SOLUTION NOTES FOR EXERCISES ON FUNCTIONS
4. Code for Standard Deviation of Cash Flows
10.10 USER-DEFINED FUNCTIONS IN Module0 AND Module1
The OPTION1 workbook contains the code for several user-defined functions, which implement most of the tree-based valuation methods discussed in this chapter. The main functions to obtain option values using either the CRR or LR binomial trees are in Module0 of the workbook. The most important functions are the BinEuroOptionValue function used on the CRRTheory sheet and the BinOptionValue function used on the Compare sheet.
In the arrays used to hold the option payoffs in the binomial trees we have allowed array dimensions to start from 0 (rather then the more usual 1) as the binomial tree time steps also begin with time period 0. This is done using the statement Option Base 0 at the top of the VBA module sheet. As an additional reminder, we also rename the module sheet to Module0.
Binomial Trees 183 The BinOptionValue function can be used to value European or American options using either the CRR (imodD1) or LR (imodD2) binomial trees. The code is most easily understood taking one type of option (a European call say, where ioptD1 and ieaD1) and one valuation method (say the CRR tree demonstrated in the CRREuro sheet). The valuation process revolves around the vector array vvec() declared as a variant type with 10 values, vvec(0) to vvec(9), as ensured by the dimensioning statement ReDim vvec(nstep).
The terminal option prices in the share price tree are derived from a formula of the form:
max[SuidniX,0] iD0,1,2, . . . ,9 In the VBA statements this is achieved using a loop:
For i = 0 To nstep
vvec(i) = Application.Max(ioptŁ(SŁ(uOi)Ł(dO(nstep - i)) - X), 0) Next i
The subsequent five lines of code drive the process of working backwards through the tree, at each stage discounting the expected option payoff, to calculate the option value:
For j = nstep - 1 To 0 Step - 1 For i = 0 To j
vvec(i) = (pŁvvec(i + 1) + pstarŁvvec(i)) / erdt Next i
Next j
There are two important points to note in the VBA code. By replacing redundant values with new values in the vvec vector as we value the option, we can limit the storage require- ments to a single vector with only nC1 elements. The other point is that for American options (ieaD2) we just need an additional line of code that chooses the maximum of the European value and the value of immediate exercise.
We describe the hedging parameters in detail in the next chapter, and so limit our comments here regarding the BinOptionGreeks function. Three of the five parameters can be estimated by using an extended binomial tree, while the remaining two parameters require two separate binomial trees to be built. The function uses the more efficient LR tree.
The Module1 sheet contains the code for the BSOptionValue function which computes the Black–Scholes formula. Detailed explanation is given in the following chapter, section 11.7.
SUMMARY
The binomial tree provides a practical way to model the share price process. The resulting discrete distribution of terminal prices can be configured to approximate to the continuous lognormal distribution assumed in the Black–Scholes analysis.
An option on the share is valued via the tree by calculating the expected option payoff (weighting each option payoff by its risk-neutral probability) and discounting the expec- tation at the risk-free rate.
The efficiency of the binomial method of option valuation (compared to the Monte Carlo sampling approach) comes from the fact that many paths in the tree recombine.
There are several different ways to configure the binomial tree using different sets of parameters (e.g. JR tree, CRR tree, LR tree). The JR tree has equiprobable up and down steps, where size of steps allows for drift and volatility. The CRR tree has related up and down step sizes reflecting volatility but no drift in mean level. To include the drift in prices, the up and down probabilities differ so that (usually) prices drift up. The LR tree at maturity is centred on the exercise price and has quite complex probability and step size parameters.
Experimentation with the different tree types for European options suggests that the LR tree produces estimates close to the Black–Scholes values using only a small number of steps.
The binomial tree method is easily adapted to include the early exercise of an option, thus allowing American options to be valued.
REFERENCES
Cox, J., S. Ross and M. Rubinstein, 1979, “Option Pricing: A Simplified Approach”, Journal of Financial Economics,7, 229–264.
Cox, J. and M. Rubinstein, 1985,Options Markets, Prentice Hall, New Jersey.
Hull, J. C., 2000,Options, Futures and Other Derivatives, Prentice Hall, New Jersey.
Jarrow, R. A. and A. Rudd, 1983,Option Pricing, Richard D. Irwin, Englewood Cliffs, NJ.
Leisen, D. P. J. and M. Reimer, 1996, “Binomial Models for Option Valuation–Examining and Improving Convergence”,Applied Mathematical Finance,3, 319–346.
Wilmott, P., S. Howison and J. Dewynne, 1996, The Mathematics of Financial Derivatives, Cambridge University Press, Cambridge.
11
The Black–Scholes Formula
Although the binomial tree provides an easier way to understand option pricing, the analytic Black–Scholes formula remains the central ingredient for European options. Its strength is in providing the option value through a formula and also in determining the hedge ratio for the replicating portfolio. In this chapter, the Black–Scholes formula is derived formally and extended to cover financial assets with a continuous dividend, thus allowing options on currencies and futures to be valued. Hedging parameters can also be derived, and these allow us to create portfolios that are invariant in value to modest changes in share price.
Hull’s (2000) textbook on Options is the best reference for most of the topics in this chapter (Chapter 11 for the derivation and explanation of Black–Scholes, Chapter 12 for the adaptation to include continuous dividends and hence how the Black–Scholes formula can be modified to price options on currencies and futures, and Chapter 13 for the ‘greeks’ and delta hedging). Models illustrating the various Black–Scholes pricing formulas are in the OPTION2.xls workbook, together with a range of useful valuation functions.