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Now Considering the Black-Scholes market model where the price of the riskless asset (bond) satisfies $$dB_t=rB_tdt,\;\; B_0 = 1$$ for some $r>0$ and the stock price evolves according to $$dS_t = µS_tdt + σS_tdW_t,\;\; S_0 = 1$$, where $µ, σ > 0$ constants and $W_t$ is a (standard) Brownian motion with fixed time horizon $T > 0$.

Again by considering an option that pays to its holder at time $T$ the amount $f(S_T )$, where $f : (0, ∞) → R$ is defined as $f(x) = x$, for $0 < x < K$, $f(x) = K$ for $x ≥ K$,

How one can derive an expression for this price in terms of the standard normal distribution function $Φ(·)$ and the given parameters $σ, r, T, K$?

Here I tried to use the option pricing theorem for the price of this option at time 0, and to derive it in similar way to the derivation of the the Black-Scholes formula for the price of a European call option, but I am stuck.

Anyone help please? Thanks!

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1 Answer 1

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Hint

This is the payoff if you're long a call at strike zero and short a call at strike $K$. (And a call at strike zero is just the underlying.)

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  • $\begingroup$ It's also a cash bond and short put position, if you like $\endgroup$
    – spaceisdarkgreen
    Commented Apr 5, 2017 at 0:02

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