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 Incoming links: Call option on a put option

T_{1}

, with underlying spot

s

H (s) = [p (T_{1}, s; K_{2}, T_{2}) - K_{1}]^{+} .

Let

D = K_{2} e^{- r (T_{2} - T_{1})}

. The underlying put option value is strictly decreasing and convex in

s > 0

, with limits

D

at zero and zero at infinity. If

0 < K_{1} < D

, there is a unique

s_{*}

at which that put value equals

K_{1}

. Hence the compound option payoff is a decreasing convex curve from

D - K_{1}

to zero on

[0, s_{*}]

, followed by a horizontal zero segment. There is an upward kink at

s_{*}

. If

K_{1} \geq D

, the payoff is identically zero.

For

K_{1} = 0

, the holder obtains the underlying put at no cost, so the payoff is its full value at

T_{1}

V (t, s) = p (t, s; K_{2}, T_{2}), t < T_{1} .

Explicitly, this is

K_{2} e^{- r (T_{2} - t)} Φ (- d_{2}) - s Φ (- d_{1})

with

d_{1}, d_{2}

evaluated over

T_{2} - t

. The intermediate choice time disappears when its exercise cost is zero.