Option pricing is determined by 5 key factors: the asset's cash price, strike price, volatility, time to expiration, and interest rates. The Black-Scholes model uses these factors to price European options, assuming the asset pays no dividends, markets are efficient, and other restricting assumptions. It models the asset's price as following geometric Brownian motion to derive a theoretical option price formula involving the standard normal distribution.

1. Option Pricing
• There are 5 determinants of Option pricing or
premiums:
1. Cash Price of Asset (S )t
2. Strike Price (K)
3. Volatility of the underlying asset’s price (σ)
4. Time to expiration (T)
5. Interest Rates (r)
These factors affect the premium / price of both
American & European options in several ways.

2. Cash Price of Asset
• Keeping all other factors constant, if cash
price of underlying asset goes up value of the
call option increases but value of the put
option diminishes.

3. Effect of Strike Price
• If all other factors remain constant but the
strike price of option increases, intrinsic value
of the call option will decrease and hence its
value will also begin to decrease.
• On the other hand, with all other factors
remaining constant, increase in strike price
will increase the intrinsic value of the put
option and it will therefore become dearer.

4. Effect of Volatility
• Volatility in the price of the underlying asset
affects both call & put options in the same
way. As higher volatility escalates the chances
of an option going in-the-money at any point
in time during the life of the contract. It
increases the risk to the option seller and
consequently makes the option, both call &
put, more expensive.

5. Effect of time to Expiration
• The effect of time to expiration on both call &
put options is similar to that of volatility on
option premiums.
• The longer the maturity of the option the
greater is the uncertainty and hence the
prices of both call & put options are higher,
keeping all other factors constant.
• Therefore, longer maturity options are always
more expensive than shorter maturity
options.

6. Effect of Interest Rates
• Higher interest rates has the same effect as
lowering the strike price, and therefore as
seen, higher interest rate will result in an
increase in the value of a call option and a
decrease in the value of a put option.

7. Black-Scholes Model for Options Pricing
• Black-Scholes model for pricing European options
published by Fischer Black & Myron Scholes in 1973 is by
far the most popular model to price an option.
• Black-Scholes model start by specifying a simple & well
known equation that the manner in which stock prices
fluctuates.
• This equation is called Geometric Brownian Motion
implies that stock returns will have a lognormal
distribution i.e., the logarithm of the stock’s return will
follow the normal (bell shaped) distribution.

8. Cont…
• Black & Scholes then propose that the price of
an option is determined by the only two
variables that are allowed to change – time
and the underlying stock price.
• While other factors such as volatility, exercise
price and risk free rate do affect the price of
the option they are not allowed to change.

9. Main assumptions are as follows:
1. The stock pays no dividend during the
option’s life.
2. European exercise terms are used.
3. Markets are efficient.
4. No commissions are charged.
5. Interest rates remain constant & known.
6. Returns are log-normally distributed.

10. The Model: (-rt)
C = SN(d1 ) – Ke N(d 2 )
Where,
C = Theoretical Call Premium
S = Current Stock Price
t = time until option Expiration
K = Option’s strike price
r = Risk-free interest rate
N = Cumulative Standard Normal Distribution
E = Exponential Term (2.7183)
d1 = 1n(S/K) + {r + s 2 /2)}t
s√t
d 2 = d1 - s√t
s = Standard Deviation of Stock returns
1n = Natural logarithm