c Ioannis Kyriakou, IF2209 Derivatives, Group coursework
General instructions
Submission deadline is 4pm on Wednesday, 23rd March 2016.
Each question counts towards 25% of the total coursework mark.
Presentation and clarity of design of your reports is as important as the correctness of your computations and output discussions where required; your reports have to be written neatly and be easily
readable to avoid undesired loss of marks.
All plots and tables containing summary of the results should be included in the written report.
Submit printed reports to the course office the usual way.
As the coursework requires some practical work on Excel to be done, in addition to your hardcopy
submission, please upload on Moodle the relevant workbook you have worked on. You should submit only ONE Excel file: please use a separate sheet in this file for each question you have to provide
Excel workings and name each sheet according to the question’s number and part. The Excel file
should be submitted by a single group member.
In summary: submit (a) a printed report to the course office which contains your responses to the
questions including tables, plots and discussions; (b) an Excel file on Moodle which shows your workings where required.
Question 1
Consider the data given in the sheet named ‘Question 1’ of workbook ‘IF2209cwk_data.xlsx’. These represent the prices of European plain vanilla call and put options on stock S of the Goldman Sachs Group, Inc.
(GS) traded on NYSE. The market option prices for maturity T = 11 months (i.e., 11/12 year) are available
for the indicated range of strikes in sheet ‘Question 1’.
Put-call parity requires that the following equation holds
Cobs Pobs = S0eδT KerT, (1)
where Cobs, Pobs are respectively the observed prices of the call and put options, δ is the continuously compounded dividend yield per annum, r the continuously compounded risk-free rate of interest per annum,
S0 the current spot price of the stock, K the strike price and T the option maturity.
a) The put-call parity (1) may be viewed as a simple linear regression with general form
y = α + βx (2)
where y corresponds to the (Cobs Pobs) price differences and x to the strike prices K.
i) By comparing (1) and (2), express the intercept α and the slope β in terms of S0, δ, r, T.
ii) Fit the linear regression model to the observed option price differences and provide estimates
for the intercept and the slope.
Page 1 of 3δT KerT, (1)
where Cobs, Pobs are respectively the observed prices of the call and put options, δ is the continuously compounded dividend yield per annum, r the continuously compounded risk-free rate of interest per annum,
S0 the current spot price of the stock, K the strike price and T the option maturity.
a) The put-call parity (1) may be viewed as a simple linear regression with general form
y = α + βx (2)
where y corresponds to the (Cobs Pobs) price differences and x to the strike prices K.
i) By comparing (1) and (2), express the intercept α and the slope β in terms of S0, δ, r, T.
ii) Fit the linear regression model to the observed option price differences and provide estimates
for the intercept and the slope.
Page 1 of 3rT, (1)
where Cobs, Pobs are respectively the observed prices of the call and put options, δ is the continuously compounded dividend yield per annum, r the continuously compounded risk-free rate of interest per annum,
S0 the current spot price of the stock, K the strike price and T the option maturity.
a) The put-call parity (1) may be viewed as a simple linear regression with general form
y = α + βx (2)
where y corresponds to the (Cobs Pobs) price differences and x to the strike prices K.
i) By comparing (1) and (2), express the intercept α and the slope β in terms of S0, δ, r, T.
ii) Fit the linear regression model to the observed option price differences and provide estimates
for the intercept and the slope.
Page 1 of 3
c Ioannis Kyriakou, IF2209 Derivatives, Group coursework
iii) Provide a scatterplot including the line of best fit. Comment on the quality of the linear fit based
on the regression output (you may provide extra supporting plots of your choice, e.g., residuals
plots, with supplementary comments).
For the regression, you may use the Regression tool offered in the Data Analysis package on Excel.
For computing the intercept and slope of the linear regression model, you may use Excel’s built-in
functions INTERCEPT and SLOPE.
b) You are given that S0 = 165.40. In addition, using your estimates for the intercept and the slope
from part (a), provide estimates for the dividend yield δ and interest rate r showing clearly your
computations.
Question 2
Consider the market call option prices (Cobs) on the same stock as in Question 1 given in the sheet named
‘Question 2’ of workbook ‘IF2209cwk_data.xlsx’.
a) Using δ = 1.53% per annum, r = 0.49% per annum, T = 11/12 year, S0 = 165.40, find for each strike
K the implied volatility σimpl satisfying the equation
Cobs CBSM(S0, K, r, δ, T, σimpl) = 0, (3)
where CBSM is the option price obtained from the Black–Scholes–Merton formula
CBSM(S0, K, r, δ, T, σimpl) = S0eδTN(d1) KerTN(d2),
with
d1 = d1(S0, K, r, δ, T, σimpl) = 1
σimplpT “ln S K 0 + r δ + σ2 impl 2 ! T# ,
d2 = d2(S0, K, r, δ, T, σimpl) = d1(S0, K, r, δ, T, σimpl) σimplpT
and N(.) the standard normal cumulative distribution function. You can use Excel’s add-in Solver to
solve numerically equation (3).
b) i) Provide in a table the implied volatilities σimpl against the strike prices K and construct the relevant plot.
ii) Fit a second-order polynomial with form
f(K) = aK2 + bK + c
to the plot in part (b.i). For this, you may use the ‘add trendline’ tool of Excel. Provide the
estimates for a, b and c.
iii) Comment.
