Financial Engineering – IC302
Autumn Term 2020/1
Seminar 5: Basic Numerical Methods for Valuing Options (Binomial Trees) Answers
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). ”
See the Excel workbook IC302-One-period binomial-Ans for the solution. The price of the one-year call option is 11.95.
(b) What is the option’s delta at these original parameter values?
The delta is 0.5. It can be calculated using the formula given in the lecture notes for the one-period binomial model:
𝑓 −𝑓 20−0 𝛥=”#= =0.5
𝑆!𝑢−𝑆!𝑑 120−80
This value is calculated in cell I3 of the workbook.
(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?
The price of the option is 17.93. With a lower strike price, the option has a higher probability of finishing in the money and therefore a larger expected payoff.
(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?
The delta is now 0.75. With a lower strike price, the option is in the money and has a higher probability of being exercised. The option price will change by more for each small change in the underlying asset price than was the case for an option with strike price 100.
(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?
You would need to go long 0.75 units of the underlying asset to hedge an option that gave the option holder the right to buy one unit of the underlying asset at 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?
An increase in the underlying asset price to 101 would increase the delta to approximately 0.77. You would need to increase slightly the size of your delta hedge. Hedging of options, unlike hedging of forwards or futures, is dynamic: we need to rebalance the hedge over time as market conditions change.
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?
See the Excel workbook IC302-Multi-period binomial-Ans for the solution. The price of the option is 11.55.
(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?
The Black-Scholes price is 10.99. The price from the CRR model is an approximation that should converge to the Black-Scholes price as the number of time steps is increased. There are too few times steps in the three-step example to generate an accurate result.
(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)?
The delta can be calculated using the formula given in the lecture notes. Using the possible stock prices and option values for the first time step in the tree, we find that the delta is:
𝑓 −𝑓
𝛥 = ” # =
𝑆!𝑢 − 𝑆!𝑑
18.20−3.67 112.24 − 89.09
= 0.6276
Note that this is only a rough estimate of the option’s delta which could be improved by using a tree with more time steps.
(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?
Approximately 500 time steps are needed for the CRR price to agree with the Black- Scholes price 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?
See the Excel workbook IC302-Multi-period binomial-Ans for the solution. The price of the option is 6.10.
(b) At which nodes of the tree (if any) would early exercise of the option be optimal?
For the original parameter values the only node at which early exercise would be optimal is at the end of the second time step (i.e. one time step before expiry) when the stock price is 79.38. At every other node in the tree the continuation value is greater than the early exercise value. You can confirm this by noting that the calculated option price given in the tree is greater than the difference between the strike price and the exercise price at all of the other nodes in the tree.
(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?
The price of the American put appears to be approximately 5.80. The algorithm converges to this result after approximately 200 time steps.
(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?
With strike price 110, the option price is 11.66, which is higher than the price 5.80 we found for the option with strike price 100. With a higher strike price than in the original example, the put option is in the money and more likely to be exercised, both at expiry and prior to expiry. You can confirm this by setting the strike price to 110 in the 3-step example, which can be done by changing the value in cell C18. Notice how this leads to early exercise of the option at the end of the first time step if the stock price is 89.09 as well as at the end of the second time step if the stock price is 79.38. When the strike price was 100, the option would only be exercised early at the second of those two nodes in the tree.