Introduction

•This chapter answers the following questions.

1. How close are the market prices of options to those predicted by the Black-Scholes-Merton  model?

2. Do traders really use the Black-Scholes-Merton model when determining a price for an  option?

3. Are the probability distributions of asset prices really lognormal?

It explains that traders do use the Black-Scholes-Merton model-but not in exactly the way that  Black, Scholes, and Merton originally intended. This is because they allow the volatility used to  price an option to depend on its strike price and time to maturity.

•A plot of the implied volatility of an option with a certain life as a function of its strike price is  known as a volatility smile. This chapter describes the volatility smiles that traders use in equity  and foreign currency markets. It explains the relationship between a volatility smile and the risk-  neutral probability distribution being assumed for the future asset price. It also discusses how  option traders use volatility surfaces as pricing tools.

•Traders do use the Black-Scholes-Merton model, but not in exactly the way that Black, Scholes,  and Merton originally intended. This is because they allow the volatility used to price an option  to depend on its strike price and time to maturity.

•In the BSM Model, volatility is an input and option price is the output.

•To get implied volatility, option price is known and volatility is inferred.

•Implied volatility isn’t based on historical pricing data on the stock. Instead, it’s what the  marketplace is “implying” the volatility of  the stock will be in the future, based on price changes  in an option. Like historical volatility, this figure is expressed on an annualized basis.

Why This Approach

•Market makers use implied volatility as an essential factor when determining what option prices  should be. However, you can’t calculate implied volatility without knowing the prices of options.  So some traders experience a bit of “chicken or the egg” confusion about which comes first:  implied volatility or option price. In reality, it’s not that difficult to understand. Usually, at-the-  money option contracts are the most heavily traded in each expiration month. If the options are  liquid then the model does not usually determine the prices of the ATM options; instead, supply  and demand become the driving forces. Then, once the at-the-money option prices are  determined, implied volatility is the only missing variable. So it’s a matter of  simple algebra to  solve for it. After implied volatility is calculated for the at-the-money contracts in any given  expiration month, pricing models and advanced volatility skews are used to determine implied  volatility for other options that are less heavily traded.

•In BSM model, volatility was assumed to be constant, but that is not the case. Volatility keeps on  changing.

•Implied volatility is typically of  more interest to  retail option  traders than  historical  volatility because it’s forward-looking.

•Using the Black Scholes option pricing model, we can compute the volatility of  the underlying  by plugging in the market prices for the options. Theoretically, for options with the same  expiration date, we expect the implied volatility to be the same regardless of  which strike price  we use. However, in reality, the implied volatility (IV) we get is different across the various  strikes. This disparity is known as the volatility skew.

•If the implied volatilities (IV) is plotted against  the strike prices, this U-shaped curve might be  obtained which resembles a smile. Hence, this  particular volatility skew pattern is better  known as the volatility smile.

Volatility Smile for Calls and Puts

•It can be shown that the implied volatility of a European call option is the same as that of a  European put option when they have the same strike price and time to maturity. This means that  the volatility smile for European calls with a certain maturity is the same as that for European  puts with the same maturity.

•As explained in FRM Part 1, put-call parity provides a relationship between the prices of  European call and put options when they have the same strike price and time to maturity. With a  dividend yield on the underlying asset of 𝑞, the relationship is

𝑝+ 𝑆₀𝑒^–qT = 𝑐 + 𝐾𝑒^–rT

As usual, 𝑐 and 𝑝 are the European call and put price. They have the same strike price, 𝐾, and  time to maturity, 𝑇. The variable 𝑆₀ is the price of the underlying asset today, and 𝑟 is the risk-free interest rate for maturity 𝑇.

•A key feature of the put-call parity relationship is that it is based on a relatively simple no-  arbitrage argument. It does not require any assumption about the probability distribution of the  asset price in the future. It is true both when the asset price distribution is lognormal and when  it is not lognormal.

•Since the put-call parity holds for the Black-Scholes-Merton model,

𝑝_BS + 𝑆₀𝑒^–qT = 𝑐_BS + 𝐾𝑒^–rT

where

𝑝_BS is the value of European put option calculated using the Black-Scholes-Merton model

𝑐_BS is the value of European call option calculated using the Black-Scholes-Merton

𝑝_mkt is the market value of European put option

𝑐_mkt is the market value of European call option

In the absence of arbitrage opportunities, put-call parity also holds for the market prices, so  that

𝑝_mkt + 𝑆₀𝑒^–qT = 𝑐_mkt + 𝐾𝑒^–rT

Subtracting these two equations, we get

pBS  ^{— 𝑝}mkt  ^{= 𝑐}BS  ^{— 𝑐}mkt

This shows that the dollar pricing error when the Black-Scholes-Merton model is used to price  a European put option should be exactly the same as the dollar pricing error when it is used to  price a European call option with the same strike price and time to maturity.

