Option Pricing without Black-Scholes Illustration

Using the inputs outlined below, what is the call and put prices both with black-scholes and without black-scholes?

• The spot and strike price is 100
• 15% volatility
• Zero interest rate
• A year to expiry

The results are pretty close, however not exact. The results are shown in the table below:

	No Black Scholes	Black Scholes
Call Price	6.599	5.979
Put Price	5.459	5.979

 

Why are there discrepancies in the pricing above?

This is down to some technical factors. For example we’ve used a discrete distribution, not a continuous one. Also, we haven’t taken the price range up to infinity. Along with this, we haven’t applied what’s referred to as the drift.

What are Option Greeks?

These are delta, sigma or vega, theta etc. and they quantify the sensitivity of the option price to incremental changes in the model inputs, other factors remaining the same.

What happens if we raise/reduce the spot price in relation to the strike?

Raise spot to 120, all other inputs remaining the same as above - this is going to increase the probability of the call being exercised, and reduce it for the put. Hence calls are more valuable and the puts less so.

	No Black Scholes	Black Scholes
Call Price	22.047	20.891
Put Price	0.765	0.891

Reduce spot to 85, all other inputs remaining the same as above - this is going to decrease the probability of the call being exercised, and increase it for the put. The Greek related to changes in option price to changes in the spot or underlying price is delta.

	No Black Scholes	Black Scholes
Call Price	1.140	0.979
Put Price	15.168	15.979

 

What happens if we change the volatility?

Increasing the volatility changes the probabilities associated with the outcomes. Those further away from the current price, have a higher probability of occurring, therefore higher potential payoffs. Hence, both calls and puts are more vulnerable. Assuming the following:

• The spot and strike price is 100
• 25% volatility
• Zero interest rate
• A year to expiry

The results are shown in the table below:

	No Black Scholes	Black Scholes
Call Price	11.288	9.948
Put Price	8.592	9.948

Reducing the volatility to 5%, all other inputs remaining the same - this reduces the price of both options. Outlying outcomes are less probable. The Greeks related to this are vega and sigma.

	No Black Scholes	Black Scholes
Call Price	2.064	1.995
Put Price	1.929	1.995

What happens to the forward price if we change the interest rate?

Changing the interest rate changes the forward price. Raising it, raises the forward price and has the same effect as raising the spot price in relation to the strike. Calls become more valuable and puts become less so. The opposite occurs if the interest rate is cut. The Greek that relates to this is rho.

What is the effect of a change in the time to expiry?

Once the option has been established, time only moves in one direction. The passage of time reduces the value of both calls and puts. The effective range of potential returns over a longer period is higher than for a shorter period. Reducing the time to expiry reduces the probabilities of the more extreme outcomes. The Greek that relates to this is theta.