Lecture 4 : Binary Model. Portfolio. Hedge. Option. Fair Price. Property of Binary Model. Fair Price, Minimal Hedge, Optimal Capital Evolution for European and American Options. Generalization of Binary Model. Option Quality. Optimal European Call Option. Optimal Option for a Special Class of Options. Optimal Option for a Wide Class of Options. Appendix : Optimal Stock Portfolio.

Binary Model

Consider the simplest model of a financial market, that is the so-called Cox-Ross-Rubinstein model, or (B,S)-model. It admits that on the market two types of assets, namely bonds and stocks, are handled with. Their prices are changed at discrete time moments n = 1, 2, …, N < ∞.

A bond is a type of an unrisky asset. Its price b_{n} at the moment n depends on that at the moment n − 1 by the simple formula

b_{n} = ρb_{n−1}, n ≥ 1,

where ρ ≥ 1 is the interest rate. Clearly, if b_{0} is a price of a bond at the starting moment, then

b_{n} = ρ^{n}b_{0}, n ≥ 0.

A stock is a type of a risky asset. Its price s_{n} at the moment n is random and depends on that at the moment n − 1 by the formula

s_{n} = ξ_{n}s_{n−1}, n ≥ 1,

where {ξ_{n}} are independent identically distributed discrete random variables taking only two values : d (down) and u (up), d < 1 < u. Clearly, if s_{0} is the price of a stock at the starting moment, then

s_{n} = s_{0}ξ_{1} · . . . · ξ_{n}, n ≥ 1.

Hence, we have a single trajectory of the bond price evolution

(b_{0}, b_{0}ρ, …, b_{0}ρ^{N}),

and 2^{N} possible trajectories of the stock price evolution

(s_{0}, s_{0}ξ_{1}, …, s_{0}ξ_{1} · . . . · ξ_{N}).

Further reasoning is sensable when d < 1 ≤ ρ < u.

Portfolio and Self-Financing

Assume for simplicity that only one type of stocks is available. We also suppose that at any moment n our capital x_{n} is splitted between bonds and stocks. Let β_{n} and γ_{n}, respectively, be the amount of bonds and the amount of stocks that we have at the moment n (these values are not necessarily integers and even positive). Thus

x_{n} = β_{n}b_{n} + γ_{n}s_{n}, n ≥ 0.

A pair π_{n} = (β_{n}, γ_{n}) forms our portfolio at the moment n. At the next moment n + 1 the prices are changing. Our capital is also changing and becomes to be equal to

β_{n}b_{n+1} + γ_{n}s_{n+1}.

At this moment we have a possibility to rearrange the capital between bonds and stocks forming a new portfolio π_{n+1} = (β_{n+1}, γ_{n+1}), which price is

x_{n+1} = β_{n+1}b_{n+1} + γ_{n+1}s_{n+1}.

We assume that we perform this rearrangement in such a way that

β_{n}b_{n+1} + γ_{n}s_{n+1} = β_{n+1}b_{n+1} + γ_{n+1}s_{n+1}.

That corresponds to the situation when we neither add nor withdraw a capital. Such a condition is called self-financing.

Therefore, our game with assets consists in rearrangement the capital, which is in our disposal, between bonds and stocks. As a result we get a trajectory describing the capital evolution

(x_{0}, x_{1}, …, x_{N}).

The sequence of portfolio π = (π_{n}) is called a strategy and it reflects our actions. If for any n the self-financing condition holds then a strategy is also called a self-financing strategy.

What is the aim of such a game ?

Hedge and Option

Assume that our goal is to make the capital at the moment N not less then a value of some function f_{N} = f_{N}(s_{0}, s_{1}, …, s_{N}). Of course, a realization of this goal depends on the starting capital and on possibility to construct a relevant strategy.

Definition 1. A strategy π = (π_{n}) for a given starting capital x_{0} > 0 and given function f_{N} = f_{N}(s_{0}, s_{1}, …, s_{N}) is said to be (x_{0}, f_{N})-hedge if

x_{N} ≥ f_{N}(s_{0}, s_{1}, …, s_{N}).  (1)

In the case when x_{N} = f_{N}(s_{0}, s_{1}, …, s_{N}) it is said that π is a minimal (x_{0}, f_{N})-hedge.

A market for which the equality x_{N} = f_{N}(s_{0}, s_{1}, …, s_{N}) can always be achieved, due to some strategy, is called complete. Otherwise, it is called uncomplete.

Define

C = inf{x_{0} > 0 : ∃ (x_{0}, f_{N}) − hedge }.  (2)

It is a minimal starting capital provided (1) utilizing a relevant strategy.

Definition 2. An option is a right to buy (or to sell) assets under the predescribed conditions.

We distinguish between a call option (a right to buy) and a put option (a right to sell). Further on we confine ourselves with considering only the first type.

The basic options are those of European type and American type. Options of the first type are realized at some predescribed moment (expiration time) N. Options of the second type are realized at any time within the predescribed period (0, N].

