---
title: "Merger Arbitrage"
book: "Strategies I: Equities and Futures"
subject: quant
language: en
chapter: 11
exercises: 8
source: https://one-course.com/books/quant/8/en/chapter/11-merger-arbitrage
---

# Chapter 11 — Merger Arbitrage

A company agrees to be bought for $40 a share in cash, and its stock trades at $38.60. The $1.40 spread, 3.6%, is what the market charges for the risk that the deal breaks, in which case the stock falls back toward the $30 it traded at before. Buying at $38.60 is selling insurance against that break: the premium is collected in small amounts deal after deal, and the claims arrive together, because deals break more often when markets fall. Mitchell and Pulvino measured it over 4 750 mergers: [merger arbitrage](#def-s1-merger-arbitrage-arb) returns are uncorrelated with the market in flat and rising markets and correlated with it in severely falling ones, like the returns from selling index puts, and still earned excess returns of about 4% a year after costs, once that nonlinearity is controlled for. This chapter prices [deal spreads](#def-s1-merger-arbitrage-spread), builds a portfolio of synthetic deals whose breaks cluster in downturns, and measures what the insurance pays. The build is `firm.mergerarb`.

## 11.1 Deal spreads and their pricing

**Definition 11.1 (Merger arbitrage, tender offer).**

*Merger arbitrage* buys the shares of a company that has agreed to be acquired, below the offered consideration, and (in a stock deal) sells short the acquirer’s shares it will receive, to earn the difference when the deal completes. A *tender offer* is an offer made directly to a company’s shareholders to buy their shares at a stated price, as opposed to a merger voted by the board and shareholders.

**Definition 11.2 (Deal spread, break price, implied deal probability).**

The *deal spread* is the offered consideration divided by the target’s price, minus one. The *break price* is the price at which the target would trade if the deal failed. The *implied deal probability* is the completion probability that makes the target’s price the discounted average of the consideration and the break price: $p = (A - B)/(K - B)$ for price $A$, offer $K$ and break price $B$, ignoring discounting.

For the $40 deal, $p = (38.60 - 30)/(40 - 30) = 0.86$: the market prices a 14% chance of a break. At that probability the trade has no expected profit ($0.86 \times 1.40 = 0.14 \times 8.60$); the arbitrageur earns only if deals break less often than the spread implies. Spreads are quoted annualised because deals take months: a 3.6% spread over 84 trading days is 11.3% a year. The synthetic deals (`firm.mergerarb.simulate_deals`) are cash deals announced at about fifty a year over the synthetic market’s ten years, at premiums of 20% to 40%, resolving after a median of 110 trading days; their [break price](#def-s1-merger-arbitrage-spread) moves with the market; their spread prices a 12% break probability; their true break probability is 6% plus 1.5 times the market’s fall over the deal’s life. The average spread at entry is 4.73%, 11.8% annualised, and the implied completion probability 88%.

**As of September 2026 — US premerger waiting periods.**

Under the Hart-Scott-Rodino premerger notification program, once the parties’ filings are complete they must wait 30 days (15 days for a cash [tender offer](#def-s1-merger-arbitrage-arb) or a bankruptcy) before consummating the deal, unless the agencies grant early termination; after a preliminary review the reviewing agency (the FTC or the Department of Justice) can let the period expire, end it early, or issue a Second Request for more information, which extends the review.

## 11.2 Break risk and its correlation with the market

| synthetic deal portfolio, years 3 to 10 | base case | breaks | spread priced | both |
| --- | --- | --- | --- | --- |
|  |  | independent | at 6% |  |
| return a year; over the 4% rate | 6.76%; 2.76% | —; 4.30% | —; $-0.47\%$ | —; 1.03% |
| volatility; Sharpe ratio (excess) | 3.26%; 0.85 | —; 1.50 | —; $-0.17$ | —; 0.46 |
| beta in months the market fell over 4% | 0.20 | 0.20 | 0.17 | 0.17 |
| beta in the other months | 0.09 | 0.11 | 0.04 | 0.06 |
| completed deals | 92.5% | 94.7% | 92.5% | 94.7% |