•As discussed in FRM Part 1, the implied volatility (𝐼𝑉) of an option contract is that value of the  volatility of the underlying instrument which, when input in an option pricing model (such as  Black–Scholes) will return a theoretical value equal to the current market price of the option. So  for example, if the implied volatility of the put option is 10%, then 𝑝_BS = 𝑝_mkt if a volatility of  10% is used in the Black-Scholes-Merton model. But since 𝑝_BS — 𝑝_mkt = 𝑐_BS — 𝑐_mkt , it  follows that 𝑐_BS = 𝑐_mkt when this volatility of 10% is used. The implied volatility of the call is,  therefore, also 10%.

•This argument shows that the implied volatility of  a European call option is always the same as  the implied volatility of a European put option when the two have the same strike price and  maturity date. This means that the volatility smile (i.e., the relationship between implied volatility  and strike price for a particular maturity) is the same for European calls and European puts.  More generally, it means that the volatility surface (i.e., the implied volatility as a function of  strike price and time to maturity) is the same for European calls and European puts. These  results are also true to a good approximation for American options.

Volatility Smile for Calls and Puts – Example

•The value of  a foreign currency is $0.60. The risk-free interest rate is 5% per annum in the  United States and 10% per annum in the foreign country. The market price of a European call  option on the foreign currency with a maturity of 1 year and a strike price of $0.59 is 0.0236.

The implied volatility of the call is   

The price 𝑝 of a European put option with a strike price of $0.59 and maturity of 1 year can  be calculated using put-call parity relationship

𝑝 + 0.60𝑒^–0.10×1 = 0.0236 + 0.59𝑒^–0.05×1

so that 𝑝 = 0.0419.

When the put has this price, its implied volatility is also 14.5%. This is what was expected from the analysis just given.

Foreign Currency options

•The volatility smile used by traders to price foreign currency options has the general form shown  in this figure. The implied volatility is relatively low for at-the money options. It becomes  progressively higher as an option moves either into the money or out of the money.

•The volatility smile in the previous figure corresponds to the implied distribution shown by the  solid line in this figure. A lognormal distribution with the same mean and standard deviation as  the implied distribution is shown by the dashed line in this figure. It can be seen that the implied  distribution has heavier tails than the lognormal distribution.

•Consider first a deep-out-of-the-money call option with a high strike price of 𝐾₂ (𝐾₂/𝑆₀well  above 1.0). This option pays off only if the exchange rate proves to be above 𝐾₂. The 𝐼𝐷 figure  shows that the probability of this is higher for the implied probability distribution than for the  lognormal distribution. Hence, it can be expected that the implied distribution will give a  relatively high price for the option. A relatively high price leads to a relatively high implied  volatility, and this is exactly is observed in the 𝐼𝑉 figure for the option. The two figures are  therefore consistent with each other for high strike prices.

•Consider next a deep-out-of-the-money put option with a low strike price of 𝐾₁, (𝐾₁/𝑆₀well  below 1.0). This option pays off only if the exchange rate proves to be below 𝐾₁. The 𝐼𝐷 figure  shows that the probability of  this is also higher for the implied probability distribution than for  the lognormal distribution. Hence, it can be expected that the implied distribution will give a  relatively high price, and consequentially a relatively high implied volatility, for this option as well.  Again, this is exactly is observed in the 𝐼𝑉 figure.

Foreign Currency Options – Empirical Results

•This table examines the daily movements in 10 different exchange rates over a 10-year period  between 2005 and 2015. The exchange rates are those between the U.S. dollar and the following  currencies: Australian dollar, British pound, Canadian dollar, Danish krone, euro, Japanese yen,  Mexican peso, New Zealand dollar, Swedish krona, and Swiss franc.

•Daily changes exceed 3 standard deviations on  1.30% of days. The log normal model predicts  that this should happen on only 0.27% of days.  Daily changes exceed 4, 5, and 6 standard  deviations on 0.49%, 0.24%, and 0.13% of  days, respectively. The lognormal model predicts that we should hardly  ever observe this happening. This table  therefore  provides evidence to support the existence of heavy tails  and the volatility smile used by traders.

Foreign Currency Options- Smile Explanation

•There following two assumptions will lead to a lognormal distribution:

1. The volatility of the asset is constant.

2. The price of the asset changes smoothly with no jumps.

•But these assumptions are not valid in practice for exchange rates since:

1. Volatility of exchange rates is not constant

2. Exchange  rates  exhibit  frequent  jumps,  sometimes  in  response  to  the  actions  of  central  banks.

•The impact of jumps and nonconstant volatility depends on the  option  maturity. As the  maturity of the option is increased, the percentage impact of a nonconstant volatility on prices  becomes more pronounced, but its percentage impact on implied volatility usually becomes less  pronounced. The percentage impact of jumps on both prices and the implied volatility becomes  less pronounced as the maturity of the option is increased. The result of all this is that the  volatility smile becomes less pronounced as option maturity increases.

