Advanced Derivatives Course Chapter 3

Week 3: First Steps Towards Continuous Processes (see also P.Wilmott, Chapter 12)

Lecture III.1 One-Step Model The goal of the next two lectures is to prepare ground for introducing conceptions of continuous time. Previously we have derived some important inequalities, which have to hold to prevent any risk-free profits. These relationships do not refer to any methods of valuing options. Given option values we can check the relations, however these formulas do not explain how these values are obtained. This is, of course, the issue of the outmost importance. There are different methods, which allow one to value the options. In what follows we shall get familiar with some very effective techniques based on the ideas of finance in continuous time.

All current methods of valuing derivatives utilize the notion of arbitrage. Option prices are obtained from conditions that preclude arbitrage opportunities. In its simplest form, arbitrage means taking simultaneous positions in different assets so that one guarantees a risk-less profit higher than the risk -less return given by, say, a government bond or a deposit in a sound bank. If such profits exist, we say that there is an arbitrage opportunity. We use this concept to obtain a practical definition of a ‘fair price’ for a financial derivative. We say that the price of a security is at a ‘fair’ level, or that the security is correctly priced, if there are no arbitrage opportunities. Such arbitrage-free option prices will be utilized as benchmarks. The mathematical environment provided by the no-arbitrage theorem is the major tool used to calculate such benchmark prices. It is instructive to start with a method, which is discrete but which can help to understand the transition to continuous approaches. By discrete we mean that any asset’s price movements can occur only in discrete finite time steps. This method is known as a binomial tree. In my lectures this popular and useful technique is presented as an intermediate step toward finance in continuous time. Continuous price movements will be introduced as the limit when time steps get shorter and shorter. As you may already know, a binominal tree is a tree that represents possible paths that might be followed by the underlying asset’s price over the life of the derivative. We start by considering a very simple situation where during the life of the derivative (as an example we consider an European call on the asset without dividends) the share price can either move up from S to a new level u ⋅ S or down from S to a new level d ⋅ S (u>1; d<1). Usually, the coefficients are chosen so that u· d = 1. This condition gives rise to recombined trees. The proportional increase in the share price when there is an up movement is u-1; the proportional decrease when there is a down movement is 1-d. If the share price moves up to u ⋅ S , we suppose that the payoff from the derivative is c+; if the share price moves down to d ⋅ S , we suppose the payoff from the derivative is c-. The situation is illustrated in Fig.3.

u⋅S

c+

S c d⋅S

c-

Fig.3 We imagine a portfolio consisting of a short position in Ä shares and a long position in one derivative. We calculate the value of Ä that makes the portfolio risk-less. If there is an up movement in the share price, the value of the portfolio at the end of the life of the derivative is

c + − u ⋅ S∆. If there is a down movement in the share price, this becomes

c − − d ⋅ S∆. The portfolio is risk-less when two are equal

c + − u ⋅ S∆ = c − − d ⋅ S∆ . The last equation allows us to find Ä. Namely,

c+ − c− ∆= . Equation 1 u⋅ S − d ⋅S In this case the portfolio is risk-less and must earn the risk-free rate. Eq.3 shows that Ä is the ratio of the change in the derivative price to the change in the share price as we move between the nodes at time T. The present value of the portfolio must be

(c

+

− u ⋅ S∆ )e − rT

(we assumed that the current time is zero, i.e. t=0). The cost of setting up the portfolio at t=0 is

c − S ⋅ ∆. It follows that

c − S ⋅ ∆ = (c + − u ⋅ S ⋅ ∆ )e − rT .

Substituting from eq.3 for Ä and simplifying, this equation reduces to

c = e − rT [ pc + + (1 − p)c − ], Equation 2 where

p=

erT − d . Equation 3 u−d

Eqs. (4) and (5) enable a derivative to be priced using a one-step binomial model. A similar formula can be derived for a European Put. Risk-Neutral Valuation The derivative pricing formula in eq. (4) does not involve the probabilities of the share price moving up or down. This is surprising and seems counterintuitive. It is natural to assume that as the probability of an upward movement in the share price increases, the value of a call option on the share increases and the value of a put option on the share decreases. This is not the case. The key reason for this is that we are not valuing the option in absolute terms. We are calculating its value in terms of the price of the underlying share. The probabilities of future up and down movements are already incorporated into the price of the share. It turns out that we do not need to take them into account again when valuing the option in terms of the share price. Although we do not need to make any assumptions about the probabilities of up and down movements to derive eq. (4), it is natural to interpret the variable p in eq. (4) as the probability of an up movement in the share price1. The variable 1-p is then the probability of a down movement and the expression

pc + + (1 − p)c − is the expected payoff from the derivative. With this interpretation of p, eq. (4) than states that the value of the derivative today is its expected future value discounted at the risk-free rate.