The base-case portfolio holds 22 deals on average, 374 over the eight years, equally weighted. It earns 2.76% a year over the rate its spreads discount at, with a volatility of 3.26% and a Sharpe ratio of 0.85. Deals completed 92.5% of the time, against the 88% the spreads implied: the gap is the premium. Deals broke 13.7% of the time when the market fell over their life (117 deals) and 4.7% when it rose. The portfolio’s beta is 0.20 in the ten months when the market fell by more than 4%, and 0.09 in the other 86: the put-like profile, weaker than in the historical data because the synthetic market had few severe falls. Two counterfactual columns separate the pieces. If breaks did not depend on the market, the same spreads would earn 4.30%: the clustering of breaks in downturns costs 1.5 points a year. If spreads priced only the flat-market break rate, the portfolio would lose 0.47% a year: the premium in the spread is exactly the price of the crash risk.

![Monthly returns of the synthetic deal portfolio against the market’s, years 3 to 10; the dashed line marks a 4% market fall. Data: s1_mergerarb.run.](https://one-course.com/images/onecourse/chapters/quant-8/s1-merger-arbitrage/fig-dc529c79ec41.svg)

***Figure 11.1.** Monthly returns of the synthetic deal portfolio against the market’s, years 3 to 10; the dashed line marks a 4% market fall. Data: `s1_mergerarb.run`.*

The worst month for the portfolio ($-3.06\%$) was a month in which the market rose 2.8%: with twenty deals, two or three breaking together is enough, and deal-specific breaks are the other half of the risk. A merger-arbitrage book diversifies that half by holding many deals; it cannot diversify the half that comes from the market.

## 11.3 Stock deals and collars

**Definition 11.3 (Stock-deal collar).**

A *stock-deal collar* fixes the exchange ratio of a stock deal while the acquirer’s price stays inside a band, and fixes the value of the consideration outside it (the ratio then adjusts), protecting the target’s shareholders against large falls in the acquirer’s price.

In a stock deal the target’s holders receive a number of acquirer shares; the arbitrageur buys the target and sells short that number of acquirer shares, so that the spread no longer depends on the acquirer’s price. A collar changes the hedge. With a ratio of 0.5 and a band of $40 to $50, an acquirer price of $42 gives consideration of $21, hedged by shorting 0.5 acquirer shares; below $40, at $36, the consideration is fixed at $20 and the hedge is none, since the value no longer moves with the acquirer. The hedge ratio is the collar’s delta (`collar_hedge`), and near the band’s edges it jumps: collar deals need rebalancing, and their spreads carry the gamma of that option.

## 11.4 Regulatory and financing risk

Deals break for reasons the market’s level does not capture. Antitrust review (the waiting periods in the dated box, and a Second Request that can extend them for months) is the best-known; financing that disappears, a shareholder vote that fails, a competing bid (which raises the price instead) and material adverse changes in the target’s business are the others. Each has its own probability and its own [break price](#def-s1-merger-arbitrage-spread), and a deal’s spread is a mix of them. The synthetic model folds them into one probability with a market-dependent part; a real desk estimates them deal by deal, reads the merger agreement, and sizes positions by what each deal can lose, not by its spread.

## 11.5 Portfolio of deals

[Merger arbitrage](#def-s1-merger-arbitrage-arb) is limited by capital as much as by opportunity. Baker and Savasoglu studied the limits to that capital in mergers and acquisitions; Mitchell and Pulvino’s put-like payoff is one reason it is limited. A deal book’s risk budget has three lines: deal-specific breaks, which diversify across twenty or more deals; market-dependent breaks, which do not; and the market exposure of the [break prices](#def-s1-merger-arbitrage-spread) themselves, which shows up as the book’s beta in falls. The last two are the insurance sold. The honest benchmark for the book is a short put on the index, not cash.

## 11.6 Strategy files

**Strategy file 11.1 — Cash-deal spread.**

**Who pays you, and why.** Target shareholders who sell after the announcement rather than wait and bear the break risk.

**Instruments and venues.** The target’s shares.

**Signal.** The annualised spread against an estimate of the break probability and [break price](#def-s1-merger-arbitrage-spread).

