Pricing a Forward-Start Put Under Black–Scholes
Summary
The document explains how to value a payoff based on the difference between a stock price at an intermediate date and its later price. It first clarifies that the stated payoff needs a positive-part operator to describe a put-like claim; under that interpretation, it is a forward-start option rather than a standard lookback option. The distinction matters because a true lookback payoff depends on the maximum or minimum price over a period.
Under geometric Brownian motion with constant interest rates, the answer applies risk-neutral valuation and the tower property to condition first on the intermediate date. The conditional claim is a Black–Scholes put with its strike set to the stock price at that date. Homogeneity then makes its value proportional to that stock price, so taking the earlier expectation yields a price proportional to the forward price. With no dividends, the document says the current spot can be inferred from the option premium. The derivation relies on the stated model assumptions and does not address a genuine path-dependent lookback option.
Key ideas
- The payoff must be clarified because a positive-part difference defines a forward-start put rather than a standard lookback.
- Conditional on the intermediate date, the claim can be valued as a Black–Scholes put struck at the then-current stock price.
- Black–Scholes homogeneity makes the conditional option value proportional to the stock price at the intermediate date.
- The resulting premium is proportional to the forward price, and with no dividends it can be related directly to spot.
Tags
Full text
# Lookback option to find stock price
# Lookback option to find stock price
Consider the payoff equation for the lookback option $\psi(T)= max(S_t-S_T)$, where $t\in[0,T]$ and $S_t$ is modeled by the geometric Brownian motion with constant parameters. Find the price of stock at current time.
I've done some research and this option seems similar to the floating strike option. But I don't know how to approach this problem yet. Any help would be appreciated.
## Answer by Quantuple (score 3)
https://quant.stackexchange.com/a/24865
Below I assume that you meant: $\psi (T) = \max (S_t - S_T, 0) $ which constitutes the payout of a forward start rather than a lookback option. If not please clarify your question...
If you are looking for the option price $V_0$, assuming a Black-Scholes diffusion (GBM + constant interest rates), you have
\begin{align*} V_0 &= P(0,T) E[ \psi (T) \vert \mathcal {F}_0] \\ & = P (0,T) E \left[ (S_t - S_T)^+ \vert \mathcal {F}_0 \right] \\ & = P (0,T) E \left[ E [ (S_t - S_T)^+ \vert \mathcal {F}_t ] \vert \mathcal {F}_0 \right] \\ & = P (0,t) E \left[ P (t,T) E [ (S_t - S_T)^+ \vert \mathcal {F}_t ] \vert \mathcal {F}_0 \right] \\ &= P (0,t) E \left[ P_{BS}(S_t; S_t, T-t) \vert \mathcal {F}_0 \right] \\ &= P (0,t) E [ S_t P_{BS}(1; 1, T-t) \vert \mathcal {F}_0 ] \\ &= P (0,t) F (0,t) P_{BS}(1; 1, T-t) \end{align*}
Where $P(s,t) = e^{-r(t-s)}$ denotes a generic discount factor and each the successive equalities come from:
- Option premium as a risk-neutral expectation
- Definition of the option's payout
- Tower property of conditional expectation
- Composition of discount factors
- Price of a put option which, under the modelling assumptions, is given by the BS formula (first argument denotes stock price, second strike level and last time to maturity)
- Space homogeneity of BS formula
- Forward price as a risk-neutral expectation ($P_{BS} (1;1,T-t)$ is deterministic)
Note that because the forward price $F (0,t) $ is directly proportional to the spot price $S_0$ you can obviously infer the spot value if you know the option premium $V_0$. Actually assuming no dividends: $V_0 = S_0 P_{BS}(1; 1, T-t) $
However, I don't see the point of doing that, except for some pure academic fun.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.