1

This probability is called martingale probability.

We now investigate the expected return from the share when the probability of an up movement is assumed to be p. According to the definition, the expected share price at time T, E(ST), is given by

E (ST ) = pSu + (1 − p) Sd

or

E( ST ) = pS (u − d ) + Sd.

Substituting from eq. (4) for p, this reduces to

E( ST ) = SerT Equation 4 showing that the share price grows on average at the risk-free rate. Setting the probability of the up movement equal to p is therefore equivalent to assuming that the return on the share equals the risk-free rate. You can compare eq.(6) with the forward (and futures) price obtained early. Now we can say that the forward (and futures) price is the expected value of the share price averaged with respect to the martingale probability. We will refer to a world where everyone is risk-neutral as a risk-neutral world. In such a world investors require no compensation for risk, and the expected return on all securities is the risk-free rate. Eq. (6) shows that we are assuming a risk-neutral world when we set the probability of an up movement to p. Eq. (4) shows that the value of the derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate. This result is an example of an important general principle in option pricing known as riskneutral valuation. This states that we can with complete impunity assume that the world is risk neutral when pricing options and other derivatives. The prices we get are correct not just in a risk-neutral world but in other worlds (in particular, in our world) as well.

Lecture III.2 American Options Up to now the options we have considered have been European. We now move on to consider how American options can be valued using a binomial tree. The procedure is to work back through the tree from the end to the beginning, testing at each node to see whether early exercise is optimal. The value of the option at the final nodes is the same as for the European option. At earlier nodes the value of the option is the greater of 1. The ‘European’ value given by equation

V = e −r (T −t) [ pV + + (1 − p)V − ].Equation 5 2. The ‘American’ payoff from early exercise. Since it is never optimal to early exercise an American call, we consider a two year American put with a strike price of €52 on a share whose current price is €50. We suppose that there are two time steps2 of one year and in each time step the share price either moves up by a proportional amount of 20% or down by a proportional amount of 20%. We also suppose that the risk-free rate is 5%. The situation is illustrated on Fig.4. At node B, eq. (7) gives the value of 1.4147, while the payoff from early exercise is negative (=-8). Clearly, early exercise is not optimal at node B and the value of the option at this node is €1.4147. At node C, eq. (7) gives 9.4636, while the payoff from early exercise is 12.0. In this case early exercise is optimal and the value of the option is €12.0. At the initial node A the value given by eq. (7) is

e −0.05×1 (0.6282 ⋅1.4147 + 0.3718 ⋅ 12) = 5.0894, while the payoff from early exercise is 2. In this case early exercise is not optimal. The value of the option is, therefore, €5.0894. 3

2

Since a one-step model is trivial for an American put, we consider a two-step model. If we considered a strategy in which we didn’t exercise the option at node C and proceeded to node D or E, we would have found a wrong price for the American put. The option would be under priced, so the issuer would be loosing money. 3

72 D 0 60 B 1.4147

48

50

E

A

4 40 C 12.0

32 F

Fig.4

20

Delta What happens to our portfolio during the lifetime of an option? Our portfolio is

= P − ∆S . In order to maintain a risk-less hedge using an option and the underlying share, we need to adjust our holdings in the share periodically. This is achieved via tuning the parameter Ä which is the number of units of the share we should short for each long option to create a risk-less hedge. The construction of a risk-less hedge is sometimes referred to as delta hedging. The delta of a call option is positive (because C+>C-), whereas the delta of a put option is negative (P+
P+ − P− ∆= + . S − S− the delta is the ratio of the change in the price of a share option to the change in the price of the underlying share. For a risk-less position an investor should buy Ä shares for each option sold. An inspection of a typical binomial tree shows that delta is liable to change during the life of an option. This means that risk-less positions do not automatically remain risk-less. They must be adjusted periodically. For example for the American put option which we just considered, the delta corresponding to share price movements over the first time step is

∆1st =

1.4147 − 12 = −0.5293. 60 − 40

Over the second time step we have to adjust Ä only for an upward movement, since a downward movement does not take place due to an early exercise. So we get

∆ 2 nd (+ ) =

0−4 = −0.1667. 72 − 48

We will return to delta hedging in later lectures. Try To Answer The Following Questions 1) 2) 3) 4)

What are the assumptions of a binomial tree model? What is the purpose of Ä hedging? How does it work? What is the idea of risk-neutral valuation? How does the back-through-the-tree pricing procedure work for American options?