = Z x

−∞

√1

2πe^−u²^/2du .

Thus, in this example the number of shares ∆(t) which should be in the portfolio at time t is

∆(t) = ∂C(S, t)

∂S S=S(t)

= Φ(ω(t)). (24)

The last formula is due to (15).

Finally, we have a simple expression for the amount Y(t) of money that should be deposited in the bank:

Y (t) = C(S(t), t) − S(t)∆(t) = −Ke^{−r(T −t)}Φ(ω(t) − σ√

T − t).

Remark 7.3. In this example, ∆(t) is always strictly positive, while the cash Y (t) is strictly negative. This means that at each moment t the amount −Y (t) is borrowed from the bank and invested in shares (together with C(S(t), t).

7.2.1 Explanation of the reason for (24) (not examinable)

To simplify the calculations and the formulae, suppose that µ = 0, that is S(t) = e_{σW (t)}. If at time t the price of the share is S(t) then at time t + h the price will be, approximately, either S(t)e^σ

√

h or S(t)e^−σ

√

h. So, the price of the derivative at time t + h will be either C(S(t)e^σ

√h, t + h) or C(S(t)e^−σ

√h, t + h). We know from the discussion of hedging for the One-Period Binomial Model that the number of shares that should be in the portfolio at time t is given by

C(S(t)e^σ

√

h, t + h) − C(S(t)e^−σ

√

h, t + h) S(t)(e^σ

√h− e^−σ^√^h) . As h → 0, we obtain

∆(t) = lim

h→0

"

C(S(t)e^σ

√

h, t + h) − C(S(t)e^−σ

√

h, t + h) S(t)(e^σ

√

h− e^−σ^√^h)

#

= ∂C(S, t)

∂S , where S = S(t).

Remark. The proof of the last relation easily follows from the fact that F (b) − F (a) = F⁰(ξ)(b − a), where ξ ∈ (a, b). You should know this formula from the Calculus I course.

Applying it to the difference in the numerator of the fraction we obtain C(Se^σ

√

h, t + h) − C(Se^−σ

√

h, t + h) = ∂C(S, t + h)

∂S S=ξ

(Se^σ

√

h − Se^−σ

√ h),

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where ξ ∈ (Se^σ

√

h, Se^−σ

√

h). It is now obvious that

∆(t) = lim

h→0

"

∂C(S, t + h)

∂S S=ξ

#

= ∂C(S, t)

∂S , where S = S(t).

8 Appendix A: Proof of Proposition 6.1

Proof of Proposition 6.1.

Firstly, rearranging the definition of ω in (13) we have that ω = ln_K^S + rT

σ√

T +1

2σ√

T ⇔ σ√

T ω = ln S

K + rT + 1 2σ²T

⇔ σ√

T ω − rT − 1

2σ²T = ln S

K . (25)

Then, by taking the derivative of the c.d.f. Φ(x) in (13) we have the formula of the p.d.f. of a standard normal random variable Z:

f_Z(x) = Φ⁰(x) = 1

√2πe^−x²^/2. (26)

Using together the equations (25)-(26), we can further conclude that Ke^−rTΦ⁰(ω − σ√

T ) = SΦ⁰(ω) . (27)

To see the latter equality, we perform the calculations K exp{−rT } Φ⁰(ω − σ√

T )⁽²⁶⁾= K exp{−rT } 1

√2πexpn

−(ω − σ√ T )² 2

o

= K 1

√2πexpn

− ω² 2

o expn

− rT + σ√

T ω − 1 2σ²To

(25)= K Φ⁰(ω) exp n

σ

√

T ω − rT − 1 2σ²T

o

(26)= K Φ⁰(ω) expn ln S

K o

= SΦ⁰(ω) .

We are now in a position to prove relations (14) - (18).

Proof of (14). We have

∂C

∂K = SΦ⁰(ω)∂ω

∂K − e^−rTΦ(ω − σ√

T ) − Ke^−rTΦ⁰(ω − σ√ T )∂ω

∂K

= ∂ω

∂K [SΦ⁰(ω) − Ke^−rTΦ⁰(ω − σ

√

T )] − e^−rTΦ(ω − σ

√ T )

(25)= −e^−rTΦ(ω − σ√ T ) .