European Options

First, we deal with European options. The standard European option, that is the so-called European call option, is a contract giving a right to buy stocks on the pregiven (strike) price K > 0 at an expiration time N. If s_{N} > K then realizing the option we gain (s_{N} −K). But if s_{N} ≤ K then there is no sense to realize the option paying an excessive price for the stocks. The function f_{N} = (s_{N} − K)^{+} is called a pay-off function. The pay-off function for European call option is quite simple and depends only on s_{N}. In general, the pay-off function f_{N} = f_{N}(s_{0}, s_{1}, …, s_{N}) can be much more complicated.

Examples.

• collar option: f_{N} = min {max{K_{1}, s_{N}}, K_{2}};

• Boston option: f_{N} = (s_{N} − K_{1})^{+} − (K_{2} − K_{1}), 0 < K_{1} < K_{2};

• look back call option: f_{N} = (s_{N} − K_{N})^{+} , K_{N} = min{s_{0}, s_{1}, …, s_{N}};

• Asian arithmetical call option: f_{N} = (s̄_{N} − K)^{+} , s̄_{N} = \frac{1}{N+1} ∑_{k=0}^{N} s_{k}.

Of course, an option costs some amount of money.

Example. Assume we buy European call option on 100 Exxon stocks with expiration time N for 700 FIM (7 FIM per stock). Let K = 60 FIM per stock and s_{0} = 65 FIM per stock. Consider three possibilities :

(a) s_{N} = 58 FIM ⟹ we do not realize the option ; our losses are 700 FIM ;

(b) s_{N} = 73 FIM ⟹ we realize the option ; our gaines are (73 − 60) · 100 − 700 = 600 FIM ;

(c) s_{N} = 66 FIM ⟹ we realize the option ; our gaines (losses) are (66 − 60) · 100 − 700 = −100 FIM .

Fair Price

How much is it reasonable to pay for an option ? Of course, it should not be too large or too small. Definition 3. The minimal initial capital x_{0}, which posession at the starting moment gives us a possibility of achievment at the expiration time N, due to some strategy, at least the capital equals to f_{N} is called a fair price of the option. The fair price is defined by formula (2).

Behaviour of the fair price C depending on s_{N}, K, N is the following :

• s_{N} is larger, C is larger ;

• K is larger, C is smaller ;

• N is larger, C is larger .

Usually, a real price of the option is calculated basing on the value of the fair price.

Property of Binary Model

Let f_{N} = f_{N}(s_{N}). To ensure the equality x_{N} = f_{N}(s_{N}) at the expiration time N, we need to have at the moment N − 1 the capital

x_{N−1} = ρ^{−1}(p_{∗}f_{N}(s_{N−1}u) + (1 − p_{∗})f_{N}(s_{N−1}d)),  (3)

where p_{∗} = \frac{ρ−d}{u−d}. Moreover,

β_{N−1} = \frac{uf_{N}(s_{N−1}d) − df_{N}(s_{N−1}u)}{ρb_{N−1}(u − d)}, γ_{N−1} = \frac{f_{N}(s_{N−1}u) − f_{N}(s_{N−1}d)}{s_{N−1}(u − d)}.

(4)

It is natural to assume that the function f_{N} is nondecreasing. Thus γ_{N−1} is always nonnegative, but β_{N−1} can be negative.

Indeed, at the moment N − 1 we have

x_{N−1} = b_{N−1}β_{N−1} + s_{N−1}γ_{N−1}.  (5)

and we would like to have

x_{N} = b_{N−1}ρβ_{N−1} + s_{N−1}ξ_{N}γ_{N−1} = f_{N}(s_{N}) = f_{N}(s_{N−1}ξ_{N}).

To ensure this equality in any of the two cases (the stock price can go up or down, that is ξ_{N} can take any of the two values u or d) we should choose at the moment N − 1 the values of β_{N−1} and γ_{N−1} such that

\{ \begin{matrix}f_{N}(s_{N−1}u) & = & ρb_{N−1}β_{N−1} + us_{N−1}γ_{N−1} \\ f_{N}(s_{N−1}d) & = & ρb_{N−1}β_{N−1} + ds_{N−1}γ_{N−1} \end{matrix} .

This pair of equations has a unique solution with respect to β_{N−1} and γ_{N−1} given by (4). Substituting (4) into (5), we obtain (3).

Fair Price, Minimal Hedge, Optimal Capital Evolution

Set

f_{N−1}(s) = ρ^{−1}(p_{∗}f_{N}(su) + (1 − p_{∗})f_{N}(sd)).

