Volatility smiles used by traders in equity and foreign currency markets will be discussed in this chapter. Moreover, the connection between the risk-neutral probability distribution assumed for asset prices of the future and a volatility smile is also explained. In addition, the discussion will also focus on the use of volatility surfaces by traders as tools of pricing.

# Why the Volatility Smile is the same for Calls and Puts

With similar strike price and maturity time the relationship between European call and put option prices is provided by put-call parity as earlier explained. The relationship shown below has a dividend yield on underlying asset of q.

$$ p+{ S }_{ 0 }{ e }^{ -qt }=c+K{ e }^{ -rT } $$

Where \(c\) and \(p\) are the European call and put prices respectively’ with similar strike price \(K\) and maturity time \(T\). Today’s price of the underlying asset is denoted by \({ S }_{ 0 }\) while maturity is \(T\) and the risk-free interest rate is denoted by \(r\).

Assumptions about the probability distribution of the future asset price are not important in this relationship either with lognormal asset price distribution or not lognormal.

Due to the fact that Black-Scholes-Merton model applies put-call parity then:

$$ { p }_{ BS }+{ S }_{ 0 }{ e }^{ -qT }={ c }_{ BS }+K{ e }^{ -rt } $$

For the market prices, put-call parity holds when arbitrage opportunities are absent such that:

$$ { p }_{ MKT }+{ S }_{ 0 }{ e }^{ -qT }={ c }_{ MKT }+K{ e }^{ -rt } $$

The difference between the two equations is:

$$ { p }_{ BS }-{ p }_{ MKT }={ c }_{ BS }-{ c }_{ MKT } $$

Where \({ p }_{ BS }\) and \({ c }_{ BS }\) are options for European puts and calls computed through the Black-Scholes-Merton model and \({ p }_{ MKT }\) and \({ c }_{ MKT }\) are the market values for these options.

The implication here is that with the same strike price and maturity time, there exists an exact similarity in the dollar pricing error when using the Black-Scholes-Merton model for pricing anEuropean put option and the dollar pricing error for pricing an European call option.

# Foreign Currency Options

If we are to draw a curve of the volatility smile traders use to price foreign currency options, the following observations will be made from its form: movement by an option either into or out of the money causes it to grow progressively higher.

Drawing a curve for the lognormal distribution with a similar mean and standard deviation as the implied distribution, the observation made will be that the implied distribution’s tail will be heavier than that of the lognormal distribution.

# Empirical Results

To determine whether volatility smile used by traders for foreign currency options means that lognormal distribution is considered to understate exchange rates’ extreme movement probability, the daily movements in several different exchange rates over a period of time is examined by the following table.

$$ \begin{array}{|c|c|c|} \hline {} & Real \quad World & Lognormal \quad Model \\ \hline >1 \quad SD & 23.2 & 31.73 \\ \hline >2 \quad SD & 4.67 & 4.55 \\ \hline >3 \quad SD & 1.30 & 0.27 \\ \hline >4 \quad SD & 0.49 & 0.01 \\ \hline >5 \quad SD & 0.24 & 0.00 \\ \hline >6 \quad SD & 0.13 & 0.00 \\ \hline \hline \end{array} $$

The existence of heavy tails and the volatility smile used by traders is supported by the table. In each exchange rates, we compute first the standard deviation of the daily percentage change to create the table.

We then observe the rate at which the actual percentage change surpasses 1 standard deviation, 2 standard deviations,etc. and finally the rate at which this occurrence would be observed if there was a normal distribution in the percentage changes.

# Reason for the Smile in Foreign Currency Options

For an asset price to have a lognormal distribution, the following conditions must be satisfied:

- The asset volatility should be constant; and
- The asset price should be smoothly changing with no jumps.

Practically, exchange rate volatility is hardly constant,and still experiences jumpsoften when responding to central banks’ actions. This therefore increases the chances of extreme outcomes.

Option maturity is the determining factor of the non-constant volatility and jumps’ effect. An increase in the option maturity leads to a corresponding rise in percentage effect of a non-constant volatility on prices, and a decline in its percentage impact on implied volatility.

# Equity Options

The following curve often referred to as volatility skew belongs to the volatility smile traders use to price equity options.

A decrease in volatility is followed by a corresponding increase in strike price. In pricing a high-strike-price option, the applied volatility is lower than the one used when pricing the low-strike-price option.

The black line in the following curve depicts the implied probability distribution corresponding to the volatility smile for equity options, whereas the red line depicts the lognormal distribution curve having similar mean and standard deviation.

The left tail of the implied distribution in the curve is heavier as compared the right tail than the lognormal distribution.

# The Reason for the Smile in Equity Options

The correlation between prices of equities is a negative one. A downward movement in equity prices is followed by an upward movement in volatility. This can be attributed to several reasons; one being that an upward movement in equity prices is followed by a corresponding decrease in leverage hence decreasing volatility and vice versa.

