The mathematical tools necessary for these first three parts are prepackaged in the appendix, in a consistent form that can be used with minimal interruption to the flow of the text. We look at futures contracts in Section 1.3 and introduce one of the central mysteries of option theory: risk neutrality.
CONVENTIONS
In relation to today's share price S0, the future value of the share can then be written. Therefore, while maintaining the short position, we have to pay the dividend to the stock lender from his own funds.
ARBITRAGE
The value of the forward contract is zero [for a rationale for this see Section 1.3(iv)]. They are quoted in different currencies, but using the spot rate S0, which expresses today's equivalence, gives iii) If a portfolio has a certain outcome (is perfectly hedged) its return must be equal to the risk-free rate.
FORWARD CONTRACTS
In section 1.2, the strength of the arbitrage arguments was illustrated with a long example using forward exchange contracts. The reason is that the forward price specified in the original contract is no longer the forward price F0T without arbitrage.
FUTURES CONTRACTS be X, so that the no-arbitrage proposition (1.1) may be written
FUTURES CONTRACTS
The amount owed at the close of the day one contract is F0T−F1T, which would normally be payable at maturity (but which can be discounted and paid up front). Without going into mechanical details, it is worth noting that for some types of futures contracts, the last part of this sequence is the delivery of the commodity against the prevailing spot price (physical settlement); others simply settle the difference between the spot price and yesterday's futures price (cash settlement). ii) Futures Price: We now consider the price of a futures contract to buy a commodity at time T.
PAYOFFS
Looking back at the payout charts in the previous sections, these apply to European options on the option's expiration date. On the other hand, the payoff charts for American options could be obtained when the holder of the option decides to exercise.
OPTION PRICES BEFORE MATURITY
The profit diagrams of such short positions are shown in Figure 2.2 and are reflections of the long positions on the pivot axis. The dotted lines in the first two graphs look a lot like profit diagrams; but they are not the same.
AMERICAN OPTIONS
It is a condition that the present value of the dividend is greater than the interest earned on the cash that would be used to exercise the option. From Figure 2.5 for a non-dividend paying put, it can be seen that the short forward diagonal is to the left of the payoff diagonal.
PUT–CALL PARITY FOR AMERICAN OPTIONS
Value now Value att=T Value att=T. of stock atX Put long;. the value of the option portfolio is always equal to or less than the proceeds of selling the future stock, regardless of the value of ST. This time it is clear that the terminal values of the options portfolio are always greater than or equal to the income of the futures contract.
COMBINATIONS OF OPTIONS
Suppose we buy a call spread for C1−C2 and a put spread for P1−P2; the put-call parity equality of the last paragraph shows that it will cost (X1−X2) e−r T. For example, the owner of the commodity bears the first $5 of loss and the insurance covers any further loss.
COMBINATIONS BEFORE MATURITY
The relationship between gamma and theta will be rigorously analyzed later in the course; but it is comforting to know that one of the most important conclusions of option theory can be confirmed by a casual glance at profit diagrams.
STOCK PRICE MOVEMENTS
Furthermore, the central limit theorem states that whatever the distribution of yi may be, the distribution of ¯yN tends to a normal distribution asN. N(mδT N, σ2δT N)=N(mT, σ2T) We can reassure ourselves that this result holds simply by theoretically dividing the time period into an arbitrarily large number of arbitrarily small segments, such that N → ∞and the central limit theorem holds .
PROPERTIES OF STOCK PRICE DISTRIBUTION
We can therefore write mδT =mδT; identical reasoning gives σδ2T =σ2δT. The mean and variance of xT can now be written. The reader would be quite right to note that this should be called the volatility of the logarithm of the stock price; but the two are closely related and for most practical purposes the same.
INFINITESIMAL PRICE MOVEMENTS
ITO’S LEMMA
This is the most important chapter in the book and must be mastered if the reader wants a solid understanding of options theory. First, they allow concepts such as risk neutrality or pseudo-probabilities to be introduced in a relatively painless way; introducing such concepts for the first time in a more generalized or ongoing context is definitely more difficult for the reader – trust me.
