## Transcribed Text

(1) The prices of three put options with strikes 40, 50, and 70, but otherwise identical, are $10,
$20, and $30, respectively.
(i) Show that the upward convexity of the price of the put with respect to the strike price
is violated in this case;
(ii) Find an arbitrage opportunity.
(2) Show that
xty 2 ,
2
(3) Compute the integral of the function f(x,y) = x2 - 2y on the region bounded by the
parabola y - (x+1)² and the line y - 5.x - 1.
(4) Which is bigger, et or ITe? Provide the proof for your answer without use of calculators!
(5) Compute
1 dac dy,
D
where
D = + + 4 < 1 }
(6) Recall that the vega of a plain vanilla European call is
- 1 e 2 ,
where
In ( + - T
d1
=
OVI
(i) Show that
S
lim vega(S) = 0 and Jim vega(S) = 0.
(ii) Show that, as a function of the spot price S of the underlying asset, vega(C) is first
increasing and then decreasing.
(iii) Find the spot price corresponding to the maximum value of vega(C).
(7) (i) Find the value of a six months down-and-out call on a non-dividend-paying asset with
price following a lognormal distribution with 30% volatility and spot price 40. The barrier
is B - 35 and the strike for the call is K - 40. The risk-free interest rate is constant at 5%.
1
(ii) Show that the value of a down-and-out call with B < K converges to the value of a
plain vanilla call with strike K, as B goes down to zero. For simplicity, assume that the
underlying asset does not pay dividends and that interest rates are zero.
(8) Consider a knock-out call option with barrier B and strike K with B - K < S(0) expiring
at time T. Derive a formula for the value of this call option. Derive the gamma and vega
of this option.
(9) (i) A European asset-or-nothing call option has a payoff equal to the asset price, as long as
the asset price exceeds the strike price K at maturity T. Use the assumptions behind the
Black-Scholes model and the risk-neutral valuation to derive the formula for the value of
such option.

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