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Remark. As stated before, ^∂C_∂K < 0 and so C decreases as K increases. This can be seen directly from C = e^−rTE(S(T ) − K)⁺ since (S(T ) − K)⁺ is a decreasing function in K.

∂C

∂S = Φ(ω) + SΦ⁰(ω)∂ω

∂S − Ke^−rTΦ⁰(ω − σ√ T )∂ω

∂S

= Φ(ω) + ∂ω

∂S [SΦ⁰(ω) − Ke^−rTΦ⁰(ω − σ√ T )]

(25)= Φ(ω) .

Remark. As stated before, ^∂C_∂S > 0 and so C increases as S increases. This should be intuitively obvious, since the price of a call option should increase as the price of the underlying share increases.

∂C

∂r = SΦ⁰(ω)∂ω

∂r + KT e^−rTΦ(ω − σ√

T ) − Ke^−rTΦ⁰(ω − σ√ T )∂ω

∂r

= KT e^−rTΦ(ω − σ√

T ) + ∂ω

∂r [SΦ⁰(ω) − Ke^−rTΦ⁰(ω − σ√ T )]

(25)= KT e^−rTΦ(ω − σ√ T ) . Proof of (17). We have

∂C

∂σ = SΦ⁰(ω)∂ω

∂σ − Ke^−rTΦ⁰(ω − σ√

T ) ∂ω

∂σ −√ T

= ∂ω

∂σ [SΦ⁰(ω) − Ke^−rTΦ⁰(ω − σ√

T )] + K√

T e^−rTΦ⁰(ω − σ√ T )

(25)= K√

T e^−rTΦ⁰(ω − σ√ T )

(27)= S√

T Φ⁰(ω) .

Remark. As stated above, ^∂C_∂σ > 0 , and so C increases as σ increases. This is plausible, since the riskier the underlying asset the more you should pay for a call option on it.

∂C

∂T = SΦ⁰(ω)∂ω

∂T + rKe^−rTΦ(ω − σ√

T ) − Ke^−rTΦ⁰(ω − σ√

T ) ∂ω

∂T − σ 2√ T

= ∂ω

∂T [SΦ⁰(ω) − Ke^−rTΦ⁰(ω − σ√

T )] + rKe^−rTΦ(ω − σ√

T ) + σ 2√

TKe^−rTΦ⁰(ω − σ√ T )

(25)= rKe^−rTΦ(ω − σ√

T ) + σ 2√

TKe^−rTΦ⁰(ω − σ√ T )

(27)= rKe^−rTΦ(ω − σ√

T ) + σ 2√

TSΦ⁰(ω) .

Remark. As stated above, ^∂C_∂T > 0 and so C increases as T increases. Again, this should not come as a complete surprise, since you expect to pay more for a call option if the maturity date is further away in the future.

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9 Appendix B: The Black-Scholes Formula

This Appendix provides a detailed derivation of the BS formula. This derivation has already been explained in the course FMI and is not examinable.

In the Black-Scholes model, share prices evolve according to a Geometric Brownian Motion. The aim of this section is to answer the following question.

Question. Given that the price of a share evolves according to a geometric Brownian motion, what is the no-arbitrage price C of the related European call option with strike price K and maturity time T ?

The answer to this question is given by the Black-Scholes Formula.

Theorem 9.1 (Black-Scholes Formula). Let C be the no-arbitrage price of a call option with strike price K and maturity time T and suppose that the underlying asset (e.g. share) is evolving according to a geometric Brownian motion with volatility parameter σ and initial price S. The continuously compounded interest rate is denoted by r. Then

C = C(S, T, K, σ, r) = SΦ(ω) − Ke^−rTΦ(ω − σ√

T ) , (28)

where

T + 1

2σ√ T

Φ(x) = Z x

−∞

√1

2πe^−u²^/2du .

We have already mentioned before that Theorem 9.1 is an easy corollary of the following statement.

