dstefan
07-08-2008, 02:32 PM
Chapter 6, Problem 2
A bull spread is made of a long position in a call option with strike K and a short position in a call option with strike K+x, both options being on the same underlying asset and having the same maturities. Let C(K) and C(K+x) be the values (at time t) of the call options with strikes K and K+x, respectively.
(i) The value of a position in 1/x bull spreads is \frac{C(K) - C(K+x)}{x}. In the limiting case when x goes to 0, show that
\lim_{x \searrow 0} \frac{C(K) - C(K+x)}{x} ~=~ -\frac{\partial C}{\partial K}(K)
(ii) Show that, in the limiting case when x \to 0, the payoff at maturity of a position in 1/x bull spreads as above is going to approximate the payoff of a derivative security that pays 1 if the price of the underlying asset at expiry is above K, and 0 otherwise.
Note: A position in \frac{1}{x} bull spreads as above, with x small, is a synthetic way to construct a cash-or-nothing call maturing at time T.
Solution:
(i) Since the value C(K) of a call option as a function of the strike K of the option is infinitely many times differentiable, the first order forward finite difference approximation of \frac{\partial C}{\partial K}(K) is
\frac{\partial C}{\partial K}(K) ~=~ \frac{C(K+x)-C(K)}{x} ~+~ O(x), ~~\mbox{as}~~ x \to 0.
We conclude that
\lim_{x \searrow 0} \frac{C(K) - C(K+x)}{x} ~=~ -\frac{\partial C}{\partial K}(K)
(ii) The payoff at maturity of the bull spread is
\max(S-K,0) - \max(S-(K+x),0) ~= \left\{ \begin{array}{cl} 0, & \mbox{if}~~ S \leq K; \\ S-K, & \mbox{if}~~ K < S \leq K+x; \\ x, & \mbox{if}~~ K+x < S. \end{array} \right.
If g_x(S) denotes the payoff at maturity of a position in 1/x bull spreads, then
g_x(S) ~=~ \left\{ \begin{array}{cl} 0, & \mbox{if}~~ S \leq K; \\ \frac{S-K}{x}, & \mbox{if}~~ K < S \leq K+x; \\ 1, & \mbox{if}~~ K+x < S. \end{array} \right.
If S \leq K, then g_x(S) = 0 for any x>0 and therefore
\label{ch6_ex2_1}
\lim_{x \searrow 0} g_x(S) ~=~ 0, ~~\forall~ S \leq K
If S > K, then g_x(S) = 1 for any x such that 0 < x < S-K, and therefore
\label{ch6_ex2_2}
\lim_{x \searrow 0} g_x(S) ~=~ 1, ~~\forall~ S \leq K
From (\ref{ch6_ex2_1}) and (\ref{ch6_ex2_2}), we conclude that, in the limiting case when x \to 0, the payoff at maturity of a position in 1/x bull spreads as above is 1 if the underlying asset expires above K, and 0 otherwise.
A bull spread is made of a long position in a call option with strike K and a short position in a call option with strike K+x, both options being on the same underlying asset and having the same maturities. Let C(K) and C(K+x) be the values (at time t) of the call options with strikes K and K+x, respectively.
(i) The value of a position in 1/x bull spreads is \frac{C(K) - C(K+x)}{x}. In the limiting case when x goes to 0, show that
\lim_{x \searrow 0} \frac{C(K) - C(K+x)}{x} ~=~ -\frac{\partial C}{\partial K}(K)
(ii) Show that, in the limiting case when x \to 0, the payoff at maturity of a position in 1/x bull spreads as above is going to approximate the payoff of a derivative security that pays 1 if the price of the underlying asset at expiry is above K, and 0 otherwise.
Note: A position in \frac{1}{x} bull spreads as above, with x small, is a synthetic way to construct a cash-or-nothing call maturing at time T.
Solution:
(i) Since the value C(K) of a call option as a function of the strike K of the option is infinitely many times differentiable, the first order forward finite difference approximation of \frac{\partial C}{\partial K}(K) is
\frac{\partial C}{\partial K}(K) ~=~ \frac{C(K+x)-C(K)}{x} ~+~ O(x), ~~\mbox{as}~~ x \to 0.
We conclude that
\lim_{x \searrow 0} \frac{C(K) - C(K+x)}{x} ~=~ -\frac{\partial C}{\partial K}(K)
(ii) The payoff at maturity of the bull spread is
\max(S-K,0) - \max(S-(K+x),0) ~= \left\{ \begin{array}{cl} 0, & \mbox{if}~~ S \leq K; \\ S-K, & \mbox{if}~~ K < S \leq K+x; \\ x, & \mbox{if}~~ K+x < S. \end{array} \right.
If g_x(S) denotes the payoff at maturity of a position in 1/x bull spreads, then
g_x(S) ~=~ \left\{ \begin{array}{cl} 0, & \mbox{if}~~ S \leq K; \\ \frac{S-K}{x}, & \mbox{if}~~ K < S \leq K+x; \\ 1, & \mbox{if}~~ K+x < S. \end{array} \right.
If S \leq K, then g_x(S) = 0 for any x>0 and therefore
\label{ch6_ex2_1}
\lim_{x \searrow 0} g_x(S) ~=~ 0, ~~\forall~ S \leq K
If S > K, then g_x(S) = 1 for any x such that 0 < x < S-K, and therefore
\label{ch6_ex2_2}
\lim_{x \searrow 0} g_x(S) ~=~ 1, ~~\forall~ S \leq K
From (\ref{ch6_ex2_1}) and (\ref{ch6_ex2_2}), we conclude that, in the limiting case when x \to 0, the payoff at maturity of a position in 1/x bull spreads as above is 1 if the underlying asset expires above K, and 0 otherwise.