**Sizing and execution.** Size by the loss if the deal breaks, not by the spread; enter after the announcement.

**Costs.** Low turnover; financing of the position.

**How it dies.** Breaks, especially in downturns; regulatory surprises.

**Horizon, capacity, infrastructure.** Months; capacity limited by deal size; legal and regulatory analysis.

**Backtest honestly.** Deals as announced, including those that broke; [break prices](#def-s1-merger-arbitrage-spread) estimated before the outcome.

**Sources.** Mitchell and Pulvino (2001): 4 750 mergers, 1963–1998, excess returns of about 4% a year after costs, put-like payoff.

**Strategy file 11.2 — Stock-for-stock hedged spread.**

**Who pays you, and why.** As for cash deals, with the acquirer’s price risk hedged.

**Instruments and venues.** The target, long; the acquirer, short in the exchange ratio.

**Signal.** The spread between the target’s price and the ratio times the acquirer’s.

**Sizing and execution.** Keep the short at the exchange ratio; adjust for dividends.

**Costs.** Borrow on the acquirer, sometimes expensive when many arbitrageurs short it.

**How it dies.** Breaks; a squeeze in the acquirer if many are short.

**Horizon, capacity, infrastructure.** Months; borrow availability.

**Backtest honestly.** Borrow fees and recalls on the acquirer; the acquirer’s move on a break.

**Sources.** Mitchell and Pulvino (2001).

**Strategy file 11.3 — Collar deals.**

**Who pays you, and why.** As for stock deals, with an option embedded in the consideration.

**Instruments and venues.** The target, the acquirer, and possibly listed options to hedge the collar’s gamma.

**Signal.** The spread against the collar’s value.

**Sizing and execution.** A hedge ratio equal to the collar’s delta, rebalanced near the band’s edges.

**Costs.** Rebalancing of the hedge.

**How it dies.** Hedging errors near the band’s edges; breaks.

**Horizon, capacity, infrastructure.** Months; an option model of the collar.

**Backtest honestly.** The collar’s terms as written, including averaging periods.

**Sources.** This chapter’s collar arithmetic; no performance record verified.

**Strategy file 11.4 — Diversified deal portfolio.**

**Who pays you, and why.** The market, for bearing concentrated break risk in downturns.

**Instruments and venues.** Twenty or more announced deals.

**Signal.** Each deal’s spread against its estimated risk.

**Sizing and execution.** Equal risk per deal; limits on deals in the same industry or with the same regulator.

**Costs.** Low.

**How it dies.** Market crashes, when breaks cluster and [break prices](#def-s1-merger-arbitrage-spread) fall together.

**Horizon, capacity, infrastructure.** Continuous; capacity set by the deal flow.

**Backtest honestly.** All deals, point in time; compare with a short index put.

**Sources.** Mitchell and Pulvino (2001); this chapter’s simulation (2.76% a year over the rate, beta 0.20 in falling months and 0.09 otherwise).

## 11.7 Tutorial: selling insurance

**Goal.** Price [deal spreads](#def-s1-merger-arbitrage-spread), simulate a portfolio of deals whose breaks depend on the market, and measure its payoff. **End state:** the table and [Figure 11.1](#fig-s1-merger-arbitrage-months).