Theorem 9.2. Let C be the no-arbitrage price of a call option with strike price K and maturity time T . Suppose that the underlying asset (say a stock) is evolving according to a GBM with volatility parameters σ, drift parameter µ, initial price S. The continuously compound interest is r. Then

C = e^−rT Z ∞

−∞

(Se^{µT +σ}^˜

√T x − K)⁺ 1

√2πe⁻^x2² dx (29)

where ˜µ = r − ^σ₂².

9.1 Example

Before proving the above theorems, let us consider the following example.

Example 9.3. Assume the Black-Scholes model and suppose a share is currently traded at a price of £30, the continuously compounded interest rate is 8% and the volatility is 0.20. Find the no-arbitrage price C of a Call option with maturity time in three months from now and strike price equal to £34.

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Solution. We have: T = 3/12 = 0.25, r = 0.08, σ = 0.20, K = 34, and S = 30 and we can compute

ω = −1.0016 and ω − σ√

t = −1.1016 . So the Black-Scholes Formula gives

C = 30Φ(−1.00) − 34e^−0.02Φ(−1.10) = 0.2386 ,

where we have used Φ(−1.00) = 1 − 0.8413 and Φ(−1.10) = 1 − 0.8643. Thus, the no- arbitrage price is £0.24.

The explanations below consist of two parts.

• Part I: contains the main idea of the proof of formulae (32) (see below) and (29). This is not completely rigorous (though it can be made rigorous), but explains a relatively deep idea.

• Part II: contains the derivation of the theorem using (29). This is elementary and completely rigorous.

9.2 Part I. The main idea of the proof of Theorem 9.2

The following proof has not been discussed in lectures and is not examinable. It is included in these notes for completeness (and for those who want to know more).

Proof of Theorem 9.2.

Remember that, by Theorem 4.4 (Convergence to Geometric Brownian motions), the Geometric Brownian motion with volatility parameter σ can be approximated by a multiperiod binomial process with initial value S(0) = S, u = exp(σpT /n), and d = exp(−σpT /n).

We assume that the upward and downward movements happen at discrete time moments iT /n, where i = 0, 1, . . . , n − 1.

Now, set h = T /n and denote by S_h(ih) the price of the share at times ih. The interest per time-period h in this model is rh. Let C_h denote the price of the corresponding Call option with underlying asset S_h. It is natural to expect that, as n → ∞ (or, equivalently, h → 0) not only S_h(ih) converges to the Geometric Brownian Motion (already known by Theorem 4.4), but also that the C_h converges to the no-arbitrage price C.

Let S(r) denote the actual price of the share at time r = ih. Note that Theorem4.4states that S_h(ih) → S(r) as n → ∞ (or, equivalently, h → 0). In particular, S_h(nh) → S(T ).

From our results for the multiperiod binomial model, the no-arbitrage price

Ch = (1 + rh)⁻ⁿE^h(Sh(nh) − K)⁺ , (30) where notation Eh emphasizes the dependence of the expectation on n (or, equivalently, h) and the (risk-neutral) probabilities corresponding to our approximation are

p = 1 + rh − d

u − d = 1 + rh − exp(−σ√ h) exp(σ√

h) − exp(−σ√ h).

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Using the first three terms of the Taylor-Maclaurin expansion of the exponential function e^x = 1 + x +x²

2! + x³

3! + · · · , we see that, for large n (or, equivalently, small h), we have

p =

1 + rh −

1 − σ√

h +¹₂σ²h + · · ·

1 + σ√

h +¹₂σ²h + · · ·

−

1 − σ√

h + ¹₂σ²h + · · ·

≈ σ√

h + r − ¹₂σ²h 2σ√

h

= 1 2

1 + r − ¹₂σ² σ

√ h

= 1

2 1 + r − ¹₂σ² σ

rT n

! .

We see that the multiperiod Binomial model equipped with risk-neutral probabilities satisfies the conditions of Theorem 4.4 and hence the S_h(ih) converges in distribution to a geometric Brownian motion with drift parameter ˜µ = r − ¹₂σ² and volatility parameter σ. We denote this limiting process by eS(t) and call it the risk-neutral Geometric Brownian motion with parameters S, ˜µ, σ, i.e.

