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Homework Risk Measures and the Greeks


1. A stock is priced at $30. The volatility of the stock is 25% per year, and the riskless
interest rate is 7%. A 3 month European call option with a strike price of 25 trades at its
Black Scholes price of $5.51.




Solution

(a) PE(0) = 5.51 - 30 + 25exp(-0.07×0.25) = $0.077
(b) The delta value is 0.952. This means the number of share equivalents is 0.952.
The leverage value is given by λ = HS/C = 0.952(30)/5.51 = 5.18
That is, if the stock increases by 1%, the option is expected to change by 5.18%.
(c) The delta of a put is
Δp = Δc - 1 = - 0.048
The option is equivalent to a short position in 0.048 shares.
The leverage is λ = -0.048(30)/0.077 = -18.7
That is, if the stock increases by 1% the option decreases by 18.7%.

d) Assume each call option controls one share. (If you assume 100 shares that’s also fine)
Since the delta value is 0.952, the trader need purchase 952.0 shares to be locally
immunized.( If you assume each option controls 100 shares then your answer is 9,520)

e) If the stock price moves $1, then the delta value will increase from its old value to a
new value which is γ higher (i.e. delta will increase by 0.0268).

Cnew = Cold + 2 Δ + γ = 5.51 + 2(0.952) + 0.0268

As the stock price increases, the trader needs to purchase stocks at a rate of γ to maintain
the immunization
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Solution:
S0 = $48, σ = 36%, r = 7%, X = $45 a) The dollar cost of the call option is$9.88×100 = $988
b) expected drop in one week:
-$4.65/52 = -$0.089423
50 × 100 × $(9.88 - 0.089423)
= $48,952.885

c) Price change
= 0.04 × 16.43 × 50 × 100
= $3,286

d) Dollar gain
= 0.005 × 24.2 × 200 × 100
= $2,420

e) Share equivalents is 0.71
B = HS - C
= 0.71 × 48 - 9.88
= $24
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3 Information on 3 month at the money European options which has a volatility of 30%
a year are shown below. The current stock price is 20.


(e) Compute the share equivalents for the position in (d). Compute the net
borrowing, and the leverage.
Solution:
(a) We set up two equations in two unknowns a
0.576 Nc - 0.424 Np = 0
3.92 Nc + 3.92 Np = 4.04
Solving these two equations we obtain Nc= 0.4370; Np = 0.5937 So
the trader needs to purchase 0.437 calls and 0.5937 puts.

(b) A straddle consists of the purchase of a call and the purchase of a put with the same
strike.
The Delta of the straddle is Delta(call) + Delta(put) = 0.152.
The vega of the straddle is the sum of the two vegas which is 7.84
This position will earn 0.152 dollars if the stock price increases by one
dollar. Since the vega value is large the straddle is very sensitive to
volatility changes.
(c) The gamma of the straddle is 0.1306+0.1306 = 0.2612
(d) The delta value of this position is 1 times delta(call) – 1 times delta(put) = 1.0
(e) The call C = H1 S – B1; Since C, S and H1 are given we can find B1
B1 = 0.576(20) – 1,38 = 10.14. Similarly P = H2S – B2 with P1= 1.02, and H2 = -0.424
so B2 = -9.5
As a result, we have the share equivalents H1-H2 = 1 and net borrowing is B1-B2 =
10.14+9.5 =19.64. The leverage is therefore 1S/(C-P) = 20/0.36 = 55.55


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4. From Hull p347


Solution
The first step is to get the delta and gamma and for part b the vega of the portfolio
-l, 000 X 0.50 - 500 X 0.80 - 2,000 X (- 0,40) - 500 X 0.70
= - 450
The gamma of the portfolio is
- 1, 000 X 2.2 - 500 X 0.6 - 2,000 X 1.3 - 500 X 1.8
= - 6, 000
The vega of the portfolio is
- 1, 000 X 1.8 - 500 X 0.2 - 2, 000 X 0.7 - 500 X 1.4
= - 4,000


(a) We require -450 + 0.6 Nc + 1Ns = 0
-6000 + 1.5Nc = 0
Solving these equations we obtain Nc = 4000 and Ns = -1950 pounds sterling ( ie a short position
in sterling)
(b ) We require -450 + 0.6 Nc + 1Ns = 0
-4000 + 0.8Nc + 0Ns = 0
Solving these two equations leads to Nc = 5000 and Ns = -2550 pounds (ie a short position in
sterling)