Equity Options

•Before the crash of 1987, there was no  marked volatility smile for equity options.  Since 1987, the volatility smile used by  traders to price equity options (both on  individual stocks and on stock indices)  has had the general form shown in this  figure. This is sometimes referred to as a  volatility skew. The volatility decreases as  the strike price increases. The volatility  used to price a low-strike-price option  (i.e., a deep-out-of-the-money put or a  deep-in-the-money call) is significantly  higher than that used to price a high-strike-price option (i.e., a deep-in-the-  money put or a deep-out-of-the-money call).

•The volatility smile for equity options  corresponds to the implied probability  distribution given by the solid line in  this figure. A lognormal distribution  with the same mean and standard  deviation as the implied distribution is  shown by the dotted line. It can be seen  that the implied distribution has a  heavier left tail and a less heavy right tail  than the lognormal distribution.

•Consider first a deep-out-of-the-money call option with a high strike price of 𝐾₂ (𝐾₂/𝑆₀well  above 1.0). This option pays off only if the stock price is above 𝐾₂. The 𝐼𝐷  figure shows that  the probability of this is lower for the implied probability distribution than for the lognormal  distribution, and so the option has a lower price when the implied distribution is used than when  the lognormal distribution is used. Hence, it can be expected that the implied distribution will  give a relatively low price for the option. A relatively low-price leads to a relatively low implied  volatility, and this is exactly is observed in the 𝐼𝑉 figure.

•Consider next a deep-out-of-the-money put option with a low strike price of 𝐾₁, (𝐾₁/𝑆₀well  below 1.0). This option pays off only if the stock price is below 𝐾₁. The 𝐼𝐷  figure shows that  the probability of this is higher for the implied probability distribution than for the lognormal  distribution. Hence, it can be expected that the implied distribution will give a relatively high  price, and consequentially a relatively high implied volatility, for this option as well. Again, this is  exactly what observe int he 𝐼𝑉 figure.

•Two explanation are offered for the volatility skew of equity options:

1. Crashophobia : The probability of a massive fall in the equities markets is greater than the  probability of a massive rise, i.e. crashes are more severe than upward spikes. So out-of-the-  money put options are more in demand, as an insurance against a massive fall, making their  price higher than the expected price.

2. Leverage : As equity prices move down, leverage increases and as a result volatility increases,  and vice versa.

Alternative Ways of Characterizing Volatility Smile

•The simple or typical volatility smile plots implied volatility against strike price. However, this  presentation can be a bit unstable because the smile tends to shift when the asset price moves. There  are four alternative approaches –

a)  Replacing the strike price with ^𝑿 : This method results in a more stable volatility smile.

b)  Replacing the strike price with ^𝑿 : The forward price should have the same maturity date as the  options.  Forward  price  are  sometimes  considered  as  a  better  reflection  of  at-the-money  option prices as it considers the theoretical expected stock price.

c)Replacing the strike price with the option’s delta : This approach enables the application  of volatility smiles to options apart from European and American calls and puts.

d)Some financial engineers choose to define the volatility smile as the relationship between implied  volatility  and  ^𝟏/T  𝐥𝐧 (x / f0). The  smile  is  then  usually  much  less  dependent  on the time to maturity.

The Volatility Term Structure and Volatility Surfaces

•The volatility term structure is a listing of implied volatilities as a function of  time to expiration  for at-the-money option contracts.

1. When short-dated volatilities are historically low, there is then an expectation that volatilities  will increase. In that case, implied volatility tends to be an increasing function of maturity.

2. When short-dated volatilities are historically high, there is then an expectation that volatilities  will decrease. In that case, implied volatility tends to be a decreasing function of maturity.

•A volatility surface is a combination of a volatility term structure with volatility smiles (i.e., those  implied volatilities away-from-the-money). The surface provides guidance in pricing options with  any strike or maturity structure. A volatility surface has usually three dimensions: Maturity, 𝐾/𝑆₀,  and Volatility Value. These volatility values are implied volatilities which are produced from the  market prices of traded options.

• The volatility term structure and volatility surfaces can be used to validate a model’s  accuracy and consistency in pricing

•An example of a volatility surface that might be used for foreign currency options is given in  this table. At any given time, some of the entries in the table are likely to correspond to options  for which reliable market data are available. The implied volatilities for these options are  calculated directly from their market prices and entered into the table. The rest of the table is  typically determined using interpolation. Once a range of volatility smiles are produced for  different tenors and expiry terms, all of the smiles on terms and tenors are joined and the smiles  are plotted together. A three-dimensional volatility surface is then produced. The table shows  that the volatility smile becomes less pronounced as the option maturity increases. As mentioned  earlier, this is what is observed for currency options. (It is also what is observed for options on  most other assets.)