1. **Pricing**: implied probability and the collar. `def implied_probability (price, offer, break_price, rate: float = 0.0 , days: float = 0.0 ): disc = math.exp(-rate * days / 252 ) return (np.asarray(price, float ) - disc * np.asarray(break_price, float )) / \ (disc * (np.asarray(offer, float ) - np.asarray(break_price, float ))) def collar_value (acquirer, ratio: float , low: float , high: float ): return ratio * np.clip(np.asarray(acquirer, float ), low, high) def collar_hedge (acquirer, ratio: float , low: float , high: float ): a = np.asarray(acquirer, float ) return np.where((a > low) & (a < high), ratio, 0.0 )` **Listing 11.1.** Implied completion probability and collar terms. code/firm/mergerarb/firm_mergerarb.py
2. **Deals**: announcement, price path, break or completion. `def simulate_deals (mkt, cfg: DealConfig | None = None , rng=None ): cfg = cfg or DealConfig() rng = rng or np.random.default_rng(11 ) mkt = np.asarray(mkt, float ) T = len (mkt) lm = np.concatenate([[0.0 ], np.cumsum(np.log1p(mkt))]) deals = [] for t0 in np.flatnonzero(rng.random(T) < cfg.per_year / 252 ): n = int (np.clip(round (cfg.days_median * math.exp(cfg.days_disp * rng.standard_normal())), 20 , 400 )) if t0 + n >= T: continue prem = rng.uniform(*cfg.premium) offer = 1.0 + prem # the undisturbed price is 1 walk = np.cumsum(cfg.idio * rng.standard_normal(n + 1 )) brk = np.exp(cfg.beta * (lm[t0:t0 + n + 1 ] - lm[t0]) + walk) tau = n - np.arange(n + 1 ) disc = np.exp(-cfg.rate * tau / 252 ) price = disc * ((1 - cfg.q) * offer + cfg.q * brk) fall = max (-(lm[t0 + n] - lm[t0]), 0.0 ) broke = rng.random() < min (cfg.base_break + cfg.crash_break * fall, 0.95 ) price[-1 ] = brk[-1 ] if broke else offer deals.append({" start " : int (t0), " end " : int (t0 + n), " offer " : offer, " price " : price, " broke " : bool (broke), " implied " : float (implied_probability(price[0 ], offer, brk[0 ], cfg.rate, n))}) return deals` **Listing 11.2.** A deal’s life, with market-dependent breaks. code/firm/mergerarb/firm_mergerarb.py
3. **Run** `run()` and its counterfactuals, `break_rates()` and `fig_mergerarb.py` .

**What to change next.** Add stock deals with collars and hedge them; let spreads widen after market falls (the market reprices $q$); cap the book’s exposure to deals in one industry.

## 11.8 Build: merger arbitrage

**Purpose.** Deal-spread pricing and a portfolio of deals with market-dependent breaks.

**Interface.** `spread(price, offer)`, `annualised(spread, days)`, `implied_probability(price, offer, break_price, rate, days)`, `collar_value(acquirer, ratio, low, high)`, `collar_hedge(acquirer, ratio, low, high)`, `DealConfig`, `simulate_deals(mkt, cfg, rng)`, `portfolio(deals, T)`.

**Rules.** Deals entered after their announcement; breaks decided by information at the deal’s end; returns in excess of the discount rate reported.

**Acceptance tests.** `code/firm/mergerarb/tests/`: spread, implied probability and annualisation by hand; collar value and hedge; deals in a flat market break at the base rate and the portfolio earns a positive return.

**Stretch.** Stock deals with acquirer hedges; competing bids; spreads that widen when the market falls.

Sources and further reading

- M. Mitchell and T. Pulvino, “Characteristics of risk and return in risk arbitrage”, *Journal of Finance* 56(6), 2001.
- M. Baker and S. Savasoglu, “Limited arbitrage in mergers and acquisitions”, *Journal of Financial Economics* 64(1), 2002.
- US Federal Trade Commission, Premerger Notification and the Merger Review Process.

## 11.9 Exercises

**Exercise 11.1 ★.**

A cash offer of $40 trades at $38.60 with a [break price](#def-s1-merger-arbitrage-spread) of $30. Compute the spread and the implied completion probability.

**Solution of Exercise 11.1.**

Spread $40/38.60 - 1 = 3.63\%$; implied probability $(38.60 - 30)/(40 - 30) = 0.86$.

**Exercise 11.2 ★.**

The deal is expected to close in 84 trading days. Annualise the spread.

**Solution of Exercise 11.2.**

$(1.0363)^{252/84} - 1 = 11.3\%$ a year.

**Exercise 11.3 ★.**

A stock deal pays 0.5 acquirer shares with a collar from $40 to $50. What is the consideration at acquirer prices of $42 and $36, and how many acquirer shares do you short per target share in each case?

**Solution of Exercise 11.3.**

At $42, inside the band, the consideration is $0.5 \times 42 = \$21$ and the hedge is 0.5 acquirer shares short. At $36, below the band, the consideration is fixed at $0.5 \times 40 = \$20$ and the hedge is zero: the value no longer depends on the acquirer.