SIMPLE EXAMPLE
There is no indication that the expected growth rate of the stock µ should be the same as the expected growth rate of the option λ. Compare equations (4.1) and (4.4): the first one illustrates the relationship between expected return and the probability that the stock price moves to Sighor Slow.
CONTINUOUS TIME ANALYSIS
So how valid is the analysis that leads to the derivation of the Black Scholes equation. In the derivation of the Black-Scholes equation (4.7), an important aspect emerged that was quickly overlooked.
DYNAMIC HEDGING
This equation can be stated alternatively: over a single hedged time period δt, the change in the value of the derivative and the hedge, net of funding costs, is zero. Or if we buy an option that is underpriced, we can generate the fair value of the option through a delta hedging procedure and thereby lock in the profit.
EXAMPLES OF DYNAMIC HEDGING
If we dynamically hedge this short option position, the hedging process (purchase or sale of shares and financing costs) will generate a cash sum equal to the fair price of the sold option. Since the portfolio is hedged, the sum of the last three columns must be zero throughout.
GREEKS
As just shown, this is due to the curvature of the curve of ft (the gamma term) and also to the fact that the entire curve shifts over time (the theta term). Rho:ρStt = ∂∂fSt tr This is a measure of the sensitivity of an option value to changes in interest rates.
INTRODUCTION
DERIVATION OF MODEL FROM EXPECTED VALUES
It is therefore sometimes stated that the factor N[d2] is the probability that ST >X, i.e. that the option is exercised. The true probability that ST >X is N[d2], but replaced by µ. iv) General Black Scholes formula for Put or Call: Remember the put-call parity relationship in section 2.2(i):.
SOLUTIONS OF THE BLACK SCHOLES EQUATION
When manipulating these formulas, we often need an option price at time . It is clear that this is obtained from equation (5.2) by simply substituting f0 → ft; S0 → St; F0T → Ft T; T →T −t.
GREEKS FOR THE BLACK SCHOLES MODEL
Replacing this with the explicit expression forn(d2) in (A) above gives n(d2)= 1. D) Differentiating the explicit expression ford1 with respect to S0 gives.
GREEKS FOR THE BLACK SCHOLES MODEL so that
ADAPTATION TO DIFFERENT MARKETS
Most of Black Scholes' assumptions in section 5.1 are quite realistic, apart from constant volatility. The theory is very simply based on the theory developed for shares: the shares simply become one unit of the foreign currency.
OPTIONS ON FORWARDS AND FUTURES
The reason can be found in equation (5.5): the cost of entering into a forward or futures contract is zero, and these instruments have no dividend yield. We found that the Black Scholes equation for the forward price option can be obtained from the general equation for the equity option.
BLACK SCHOLES EQUATION REVISITED
This is obviously not true, so PA=X−S0 is not a solution of the Black Scholes equation. iii). On the other hand, if the stock price increases to S0∗+δS0, the value of the portfolio increases with it.
BARONE-ADESI AND WHALEY APPROXIMATION
In each of these limits of the variables S0andT, the last term of equation (6.4) can be set to. The results are generally quite good and are consistent with the nature of the approximation made: when is large or small, subsection (iii) shows that the last term in equation (6.4).
PERPETUAL PUTS
AMERICAN OPTIONS ON FUTURES AND FORWARDS
We compare the payoff of an American option at a point in time (if it had been exercised at) with the fair value of a European option at the same time. The arguments in Section 2.3 show that this leads to the conclusion that it would never pay to exercise an American call option on the futures price; however, if the American option were never exercised, its value would be equal to the value of the corresponding European option.
RANDOM WALK AND THE BINOMIAL MODEL
This is one of the most important chapters of this book, so it's worth giving an outline of where we're going. This converges to a log-normal distribution for stock price movements when the number of steps is large; the option price calculated using the binomial model will therefore converge to an analytical formula based on the lognormal assumption for the stock price movement.
