1. Prepared by : Antonios Perla Bou Khalil Joelle Hachem Alaa Droubi Ali
Saint Joseph University
Volatility Smiles
Prepared by :
Antonios Perla
Bou Khalil Joelle
Hachem Alaa
Droubi Ali

2. The most widely known model for option pricing is the black Scholes model, we will discuss in our presentation the following point regarding the model :
First we will explain the black Scholes model.
Second will discover how close are the market prices of options to those predicted by the black scholes model?

3. The black scholes model calculates the theoretical price of the options taking into consideration its intrinsic value and the time value. How ever the actual price is the premium which is priced on the market which is derived from the supply and demand.
The intrinsic value can be obtained using the current price of the option and the predetermined strike price.
The time value takes into consideration the time to expiration, the risk free rate, the volatility of the underlying asset, and the dividend paid by the stock “if any”
The calculation of the theoretical price using the black scholes model is based on the following assumptions :
The volatility is constant.
The return are normally distributed.
The standard deviation can be estimated using the historical data

4. Theoretical price

5. Time to expiration : the time can effect the price of the call/put option.
The risk free rate will affect the price of the option since it will be used to calculate the present value of our inflow/outflow in case our long position is respectively put/call.
In case a company declares that it will pay dividends this will lead to a decrease in the value of the stocks of this company, since the cash flow of the company will decrease.
Time to expiration
Call
Effect
increase
It will postpone the outflow/ more time to invest our money in the market
Increase value of the option
It will reduce the actualized value of the future outflow
It will increase the probability that the asset volatility will fluctuate in our favor
Decrease
It will accelerate the outflow/ our money cannot be invested
Decrease value of the option
It will increase the actualized value of the future outflow
It will reduce the probability that the asset volatility will fluctuate in our favor
Time to expiration
Put
Effect
increase
It will postpone the inflow
Decrease value of the option
It will reduce the actualized value of the future inflow
It will increase the probability that the asset volatility will fluctuate in our favor
Increase value of the option
Decrease
It will accelerate the inflow
It will increase the actualized value of the future inflow
It will reduce the probability that the asset volatility will fluctuate in our favor

6. Volatility, the probability that the price of the underlying asset will fluctuate in our favor will increase in case the volatility of the asset is high.
Asset :A
Asset : B
The price of asset B has more distribution a higher standard deviation thus a higher volatility.
The prices of asset B has a smaller distribution a smaller standard deviation thus a lower volatility
As a result the price of the option having asset B as an underlying worth more than the option having Asset A as an underlying asset.

7. Black Scholes Merton Model
The black Scholes model is used to price put and call options. Using the following formula :

9. N(d1) is the future value of the stock if, and only if the stock price is above the strike price at expiration. If and only if, the option expires in the money, N(d1) is the probability of how far into the money the stock price will be. Thus N(d1) is the expected value of the stock, multiplied by the probability that the stick price will be at or above the strike price.
N(d2) The probability that the stock price will be at or above the strike price when the option expires

10. The central limit theorem tells us that if we graph enough periodic daily Returns of an asset, the graph should form a normal distribution, bell shaped graph. Thus the black scholes model assumes that the future returns of the asset will be normaly distributed which is equal to the historical standard diviation.

11. In order to determine the probability that S>k at maturity
In order to determine the probability that S>k at maturity. We should see how much of growth rate (ln k/S) will it take in order to have a current price higher than the Strike price.

12. The probability or odds of the growth rate is represented by a normal distribution graph
If 30 % percent of total area under the curve is on the right side of the Zscore than there is a 30% percent probability that the chance the stock price will be at or above the strike price when the option expires. We want the total area of the right of the Z score.

13. In order to determine if the black and scholes model will be able to estimate the actual price of the option in the market, we chose to calculate the option price of face book stock, having 1 month till expiration. Kindly refer to the excel in the file. black scholes model.xlsx
As we can see the black and scholes model gave us a pricing of 8.12 however the market price is 8.25, this difference is due to the fact that the black scholes model assumes that the volatility is constant. The return are normally distributed and The standard deviation can be estimated using the historical data. However the volatility in the market is not constant, and the future volatility can highly deviate from the historical volatility it’s implied by the supply and demand of the market and finally the growth rate of the asset is not normally distributed in the real word.

14. IMPLIED VOLATILITY

15. Volatility is the most critical parameter for option pricing -- option prices are very sensitive to changes in volatility. Volatility however cannot be directly observed and must be estimated.
Whilst implied volatility -- the volatility of the option implied by current market prices -- is commonly used the argument that this is the "best" estimate.

