Replicating a Piecewise Linear Derivative with Options

Summary

The document shows how to price a European derivative with payoff equal to the sum of two absolute-value terms by expressing its piecewise linear payoff using the underlying asset, a cash position, and options at the payoff’s kink strikes. The general method is to rewrite absolute values using positive-part payoffs, then apply no-arbitrage pricing to each component. If only calls are quoted, put-call parity can convert put values into call values plus underlying and cash positions.

The response’s derivation identifies puts at both kink strikes, but its displayed pricing formula instead lists a put at the lower strike and a call at the higher strike. That conflicts with the preceding payoff decomposition, so the final leg list should be checked before use. The document provides an algebraic replication argument rather than market data or empirical tests. Its pricing expression also assumes the relevant discounting and dividend yield inputs are available and consistent with the option market.

Key ideas

• Piecewise linear payoffs can be decomposed into underlying, cash, and option positions.
• Absolute-value payoffs can be rewritten with positive-part payoff terms.
• No-arbitrage prices each component of a replicating portfolio.
• Put-call parity allows put values to be expressed using calls, the underlying, and cash.
• The response’s final option legs conflict with its own decomposition and require verification.

Tags

# Finding todays price of a derivative


# Finding todays price of a derivative












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.

## Answer by Kevin (score 2, accepted)

https://quant.stackexchange.com/a/48901

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.