THE BINOMIAL NETWORK
This series of calculations allows the current value of the option, f0, to be calculated from the option's payoff values, fm,T(ST); this is commonly called “rolling back down the tree.” ii) Jarrow and Rudd: There remains the question of our choice of u,d enp. The centerline of the grid now has the equation Scenter=S0e(r−q)t, which is the equation for the forward speed (known as the forward curve).
APPLICATIONS
Set up the tree and calculate the values of each Stay option terminal values. Assume that the hard part of the tree is the first two steps in calculating an option price.
FINITE DIFFERENCE APPROXIMATIONS
The left-hand side of the heat equation, on the other hand, cannot be uniquely approximated. The resulting difference equation is vi) Richardson: The previous two methods cause forward and backward skews on the time axis, so a simple solution might be to average the two:.
CONDITIONS FOR SATISFACTORY SOLUTIONS
On the other hand, if we reduce the grid so that α=δt/(δx)2 =constant, β would tend to zero and the finite difference equation would be consistent with the heat equation. : Suppose we set up a discretization scheme to solve the heat equation; we have calculated all the numbers by hand to four decimal places and are satisfied with the answers.
EXPLICIT FINITE DIFFERENCE METHOD
In the conventions of the heat equation, t = 0 means “at the beginning” in the calendar sense; this is the moment when the initial conditions (temperature distribution in a long thin conductor) are imposed. This is precisely the Jarrow-Rudd version of the binomial model, summarized in equation (7.6).
IMPLICIT FINITE DIFFERENCE METHODS
8 Numerical solutions of the Black Scholes equation In terms of x,tandk defined in equation (8.2), this can be written. From the boundary conditions, we also know the values at the upper and lower edges of the grid,.
A WORKED EXAMPLE
8 Numerical solutions of the Black Scholes equation. C) Grid spacing in the x direction comes from the definition of α, i.e. δx=√ δt/α=. We begin by recognizing that the value of an American option is only a solution to the Black Scholes equation when the stock price is above the strike;
COMPARISON OF METHODS
We assume that the dividend is paid at a moment fortn+1, so that each of the pointsnexmn+1 falls by an amountδx=lnSmn −ln(Smn −Q)=ln[Smn/(Smn −Q)]. The results are shown in Figure 8.13 and appear to be surprisingly close in form to the Crank Nicolson results.
INTRODUCTION
Historical volatility or realized volatility. This is the volatility of the underlying share price observed in the market. In Figure 9-1, we plot the implied volatility of a series of traded call options with the same maturity but different strike prices; the share price was 100.
LOCAL VOLATILITY AND THE FOKKER PLANCK EQUATION
This gives a higher option price for a put option with a lower strike and the same maturity, allowing for a potential arbitrage. From this smooth implied volatility surface we can immediately derive a smooth "market price" surface simply by using the Black Scholes model.
So equation (9.3) relates the local volatility surface directly to the shape of the probability density. Assumptions can be made, for example that the probability density is only a small perturbation from the lognormal form; a series called the Edgworth expansion (analogous to Taylor expansions for analytic functions) can then be used to derive the volatility surface.
FORWARD INDUCTION
Take, for example, the Arrow Debreu security, which pays $1 if the top right node of the tree in Figure 9.5 is reached. This is true if the probabilities and interest rates change throughout the tree, or if the time steps or price spreads in the tree are variable. iv).
TRINOMIAL TREES
TRINOMIAL TREES With m = 1, equations (9.8) become (to O[δt])
With this choice, it appeared in appendix A.9 that the finite difference method was formally equivalent to a trinomial tree. It therefore seems sensible to make the same choice, written in the notation of this section as u= d−1=eσ√3δt.
