Question

Consider a stock with a current value of $76 and a standard deviation of returns equal to 30%. Suppose there is a call option on this stock with a maturity of 6 months and a strike price of $70. The risk-free rate is 3%.

1. What is the value of this option?
2. If the stock price increases by $1, what is the change in the option value? Provide some intuition to this change (e.g., is it a high or a low value?)
3. Explain the concept of option delta and its use cases.

Answer 1

Question 1: Value of the option

In order to calculate option value, lets understand few terminologies -

a. Strike Price, K is price at which call option can be exercises. Here K = 70

b. Spot Price, S is current price of the stock. Here S = 76

Call option definition - Call option refers to an option to buy a stock at strike price. This option is given to buyer or person who has bought the option. Seller has an obligation to transact at strike price.

Step 1 = Formula for Value of call option = Max [(PV(S)-K),0] -> PV stands for present value of stock

This means on any given point, option's value for buyer will be present value of stock minus strike price or zero. As buyer has an option he will never exercise the option having negative value. Hence its floored at zero.

Step 2 = Putting value in formula - Value of option = PV(76) - 70

Formula to calculate PV(76) = Spot price/(1+risk free rate)^timeperiod

Here Spot price = 76, Risk Free rate = 3% time period = 6/12=0.5

PV(76) = 76/(1+3%)^0.5 = 74.88

Step 3 = Putting value of PV(76) in step 1, we get value of option as = 74.88-70 = 4.88

This solves question 1.

Question 2

The concept of option delta is explained as below:

Option delta is defined as change in option price based on $1 or 1% change in underlying asset price. In above context, if the underlying price of stock increases to $71 or decreases to $69, what will be change in option price - is defined by option delta.

Option Delta can take values from -1 to 1. For call option it ranges from 0 to 1 and for put option it ranges from -1 to 0.

For call option, when delta is 0 its considered as out of money - buyer will not exercise the option;

when delta is 0.5 its considered at money - buyer will be indifferent

When delta is 1 its considered deep in money - buyer will be in profit by exercising the option

For given scenario, as option has a positive value, increase stock price by $1 will make the option more worthy and hence delta will move from 0.5 to 1. Hence it will increase and buyer of option will be happy as he can see his option in the money.