**Exercise 11.4 ★★.**

Show that buying the $40 deal at $38.60 has zero expected profit at the implied probability, and state what the arbitrageur must believe to trade.

**Solution of Exercise 11.4.**

Expected profit $0.86 \times (40 - 38.60) - 0.14 \times (38.60 - 30) = 1.204 - 1.204 = 0$. To trade, the arbitrageur must believe the break probability is below 14% (or the [break price](#def-s1-merger-arbitrage-spread) above $30), after discounting and costs.

**Exercise 11.5 ★★.**

Why does a deal portfolio’s beta rise in falling markets even if its deals are unrelated to each other?

**Solution of Exercise 11.5.**

Each deal’s [break price](#def-s1-merger-arbitrage-spread) moves with the market, so a market fall marks every open position down at once, whether or not any deal breaks; and more deals break when the market falls. Both are common to all deals and do not diversify.

**Exercise 11.6 ★★.**

What does a Second Request do to a deal’s spread, and why?

**Solution of Exercise 11.6.**

It extends the review by months and signals antitrust concern: the break probability rises and the time to completion lengthens, so the spread widens and its annualised return falls for the same absolute spread.

**Exercise 11.7 ★★★.**

*Coding.* Run `run(0.0, 0.12, 0.06)` and `run(1.5, 0.06, 0.06)`. Explain what each counterfactual isolates.

**Solution of Exercise 11.7.**

With breaks independent of the market the portfolio earns 4.30% over the rate (Sharpe ratio 1.50): the clustering of breaks in downturns costs 1.5 points a year. With spreads priced at the flat-market break rate it loses 0.47% (Sharpe ratio $-0.17$): the premium in the base-case spread is what pays for the crash-dependent breaks.

**Exercise 11.8 ★★★.**

*Find the flaw.* “[Merger arbitrage](#def-s1-merger-arbitrage-arb) has a Sharpe ratio near one and a beta near zero; it is market neutral.”

**Solution of Exercise 11.8.**

Its beta near zero is an average: close to zero in rising and flat markets and higher in falling ones (0.20 against 0.09 here, larger in the historical data), like a short put. It is short crash risk, and a Sharpe ratio built on months without crashes overstates its quality.

## 11.10 Problem: Selling Insurance

**Problem 11.1.**

Weekend problem — the premium and the claims

The chapter’s synthetic deals and the public record.

**Part I — Pricing.**

1. Define [merger arbitrage](#def-s1-merger-arbitrage-arb) , a [tender offer](#def-s1-merger-arbitrage-arb) , the [deal spread](#def-s1-merger-arbitrage-spread) and the [break price](#def-s1-merger-arbitrage-spread) .
2. Compute the implied probability of the opening deal.
3. Why are spreads annualised?
4. Describe the synthetic deals.

**Part II — The portfolio.**

5. Give its return, excess return, volatility and Sharpe ratio.
6. Compare the implied and realised completion rates.
7. How do break rates depend on the market?
8. Give the betas in falling and other months.

**Part III — The pieces.**

9. What do the two counterfactuals show?
10. Describe the worst month and what it says about deal-specific risk.
11. How are stock deals and collars hedged?
12. What are the non-market reasons deals break?

**Part IV — The verdict.**

13. State the *named result* : the portfolio’s beta in down and up markets and its implied completion probability against realised completion.
14. What did Mitchell and Pulvino find?
15. What is the right benchmark for a deal book?
16. How would you size individual deals?
17. What does the HSR waiting period mean for a deal’s timeline?
18. Which risk can a deal book diversify, and which not?
19. When would you add to the book: after a market fall or after a rise?
20. In one sentence: what does a merger arbitrageur sell?

