last updated: October 16, 2024

Stochastic Processes and Financial Mathematics
(part two)

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16.4 Exercises on Chapter 16

On parity relations
  • 16.1

    • (a) Draw graphs of the functions \(\Phi ^{cash}(S_T)\), \(\Phi ^{stock}(S_T)\), \(\Phi ^{call}(S_T)\) and \(\Phi ^{put}(S_T)\) defined in Section 16.1, as functions of \(S_T\).

    • (b) Verify that the put-call parity relation (16.1) holds.

  • 16.2 Use the put-call parity relation to show that the price at time \(0\) of a European put option with strike price \(K\) and exercise date \(T\) is given by \(Ke^{-rT}\mc {N}(-d_2)-S_0\mc {N}(-d_1)\), where \(d_1\) and \(d_2\) are from (15.23).

  • 16.3 Write down a constant portfolio, which may consist of cash, stock and/or European call and put options, that would replicate the contingent claim

    \[ \begin {cases} S_T-1 & \text { if }S_T>1,\\ 0 & \text { if }S_T\in [-1,1]\\ -1-S_T & \text { if }S_T<-1. \end {cases} \]

  • 16.4 Let \(A\geq 0\) and \(K\geq 0\) be deterministic constants. Consider the contingent claim, with exercise date \(T\),

    \[\Phi (S_T)= \begin {cases} K & \text { if }S_T\leq A,\\ K+A-S_T & \text { if }A\leq S_T\leq K+A,\\ 0 & \text { if }K+A<S_T. \end {cases} \]

    • (a) Sketch (for general \(A\geq 0\)) a graph of \(\Phi (S_T)\) as a function of \(S_T\). Find a constant portfolio consisting of European put options, with exercise dates and strike prices of your choice, that replicates \(\Phi (S_T)\).

    • (b) Find a constant portfolio consisting of cash, stock, and European call options (with exercise dates and strike prices of your choice) that replicates \(\Phi (S_T)\).

    • (c) In parts (a) and (b) we found two different replicating portfolios for \(\Phi (S_T)\). However Corollary 15.3.3 claimed that ‘replicating portfolios are unique’. Why is this not a contradiction?

  • 16.5 Let \(A\) and \(B\) be deterministic constants with \(A<B\). Consider the contingent claim

    \[ \Phi ^{bull}(S_T)= \begin {cases} B & \text { if }S_T>B,\\ S_T & \text { if }A\leq S_T\leq B,\\ A & \text { if }S_T<A. \end {cases} \]

    This is known as a ‘bull spread’. Find a constant portfolio consisting of cash and call options that replicates \(\Phi ^{bull}(S_T)\).

On the Greeks and delta/gamma hedging
  • 16.9 Let \(\beta >2\). Find the values of all the Greeks, at time \(t\in [0,T]\), for the derivative with contingent claim \(\Phi (S_T)=S_T^\beta \). (Hint: You will need part (b) of 15.4.)

  • 16.10 At time \(t\) you hold a portfolio \(h\) with value \(F(t,S_t)\), for which (at time \(t\)) \(\Delta _F=2\) and \(\Gamma _F=3\).

    • (a) You want to make this portfolio delta neutral by adding a quantity of the underlying stock \(S_t\). How much should you add? What is the cost of doing so?

    • (b) You want to make this portfolio both delta and gamma neutral, by adding a combination of the underlying stock \(S_t\) as well a second financial derivative with value \(D(t,S_t)\), for which \(\Delta _D=1\) and \(\Gamma _D=2\). How much of each should you add? What is the cost of doing so?

  • 16.11 Consider trying to gamma hedge a portfolio with value \(F(t,S_t)\), by adding in an amount \(w_t\) of a financial derivative with value \(W(t,S_t)\) and an amount \(z_t\) of a financial derivative with value \(Z(t,S_t)\).

    • (a) An excitable mathematician suggests the following idea:

      First, delta hedge using \(W\): add in \(w_t=-\frac {\Delta _F}{\Delta _W}\) of the first derivative to make our portfolio delta neutral. Then, add in a suitable amount \(z_t\) of \(Z\) to make the portfolio gamma neutral.

      Why does this idea not work?

    • (b) Consider the case in which \(Z(t,S_t)=S_t\). In this case, solve the equations (16.4) to find explicit expressions for \(w_t\) and \(z_t\).

    • (c) Does the following idea work? Explain why, or why not.

      First, add in an amount \(w_t\) of the first derivative to make the portfolio gamma neutral. Then, add in a suitable amount \(z_t\) of stock to make the portfolio delta neutral.

Challenge Questions
  • 16.12 Use the Black-Scholes formula (15.23) to verify that, in the case of a European call option with strike price \(K\) and exercise date \(T\), the Greeks are given by

    \[\Delta =\mc {N}(d_1),\hspace {5pc} \Gamma =\frac {\phi (d_1)}{s\delta \sqrt {T-t}},\]

    \[\rho =K(T-t)e^{-r(T-t)}\mc {N}(d_2),\hspace {1.7pc} \Theta =-\frac {s\phi (d_1)\sigma }{2\sqrt {T-t}}-rKe^{-r(T-t)}\mc {N}(d_2),\hspace {1.7pc} \mc {V}=s\phi (d_1)\sqrt {T-t}.\]

    Here, \(\phi (x)=\frac {1}{\sqrt {2\pi }}e^{-x^2/2}\) is the p.d.f. of a \(N(0,1)\) distribution, and \(\mc {N}(x)\) is its c.d.f.

    Use the put-call parity relation to find the values of the Greeks in the case of a European put option.