Building Synthetic Options on Cointegrated Stock Portfolios
Summary
The document considers three cointegrated stocks with known portfolio weights and a mean-reversion view: when the weighted portfolio falls unusually low, a trader may expect it to recover. It first describes matching the portfolio’s directional exposure by holding weighted stock shares, then outlines a simple option construction that aligns each option’s delta with its stock’s portfolio weight. Calls are used for positively weighted stocks and a put for the negatively weighted stock, creating a long-delta, positive-gamma position resembling a call on the portfolio.
The question asks how to extend this construction to nonlinear payoff shapes, such as a one-by-two ratio spread or butterfly, using options on the constituent stocks. It provides no solution, pricing analysis, or evidence that a particular combination replicates the portfolio option payoff. The main limitation is that matching deltas alone does not establish a full payoff replication: the document leaves the choice of strikes, quantities, and treatment of changing correlations or hedge exposures unanswered.
Key ideas
- A cointegrated portfolio’s stock weights can guide the construction of a directional position in its constituents.
- Matching option deltas to portfolio weights gives an initial long-delta position with positive gamma.
- Calls on positively weighted stocks and a put on a negatively weighted stock can approximate a call-like exposure to the portfolio.
- The document poses, but does not answer, how to extend the method to ratio spreads or butterflies.
Tags
Full text
# Given a cointegrated portfolio of stocks, how to build a synthetic option position by using the options on the stocks # Given a cointegrated portfolio of stocks, how to build a synthetic option position by using the options on the stocks ### The setting Let stocks $A$, $B$, and $C$ are cointegrated. Moreover, we know the weights of the cointegrated portfolio (scaled so that the absolute value of the maximum weight is $0.5$): - $w_A = 0.15$ - $w_B = 0.35$ - $w_C = -0.5$ And one fine day, the cointegrated portfolio - a stationary time series, of course - goes below its first percentile: time to buy the portfolio and make some statistical arbitrage! Had a trader to go long on this portfolio, he could just buy $nw_A$ shares, buy $nw_B$ shares, and short $n|w_C|$ shares provided that the ratio between the weights reflected the balance stated by the cointegration relation ($n > 0$ is any integer). Had the same trader to start a long Delta position on this portfolio, he could use the weights $w_i$ as a reference to pick the options he liked the most and set the Delta, $\Delta_i$, of each position $i$ equal to $w_i$: the simplest Gamma-positive position would be to buy Call options on $A$ and $B$ with $\Delta_i = w_i$ and to buy Put options on $C$ with $\Delta_C = w_C$ and this would be like buying a Call option on the cointegrated portfolio. ### The question Let you want to build a position on the cointegrated portfolio that is more complex than a single (synthetic) naked Call/Put option like a ratio spread 1x2 or a butterfly; the only constrain is that the "view" shall be the same, that is, the portfolio is too cheap and we shall bet on the mean reversion. For example, let $s_t$ be the cointegrated portfolio like in the picture below and you want to build this payoff (1x2 ratio spread): How would you do that? Which options on $A$, $B$, and $C$ would you pick and why?
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.