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i want to prove BSM

Dec 25, 2023

i want to prove BSM

Solution by Steps

step 1

To prove the Black-Scholes-Merton (BSM) formula, we start by setting up the Black-Scholes partial differential equation (PDE) which is derived from the concept of a "risk-neutral" portfolio that includes the stock and the option

step 2

The Black-Scholes PDE is given by:

\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0

where

V

is the option price,

S

is the stock price,

t

is time,

\sigma

is the volatility of the stock price, and

r

is the risk-free interest rate

step 3

Solve the Black-Scholes PDE using the boundary conditions for a European call or put option. For a call option, the boundary condition at expiration is

V(S,T) = \max(S - K, 0)

, where

K

is the strike price and

T

is the expiration time

step 4

Applying the boundary condition and solving the PDE, we obtain the Black-Scholes formula for a European call option:

C(S, t) = SN(d_1) - Ke^{-r(T-t)}N(d_2)

where

N

is the cumulative distribution function of the standard normal distribution, and

d_1

and

d_2

are given by:

d_1 = \frac{\ln(\frac{S}{K}) + (r + \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}

d_2 = d_1 - \sigma\sqrt{T-t}

Answer

The Black-Scholes-Merton formula for a European call option is

C(S, t) = SN(d_1) - Ke^{-r(T-t)}N(d_2)

with

d_1

and

d_2

as defined above.

Key Concept

Black-Scholes-Merton Formula

Explanation

The BSM formula provides a theoretical estimate for the price of European call and put options based on the Black-Scholes PDE, which assumes a lognormal distribution for stock prices and no-arbitrage conditions in the market.