16. Implied volatility of face book stocks.volatility smile.xls
We used the following data from Nasdaq in order to calculate the implied volatility of the face book’s stock.

17. The following parameters where extracted from the market.

18. The graph in the previous slide shows us a volatility skew
The graph in the previous slide shows us a volatility skew. In demonstrate that the volatility decreases as the strike price increases.
Thus the volatility used to price a low strike option is significantly higher than that used to price a high strike option.
The volatility smile for equity options correspond to the implied volatility distribution, however the black Scholes model uses a lognormal distribution

19. Reasons and explanation for a volatility skew.
As a company’s equity declines in value, the company’s leverage increases this means the equity becomes more risky and the investors become less confident to hold the stocks which lead in an increase in the volatility accordingly the option prices increases

20. Reasons and explanation for a volatility skew.
As a company’s equity increases in value, the company’s leverage decreases this means the equity becomes more risky and the investors become more confident to hold the stocks which lead in an decrease in the volatility accordingly the option prices decreases

21. The volatility smile for equity options corresponds to the implied probability distribution given by the solid line in the below figure, a lognormal distribution with the same mean and standard deviation as the implied distribution is shown by the dotted lines. It can be seen the implied distribution has a heavier left tale and a less heavy light tale than the lognormal distribution.
We expect the implied distribution to give a relatively low price for the option as observed on the left of the graph.
However we expect the implied distribution to give a higher prices from the lognormal distribution for the option as observed on the right of the graph.

22. Foreign currency options
In the following we will discuss the difference between the implied probability and the lognormal distribution.
The implied volatility is relatability low at the money options, it becomes progressively higher as an option moves either into the money or out of the money

23. We refer to the previous graph as the implied distribution
We refer to the previous graph as the implied distribution. The volatility smile shown previously correspond to the implied distribution shown in the solid line in the below figure. A lognormal distribution with the same mean and standard deviation as the implied distribution is shown by the dashed line below, it can be seen the implied distribution has heavier tail than the lognormal distribution
We therefore expect that the implied distribution to give a higher price for the option as observed on the right and left of the graph.

24. Empirical results
Traders for foreign currency options consider that the lognormal distribution understate the probability of extreme movements in exchange rate.
To test if they are right this table examines the daily movements in 12 different exchange rates over a 10 year period
The table provides evidence to support the existence of heavy tails and the volatility smile, just by comparing the exchange rates in the real world and in the normal distribution.
The conditions for the assets price to have a lognormal distribution are that the volatility of the asset is constant and the asset price changes smoothly with no jumps, in practice neither of these conditions are satisfied for the exchange rates

25. Greek letters :
Greek letters is the dimensions of risk involved in taking a position in an option (or other derivative). Each risk variable is a result of an imperfect assumption or relationship of the option with another underlying variable. Various sophisticated hedging strategies are used to neutralize or decrease the effects of each variable of risk.
Greek letters provide a way to measure the sensitivity of an option's price to quantifiable factors. These terms may seem confusing and intimidating to new option traders, but broken down, the Greeks refer to simple concepts that can help you better understand the risk and potential reward of an option position :
Delta : measures the rate of change of the theoretical option value with respect to changes in the underlying asset's price. Delta is the first derivative of the value  of the option with respect to the underlying instrument's price .
Vega : measures sensitivity to volatility. Vega is the derivative of the option value with respect to the volatility of the underlying asset.
Theta : measures the sensitivity of the value of the derivative to the passage of time.
Rho : measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk free interest rate.
Gamma : measures the rate of change in the delta with respect to changes in the underlying price. Gamma is the second derivative of the value function with respect to the underlying price.

26. volatility smile complicate the calculation of Greek letters
volatility smile complicate the calculation of Greek letters. There is a relationship between  the implied volatility and  K/S. as the price of the underlying asset changes, the implied volatility of the option change. (FORMULE)
Delta is a partial of volatility respect to stock price. It’s the amount of stock we have to short . It represent the sensitivity to changes in the underlying asset.(TENE FORMULE)
  Volatility is decreasing function  of K/S. That s mean the implied volatility increase when the asset price increase.

27. Binomial trees
The binomial model breaks down the time to expiration into potentially a very large number of time intervals, or steps.
A tree of stock prices is initially produced working forward from the present to expiration.
At each step it is assumed that the stock price will move up or down by an amount calculated using volatility and time to expiration.
This produces a binomial distribution, or recombining tree, of underlying stock prices.
The tree represents all the possible paths that the stock price could take during the life of the option.
At the end of the tree - i.e. at expiration of the option all the terminal option prices for each of the final possible stock prices are known as they simply equal their intrinsic values.

