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Derivatives

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Random Walk

Probability

Recurrence Relations

Discrete Mathematics

Triangular Grid

Random Walk

Probability

Recurrence Relations

Discrete Mathematics

We define $\text{Vega}_{1\%}$ as the variation of an options value for a variation of $0.01$ of its annualized implied volatility $\sigma_\text{implied}$ .

You are short European put options with strike

K_A=\$123

on stock A expiring in 1 year for

\$1000

of

\text{Vega}_{1\%}

,

You are long European call options with strike

K_B=\$456

on stock B expiring in 1 year for

\$1000

of

\text{Vega}_{1\%}

.

In this market,

There are 256 trading days in a year,

Stocks do not pay dividends,

Interest rates, execution costs, margin costs, etc. are

0

,

You can perfectly delta hedge your positions (fractional shares).

At the last close,

You flattened your delta exposure in both stocks,

Stock A closed at

S_A= \$ 123

,

Stock B closed at

S_B= \$ 456

,

Option A closed at

\sigma_{\text{A, implied} } =15\%

Option B closed at

\sigma_{\text{B, implied} }= 35\%

Today, the implied volatility of the options remained constant and the same as last close. Both stocks traded in a range and you rebalanced your delta hedge $4$ times during the day, $1\%$ apart in stock A and $11$ times, $1\%$ apart for stock B.

Using the Black-Scholes framework, what is your PnL (in $\$$ ) at the end of the day?

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