CONTINUOUS TIME ANALYSIS

4 Principles of Option Pricing

REAL WORLD RISK-NEUTRAL WORLD

1. We start with a knowledge of the true prob- abilities (0.7 and 0.3 in our example). Al- ternatively, if we only know the expected growth rate we use equation (4.1):

(1+µ)S0 =0.7Shigh+0.3Slow

2. The probabilities of achieving S_high and Sloware just the same as achieving fhighand flow. The true expected value of f1 monthis E[f1 month]real world=0.7fhigh+0.3flow

3. The present expected value of the deriva- tive is given by discounting the future ex- pected value byλ, the expected growth rate of the derivative:

f0= 1

(1+λ)E[f1 month]real world

4. Unfortunately, neitherµnorλare known in most circumstances so this method is useless.

Calculate the pseudo-probabilities from equa- tion (4.4) :

(1+r)S₀= pS_high+(1−p)S_low

Pretend that the probabilities of achiev- ing S_high and S_low (and therefore also f_high and f_low) are the pseudo-probabilities. The pseudo-expectation is then

E [f_{1 month}]_pseudo=p f_high+(1−p)f_low Equation (4.3) shows that f₀ is just E[f_{1 month}]_pseudodiscounted back at the inter- est rate:

f₀= 1

(1+r)E[f_{1 month}]_pseudo This allows us to obtain f0 entirely from ob- servable quantities.

Astonishingly, we have suddenly found a way of calculating f₀ in terms of known or observable quantities, yet only a page or two back, it looked as though the problem was insoluble since we had no way of calculating the returnsµandλ. The log-jam was broken by an arbitrage argument which hypothesized that an option could be hedged by a certain quantity of underlying stock. The principle is exactly the same as for a forward contract, explained in Section 1.3. Remember, this approach can only be used if the underlying commodity can be stored, otherwise the hedge cannot be set up: equities, foreign exchange and gold work fine, but tomatoes and electricity need a different approach; this book deals only with the former category.

4.2 CONTINUOUS TIME ANALYSIS

The value of the portfolio at timetmay be written ft−St . The increase in value of this portfolio over a small time intervalδt, during whichStchanges byδSt, may be written

δft−St −Stq δt

The first two terms are obvious while the last term is just the amount of dividend which we must pay to the stock lender from whom we have borrowed stock in the time intervalδt, assuming a continuous dividend proportional to the stock price.

The quantity is chosen so that the short stock position exactly hedges the derivative over a small time intervalδt; this is the same as saying that the outcome of the portfolio is certain.

The arbitrage arguments again lead us to the conclusion that the return of this portfolio must equal the interest rate:

δft−δSt −Stq δt ft−St

=rδt or

δft−δSt +(r−q)St δt =r ftδt (4.5) These equations are the exact analogue of equations (4.2) for the simple high–low model of the last section.

(ii) As they stand, equations (4.5) are not particularly useful. However, it is assumed thatStfollows a Wiener process so that small movements are described by the equation

δSt

St =(µ−q)δt+σδWt

We can now invoke Ito’s lemma in the form of equation (3.12) and substitute forδftandδSt

into the first of equations (4.5) to give ∂f1

∂t +(µ−q)St

∂ft

∂St

+1

2σ²S_t²∂²ft

∂St²

δt+σSt

∂ft

∂St

δWt

−St[(µ−q)δt+σδWt] −Stq δt =(ft−St )rδt (4.6) Recall that the left-hand side of this equation is the amount by which the portfolio increases in value in an intervalδt; but by definition, this amount cannot be uncertain in any way because the derivative is hedged by the stock. Therefore it cannot be a function of the stochastic variable δWt, which means that the coefficient of this factor must be equal to zero. This gives

∂ft

∂St

= (4.7)

We return to an examination of the exact significance of this in subsection (vi) below.

(iii)Black Scholes Differential Equation:Setting the coefficient ofδWt to zero in equation (4.6) leaves us with the most important equation of option theory, known as the Black Scholes equation:

∂ft

∂t +(r−q)St

∂ft

∂St

+1

2σ²S_t²∂²ft

∂St²

=r ft (4.8)

39

4 Principles of Option Pricing

Any derivative for which a neutral hedge can be constructed is governed by this equation;

and all formulas for the prices of derivatives are solutions of this equation, with boundary conditions depending on the specific type of derivative being considered. The immediately remarkable feature about this equation is the absence ofµ, the expected return on the stock, and indeed the expected return on the derivative itself. This is of course the continuous time equivalent of the risk-neutrality result that was described in Section 4.1(iv).

When the Black Scholes equation is used for calculating option prices, it is normally pre- sented in a more directly usable form. Generally we want to derive a formula for the price of an option at timet =0, where the option matures at timet =T. Using the conventions of Section 1.1(v), we write∂f₀/∂t⇒ −∂f₀/∂T so that the Black Scholes equation becomes

∂ft

∂T =(r−q)St

∂ft

∂St

+1

2σ²S_t²∂²ft

∂S_t² −r ft (4.9)

(iv)Differentiability:For what is a cornerstone of option theory, the Black Scholes differential equation has been derived in a rather minimalist way, so we will go back and examine some issues in greater detail. First, we need to look at some of the mathematical conditions that must be met.

