Today's market prices for European call options $c(T;K)$ and put options $p(T;K)$ with maturity T and any strike K. Let $B_t = e^{rt}$ be the price of the risk-free bond and St the price of the stock. a) Let $f(x) = |5-x| + |10-x|$. How would you calculate today's price of the derivative with payoff $f(S_T)$ at maturity T (in terms of call and put prices)? What if you can observe only market prices for European call options? I know that if I plot the function f(x), I get a payoff diagram that is a strangle. But I'm not quite sure on how I should proceed from here. I am also aware that the strike for one long call option is 5 and put option is 10.
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$\begingroup$ This thing is equivalent to a combination of 4 options: two puts and two calls.... $\endgroup$– Alex CSep 25, 2019 at 2:59
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$\begingroup$ Hi Alex, it would be great if you could expand on that.. $\endgroup$– AnonSep 25, 2019 at 3:56
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Recall that $|x|=\max\{x,-x\}=2\max\{x,0\}-x$. Thus, \begin{align*} f(x)&=|5-x|+|10-x| \\ &= 2\max\{5-x,0\} +x-5 + 2\max\{10-x,0\} +x-10 \\ &= 2x-15+ 2\max\{5-x,0\} + 2\max\{10-x,0\} \\ \end{align*} Thus, by no-arbitrage, the time $t$ price of $f(S_T)$ is given by $$V(t,S_t)= 2S_te^{-q(T-t)}-15e^{-r(T-t)} + 2P(S_t,5,T) +2C(S_t,10,T).$$
If all you have available are call options, use the put-call parity to transforms puts into corresponding call options: $$P(S_t,K,T) = Ke^{-r(T-t)}-Se^{-q(T-t)}+C(S_t,K,T).$$
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$\begingroup$ how would it change if the stock doesnt pay a dividend? $\endgroup$– AnonSep 28, 2019 at 14:17
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