28. Next the option prices at each step of the tree are calculated working back from expiration to the present. The option prices at each step are used to derive the option prices at the next step of the tree using risk neutral valuation based on the probabilities of the stock prices moving up or down, the risk free rate and the time interval of each step.
Any adjustments to stock prices (at an ex-dividend date) or option prices (as a result of early exercise of American options) are worked into the calculations at the required point in time.
At the top of the tree you are left with one option price.

29. A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram representing different possible paths that might be followed by the stock price over the life of an option.
The underlying assumption is that the stock price follows a random walk.
The binomial tree can be used to value options using both no-arbitrage arguments and a principle known as risk-neutral valuation

30. One-step Binomial Model and a no-arbitrage argument
We set up a portfolio of the stock and the option in such a way that there is no uncertainty about the value of the portfolio at the end of a determined period.
We then argue that because the portfolio has no risk, the return it earns must equal the risk-free interest rate.
So we can work out the cost of setting up the portfolio and therefore the option’s price. Because there are two securities (the stock and the stock option) and only two possible outcomes, it is always possible to set up the riskless portfolio.

31. Generalization
During the life of the option the stock price can either move up from So to a new level, So u, where u>1, or down from So to a new level. So d, where d<1.
The percentage increase in the stock price when there is an up movement is u-1.
The percentage decrease when there is a down movement is 1-d.

32. If there is an up movement in the stock price, the value of the portfolio at the end of the life of the option is:
𝑆𝑜𝑢∆−𝑓𝑢
If there is a down movement in the stock price the value becomes
𝑆𝑜𝑑∆−𝑓𝑑
ƒu: payoff from the option if the stock price moves up to So u
ƒd: payoff from the option if the stock price moves down to So d
The portfolio is riskless when S0u∆ – ƒu = S0d∆ – ƒd or
∆= 𝑓𝑢−𝑓𝑑 𝑆𝑜𝑢−𝑆𝑜𝑑
The portfolio is riskless if the value of delta is chosen so that the final value of the portfolio is the same for both alternatives of the binomial tree at the end of the life of the option.

33. Substituting for ∆ we obtain
ƒ = [p ƒu + (1 – p) ƒd] e-rT
ƒ: option price
The expression p ƒu + (1 – p) ƒd is the expected payoff from the option.
The states that the value of the option today is its expected future payoff discounted at the risk free rate..
The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate.

34. Risk neutral valuation
The variables p and (1– p) can be interpreted as the risk-neutral probabilities of up and down movements in the stock price.
The stock price grows on average at the risk free rate. Setting the probability of the up movement equal to p is therefore equivalent to assuming that the return on the stock equals the risk-free rate.
In a risk-neutral world all individuals are indifferent to risk.
In such a world, investors require no compensation for risk, and the expected return on all securities is the risk free interest rate.

35. Matching volatility with u and d
One way of matching the volatility is to set where σ is the volatility and Δt is the length of the time step.
𝑢= 𝑒 𝜎 ∆𝑡
𝑑= 1 𝑢 = 𝑒 𝜎 ∆𝑡
This is the approach used by Cox, Ross, and Rubinstein

36. Volatility is the same in the real world and the risk-neutral world
We can therefore measure volatility in the real world and use it to build a tree for the an asset in the risk-neutral world
The probability of an up movement in the real world is denoted by p* and, consistent with our earlier notation, in the risk-neutral world this probability is p.t
𝑝∗= (𝑒 𝑢∆𝑡 −𝑑) (𝑢−𝑑)
If we behave as though the world is risk neutral ,the variable p is given by equation
𝑝 = a−d u−d
Where 𝑎= 𝑒 𝑟∆𝑡

37. When a single large jump is anticipated
Effect of a single large jump. The solid line is the true distribution, the dashed line is the lognormal distribution.

38. Change in stock price in 1 month
The true probability distribution is bimodal not log normal. The effect of a bimodal stock price distribution is shown when we consider the extreme case where the distribution is binomial.

39. For that we suppose that:
The stock price is currently 50$ and that it is known that in 1 month it will be either 42$ or 58$. Suppose further that the risk free rate is 12% per annum. Option can be valued using binomial model. in this case u=1.16,d=0.84,a=1.0101, and p=0.5314
𝑝 = a−d u−d 𝑝 = − −0.84 =0.5314

40. The results from valuing a range of different options are shown in the table below
First column: Alternative strike prices
Second column shows prices of 1 month European call options
Third column: The price of one month European put option prices
Fourth column: implied volatilities.

41. The figure displays the volatility smile which is actually a “frown” with volatilities declining as we move out of or into the money. The volatility implied from an option with a strike price of 50 will overprice an option with a strike price of 44 or 56.