	K/S₀
	0.90	0.95	1.00	1.05	1.10
1 month	14.2	13.0	12.0	13.1	14.5
3 month	14.0	13.0	12.0	13.1	14.2
6 month	14.1	13.3	12.5	13.4	14.3
1 year	14.7	14.0	13.5	14.0	14.8
2 year	15.0	14.4	14.0	14.5	15.1
5 year	14.8	14.6	14.4	14.7	15.0

Minimum Variance Delta

•The equity options have a volatility smile dictated by two phenomena:

1. As the equity price increases (decreases), ^𝑲/S0 decreases (increases) and the volatility increases (decreases). In other words, the option moves up the curve when the equity price increases and down the curve when the equity price decreases.

2. There is a negative correlation between equity prices and their volatilities. When the equity  price increases, the whole curve tends to move down; when the equity price decreases, the  whole curve tends to move up.

•It turns out that the second effect dominates the first, so that implied volatilities tend to move  down (up) when the equity price moves up (down). The delta that  takes this relationship  between implied volatilities and equity prices into account is referred to  as the  minimum  variance delta.

•Minimum variance delta is given by:

\Delta_{MV} = \frac{\partial f_{BSM}}{\partial S} + \frac{\partial f_{BSM}}{\partial \sigma_{imp}} \frac{\partial E(\sigma_{imp})}{\partial S}

where 𝑓_BSM is the Black-Scholes-Merton price of  the option, 𝜎_imp is the option’s implied  volatility, 𝐸 (𝜎_imp) denotes the expectation of 𝜎_imp as a function of the equity price, 𝑆.

This gives

\Delta_{MV} = \Delta_{BSM} + v_{BSM} \frac{\partial E(\sigma_{imp})}{\partial S}

Where  Δ_BSM  and  𝑣_BSM are  the  delta  and  vega  calculated  from  the  Black-Scholes-Merton (constant volatility) model. Because 𝑣  is positive and, as we have just explained 𝜕𝐸 ^σimp is negative, the minimum variance delta is less than the Black-Scholes-Merton delta.

When A Single Large Jump is Anticipated

•Let us now consider an example of how an unusual volatility smile might arise in equity  markets. Suppose that a stock price is currently $50 and an important news announcement due  in a few days is expected either to increase the stock price by $8 or to reduce it by $8. (This  announcement could concern the outcome of a takeover attempt or the verdict in an important  lawsuit.) The probability distribution of the stock price in, say, 1 month might then consist of a mixture of two lognormal distributions, the first  corresponding to favorable news, the second to  unfavorable news. The situation is illustrated in  this figure. The solid line shows the mixture-of-  lognormal distributions for the stock price in 1  month; the dashed line shows a lognormal  distribution with the same mean and standard  deviation as this distribution.

•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. Options can be  valued using the binomial model. In this case 𝑢 = 1.16, 𝑑 = 0.84, The results from valuing a  range of  different options are shown in the given table (The implied volatility of  a European  put option is the same as that of  a European call option when they have the same strike price  and maturity.)

 

Strike price ($)	Call price ($)	Put price ($)	Implied volatility (%)
42	8.42	0.00	0.0
44	7.37	0.93	58.8
46	6.31	1.86	66.6
48	5.26	2.78	69.5
50	4.21	3.71	69.2
52	3.16	4.64	66.1
54	2.10	5.57	60.0
56	1.05	6.50	49.0
58	0.00	7.42	0.0

 

•This figure displays the volatility smile from the previous table. It is actually a “frown” (the  opposite of that observed for currencies) 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.

Summary

•The Black-Scholes-Merton model and its extensions assume that the probability distribution of the  underlying asset at any given future time is lognormal. This assumption is not the one made by traders.  They assume the probability distribution of an equity price has a heavier left tail and a less heavy right tail  than the lognormal distribution. They also assume that the probability distribution of  an exchange rate has  a heavier right tail and a heavier left tail than the lognormal distribution.

•Traders use volatility smiles to allow for nonlognormality, The volatility smile defines the relationship  between the implied volatility of an option and its strike price. For equity options, the volatility smile tends  to be downward sloping. This means that out-of-the-money puts and in-the-money calls tend to have high  implied volatilities whereas out-of-the-money calls and in-the-money puts tend to have low implied  volatilities. For foreign currency options, the volatility smile is U-shaped. Both out-of-the-money and in-  the-money options have higher implied volatilities than at-the money options.

•Often traders also use a volatility term structure. The implied volatility of  an option then depends on its  life. When volatility smiles and volatility term structures are combined, they produce a volatility surface.  This defines implied volatility as a function of both the strike price and the time to maturity.