The volatility feedback effect is the other reason; an increase in volatility due to external factors causes a higher return requirement by investors thereby causing the stock prices to fall.

The implication of the negative correlation is that volatility increases due to falling stock prices thus enabling further declines. The decrease in volatility is preceded by rising stock prices hence decreasing the chances of prices of stocks to rise.

# Alternative Ways of Characterizing the Volatility Smile

The relationship between implied volatility and strike price \(K\) depends on the price of the asset.

Computing the volatility smile as the relationship between implied volatility and \({ K }/{ { F }_{ 0 } }\) is an aim to refine this (\({ F }_{ 0 }\) is the asset’s forward price for contracts maturing at similar times as the considered options).

Connection between delta of the option and implied volatility is another definition of volatility smile. With this approach, it is often possible for options other than European and American calls and puts to be applied with volatility smiles.

# The Volatility Term Structure and Volatility Surfaces

Strike price and time to maturity are what traders allow implied volatility to depend on. In case short-dated volatility becomes historically low, then implied volatility has a tendency of being an increasing maturity function, the reason being that an increase in the volatility function is expected.

Likewise, in case short-dated volatility becomes historically high, then there is a tendency by implied volatility to be a decreasing maturity function.

An option can be priced with any strike price and maturity using appropriate volatilities tabulated by a combination of volatility smiles and volatility term structure through volatility surfaces. Engineers will search for the appropriate volatility in the table during valuation of a new option.

When contracts and options mature at the same time, \(T\) is defined as maturity time and \({ F }_{ 0 }\) the forward price of the asset.

Rather than the relationship between implied volatility and \(K\), volatility smile is often defined as the relationship between implied volatility and:

$$ \frac { 1 }{ \sqrt { T } } ln\left( \frac { K }{ { F }_{ 0 } } \right) $$

# Minimum Variance Delta

The assumption in delta formulas and other Greek alphabets is that a change in other prices has no effect on implied volatility whereas this is not usually the case. To identify the phenomenon in the volatility smile curve previously discussed:

- A decrease in equity price is followed by an increase in \({ K }/{ { S }_{ 0 } }\) and a decrease in volatility.
- Prices of equity and their volatility are negatively correlated. A downward movement in volatility is preceded by an upward movement in price and vice versa. The delta accounting for this relationship is called the minimum variance delta:

$$ { \Delta }_{ MV }=\frac { \partial { f }_{ BSM } }{ \partial S } +\frac { \partial { f }_{ BSM } }{ \partial { \sigma }_{ imp } } \frac { \partial E\left( { \sigma }_{ imp } \right) }{ \partial S } $$

Where, \({ f }_{ BSM }\) is the option’s Black-Scholes-Merton price, \({ \sigma }_{ imp }\) is the implied volatility of the option and \(E\left( { \sigma }_{ imp } \right) \) is its expectation as an equity price function. Thus:

$$ { \Delta }_{ MV }={ \Delta }_{ BSM }+{ V }_{ BSM }\frac { \partial E\left( { \sigma }_{ imp } \right) }{ \partial S } $$

Where \({ \Delta }_{ BSM }\) And \({ V }_{ BSM }\) are the delta and Vega computed from the Black-Scholes-Merton model.

# The Role of the Model

Should traders decide to change from the Black-Scholes-Merton model to another on more plausible, the quoted dollar prices will remain constant,but there would be changes in the volatility surface and the smile’s shape models used are depended upon by hedging strategies and hence Greek letters.

Lack of active trades by similar derivatives in the market implies that models will be most effective on derivatives pricing.

# Anticipation of a Singular Large Jump

Imagine a stock currently valued at $100 but could either gain $20 or lose $20 of its value when some important anticipated news is made in a few days. A combination of two lognormal distributions, one depicting the favorable news and the second unfavorable news, make up the stock price’s probability distribution.

The true probability distribution is not lognormal but bimodal. In extreme cases with only two probable future prices of stocks is an important consideration if we are interested in determining the impact of a bimodal stock price distribution.