DERMAN KANI IMPLIED TREES
The first step is therefore to repeat the calculations of subsection (ii) for each column of nodes in the tree, using the interpolated “market prices”. For the trinomial tree, an exactly analogous relationship applies: if we take the upper branch in the diagram, the formula for λ6 m3 has the simple form.
VOLATILITY SURFACES
It has already been pointed out that option prices change rapidly with the strike price, so it is better to interpolate between implied volatilities. Therefore, the procedure can be as follows: rTake the 4-month prices from Table 9.2 and convert them to implied volatilities using the Black Scholes formula. rUse the cubic line to interpolate between these instabilities. rUse Black Scholes to return implied volatilities interpolated into option prices forX = 80.81,. Use the cubic linear interpolation method of Appendix A.10 to find the value of ∂2C/∂X2 at each pointX =80,81,. ii) Instantaneous volatilities: Steps taken to obtain ∂2C/∂X2 values for 4 months to maturity and a relatively dense set of points X =80.81,.
APPROACHES TO OPTION PRICING
In its simplest forms ("middle of the range" or trapezoidal rules), this method amounts to the same thing as method (4) above, i.e. Repeat the operation many thousands of times (N), making sure that the random paths are all taken from a distribution that adequately reflects the distribution of St. The Monte Carlo estimate of the =0 value of the option is fair. iii) The curse of dimensionality: The option pricing approaches (1) to (6) above share a major shortcoming: they cannot handle path-dependent options in a general way.
BASIC MONTE CARLO METHOD
The most important point is that the Monte Carlo error does not depend on the number of dimensions of the problem. With the Monte Carlo method, we can calculate the payoff at each point on our simulated path, but there is no way to compare it to the option value along the way.
RANDOM NUMBERS
We can convert standard uniform random numbers produced by a random number generator to normal standard random numbers with the transformation. Ifx1 dhex2 are two independent standard uniform random numbers, then z1andz2 are two independent standard normal random numbers, where.
PRACTICAL APPLICATIONS
A demonstration of the effectiveness of this technique is given in Table 10.2 for the following options:. If the standard errors (SE) of the geometric and arithmetic results are approximately the same, the condition for improving the results with the control variable technique becomes ρ > 1; correlation with.
QUASI-RANDOM NUMBERS
The simplest series to produce are the simple Halton numbers that we will use for illustration. The simplest procedure is therefore to use another prime number as the basis of the Halton numbers used for each dimension, e.g.
EXAMPLES In our numerical example
EXAMPLES
On the other hand, if you are building a reusable model, you should always use quasi-Monte Carlo if possible. The low deviation sequence was based on Halton numbers with bases 2 and 3; graph of price vs
FORWARD START OPTIONS
Let f(n S0,n X) be the value of the stock option, where S0 is the stock price and X is the payout. If today we buy f(1, α,T −τ) units of stock at the price S0f(1, α,T −τ), the value of this stock at time τ will be Sτf(1, α,T−τ); however, this is equal to the future value of the forward initial option.
CHOOSERS
The form of the second term in the profit is that of a put option that matures in timeτ. iii) Complex Picker: The picker concept can be extended very simply so that put and call options have different prices and maturities. Unfortunately, the mathematics of pricing does not extend so simply, and we therefore postpone this until Section 14.2.
SHOUT OPTIONS
BINARY (DIGITAL) OPTIONS
ST Figure 11.2 Asset or nothing option (iii) Asset or nothing option: On the same reasoning as. But then where would he get the reward, when the bet is exercised against him.
POWER OPTIONS
Let's say that at the maturity of the bundle, the stock price is ST =X+nh, which is above X. If it is small, the payoff of this bundle is. A power option can be simulated with a bundle of options, with the approximation becoming accurate as the spacing between the option hits shrinks to zero.
EXCHANGE OPTIONS (MARGRABE)
Before diving into the details of specific options, we need to take a broad overview of the principles underlying this chapter. One final piece of information is needed: a value for σQ, the volatility of the composite asset Qt=St(1)/St(2)St(2).
