Answer to Question #190287 in Financial Math for Red

Question

Question 5 [9 marks]. The price S(t) of a share follows the geometric Brownian motion S(t) = Se µt+σW(t) . The price of the share at t = 0 is S = £25 and µ = 0.1. The continuously compounded interest rate is 8%. However, the volatility σ is not known.

(a) Explain the definition of implied volatility. [3]

(b) A European call option on the above share with the strike price K = £23 and expiration time of 6 months has the market price £3.1. Does the equation defining the implied volatility have a solution? [6] Hint. You are not supposed to solve the equation defining the volatility to answer this question.

Answer 1

(a)

Implied volatility concept is used in case of options. It is a value of volatility of the underlying asset over a time period largely influenced by market demand and supply ; i.e when the expectations rises then demand of options also boosts up, this results in rise in implied volatility. High implied volatility means high price of assets. 

(b)

$S(t) = Se µt+σW(t)$

The above equation finds the value of S(t) that has lognormal distribution. The equation consist of strike price, mean, expected price, time period, and the implied volatility of asset. Change in implied volatility will also change the value of price of the share. Say, if the implied volatility of share is high then the price of the share will also increase in the stock market. This equation shows the inter connection between implied volatility and the price of stock.