We would like to have x_{N−1} = f_{N−1}(s_{N−1}). Now consider a new European option with the pay-off function f_{N−1} and the expiration time N − 1. Similarly, we get

x_{N−2} = ρ^{−1} (p_{∗}f_{N−1}(s_{N−2}u) + (1 − p_{∗})f_{N−1}(s_{N−2}d)) ,

or equivalently

x_{N−2} = ρ^{−2} (p_{∗}^{2}f_{N}(s_{N−2}u^{2}) + 2p_{∗}(1 − p_{∗})f_{N}(s_{N−2}ud) + (1 − p_{∗})^{2}f_{N}(s_{N−2}d^{2})) ,

and

β_{N−2} = \frac{uf_{N−1}(s_{N−2}d) − df_{N−1}(s_{N−2}u)}{ρb_{N−2}(u − d)}, γ_{N−2} = \frac{f_{N−1}(s_{N−2}u) − f_{N−1}(s_{N−2}d)}{s_{N−2}(u − d)}.

Set then

f_{N−2}(s) = ρ^{−2}(p_{∗}^{2}f_{N}(su^{2}) + 2p_{∗}(1 − p_{∗})f_{N}(sud) + (1 − p_{∗})^{2}f_{N}(sd^{2}))

and consider a new European option with the pay-off function f_{N−2} and the expiration time N − 2 etc. Note that at the each stage we have a unique choice of portfolio.

In general, if we want to ensure the equality x_{N} = f_{N}(s_{N}), we should have at the moment n = 0, 1, …, N − 1, a capital

x_{n} = f_{n}(s_{n}) = ρ^{−(N−n)} N−n ; ∑ ; j=0 (\begin{matrix}N − n \\ j \end{matrix}) p_{∗}^{j}(1 − p_{∗})^{N−n−j}f_{N}(s_{n}u^{j}d^{N−n−j})

(6)

and the portfolio π_{n} = (β_{n}, γ_{n}) given by

β_{n} = \frac{uf_{n+1}(s_{n}d) − df_{n+1}(s_{n}u)}{ρb_{n}(u − d)} , γ_{n} = \frac{f_{n+1}(s_{n}u) − f_{n+1}(s_{n}d)}{s_{n}(u − d)} . (7)

Fair Price, Minimal Hedge, Optimal Capital Evolution (continuation)

In particular, if we set n = 0 then we obtain a starting capital

x_{0} = C = ρ^{−N} N ; ∑ ; j=0 (\begin{matrix}N \\ j \end{matrix}) p_{∗}^{j}(1 − p_{∗})^{N−j}f_{N}(s_{0}u^{j}d^{N−j}).  (8)

Formula (8) gives the fair price of European option with the pay-off function f_{N} = f_{N}(s_{N}). There exists a unique minimal hedge π = (π_{n}), namely (C, f_{N})-hedge. The elements π_{n} = (β_{n}, γ_{n}) of the minimal hedge are given by (7). Optimal capital evolution leading from x_{0} to x_{N} = f_{N}(s_{N}) is unique and is given by (6).

Similarly, we can obtain the relevant formulae for the fair price, minimal hedge and optimal capital evolution in the case when the pay-off function is of the general form f_{N} = f_{N}(s_{0}, s_{1}, …, s_{N}) (Exercise 4).

Example. Assume that b_{0} = 1, ρ = 1, s_{0} = 150, u = 1.2, d = 0.6, f_{N} = (s_{N} − K)^{+}, N = 1, K = 150. What is the fair price of the option, minimal hedge and optimal capital evolution ? In this case the crucial parameter equals to p_{∗} = \frac{1−0.6}{1.2−0.6} = \frac{2}{3}. According to (8),

C = (1 − p_{∗})f_{N}(s_{0}d) + p_{∗}f_{N}(s_{0}u) = \frac{1}{3} · 0 + \frac{2}{3} · 30 = 20.

According to (7),

β_{0} = \frac{1.2 · 0 − 0.6 · 30}{1 · 1 · (1.2 − 0.6)} = −30, γ_{0} = \frac{30 − 0}{150(1.2 − 0.6)} = \frac{1}{3}.

That is why the minimal hedge, or our strategy, is π_{0} = (−30, \frac{1}{3}). Optimal capital evolution : x_{0} = 20, x_{1} = (s_{1} − 150)^{+}.

Generalization of Binary Model

Assume that the random variables (ξ_{n}) are not discrete but can take any value from the interval [d, u], that is ξ_{n} ∈ [d, u], n = 1, …, N. For simplicity, consider the case when the pay-off function is of the form f_{N} = f_{N}(s_{N}) and suppose f_{N} to be convex.

Again, the minimal capital we need to have at the moment N − 1 to guarantee the inequality x_{N} ≥ f_{N}(s_{N}) is given by (3) and π_{N−1} = (β_{N−1}, γ_{N−1}) is given by (4).