S(t) = S exp(˜˜ µt + σW (t)) = S exp

r − 1

2σ²

t + σW (t)

. In particular, it follows that

n→∞lim Eh(Sh(nh) − K)⁺ = E( eS(T ) − K)⁺ . (31) As mentioned above, proving (31) constitutes the main part of the proof. Beyond this point, everything is perfectly rigorous and easy. Since

n→∞lim (1 + rh)⁻ⁿ = lim

n→∞

1 + r T n

−n

= e^{−r T} ,

we obtain from (30) the formula for the no-arbitrage price C of a Call option in the Black- Scholes model:

C = e^−rT E(S(T ) − K)e ⁺ . (32) By the definition of the Geometric Brownian motion, we have eS(T ) = SeµT +σW (T )^˜ . Remember that W (T ) ∼ N (0, T ) and therefore we can write W (T ) =√

T Z, where Z ∼ N (0, 1). Hence C = e^−rT E

h

Se^{µ T +σ}^˜

√T Z − K+i

and we can re-write (32) in a more explicit form:

C = e^−rT Z +∞

−∞

Se^{µ T +σ}^˜

√

T z− K+ 1

√2πe^−z²^/2dz . (33) (Note that here, we use the definition of the Expectation of g(Z) and the p.d.f. of the standard Normal random variable Z, with g(z) = (Se^{µ T +σ}^˜

√

T z − K)⁺.)

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9.3 Part II. Derivation of the Black-Scholes formula

The proof presented below has been explained in lectures and you are supposed to understand it.

Proof of Theorem 9.1.

It remains to show that the desired expression of C in (28) is equivalent to (33) obtained in Part I. To this end, note that the following lemma holds true:

Lemma 9.4. The following inequalities are equivalent:

Se^{µT +σ}^˜

√

T z − K > 0 ⇔ z > σ√

T − ω . Proof. Observe that

Se^{µ T +σ}^˜

√

T z− K > 0 ⇔ Se^(r−σ²^{/2) T +σ}

√

T z− K > 0

⇔ (r − σ²/2) T + σ

√

T z > ln K/S

⇔ z > ln K/S − (r − σ²/2) T σ√

T

⇔ z > σ√

T − ln S/K + r T σ√

T −1

2σ√ T

⇔ z > σ√ T − ω

Using the above lemma and setting a = σ√

T − ω, we have from (33) that C = e^{−r T} 1

√2π Z +∞

a

(S e^{µ T +σ}^˜

√

T z− K)e^−z²^/2dz (34) and we have

C = e^{−r T} 1

√2π Z +∞

a

S e^{µ T +σ}^˜

√

T ze^−z²^/2dz − e^{−r T} K 1

√2π Z +∞

a

e^−z²^/2dz . (35) SinceR∞

a e^−z²^/2dz =R−a

−∞e^−z²^/2dz for any real number a, the second integral is given by e^−rTK 1

√2π Z +∞

a

e^−z²^/2dz = e^−rTK Z −a

−∞

√1

2π e^−z²^/2dz

= Ke^−rTΦ(−a) = Ke^−rTΦ(ω − σ√ T ) . We now deal with the first integral in (35), which is given by

√S 2π

Z +∞

a

e^{−rT +˜}^{µT +σ}

√

T z−z²/2dz ^(∗)= S

√2π Z +∞

a

e^−σ²^{T /2+σ}

√

T z−z²/2dz

= S

√2π Z +∞

a

e^−(σ

√

T −z)²/2dz

(∗∗)= − S

√2π Z −∞

ω

e^−y²^/2dy = S

√2π Z ω

−∞

e^−y²^/2dy

= SΦ(ω) ,

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where (∗) holds because ˜µ = r −¹₂σ² and (∗∗) is due to the change of variables y = σ√ T − z.

It thus follows from (35) that

C = SΦ(ω) − Ke^−rTΦ(ω − σ

√ T ).

MTH6155 Financial Mathematics II Weeks 3, 4, and 5 (part 1)