Page 2 of 3δTN(d1) KerTN(d2),
with
d1 = d1(S0, K, r, δ, T, σimpl) = 1
σimplpT “ln S K 0 + r δ + σ2 impl 2 ! T# ,
d2 = d2(S0, K, r, δ, T, σimpl) = d1(S0, K, r, δ, T, σimpl) σimplpT
and N(.) the standard normal cumulative distribution function. You can use Excel’s add-in Solver to
solve numerically equation (3).
b) i) Provide in a table the implied volatilities σimpl against the strike prices K and construct the relevant plot.
ii) Fit a second-order polynomial with form
f(K) = aK2 + bK + c
to the plot in part (b.i). For this, you may use the ‘add trendline’ tool of Excel. Provide the
estimates for a, b and c.
iii) Comment.
Page 2 of 3rTN(d2),
with
d1 = d1(S0, K, r, δ, T, σimpl) = 1
σimplpT “ln S K 0 + r δ + σ2 impl 2 ! T# ,
d2 = d2(S0, K, r, δ, T, σimpl) = d1(S0, K, r, δ, T, σimpl) σimplpT
and N(.) the standard normal cumulative distribution function. You can use Excel’s add-in Solver to
solve numerically equation (3).
b) i) Provide in a table the implied volatilities σimpl against the strike prices K and construct the relevant plot.
ii) Fit a second-order polynomial with form
f(K) = aK2 + bK + c
to the plot in part (b.i). For this, you may use the ‘add trendline’ tool of Excel. Provide the
estimates for a, b and c.
iii) Comment.
Page 2 of 3
c Ioannis Kyriakou, IF2209 Derivatives, Group coursework
Question 3
a) A European plain vanilla call option on a dividend-paying stock S has strike price 160 and matures in
one year. The current price of the underlying stock is 165.40. The annual stock price volatility is σ =
22.48%, the continuously compounded dividend yield is δ = 1.53% per annum and the continuously
compounded risk-free rate of interest is r = 0.49% per annum. Consider a two-period binomial model
for the stock with u = exp(σp∆t) (stock price move-up factor per period), d = 1/u (stock price movedown factor per period) and risk neutral probability of an upward movement q = (exp((r δ)∆t)
d)/(u d).
i) Compute u, d, q.
ii) Construct the binomial tree of the stock prices showing clearly your calculations in your report.
iii) Construct the binomial tree of the option values showing clearly your intermediate calculations
in your report. What is the time-0 price of the option?
b) A one-year European cash-or-nothing call option on stock S has the following payoff at maturity T:
DT = 5, if 0, if S ST T > 160 160 .
Using the two-period binomial model for the stock from part (a), compute the time-0 price of the
cash-or-nothing call option showing clearly your intermediate calculations.
c) A one-year European gap call option on stock S has the following payoff at maturity T:
GT = ST 155, if 0, if S ST T > 160 160 .
Without performing further binomial-model calculations, obtain the time-0 price of the gap call option justifying your answer.
Question 4
Consider the following European plain vanilla options: (1) a call with strike price K = 160, (2) a put with
strike price K = 160, (3) a call with strike price Kc = 165, and (4) a put with strike price Kp = 155. All
options have the same non-dividend-paying underlying stock and mature after one year.
a) Assuming current stock price 160, stock price volatility 22% and continuously compounded risk-free
rate of interest 0.49%, compute the prices of options (1)–(4) using the Black–Scholes–Merton formula
showing clearly all your computations.
b) Assume a long position in options (1) and (2) and a short position in options (3) and (4). An iron
butterfly is an option strategy that involves the aforementioned positions.
i) Construct the table of the payoff profile of this strategy at maturity.
ii) Calculate the implementation cost of this strategy.
c) Provide an alternative way of obtaining iron butterfly’s payoff profile that combines short and long
positions in only call options with strike prices K, Kc, Kp plus some constant cash amount A which
you need to identify.
Page 3 of 3
Call Options Put Options
Strike Cobs Strike Pobs
60 103.1 60 0.23
70 98.23 70 0.32
80 83.44 80 0.49
90 76 90 0.65
95 68.09 95 1.15
100 65.45 100 1.19
105 57.4 105 1.76
110 52.2 110 1.5
115 51.46 115 2.05
120 46 120 2.35
125 41.78 125 2.91
130 37.4 130 3.81
135 33 135 4.5
140 28.68 140 5.51
145 25.64 145 6.95
150 22.05 150 8.5
155 19.48 155 10.27
160 15.8 160 12.2
165 13.2 165 14.99
170 10.8 170 17.35
175 8.53 175 21.25
180 7.18 180 24.3
185 5.55 185 30.45
190 4.5 190 31.9
195 3.3 195 38.2
200 2.6 200 39
205 2.06 205 34
210 1.73 210 48.75
215 1.42 215 40.7
220 1.04 220 51.1
230 0.6 230 69.9
240 0.35 240 81.8
255 0.2 255 93.2
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