Indeed, if ξ_{N} = d or ξ_{N} = u then a portfolio π_{N−1} = (β_{N−1}, γ_{N−1}) given by (4) guarantees the equality x_{N} = f_{N}(s_{N}), and the minimal capital we need to have at the moment N − 1 is given then by (3). If ξ_{N} ∈ (d, u) then utilizing the same portfolio, due to the convexity of f_{N}, we have

x_{N} = ρb_{N−1}β_{N−1} + ξ_{N}s_{N−1}γ_{N−1}

= \frac{uf_{N}(s_{N−1}d) − df_{N}(s_{N−1}u)}{u − d} + ξ_{N} \frac{f_{N}(s_{N−1}u) − f_{N}(s_{N−1}d)}{u − d}

= f_{N}(s_{N−1}d)\frac{u − ξ_{N}}{u − d} + f_{N}(s_{N−1}u)\frac{ξ_{N} − d}{u − d}

≥ f_{N} (s_{N−1}d\frac{u − ξ_{N}}{u − d} + s_{N−1}u\frac{ξ_{N} − d}{u − d}) = f_{N}(s_{N−1}ξ_{N}) = f_{N}(s_{N}).

Clearly, the capital x_{N−1} given by (3) is minimal to ensure x_{N} ≥ f_{N}(s_{N}).

Set

f_{N−1}(s) = ρ^{−1}(p_{∗}f_{N}(su) + (1 − p_{∗})f_{N}(sd)).

We would like to have x_{N−1} ≥ f_{N−1}(s_{N−1}). Now consider a new European option with the pay-off function f_{N−1} and the expiration time N − 1. Repeating the same arguing as above, at the moment n we have x_{n} ≥ f_{n}(s_{n}), where f_{n}(s_{n}) is given by (6) and π_{n} = (β_{n}, γ_{n}) is given by (7).

In particular, the fair price of the option is again given by (8).

American Options

This type of options can be realized at any time moment n ∈ (0, N]. That is why, here a sequence of the pay-off functions f_{n} = f_{n}(s_{0}, s_{1}, …, s_{n}), n = 1, …, N, is given. What should be the minimal starting capital (fair price) to ensure fulfilling the inequalities x_{n} ≥ f_{n}(s_{0}, s_{1}, …, s_{n}) for all n = 1, …, N ?

Basing on the formulae for the fair price, minimal hedge and optimal capital evolution for European options, one can propose the following algorithm for calculation the relevant values for American options. Here we confine ourselves only with the case when f_{n} = f_{n}(s_{n}), n ≥ 1, but the algorithm can be extended also to the general case.

1. Calculate the fair price, minimal hedge and optimal capital evolution for European option with the pay-off function f_{N} according to (6)-(8).

2. For any n = 1, …, N − 1, compare the calculated values x_{n} with f_{n} for all possible trajectories (s_{0}, s_{1}, …, s_{n}).

3. If in any case x_{n} ≥ f_{n} then the evolution of the capital calculated on the first step is optimal and correspondingly x_{0} is the fair price and relevant strategy is a minimal hedge.

4. If there exist trajectories and integers n such that x_{n} < f_{n}, come back to the first step with the pay-off function f_{n_{1}}, where

n_{1} = max{k : 1 ≤ k < N, ∃ trajectory (s_{0}, s_{1}, …, s_{k}) : x_{k} < f_{k}}.

Really, this algorithm is stopped in a finite number of steps. The last calculated value of x_{0} is the fair price. However, the last calculation of optimal capital evolution and minimal hedge gives only a part of the whole optimal capital evolution and the whole minimal hedge.

Option Quality

Consider again European option with a pay-off function f_{N} = f_{N}(s_{N}) within the binary model. Clearly, formula (8) giving the fair price of the option can be rewritten as

C = ρ^{−N}E_{p_{∗}}(f_{N}(s_{N})),

where

E_{p_{∗}}(f_{N}(s_{N})) = N ; ∑ ; j=0 (\begin{matrix}N \\ j \end{matrix}) p_{∗}^{j}(1 − p_{∗})^{N−j}f_{N}(s_{0}u^{j}d^{N−j})

is an expectation of the random variable f_{N}(s_{N}) under the assumption P (ξ_{n} = u) = p_{∗}, n ≥ 1.

Assume at the starting moment that P (ξ_{n} = u) = p, n ≥ 1. Then ρ^{−N}E_{p}(f_{N}(s_{N})) is an expectation of our gains discounted by the interest rate. Since for given ρ, u, d, N, p our expected gains are larger when ρ^{−N}E_{p}(f_{N}(s_{N})) is larger and when ρ^{−N}E_{p_{∗}}(f_{N}(s_{N})) is smaller, we define an option quality as follows.

Definition 4. Let P (ξ_{n} = u) = p, n ≥ 1, while p_{∗} = \frac{ρ−d}{u−d}, and f_{N} = f_{N}(s_{N}) be the pay-off function of a given option. The functional

Q(f_{N}) = E_{p}(f_{N}(s_{N})) − E_{p_{∗}}(f_{N}(s_{N})) = E_{p}(f_{N}(s_{N})) − Cρ^{N}

is called a quality of the option. The larger it is, the better the option is.

It is easy to see that the functional Q(f_{N}) is linear.

Intuitively it is clear, that the function f_{N} should be nondecreasing and nonnegative. And due to (9) and (10) (see on the next page) it can be shown that for such functions Q(f_{N}) ≥ 0 if p > p^{∗} (Exercise 9).