# Determining Implied Risk-Neutral Distributions from Volatility Smiles

The following equation determines the price of a European call option on an asset with strike price \(K\) and maturity \(T\):

$$ c={ e }^{ -rT }\int _{ { S }_{ T }=K }^{ \infty }{ \left( { S }_{ T }-K \right) g\left( { S }_{ R } \right) d{ S }_{ T } } $$

Where \(g\) is the risk-neutral probability density function of \({ S }_{ T }\), ris the interest rate and \({ S }_{ T }\) is the asset price at time \(T\)’s. A single differentiation with respect to \(K\) gives:

$$ \frac { \partial c }{ \partial K } =-{ e }^{ -rt }\int _{ { S }_{ T }=K }^{ \infty }{ g\left( { S }_{ T } \right) d{ S }_{ T } } $$

Another differentiation with \(K\) gives:

$$ \frac { { \partial }^{ 2 }c }{ \partial { K }^{ 2 } } ={ e }^{ -rt }g\left( K \right) $$

Therefore,the probability density function \(g\) is given by:

$$ g\left( K \right) ={ e }^{ rt }\frac { { \partial }^{ 2 }c }{ \partial { K }^{ 2 } } $$

Assuming that \(T\)-year European call options with strike prices \(K\)-\(\delta \),\(K\),and \(K+\delta\) have prices \({ b }_{ 1 }\),\({ b }_{ 2 }\) and \({ b }_{ 3 }\) respectively, then if \(\delta\) is small, an estimate of \(g\left( K \right)\) is:

$$ g\left( K \right) ={ e }^{ rt }\frac { { b }_{ 1 }+{ b }_{ 3 }-2{ b }_{ 2 } }{ { \delta }^{ 2 } } $$

# Practice Questions

1) A foreign currency is valued at $200.71. The foreign currency has a European call option market price of $13.55 and a strike price of $225. In the US, the risk-free interest rate is 4% per annum and 7% per annum in the foreign country. Determine the price of a European put option with a 1-year maturity for the foreign currency.

- $14.68
- $13.55
- $42.59
- $15.48

The correct answer is **C**.

The price \(p\) of a European put option with a strike price of $1.60 and 1-year maturity must satisfy the following equation:

$$ p+{ S }_{ 0 }{ e }^{ -qT }=c+K{ e }^{ -rT } $$

From the question, we are provided that:

\({ S }_{ 0 }=$200.71\), \(q=16\%\), \(T=1\), \(c=13.55\), \(K=$225\) and \(r=7\%\)

Therefore:

$$ p+200.71\times { e }^{ -0.07\times 1 }=13.55+225{ e }^{ -0.04\times 1 } $$

$$ \Rightarrow p=42.59 $$

2) \(g\left( { S }_{ T } \right) \) is a constant between \({ S }_{ T } =8\) and \({ S }_{ T }=10\). We are also informed that, a non-dividend paying stock has a price of $11.3 and a risk-free interest rate of 3%. The implied volatilities of 6-month European options with strike prices of $8, $9 and $10 are 13%, 12% and 10.5% respectively. From Deriva Gem their prices are $2.35, $2.25 and $2.14 respectively with \(K=7\) and \(\delta=0.32\). Compute the value of \({ g }_{ 1 }\) by interpolating to get the implied volatility of a 6-month option with a strike price of $8.50 as 12.6%.

- 3.22
- 3.17
- 1.02
- 5.15

The correct answer is **B**.

Recall that:

$$ g\left( K \right) ={ e }^{ rt }\frac { { b }_{ 1 }+{ b }_{ 3 }-2{ b }_{ 2 } }{ { \delta }^{ 2 } } $$

From the problem, we have:

\(r=0.03\), \(t=0.5\), \({ b }_{ 1 }\), \({ b }_{ 2 }\) and \({ b }_{ 3 }\) are $2.35, $2.25 and $2.14 respectively.

Therefore:

$$ g\left( { S }_{ 1 } \right) ={ e }^{ 0.5\times 0.03 }\times \frac { \left( 2.35+2.25-2\times 2.14 \right) }{ { 0.32 }^{ 2 } } $$

$$ \Rightarrow g\left( { S }_{ 1 } \right) =3.17 $$

3) Suppose that a foreign country \(X\) has its currency valued at $0.656. Suppose further that the European call and put options computed by Black-Scholes-Merton model are 0.0249 and 0.0501 respectively. Compute the market price of the call option if the market price of the put option is 0.0317.

- 0.0056
- 0.0025
- 0.0337
- 0.0654

The correct answer is **A**.

For Black-Scholes-Merton model and in the absence of arbitrage opportunities, the put-call parity satisfies:

$$ { p }_{ BS }+{ S }_{ 0 }{ e }^{ -qT }={ c }_{ BS }K{ e }^{ -rt } $$

For the market prices, put-call parity holds when arbitrage opportunities are absent such that:

$$ { p }_{ MKT }+{ S }_{ 0 }{ e }^{ -qT }={ c }_{ MKT }K{ e }^{ -rt } $$

The difference between the two equations is:

$$ { p }_{ BS }-{ p }_{ MKT }={ c }_{ BS }-{ c }_{ MKT } $$

From the question we have that:

\({ c }_{ BS }=0.0249\), \({ p }_{ BS }=0.051\) and \({ p }_{ MKT }=0.0317\)

Thus:

$$ 0.051-0.0317=0.0249-{ c }_{ MKT } $$

$$ \Rightarrow { c }_{ MKT }=0.0056 $$