Put option black scholes model

Question

I have the following problem with a put option in a black scholes setting.

Consider a put option with time to maturity $T = 1$ and the underlying return $R$ following the log-normal model with volatility $σ\sqrt{T}$, and assume that the risk-free return is $r = 0.04$. First, I assume that $r_g = 0.06$ and $σ_1 = 0.25$.

I want to compute value of this option using the Black-Scholes formula. However, I can't figure out what to do?

When turning to Monte Carlo, I let $σ_1$ vary between $0.25, 0.3, 0.35$. I now want to recompute the value of this option using Monte Carlo method and compare them.

Assume now that $σ_1 = 0.25$ and that there is another asset with $σ_2 = 0.35$ and that the two assets are correlated with correlation coefficient $ρ = −0.9$ and weights $w1, w2 = 0.5$.

I want to calculate the premium of the put option.

Thanks in advance

Answer 1

The formula for computing the Black and Scholes price at time $t$ for a put with maturity $T$, strike price $K$ and assuming a lognormal underlying asset following

$$ dS_t = \mu S_tdt + \sigma S_tdW_t $$

is

$$ p_t = Ke^{r(T-t)}N(-d_2)-S_tN(-d_1), $$

with

• $$N(x)=\int_{-\infty}^{x}e^{-\frac{u^2}{2}}du$$
• $$d_1=\frac{\ln(\frac{S_t}{K})+(r+\frac{\sigma^2}{2})(T-t)}{\sigma \sqrt{T-t}}$$
• $$d_2=\frac{\ln(\frac{S_t}{K})+(r-\frac{\sigma^2}{2})(T-t)}{\sigma \sqrt{T-t}}$$