Further on, we confine ourselves with the case p > p_{∗} and drop the index N from the pay-off function f_{N} for simplicity.

Optimal European Call Option

Consider a class of European call options when f is from

F = {f(s) = (s − K)^{+}, 0 ≤ s ≤ s_{0}u^{N}, 0 < K < s_{0}u^{N}}.

We say that the option with the pay-off function f^{∗} ∈ F is optimal if it maximizes Q(f) on F.

Denote z_{j} = s_{0}u^{j}d^{N−j} and

a_{j} = (\begin{matrix}N \\ j \end{matrix}) p^{j}(1 − p)^{N−j} − (\begin{matrix}N \\ j \end{matrix}) p_{∗}^{j}(1 − p_{∗})^{N−j}, j = 0, 1, …, N.

Then

Q(f) = N ; ∑ ; j=0 a_{j}f(z_{j})  (9)

and it is easy to see that

N ; ∑ ; j=0 a_{j} = 0.  (10)

Let us compare the quality of two European call options with the pay-off functions

f_{1}(s_{N}) = (s_{N} − K_{1})^{+}  and  f_{2}(s_{N}) = (s_{N} − K_{2})^{+} .

If K_{1}, K_{2} ∈ [0, z_{0}] then Q(f_{1}) = Q(f_{2}). Indeed, in this case

f_{1}(z_{j}) = z_{j} − K_{1}, f_{2}(z_{j}) = z_{j} − K_{2}, j = 0, 1, …, N.

Therefore, due to (10)

Q(f_{1}) = N ; ∑ ; j=0 a_{j}(z_{j} − K_{1}) = N ; ∑ ; j=0 a_{j}z_{j} = N ; ∑ ; j=0 a_{j}(z_{j} − K_{2}) = Q(f_{2}).

Optimal European Call Option (continuation)

Now we shall show that any option with the pay-off function

f(s_{N}) = (s_{N} − K)^{+}, K > z_{0}

is worse than that with the pay-off function

g(s_{N}) = (s_{N} − z_{0})^{+} = s_{N} − z_{0}.

If p > p_{∗} then a_{j}, j = 0, 1, …, N, are negative first and then become to be positive. Let

j_{0} = min{0 ≤ j ≤ N : a_{j_{0}} > 0}.

The values f(z_{j}) = (z_{j} − K)^{+}, j = 0, 1, …, N are equals to zero first and then become to be positive. Let

k_{0} = min{0 ≤ j ≤ N : z_{k_{0}} > K}.

Let k_{0} ≤ j_{0} that implies K < z_{j_{0}}. We have

N ; ∑ ; j=j_{0} a_{j}(z_{j}−K)^{+} = N ; ∑ ; j=j_{0} a_{j}(z_{j}−K) = N ; ∑ ; j=j_{0} a_{j}(z_{j}−z_{0})−(K−z_{0}) N ; ∑ ; j=j_{0} a_{j}. (11)

On the other hand,

(z_{j} − K)^{+} ≥ (z_{j} − z_{0}) − (K − z_{0}),  j = 0, 1, …, N  ⟹
j_{0}−1 ; ∑ ; j=0 a_{j}(z_{j} − K)^{+} ≤ j_{0}−1 ; ∑ ; j=0 a_{j}(z_{j} − z_{0}) − (K − z_{0}) j_{0}−1 ; ∑ ; j=0 a_{j}.  (12)

From (10)–(12) it follows that

Q(f) = N ; ∑ ; j=0 a_{j}(z_{j} − K)^{+} ≤ N ; ∑ ; j=0 a_{j}(z_{j} − z_{0}) = Q(g).

In particular,

Q(g) ≥ Q(f̃),  where f̃(s_{N}) = (s_{N} − z_{j_{0}−1})^{+},  (13)

since here k_{0} = j_{0}.

Optimal European Call Option (continuation)

Let k_{0} > j_{0} that implies K ≥ z_{j_{0}}. In this case,

j_{0}−1 ; ∑ ; j=0 a_{j}(z_{j} − z_{j_{0}−1})^{+} = j_{0}−1 ; ∑ ; j=0 a_{j}(z_{j} − K)^{+} = 0,  (14)

N ; ∑ ; j=j_{0} a_{j}(z_{j} − z_{j_{0}−1})^{+} > N ; ∑ ; j=j_{0} a_{j}(z_{j} − K)^{+}.  (15)

Relations (14) and (15) yield

Q(f̃) > Q(f).  (16)

From (13) and (16) it follows that

Q(g) > Q(f).

Summary. If f(s_{N}) = (s_{N} − K)^{+} then Q(f) is a nonincreasing function of K > 0 (Exercise 11). It takes the same maximal value for all K ≤ z_{0}.

Hence, an optimal European call option is any option with the pay-off function having K ≤ z_{0}. But in practice, it is senseless to have K ≤ z_{0}. The reasonable values of K are z_{0} < K < z_{N}. For such K there is no optimal European call option. It is because the class F is rather poor. Although the smaller K, the better European option is. Further on, we shall consider a reacher class of European options.

