European Put Value When the Underlying Price Reaches Zero
Summary
The document explains the European put’s value when its underlying stock price has reached zero before expiration. Under a geometric Brownian motion model such as Black–Scholes, zero is an absorbing state: the stock remains at zero. The put’s future payoff is then known to be the strike, so the option behaves like a riskless payment of that amount at maturity.
Discounting the strike payment by the maturity discount factor gives the put’s value at the valuation time. This addresses the concern that the discounted strike might only be an upper bound if the stock could later recover. The result depends on the assumed process and the stock actually being at zero; it does not apply to models in which the price can leave zero or to situations where zero is only an approximate observed price.
Key ideas
- In geometric Brownian motion, a stock price that reaches zero remains there.
- With the stock fixed at zero, a European put pays the strike with certainty at maturity.
- The certain strike payoff is valued by discounting it to the current time.
- The conclusion depends on an absorbing-zero price process and a truly zero stock value.
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Full text
# European put price when stock price is 0 before maturity # European put price when stock price is 0 before maturity According this answer, https://quant.stackexchange.com/a/39298/29108, the European put price (with maturity $T$) at time $t$ for a stock whose current price is $0$ should be the strike $K$ discounted from $T$ to $t$. So $P(t,T)K = p$. Is this exactly true? I had thought that the value $P(t,T)K$ should be an upper bound on the put price, as there is still a chance the stock price increases in the time before maturity. And if the European put price is basically growing at the risk free rate, doesn't this remove the probability of stock price movements? Any help would be appreciated! ## Answer by Slade (score 1, accepted) https://quant.stackexchange.com/a/44069 Thanks to @Alex C for clarifying. His answer is in the comments, but I wanted to have the question answered for others in the future. Assuming a geometric brownian motion as the stock price process (like in Black-Scholes), when the stock price becomes zero, it will stay at zero 'forever'. So basically the stock price is known in the future to be $0$, and thus an European Put option on the stock with strike $K$ and maturity date $T$, is essentially a zero-coupon bond delivering $K$ on date $T$, since both are riskless investments. So discounting the known, riskless payoff of the option, $K - 0$, to the present, the option value (just like the value of the zero-coupon bond) is $P(t,T)K$.
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