It is clear from any graph of stock price against time thatStis not a smoothly varying function of time. It is really not the type of function that can be differentiated with respect to time. So just how valid is the analysis leading up to the derivation of the Black Scholes equation? This is really not a simple issue and is given thorough treatment in Part 4 of the book; but for the moment we content ourselves with the following commonsense observations:

r

_Stand the derivative price fSttare both stochastic variables. In this subsection we explicitly show the dependence of fSttonStfor emphasis.

r

_{BothStandfStt}are much too jagged for dSt/dtor for dfStt/dtto have any meaning at all, i.e.

in the infinitesimal time interval dt, the movements ofδStandδSS_ttare quite unpredictable.

r

However, partial derivatives are another matter. If you know the time to maturity and the underlying stock price, there is a unique value for a given partial derivative. These values might be determined either by working out a formula or by devising a calculation procedure;

but you will be able to plot a unique smooth curve of fS_ttvs.St for a given constantt, and also a unique curve for fS_ttvs.tfor a given constantSt.

r

The derivation of the Black Scholes equation ultimately depends on Ito’s lemma which in turn depends on a Taylor expansion of fS_tt to first order in tand second order in St. Underlying this is the assumption that the curves for fS_tt againsttandStare at least once differentiable with respect totand twice differentiable with respect toSt.

r

A partial derivative is a derivative taken while holding all other variables constant. dfStt/dt and∂fStt/∂tmean quite different things. Consider the following standard result of differential calculus:

dfS_tt

dt ≡∂fS_tt

∂t +∂fS_tt

∂St

∂St

∂t

We have already seen that the first two partial derivatives on the right-hand side of this identity are well defined. However∂St/∂t is just a measure of the rate at which the stock price changes with time, which is random and undefined; thus dfS_tt/dtis also undefined.

r

In pragmatic terms, this is summed up as follows: we know that the stock price jumps around in a random way and therefore cannot be differentiated with respect to time; the same is

4.2 CONTINUOUS TIME ANALYSIS

true of the derivative price. However, the derivative price is a well-defined function of the underlying stock price and can therefore be differentiated with respect to price a couple of times. The derivative price is also a well-defined function of the maturity, so that it can be partiallydifferentiated with respect to time, while holding the stock price constant.

(v) The concept of arbitrage is the foundation of option theory. It has been assumed that we can construct a little portfolio consisting of a derivative and a short position of S_tt units of stock, such that the short stock position exactly hedges the derivative for any small stock price movements; this is referred to as aninstantaneous hedge. The dependence of S_tton the stock price is explicitly expressed in the notation. This portfolio has a value which may be written

VS_tt= fS_tt− S_ttSt (4.10)

The fact that the short stock position hedges the derivative does not mean that the movement in one is equal but opposite to the movement in the other: it merely means that the move inVS_ttis independent of the size of the stock price moveδStover a small time intervalδt. The normal sign conventions are followed when interpreting the last equation, e.g. if fS_tt is negative, a short option position (option sold) is indicated; if S_ttis negative, so that− S_ttStis positive, a long position is taken in the stock.

At this point it needs to be made clear that there are alternative (but equivalent) conventions used in describing instantaneous hedging. The reader needs to be at home with the different ways of looking at the problem since the approaches in the literature are quite random, with authors sometimes switching around within a single article or chapter.

Hedging:In equation (4.10), VStt is the value of the portfolio and is therefore the amount of money paid out or received in setting up the portfolio; but we normally look at the set-up slightly differently. It is easier to keep tabs on values if it is assumed that we start any derivatives exercise with zero cash. If we need to spend cash on a portfolio, we obtain it by borrowing from a so-called cash account; alternatively, if the portfolio generates cash, we deposit this in the same cash account. Equation (4.10) may then be written

BStt+ fStt− SttSt =0 (4.11) whereBSttis the level of the cash account, negative for borrowings and positive for deposits.

Except where explicitly stated otherwise, it is assumed that interest rates are constant.

Replication:While option theory can be developed perfectly well with the above conventions, many students find it easier to picture the set-up slightly differently. Rewrite equation (4.10) as follows:

fStt = SttSt+BStt (4.12)

Instead of thinking in terms of hedging this can be interpreted as representing a replication.

We would say that within a very small time interval, a derivative whose price is fSttbehaves in the same way as a portfolio consisting of Sttunits of stock andBStt units of cash. In this approach it is again assumed that we start with zero wealth so that any cash needed has to be borrowed (indicated by negativeBStt) and any surplus cash is deposited (indicated by positive BStt). For example, SttSt is always positive for a call option and always larger than fStt, so BS_tt is always negative, indicating a cash borrowing. On the other hand, S_ttis negative for a put option, indicating that the replication strategy requires a short stock position and thatBS_tt

is positive, i.e. surplus cash is generated by the process.