**Solution of Problem 11.1.**

1. Buying targets below the consideration (hedging stock deals) to earn the spread on completion; an offer made directly to shareholders; offer over price minus one; the price if the deal fails.
2. 0.86.
3. Deals take months; the same spread over different horizons is a different return.
4. About fifty cash deals a year, premiums of 20–40%, a median of 110 days, [break prices](#def-s1-merger-arbitrage-spread) moving with the market, spreads pricing a 12% break probability, true break probability 6% plus 1.5 times the market’s fall.
5. 6.76% a year, 2.76% over the rate, volatility 3.26%, Sharpe ratio 0.85.
6. 88% implied, 92.5% realised.
7. 13.7% when the market fell over the deal’s life, 4.7% when it rose.
8. 0.20 in months the market fell over 4%, 0.09 otherwise.
9. Independent breaks earn 4.30%: clustering costs 1.5 points; spreads priced at 6% lose 0.47%: the premium pays for the crash risk.
10. $-3.06\%$ in a month the market rose 2.8%: a few deal-specific breaks together.
11. Short the exchange ratio of acquirer shares; with a collar, short its delta, which jumps at the band’s edges.
12. Antitrust review, financing, shareholder votes, competing bids, material adverse changes.
13. **Named result.** The portfolio’s beta is 0.20 in months when the market falls more than 4% and 0.09 in other months; the spreads imply an 88% completion probability and 92.5% of deals complete.
14. Returns correlated with the market in severe falls and not otherwise, like a short index put; about 4% a year in excess returns after costs, 1963–1998.
15. A short out-of-the-money index put.
16. By the loss if it breaks, relative to the book.
17. At least 30 days (15 for cash [tender offers](#def-s1-merger-arbitrage-arb) ) after complete filings, longer with a Second Request.
18. Deal-specific breaks; not market-dependent breaks and [break prices](#def-s1-merger-arbitrage-spread) .
19. After a fall, when spreads are wide, if the capital can bear further falls.
20. Insurance against deals breaking, which pays most in bad markets.

## 11.11 Interview questions

**Interview question 11.1 ★ trader, researcher.**

What is [merger arbitrage](#def-s1-merger-arbitrage-arb), and where does its return come from?

**Solution of Interview question 11.1.**

Buying the target of an announced acquisition below the offer (and shorting the acquirer in stock deals) to earn the spread when the deal closes. The return is a premium for bearing break risk, much of which comes in market downturns.

**Interview question 11.2 ★★ trader.**

How do you hedge a stock-for-stock deal with a collar?

**Solution of Interview question 11.2.**

Short the collar’s delta in acquirer shares: the exchange ratio inside the band, zero outside it where the value is fixed; rebalance as the acquirer’s price approaches the band’s edges, where the delta jumps, or buy options that offset the collar’s gamma.

**Interview question 11.3 ★★ risk.**

Why is a merger-arbitrage book compared to a short put?

**Solution of Interview question 11.3.**

Its payoff is near zero exposure in normal markets and losses in crashes, when deals break and [break prices](#def-s1-merger-arbitrage-spread) fall together: small steady premiums with rare large correlated losses, the profile of a short put.

**Interview question 11.4 ★★ researcher.**

How would you estimate a deal’s break probability and [break price](#def-s1-merger-arbitrage-spread)?

**Solution of Interview question 11.4.**

[Break price](#def-s1-merger-arbitrage-spread) from the undisturbed price moved by the market and the sector since the announcement, or from peers; break probability from deal features (regulatory overlap, financing, hostility, vote), base rates of similar deals, and the market-implied probability as a prior.

**Interview question 11.5 ★★ trader.**

A deal’s spread doubles overnight with no news. What might have happened?

**Solution of Interview question 11.5.**

Rumours of regulatory trouble or financing problems, a market fall moving [break prices](#def-s1-merger-arbitrage-spread), forced selling by other arbitrageurs, or a borrow problem in the acquirer for a stock deal.

**Interview question 11.6 ★★★ researcher.**

Derive the implied completion probability with discounting at rate $r$ over $\tau$ years, and show how the arbitrageur’s expected excess return depends on the gap between the implied and the true probability.

**Solution of Interview question 11.6.**

With price $A = e^{-r\tau}(pK + (1 - p)B)$, the implied probability is $p = (Ae^{r\tau} - B)/(K - B)$. If the true probability is $\pi$, the expected payoff is $\pi K + (1 - \pi)B$ and the expected excess return is $(\pi - p)(K - B)/(Ae^{r\tau})$: proportional to the gap between the true and implied probabilities and to the spread between the offer and the [break price](#def-s1-merger-arbitrage-spread).