STOCK PRICE IN AN ECONOMY

QUESTION

Finance (Derivative Securities)
Semester 1 2012

ASSIGNMENT 2

Note: a variant of these questions was in a previous final exam paper.

Students are referred to the unit outline which provides detailed information on the assessments,
including the due date and time which will be strictly enforced. Students are reminded that their
assignment should be their own work. Students should not simply cut and paste from an internet
search as that is easily discovered, risks plagiarism, and rarely answers the question. Students are
encouraged to visit the lecturer during consultation (Mon 09:00-12:00; Wed 09:00-12:00) if they
wish to discuss their answers. Both questions must be answered. Answers are mathematical and
hence a marking rubric is not required.

Questions

Let
( )
CK
denote a European vanilla Call option with strike price
K
. Assume that all options are
identical except for strike price, and strike prices satisfy
KKK<<
and
2K KK= +
.

123
Question 1 [5 marks]
What are the no-arbitrage lower bound, and the no-arbitrage upper bound, of the vertical spread
( ) ( )
CK CK?

12
Question 2 [10 marks]
Derive the functional relationship between the no-arbitrage values of the two vertical spreads,
( ) ( )
CK CK
and

(
) ( )
CK CK.

12
23
2 13

SOLUTION

Question 1

In the given situation it is given that K₁<K₂<K₃ thus the relation between CK₁, CK₂ and CK₃ will be CK₁ > CK₂ > CK₃. In the given situation as

Let the current Price of the stock be S₀ and at the time of expiry is S_t

Thus the Payoff for a given call is given by S_t – K_i if the call option is exercised and when S_t < K_i the call option will not be exercised. Thus the profit/ loss in case of exercising the call option will be given by S_t – K_i – CK_i

Thus the upper bound will be given by CK_i<=S₀ (Mercurio, 2010). Or it is the cost of the call option is equal to the current price of the stock.

Thus one stock is bought at current price S₀ and the call is sold at price C. Thus there will be minimum advantage of either K_i or S_y which will be greater than or equal to zero.

Thus for both the call options the upper bound is given by

Min (K₁,S_t) and Min (K₂,S_t). Since it is given that K₂= (K₁+K₃)/2 we can say that the upper bound for C(K₂) is Min ((K₁+K₃)/2, S_t) (Kothari, 2010)

Similarly the lower bound for the call option where in the call price is the present value of S_t– K_i which give a maximum of S_t and K_i.

Thus in case of the of lower bound the maximum advantage is given as Max( S_t, K_i)

Thus for both the call options the Lower bound is given by Max ((K₁+K₃)/2, S_t)

Max (S_t,K₁) and Max( S_t,K₂). Since it is given that K₂= (K₁+K₃)/2 we can say that the upper bound for C(K₂) is Max (S_t, (K₁+K₃)/2). Thus the upper bound will be Min (K₁,S_t)  –  Min ((K₁+K₃)/2, S_t) and the lower bound is given by Max (K₁, S_t) – Max ((K₁+K₃)/2, S_t)

Question 2

The functional relationship between C(K₁) – CK(₂) and C(K₂) – CK(₃). The lower and upper bounds can be derived as shown in the above question as shown in the above question.

Thus the lower bound for C(K₁) – CK(₂) will be Max (K₁, S_t) – Max ((K₁+K₃)/2, S_t) and the lower bound of C(K₂) – CK(₃) will be Max ((K₁+K₃)/2, S_t) – Max (K₃, S_t).

Thus the relation between the lower bound will be

(Lower Bound) of C(K₁) – CK(₂) = Max (K₁, S_t) – Max (K₃, S_t). – (Lower Bound) of C(K₂) – CK(₃)

Also the Upper Bound will be given by Min (K₁,S_t)  –  Min ((K₁+K₃)/2, S_t) for C(K₁) – CK(₂) and Min ((K₁+K₃)/2, S_t) – Min (K₃,S_t)

Thus the upper bound of C(K₁) – CK(₂) = Min (K₁,S_t) – Min (K₃,S_t)   – (Upper Bound) of C(K₂) – CK(₃)

References:

Vinod Kothari. (2010). Option pricing. Available: http://vinodkothari.com/tutorials/option%20pricing.pdf. Last accessed 07th May 2012

Fabio Mercurio. (2010). No-Arbitrage Conditions for a Finite Options System. Available: http://www.fabiomercurio.it/NoArbitrage.pdf. Last accessed 07th May 2012.

Sergei Fedotov. (2010). Introduction to Financial Mathematics. Available: http://www.maths.manchester.ac.uk/~sf/20912lecture6.pdf. Last accessed 07th May 2012

KH59

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