Expected date of exercise - American put

Question

I am interested in an analytic or computational estimate of the expected date of exercise of an American put. Are there research papers (or discussions on this site) estimating the expected date upon which the holder of an American put would early exercise?

The potential dates are bounded $0 \le t \le T$, where $t=0$ is the current date and $t=T$ is the expiry of the option. What is $\textrm{E}\left(t\right)$?

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I have QuantLib available if this expected date of exercise problem is solved/solvable with that toolkit.

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Revision 1

I am interested in the expected date of termination $ \textrm{E}\left( t_\textrm{termination}\right) $ for an American put contract.

$$ 0 \le \textrm{E}\left( t_\textrm{termination}\right) = \begin{cases} t_\textrm{exercised} \le T &\text{if put is exercised}\\ T & \text{otherwise} \end{cases} $$

My original version of this question is problematic. An American put may be exercised up to the expiration date $T$ of the contract or not at all.

Answer 1

Solution concept

Assuming otherwise identical parameters, the difference in value at time $t=0$ between an American put $P_0$ and a European put $p_{~0}$ is attributable to interest earned on the strike from expected early exercise date $\bar{t}$ to expiry date $T$.

If both the American and European put are exercised, the American put holder will have the future value of the early exercised strike $K e^{r\left(T-\bar{t}\right)}$, while the European put holder will receive the strike $K$.

$$ \begin{align} P_0 - p_{~0} &= K e^{r\left(T-\bar{t}\right)} - K \\ &\approx K \left(1 +rT -r\bar{t}\right) -K \\ \frac{P_0 - p_{~0} }{K}&\approx rT - r\bar{t} \end{align} \\ \boxed{ \bar{t} \approx T - \frac{P_0 - p_{~0} }{rK} } $$

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Interpretation

For a given pair of American and European puts

• If $P_0=p_{~0}$, expected termination of American put is $\bar{t} = T$.
• If $P_0>p_{~0}$, expected termination of American put is $0 \le \bar{t} \lt T$.

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R Code

library(RQuantLib)

S <- 100     # underlying
T <- .5      # expiry
K <- 100     # strike
r <- 0.0533  # risk free rate
d <- 0       # dividend rate
sigma <- .40  # volatility

P0 <- AmericanOption(
  "put",
  underlying = S,
  strike = K,
  dividendYield = d,
  riskFreeRate = r,
  maturity = T,
  volatility = sigma
)
p0 <- EuropeanOption(
  "put",
  underlying = S,
  strike = K,
  dividendYield = d,
  riskFreeRate = r,
  maturity = T,
  volatility = sigma
)

t_bar <- T - (P0$value - p0$value) / (r * K)

cat( sprintf('P0 = %8.3f, t_bar = %6.3f\np0 = %8.3f, T     = %6.3f',
  P0$value,t_bar,p0$value,T))
#> P0 =   10.066, t_bar =  0.456
#> p0 =    9.832, T     =  0.500

^{Created on 2024-06-24 with reprex v2.1.0}

Answer 2

Not an answer, but a comment on how to possibly get the expected date of exercise $E(t)$.

From what I understand, an American put is exercised early only when the spot is at zero and the put is at its max value. Otherwise, there is always time left in the option for the spot to decrease further and generate more intrinsic value for the put. It would make more sense to trade away the put option.

If one would want to determine when the spot hits zero on average/expectation, one would have to simulate the spot price evolution of the underlying and see how often it hits zero (and when the holder exercises the American put). The rest of the simulations, I suppose the holder keeps it till maturity. The expected time would then be a weighted average of the number of times the spot hits zero times their relevant time and the number of times the spot is kept till maturity and time $T$.

Hopefully this makes sense? Happy to clarify and to discuss what you think.