Birla Institute of Technology and Science, Pilani

First Semester: 2024-2025

Part – B (Open Book)

Course Name & No.: Derivatives and Risk Management (ECON F354 / FIN F311)
Maximum Marks: (90 Marks) 30% Weight-age Date: 13/12/ 24
Duration: 2 Hrs. Comprehensive Exam

Q1. Let the initial stock price be S(0) = 100 and volatility be 22.5%. Let the continuously compounded
interest rate be r = 0.05 per annum. Consider a two-period binomial model for the stock price with both
periods of length one year to answer the following.
Note: Round up the up and down factors to two decimals.
A) As a market maker, you have to price an option contract for the investor, which has a provision
that the investor can choose if the option is a put or a call after one year independently at each
node. The strike price for this option is Rs.100, and the expiry is two years. Assume no
dividends are paid. (10 Marks)
% !" $&
SOLUTION: 𝑢 = 𝑒 !√# = 1.25 ; 𝑑 = 𝑒 $!√# = 0.80; 𝑝 = '$& = 0.558
Let’s say we choose call in the up node, c = e -0.05 (0.558x56.25 + 0.442x0) = 29.85
Let’s say we choose put in the up node, p = e-0.05 (0.558x0 + 0.442x0) = 0

Let’s say we choose call in the down node, c = e-0.05 (0.558x0 + 0.442x0) = 0
Let’s say we choose put in the up node, p = e-0.05 0.558x0 + 0.442x36) = 15.13

Therefore, we will choose call and put at the up and down node respectively.
Thus, the value of the contract today will be
V = e-0.05 (0.558x29.85 + 0.442x15.13) = 22.2077

B) As a market maker, you must price an American put option if there is a dividend yield of 1.5%,
expiry in two years and a strike price of 105. (10 Marks)
% (!$%)" $&
SOLUTION: New 𝑝 = '$& = 0.5236
The payoff at t=2 will be, (2,2) = 0; (2,1) = 5; (2,0) = 41

The value of payoffs at nodes (1,1) = 2.26 and (1,0) = 21.07

Vs The exercise values at nodes (1,1) = 0 and (1,0) = 25

Thus, we will select the payoffs as (1,1) = 2.26 and (1,0) = 25 and value the American put
P = e-0.05 (0.5236x2.26 + 0.4764x25) = 12.4548

Q2. The following are prices of options traded on Microsoft Corporation, which pays no dividends.
Call Put
K = 85 K = 90 K = 85 K = 90
1-Month 2.75 1 4.5 7.5
3-Months 4.0 2.75 5.75 9.0
6-Months 7.75 6.0 8.0 12.0
The stock trades at $83; the annualised riskless rate is 3.8%. The volatility of stock prices (based on
historical values) is 30% (5x4 = 20 Marks)
A) Estimate the value of a three-month call with a strike price of 85 using the Black Scholes Model.
SOLUTION: d1 = -0.0204; d2 = -0.1704; N(d1) = 0.49186; N(d2) = 0.43234; c = 4.4224
B) Using inputs, specify how you would replicate this call.
SOLUTION: C = Put + Stock – ZCB = Buy put@85, Buy stock @83 and Short a ZCB @84.196
C) Using put-call parity, estimate the value of a three-month put with a strike price of 85.
SOLUTION: p = c + Ke-rt - So = 4.4224 + 84.196 – 83 = 5.618
D) Find the probability of a 6-month Put @ 90 getting exercised if the beta of Microsoft is 1.2 and
the market risk premium is 2%.
SOLUTION: exp(r)=3.8+(1.2x2)= 6.2%; Exercise Probability= N(-d2) = 0.63368 or 63.37%

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Q3. A) Design an option strategy (using the above Microsoft options) that will have a consistent payoff
after 6 months. Show the payoff and profit table and a well-labelled plot for the strategy. (10 Marks)
@t=0 ST < 85 85<ST<90 ST > 90
Call @ 85 -7.75 0 ST-85 ST-85
Call @ 90 6 0 0 90-ST
Put @ 85 8 ST – 85 0 0
Put @ 90 -12 90 – ST 90-ST 0
Payoff 0 5 5 5
Profit -5.75 -0.75 -0.75 -0.75

B) If, as an investor, you are excited about the increasing volatility in the market and expect a sharp rise
in the volatility in the next 6 months. Create a strategy to benefit from the expectation using the above
Microsoft options. Show the payoff and profit table and a well-labelled plot for the strategy. (10 Marks)
@t=0 ST < 85 ST = 85 ST > 85
Call @ 85 -7.75 0 0 ST-85
Put @ 85 -8 85-ST 0 0
Payoff 0 85-ST 0 ST-85
Profit -15.75 69.25-ST -15.75 ST-100.75

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Q4. While checking the market listings, your trader friend discovered that the value of ATM call options
with a strike price of Rs. 50 and a maturity of 1 year is Rs. 5, while the value of ATM put options is Rs.
4. Additionally, your friend learned that Rs. 45 invested in a bond will grow to Rs. 50 after 1 year. With
this information, your friend has good reasons to celebrate. Justify (20 Marks)

SOLUTION: using put-call parity equation

LHS is 45 + 5 = 50 < RHS: 50 + 4 = 54
Using the arbitrage strategy,
we short the put and stock, buy the call (54-5 = 49), and invest 49 for 1 year.

After 1 year,
we cover short position using either call or put @ the cost of 50.

This leaves us with a procit of 4 in PV terms

Q5. For Microsoft stocks, the following are additional options trading 3-months Call and Puts @ 75
and 80. The 80 & 75 calls have time values of 3.75 and 2.5, respectively. Draw profit tables for the
following strategies exploiting the two scenarios below: (10 Marks)
A) Strategy with minimum investment for benefiting from no-volatility.

Call@80 is valued at 3 + 3.75 = 6.75 and Call @75 at 8 + 2.5 = 10.5

Strategy is Long Buttercly, Buy calls @ 80 and 90 and sell 2 calls @85

@t=0 ST < 80 80<ST<85 85<ST< 90 ST>90

Call @ 80 -6.75 0 ST-80 ST-80 ST-80
Calls @ 85 8 0 0 170-2ST 170-2ST
Call @ 90 -2.75 0 0 0 ST-90
Payoff 0 0 ST-80 90-ST 0
Profit -1.5 -1.5 ST-81.5 88.5-ST -1.5

B) Strategy with built-in cushion for benefiting from high volatility.

Put@80 is valued at 2.99 and Put @75 at 1.79

Strategy is Short Condor,
Short put@75 and long put@80; long call@85 and short call@90
@t=0 ST < 75 75<ST<80 80<ST< 85 85<ST< 90 ST>90
Put @ 75 1.79 ST-75 0 0 0 0
Put @ 80 -2.99 80-ST 80-ST 0 0 0
Call @ 85 -4 0 0 0 ST-85 ST-85
Call @ 90 2.75 0 0 0 0 90-ST
Payoff 0 5 80-ST 0 ST-85 5
Profit -2.45 2.55 77.55-ST -2.45 ST-87.45 2.55

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