MAXIMUM OF TWO ASSETS
MAXIMUM OF THREE ASSETS
The effect of a change of variables in the manner of equations (A1.7) and the use of the normal bivariate definitions of equation (A1.12) gives. The techniques of this section and the last section can be extended to a larger number of assets (Johnson, 1987); the formula for fmaxw will then involve higher-order multivariate normal functions.
RAINBOW OPTIONS
BLACK SCHOLES EQUATION FOR TWO ASSETS
12 Two Funding Options . 12.5 THE BLACK SCHOLES EQUATION FOR TWO FUNDS FLOWS) must be equal to the risk-free return:.
BLACK SCHOLES EQUATION FOR TWO ASSETS flows) must equal the risk-free return
BINOMIAL MODEL FOR TWO ASSET OPTIONS
BINOMIAL MODEL FOR TWO ASSET OPTIONS Using the approach of Section 7.1(iv), the Wiener processes for x t and y t are written
Descend these through the tree in the normal way, remembering that values at four nodes are needed for each step back (instead of two in the single asset tree); the probabilities are all set to p=14. An example is given in Chapter 10 of pricing a two-asset spread option using quasi-Monte Carlo.
INTRODUCTION
DOMESTIC CURRENCY STRIKE (COMPO)
In fact, the New York office claims that Frankfurt does not need to be involved at all, since the shares are listed on both the Frankfurt and New York exchanges at the same time. This is no different than any other stock listed in New York, and the fact that the ultimate underlying company is German is irrelevant; therefore, the option is no different from any US domestic call option.
FOREIGN CURRENCY STRIKE
FOREIGN CURRENCY STRIKE: FIXED EXCHANGE RATE (QUANTO)
So the rule for finding the price of a call option calculated in US dollars is simple. This does not allow us to write Ft= ft; but if we can set up a Black Scholes type equation for Ft, the initial conditions will be the same as for ft.
SOME PRACTICAL CONSIDERATIONS
Here we use a modification of the Black Scholes equation for the quotient of two $ securities, (S&P)t and (DAX)t/φt. Note that each of the above options involves correlations between three or four random variables.
OPTIONS ON OPTIONS (COMPOUND OPTIONS)
At timeτ, the price of the underlying call is given by the curve shown in Figure 14.1. This is shown as the solid line in Figure 14.2, together with the compound option price before expiration (dotted curve).
COMPLEX CHOOSERS
EXTENDIBLE OPTIONS
It will be clear to the reader who has studied section A.1 of the appendix that there are different ways of expressing the answers, so comparison by terms with results given in other publications can be difficult: for example, an expression equal to the value of a call option with strikeS∗∗and maturityT; some other authors show the call option with strikeK instead, and other expressions are slightly modified (Longstaff, 1990). A first-time reader may feel intimidated by the size of the formulas and the number of large integrals; but you should take a step back to understand the underlying principles rather than drowning in the details.
SINGLE BARRIER CALLS AND PUTS
This will involve a transformation of the variable ST to either of the variableszT orzT, defined in the last subsection by. However, there is still a difference in the term [J]: the signs of the arguments of the cumulative normal functions are reversed.
GENERAL EXPRESSIONS FOR SINGLE BARRIER OPTIONS
15 Obstacles: simple European options . and −1 for sales); this was explained in Section 5.2(iv) where we wrote a general Black Scholes formula that could be used for either a put or a call. This is mainly due to the fact that the limits of integration were ZKto+∞ in the first case and −∞to ZK in the second; the difference arises because the share price had to fall to reach the barrier in the first case, but rose in the second.
SOLUTIONS OF THE BLACK SCHOLES EQUATION
[0,ex−X] can be replaced by ex−X because this is always positive in the region of integration. It then remains only to follow the computational procedures set out in Section 5.3 to calculate this integral; not surprisingly, the answer is the same as that in Table 15.1.
TRANSITION PROBABILITIES AND REBATES
BINARY (DIGITAL) OPTIONS WITH BARRIERS