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4 Principles of Option Pricing

Replication is perhaps more intuitive as an approach, but people with a trading background tend to be more comfortable with the hedging for obvious reasons. One point which sometimes causes puzzlement should be mentioned: equations (4.11) and (4.12) seem to express the same idea, so where does the sign change in the BS_tt term come from? The answer is that the two equations do not represent quite the same thing. In fact hedging an option might be best described as replicating a short option, rather than the option itself. It is completely straightforward to develop option theory using either approach, but the reader is warned that mistakes are likely to occur if it is not absolutely clear which method is being used at a given time.

To illustrate this alternative approach, we now recast the analysis leading up to the Black Scholes equation in terms of replication rather than hedging. The option can be replicated by a portfolio of stock and cash: ft = tSt+Bt, where once again we ease the notation by using the suffixtto indicate dependence on bothtandSt. In a small time interval, the change in value is given by

δfStt = SttδSt+ SttStqδt+BSttrδt

The middle term on the right-hand side is again the dividend throw-off, while the last term is just the interest earned or incurred on the cash account. Substituting forBt from equation (4.12) gives

δft = tδSt+ tStqδt+(ft− tSt)rδt (4.13) which is just equation (4.5). The rest of the argument is the same as before, leading directly to the Black Scholes differential equation.

(vi)Graphical Representation of Delta:In the derivation of the Black Scholes equation (4.7), an important aspect emerged and was quickly passed over. We now return to the equation

∂ft

∂St

=

In the spirit of the last subsection, we assume that we can obtain a formula for ftas a function ofSt; the curve of this function is shown in Figure 4.1. This illustrates the replication approach to studying options which was described in the last section.

is clearly the slope of the curve of fStt and the equation of the tangent to the curve is y= SttSt+BwhereBis some as yet to be defined point on they-axis. Over a very small rangeδSt, the properties of the curve (derivative) can be approximated by those of the tangent (replication portfolio). This is completely in line with the precepts of differential calculus.

ft

ft

St

St

t t

S −B

0

Bt

−

D

d d

Figure 4.1 Delta

4.2 CONTINUOUS TIME ANALYSIS

(vii)Risk Neutrality in Continuous Time:Let the expected return on an equity stock beµand the return on a derivative beλ. The Wiener process governing the stock price movements is

δSt

St

=(µ−q)δt+σd Wt

and by definition

E[δft] ft

=λδt

Using Ito’s lemma [equation (3.12)] forδftgives E ∂ft

∂t +(µ−q)St

∂ft

∂St

+1

2σ²S_t²∂²ft

∂St²

δt+σSt

∂ft

∂St

δWt

=λftδt Now useE[δWt]=0 for the only stochastic term in the last equation to give

∂ft

∂t +(µ−q)St

∂ft

∂St +1

2σ²S²_t∂²ft

∂St²

=λft

This is not of much use in pricing derivatives since we have no way of findingµorλ. However, suppose we now get onto our magic carpet and fly back to the fantasy world described in the last section, where investors are insensitive to risk and therefore accept a risk-free rate of return ron all investments including our derivative and its underlying stock. We would then be able to putµ=λ=rin the last equation to retrieve the Black Scholes equation which can be solved in terms of observable quantities.

This result is just the continuous time equivalent of the result which was obtained for the simple high/low model of Section 4.1. The no-arbitrage condition again leads us to the conclusion that an option price computed in the risk-neutral imaginary world would have the same value as an option price computed in the real world, if we happened to know the values ofµandλ. We can formally write this result as

fS₀0= e^−λtE[fS_tt]real world

fS₀0= e^{−r t}E[fS_tt]risk-neutral world

(viii)Approaches to Option Pricing:The main purpose of the preceding theory is to find a way of pricing options. Two approaches have emerged from this chapter: we derived the Black Scholes equation which applies to any derivative of a stock price. The option price can therefore be obtained by solving this equation subject to the appropriate boundary conditions. The main drawback of this approach is that the equation is very hard to solve analytically in most cases.

A later chapter will be dedicated to finding approximate numerical solutions to the equation.

In Section 3.1 the central limit theorem was used to derive a probability distribution function for the stock price in timeSt. This was a function ofµ, the stock’s rate of return. But in the last subsection it was shown that the option may be priced by first making the substitutionµ→r and deriving a pseudo-distribution forSt(i.e. the distributionSt would have ifµ were equal tor). From this pseudo-distribution and a knowledge of the payoff function of the option, a pseudo-expected terminal value can be calculated for the option; if this is discounted back at the risk-free rater, we get the true present fair value of the option.

On the face of it, this seems the simpler approach; it certainly is for simple options, but it will become apparent later in this course that the probability distribution can be very difficult

43

4 Principles of Option Pricing

to derive in more complex cases. In fact it is shown in Appendix A.4(i) that deriving a formula for the probability density function is mathematically equivalent to solving the Black Scholes equation.

Other powerful approaches to option pricing are developed later in this book, but for the moment we concentrate on these methods. They are applied to simple European put and call options in the next chapter, but for the moment we continue with the development of the general theory which will be applied throughout the rest of the book.