Optimal Option for a Special Class of Options

Consider a special class of European options with the pay-off functions f_{N} = f(s_{N}), where f is from

F_{0} = \{f(s) = \frac{s_{0}u^{N}}{s_{0}u^{N} − K} (s − K)^{+} , 0 ≤ s ≤ s_{0}u^{N}, 0 < K < s_{0}u^{N} \} .

In other words,

f(s_{N}) = f_{K}(s_{N}) = \frac{z_{N}}{z_{N} − K} (s_{N} − K)^{+} , 0 < K < z_{N}.

Clearly,

Q(f_{K}) = \frac{z_{N}}{z_{N} − K} N ; ∑ ; j=k_{0} a_{j}(z_{j} − K)

and

\frac{∂Q(f_{K})}{∂K} = \frac{z_{N}}{(z_{N} − K)^{2}} N ; ∑ ; j=k_{0} a_{j}[z_{j} − K − z_{N} + K]

= −\frac{z_{N}}{(z_{N} − K)^{2}} N ; ∑ ; j=k_{0} a_{j}(z_{N} − z_{j}).

Note that the derivative \frac{∂Q(f_{K})}{∂K} is not a continuous function of K ∈ (0, z_{N}). It is continuous if K ∈ (z_{i−1}, z_{i}), i = 0, 1, …, N (z_{−1} = 0). The points of its discontinuity are z_{i}, i = 0, 1, …, N − 1. Moreover, the derivative keeps the same sign within each of the intervals (z_{i−1}, z_{i}), i = 0, 1, …, N. This sign is opposite to that of the expression

N ; ∑ ; j=i a_{j}(z_{N} − z_{j}).  (17)

Optimal Option for a Special Class of Options (continuation)

It is not difficult to understand that

N ; ∑ ; j=0 a_{j}(z_{N} − z_{j}) < 0 ,  N ; ∑ ; j=j_{0} a_{j}(z_{N} − z_{j}) ≥ 0

and the expression (17) increases as a function of i.

Define

j_{∗} = min {1 ≤ i ≤ N : N ; ∑ ; j=i a_{j}(z_{N} − z_{j}) ≥ 0} .

If (17) does not equal to zero for any i then the derivative \frac{∂Q(f_{K})}{∂K} is positive within the interval (0, z_{j_{∗}}) and negative within the interval (z_{j_{∗}}, z_{N}). In such a case the maximum of Q(f_{K}) is attained at z_{j_{∗}}.

If (17) equals to zero for some i then i = j_{∗} and the derivative \frac{∂Q(f_{K})}{∂K} equals to zero within the interval [z_{j_{∗}−1}, z_{j_{∗}}) and the maximum of Q(f_{K}) is attained at any point from [z_{j_{∗}−1}, z_{j_{∗}}).

Optimal Option for a Wide Class of Options

Consider a wide class of European options with the pay-off functions f_{N} = f(s_{N}), where f is from

F_{1} = {f : [0, z_{N}] ↦ [0, z_{N}], f(x) ≤ x, f(0) = 0,

f is continuous, convex and nondecreasing}.

Clearly, F_{0} ⊂ F_{1} and F ⊂ F_{1}.

Our goal is to find an optimal option within F_{1} in the sense of maximization Q(f).

It turns out that the optimal option within F_{1} coincides with that within F_{0}. It is due to the following fact :

∀f ∈ F_{1} ∃f_{K} ∈ F_{0} : Q(f_{K}) ≥ Q(f).

Let f ∈ F_{1} be given. It is always possible to choose f_{K} ∈ F_{0} such that

f_{K}(z_{j_{0}}) = f(z_{j_{0}}).

Moreover, there exists only one such a function, namely with

K = \frac{z_{N}(z_{j_{0}} − f(z_{j_{0}}))}{z_{N} − f(z_{j_{0}})}.

Note that for such a choice,

f_{K}(z) ≥ f(z) ∀z ≥ z_{j_{0}},  (18)
f_{K}(z) ≤ f(z) ∀z < z_{j_{0}}.  (19)

Optimal Option for a Wide Class of Options (continuation)

Indeed, assume for a moment that (18) does not hold, that is

∃ z^{∗} > z_{j_{0}} : f_{K}(z^{∗}) < f(z^{∗}).

Since f(z_{N}) ≤ f_{K}(z_{N}),

∃ z^{∗∗} ∈ (z^{∗}, z_{N}] : f_{K}(z^{∗∗}) = f(z^{∗∗}).

We have z_{j_{0}} < z^{∗} < z^{∗∗} and hence

∃ γ > 0 : z^{∗} = γz_{j_{0}} + (1 − γ)z^{∗∗}.

Since f_{K} is linear within the interval [z_{j_{0}}, z^{∗∗}], we also have

γf_{K}(z_{j_{0}}) + (1 − γ)f_{K}(z^{∗∗}) = f_{K}(z^{∗}).

