Financial Engineering – IC302
Autumn Term 2020/1
Seminar 5: Basic Numerical Methods for Valuing Options (Binomial Trees) Questions
1. Use the Excel workbook IC302-One-period binomial-Que to answer this question.
(a) Complete the workbook by entering an appropriate formula in each of the blue-shaded cells. Use the lecture notes as a guide. What is the price of the option for the parameter values given in the workbook?
For reference, the parameter values are shown in the table below:
Variable
Variable name in workbook
Value
𝑆!
S_now
100
𝑢
up
1.2
𝑑
down
0.8
𝑟
risk_free
0.05
𝑇
T
1
𝐾
K
100
The variable names used elsewhere in the workbook are related to those
used in the lecture in similar ways (e.g. S_up corresponds to 𝑆!𝑢, C_up
corresponds to 𝑓 , and so on). ”
(b) What is the option’s delta at these original parameter values?
(c) Use the model to calculate the price of a one-year European call option for the same parameter values as in the original problem but an exercise price of 90. What explanation can you offer for the impact that changing the exercise price has on the option’s value?
(d) What is the delta of the option with exercise price 90? What explanation can you offer for the effect that changing the strike price has on the delta?
(e) According to the estimated delta in part (d), what position in the underlying asset would you need in order to hedge a short position in one option with strike price 90?
(f) Suppose now that the underlying asset price increases to 101. According to the model, how would this affect the delta? In what way would you need to adjust your hedge if you were delta hedging a short position in one option?
2. Use the Excel workbook IC302-Multi-period binomial-Que to answer this question. This workbook uses the CRR model to value European and American call and put options in a three-period example and a full VBA implementation.
(a) Use the workbook to calculate the value of a European call option by entering an appropriate formula in each of the blue-shaded cells in the sections labelled ‘Multi-period binomial model (3-step example parameters)’ and ‘European call option (3-step example)’. Use the lecture notes as a guide. What is the price of the option for the parameters originally given in the workbook?
(b) Compare this to the Black-Scholes-Merton price for the option for the same parameter values. This is calculated in the section of the worksheet labelled ‘Black-Scholes model (VBA function for European call options)’. Why are the two values not the same?
(c) According to the calculations in part (a), what is the delta of the option at the initial stock price (i.e. at the root of the binomial tree)?
(d) The section of the worksheet labelled ‘Multi-period binomial model (full VBA implementation for European call option’ implements the CRR model for an arbitrary number of time steps. The accuracy of the approximation should increase with the number of time steps, which can be varied by changing the value in cell L40. Approximately how many time steps are required for the CRR price to converge to the Black-Scholes price for the option to the nearest penny?
3. Use the Excel workbook IC302-Multi-period binomial-Que to answer this question. This workbook uses the CRR model to value European and American call and put options in a three-period example and a full VBA implementation.
(a) Use the workbook to calculate the value of an American put option by entering an appropriate formula in each of the blue-shaded cells in the section labelled ‘American put option (3-step example)’. Use the lecture notes as a guide. What is the price of the option for the parameters originally given?
(b) At which nodes of the tree (if any) would early exercise of the option be optimal?
(c) Use the section of the worksheet labelled ‘Multi-period binomial model (full VBA implementation for American put option)’ to calculate a more accurate approximation to the option value. What is this value, and approximately
how many time steps are required for the algorithm to converge to this result?
(d) Use the section of the worksheet labelled ‘Multi-period binomial model (full VBA implementation for American put option)’ to calculate the value of a one-year American put for the original parameter values but with strike price equal to 110. What explanation can you offer for the difference between this value and the option value for strike price 100?