Due to f_{K}(z_{j_{0}}) = f(z_{j_{0}}) and f_{K}(z^{∗∗}) = f(z^{∗∗}),

γf(z_{j_{0}}) + (1 − γ)f(z^{∗∗}) = f_{K}(z^{∗}) < f(z^{∗})

because of the assumption. But in such a case, the function f is not convex and we obtain a contradiction.

In a similar way it can be proved that also (19) holds.

From (18) and (19) it follows that

N ; ∑ ; j=j_{0} a_{j}f_{K}(z_{j}) ≥ N ; ∑ ; j=j_{0} a_{j}f(z_{j}),

j_{0}−1 ; ∑ ; j=0 a_{j}f_{K}(z_{j}) ≥ j_{0}−1 ; ∑ ; j=0 a_{j}f(z_{j})

and, as a consequence,

Q(f_{K}) = N ; ∑ ; j=0 a_{j}f_{K}(z_{j}) ≥ N ; ∑ ; j=0 a_{j}f(z_{j}) = Q(f).

Questions and Exercises after Lecture 4

1. Describe all possible trajectories for the bond price evolution and the stock price evolution for N = 3, b_{0} = 50, ρ = 1.1, s_{0} = 100, u = 1.3, d = 0.8.

2. Describe all possible losses and gaines for the example from page 5 in the case of

(a) collar option with K_{1} = 65, K_{2} = 71;

(b) Boston option with K_{1} = 60, K_{2} = 65;

(c) look back call option with N = 1;

(d) Asian arithmetical call option.

3. Describe all possible losses and gaines for the case of buying European call option for 10 FIM when K = 80 FIM, s_{0} = 75 FIM, s_{N} = 79 FIM, 85 FIM, 95 FIM.

4. Give a formula for the fair price of European option with the pay-off function f_{N} = f_{N}(s_{0}, s_{1}, …, s_{N}).

5. Calculate the fair price, minimal hedge and optimal capital evolution for European call option taking the conditions of the example from page 9 when

(a) N = 2;

(b) K = 160.

6. Calculate the fair price, minimal hedge and optimal capital evolution for European call option when b_{0} = 100, ρ = 1.1, s_{0} = 50, u = 1.2, d = 0.8, K = 55, N = 1, 2.

7. Does the real capital evolution in the case when (ξ_{n}) can take any value from the interval [d, u] differ from the calculated optimal one given by (6) ? If yes then why and what can you propose for increasing the gains ?

8. Calculate the fair price, minimal hedge and optimal capital evolution for American call option taking the conditions of Exercise 5a with f_{n} = (s_{n} − K)^{+}, n = 1, 2.

9. Show that if p > p_{∗} then

Q(f_{N}) = E_{p}(f_{N}(s_{N})) − E_{p_{∗}}(f_{N}(s_{N})) ≥ 0

for any nondecreasing and nonnegative function f.

10. Calculate the values a_{j}, z_{j}, j = 0, 1, …, N and Q(f_{N}) for the example from page 9 taking p = 0.75 and N = 2, 3.

11. Complete the proof of the following statement : if f(s_{N}) = (s_{N} − K)^{+} then Q(f) is a nonincreasing function of K > 0.

12. Find an optimal European option with the pay-off function f_{N} = f(s_{N}), where f is from the class F_{0} when b_{0} = 1, ρ = 1, s_{0} = 150, u = 1.2, d = 0.6, p = 0.75, N = 2, 3. Draw a graph of the derivative \frac{∂Q(f_{K})}{∂K}.

13. Find an optimal European option with the pay-off function f_{N} = f(s_{N}), where f is from the class F_{1} when b_{0} = 100, ρ = 1.1, s_{0} = 50, u = 1.2, d = 0.8, p = 0.8, N = 2, 3.

Appendix : Optimal Stock Portfolio

Consider a financial stock market and suppose the prices of stocks to be changed at discrete time moments n = 1, 2, ….

Denote by ξ_{i} a relevant return for a stock of the i-th company in a given period of time. More exactly, ξ_{i} is a ratio of an absolute return and the price of a stock at the starting moment. If we speak about a future period of time then, of course, the values {ξ_{i}} are unknown.

Assume that at the starting moment we possess a capital X and decide to share it between stocks of m companies in the following manner : the capital X_{i} ≥ 0 is for the stocks of the i-th company, i = 1, 2, …, m. Of course, X = X_{1} + … + X_{m}.

Denote x_{i} = \frac{X_{i}}{X} ≥ 0, i = 1, 2, …, m. Then x_{i} is a part of our capital which is invested into the stocks of the i-th company, x_{1} + … + x_{m} = 1. The collection of those numbers {x_{1}, x_{2}, …, x_{m}} forms our portfolio.

The return for the whole portfolio in a given period of time equals to

ξ = m ; ∑ ; i=1 X_{i}ξ_{i} = X m ; ∑ ; i=1 x_{i}ξ_{i}.

Since (ξ_{i}), and as a consequence ξ, are random, it is reasonable to characterize the portfolio with the expected return

E(ξ) = X m ; ∑ ; i=1 x_{i}E(ξ_{i}).

The portfolio is better if the expected return for it is higher.

A risk is another important characteristic of the portfolio. It is reasonable to propose the variance of the return V ar(ξ) as a measure of the risk of the portfolio. Clearly,

V ar(ξ) = X^{2}V ar ( m ; ∑ ; i=1 x_{i}ξ_{i}) = m ; ∑ ; i,j=1 x_{i}x_{j}Cov(ξ_{i}, ξ_{j}).

The portfolio is better if V ar(ξ) is smaller.

Suppose that we are able to estimate somehow the expected relevant returns E(ξ_{i}), i = 1, …, m for the stocks in a given period of time. Denote those estimators, correspondingly, by a_{1}, a_{2}, …, a_{m}. Then the estimated expected return for the whole portfolio in a given period of time equals to

X m ; ∑ ; i=1 a_{i}x_{i}.

It is reasonable to choose such a portfolio for which the estimated expected return is maximal, that is

f(x) = m ; ∑ ; i=1 a_{i}x_{i}  ⟶  max ; x_{i}≥0,i=1,...,m,∑_{i} x_{i}=1 .  (20)

Suppose that we are also able to estimate the covariance of relevant returns. Denote by A_{ij} the estimator for Cov(ξ_{i}, ξ_{j}), i, j = 1, …, m. Then the estimated variance of the return for the whole portfolio in a given period of time equals to

m ; ∑ ; i,j=1 A_{ij}x_{i}x_{j}.

It is reasonable to choose such a portfolio for which the estimated variance of the return is minimal, that is

f(x) = m ; ∑ ; i,j=1 A_{ij}x_{i}x_{j}  ⟶  min ; x_{i}≥0,i=1,...,m,∑_{i} x_{i}=1 .  (21)

We can also propose another criteria for choosing the portfolio basing on combination both the expected return component and the risk component. Let b ≥ 0 be given. We can try to balance somehow those two components proposing to choose a portfolio according to the solution of the following minimization problem :

f(x) = m ; ∑ ; i,j=1 A_{ij}x_{i}x_{j} − b m ; ∑ ; i=1 a_{i}x_{i}  ⟶  min ; x_{i}≥0,i=1,...,m,∑_{i} x_{i}=1 . (22)

Of course, if b = 0 then (22) is the same as (21), while if b is sufficiently large, then the portfolio chosen in accordance with (22) is the same as that with (20).

At last one can propose to combine both the expected return component and the risk component in another way. Namely, let us choose a portfolio according to the solution of the next minimization problem :

f(x) = m ; ∑ ; i,j=1 A_{ij}x_{i}x_{j}  ⟶  min ; x_{i}≥0,i=1,...,m,∑_{i} x_{i}=1,∑_{i} a_{i}x_{i}=a  (23)

for given a ∈ (min a_{i}, max a_{i}). Here we propose to choose a portfolio having the minimal estimated risk among all those having the same level a of the estimated expected return.

Of course, the problems (20)–(23) are typical optimization problems. For their solving the methods considered in Optimization Unit can be used. More exactly, we should apply here the methods for solution of optimization problems with equality and inequality constraints. As an example, recall a necessary condition and a sufficient condition for an extremum adopted to the problems (20)–(22).

Necessary condition for an extremum. If x^{∗} is a local minimum (maximum) point for f then there exist λ^{∗} and μ_{1}^{∗}, …, μ_{n}^{∗} such that

\frac{∂f(x^{∗})}{∂x_{i}} + λ^{∗} − μ_{i}^{∗} = 0,  i = 1, …, n,  (24)
μ_{i}^{∗} ≥ 0 (≤ 0), μ_{i}^{∗}x_{i}^{∗} = 0, x_{i}^{∗} ≥ 0, i = 1, …, n, n ; ∑ ; i=1 x_{i} = 1. (25)

Sufficient condition for an extremum. Let x^{∗} is such that (24) and (25) hold. If for any z ≠ 0 satisfying simultaneously to each of the conditions

n ; ∑ ; i=1 z_{i} = 0,

z_{i} = 0, if x_{i}^{∗} = 0 and μ_{i}^{∗} > 0 (< 0),
z_{i} ≤ 0, if x_{i}^{∗} = 0 and μ_{i}^{∗} = 0,

the inequality

z′Hess f(x^{∗})z > 0 (< 0)

holds then x^{∗} is a local minimum (maximum) point for f.

Of course, the maximization problem (20) is quite simple and can be solved without any optimization techniques. Indeed, if i_{∗} is such index that a_{i_{∗}} = max a_{i} then the optimal solution is : x_{i_{∗}}^{∗} = 1 and x_{i}^{∗} = 0 for all other i ≠ i_{∗}.

In general, other problems are not so simple and at least for large n one should use numerical methods for their solution, which, unfortunately, are out of the scope of the course.

The estimators for the expected relevant returns of the stocks as well as for the covariances of relevant returns can be obtained, for example, on the basis of the past stock quotations.