The Black-Scholes-Merton Model

After completing this reading you should be able to:

• Explain the lognormal property of stock prices, the distribution of rates of return, and the calculation of expected return.
• Compute the realized return and historical volatility of a stock.
• Describe the assumptions underlying the Black-Scholes-Merton option pricing model.
• Compute the value of a European option using the Black-Scholes-Merton model on a non-dividend-paying stock.
• Define implied volatilities and describe how to compute implied volatilities from market prices of options using the Black-Scholes-Merton model.
• Explain how dividends affect the decision to exercise early for American call and put options.
• Compute the value of a European option using the Black-Scholes-Merton model on a dividend-paying stock.
• Describe warrants, calculate the value of a warrant, and calculate the dilution cost of the warrant to existing shareholders.

Suppose we have a random variable X. This variable will have a lognormal distribution if its natural log (ln X) is normally distributed. In other words, when the natural logarithm of a random variable is normally distributed, then the variable itself will have a lognormal distribution.

The two most essential characteristics of the lognormal distribution are as follows:

• It has a lower bound of zero, i.e., a lognormal variable cannot take on negative values.
• The distribution is skewed to the right, i.e., it has a long right tail.

These characteristics are in direct contrast to those of the normal distribution, which is symmetrical (zero-skew) and can take on both negative and positive values. As a result, the normal distribution cannot be used to model stock prices because stock prices cannot fall below zero. The lognormal distribution is also used to value options.

The Lognormal Property of Stock Prices

A crucial part of the BSM model is that it assumes stock prices are log-normally distributed. Precisely,

$$ \text{ln}{ \text{S} }_{ \text{T} }\sim \text{N}\left( \text{ln}{ \text{S} }_{ 0 }+\left( \mu -\cfrac { { \sigma }^{ 2 } }{ 2 } \right) \text{T},{ \sigma }^{ 2 }\text{T} \right) $$

Where:

\({\text{S}}_{\text{T}}\)=stock price at time T

\({\text{S}}_{\text{0}}\)=stock price at time 0

\(\mu \)=expected return on stock per year

\(\sigma \)=annual volatility of the stock price

Note: The above relationship holds because mathematically, if the natural logarithm of a random variable \(x\), ln(\(x\)) is normally distributed, then \(x\) has a lognormal distribution. It’s also imperative to note that the BSM model assumes stock prices are lognormally distributed, with stock returns being normally distributed. Specifically, continuously compounded annual returns are normally distributed with:

$$ \text{a mean of} \left[ \mu -\cfrac { { \sigma }^{ 2 } }{ 2 } \right] \text{and a variance of} \quad \cfrac { { \sigma }^{ 2 } }{ \text{T} } $$

Example: Mean and standard deviation given a lognormal distribution

ABC stock has an initial price of $60, an expected annual return of 10%, and annual volatility of 15%. Calculate the mean and the standard deviation of the distribution of the stock price in six months.

$$ \begin{align*} & \text{ln}{ \text{S} }_{\text{T}} \sim \text{N}\left( \text{ln}{ \text{S} }_{ 0 }+\left( \mu -\cfrac { { \sigma }^{ 2 } }{ 2 } \right) \text{T},{ \sigma }^{2}{ \text{T} } \right) \\= & \text{N}\left[ \text{ln}60+\left( 0.10-\cfrac { { 0.15 }^{ 2 } }{ 2 } \right) 0.5,{ 0.15 }^{ 2 }\times 0.5 \right] \\& \text{ln}{ \text{S}}_\text{T}\sim \text{N}\left[ 4.139,0.011 \right] \\ & (\text{In this case,we have } \sigma=\sqrt{0.011}=0.1049) \end{align*} $$

Sometimes, the Global Association of Risk Professionals (GARP) may want to test your understanding of the lognormal concept by involving confidence intervals. Since \(\text{lnS}_{\text T}\) is log-normally distributed, 95% of values will fall within 1.96 standard deviations of the mean. Similarly, 99% of the values will fall within 2.58 standard deviations of the mean. For example, to obtain the 99% confidence interval for stock prices using the above data, we will proceed as follows:

$$ \text{ln}{ \text{S}}_{\text{T}} \sim \text{N}\left[ 4.139,0.011 \right] \\ \text{ln}{ \text{S}}_{\text{T}}=\mu \pm { \text{z} }_{ \alpha }\times \sigma \\ (\text{In this case,we have } \mu=\sqrt{0.011}=0.1049)\\ 4.139-{ \text{z} }_{ \alpha }\times \sigma <\text{ln}{ \text{S}}_{\text{T}}<4.139+{ \text{z} }_{ \alpha }\times \sigma \\ \text e^{4.139-\text z_\alpha \times \sigma}< \text S_{\text T} < \text e^{4.139+\text z_{\alpha} \times \sigma}\\  47.86 < { \text{S} }_{ \text{T} } < 82.24\\ \left[ { \text{e} }^{ \left( \text{lnx} \right) }=\text{x} \right] $$

Expected Stock Price

Using the properties of a lognormal distribution, we can show that the expected value of \({\text{S}}_{\text{T}}, {\text{E}}({\text{S}}_{\text{T}} )\), is:

$$ {\text{E}}({\text{S}}_{\text{T}} )={ \text{S} }_{ 0 }{ \text{e} }^{ \mu \text{T} } $$

\(\mu\)=expected rate of return

Example: Expected value of a stock

The current price of a stock is $40, with an expected annual return of 15%. What is the expected value of the stock in six months?

Solution

$$ \text{Expected stock price}={$40}{ \text{e} }^{ 0.15 \times 0.5}=$43.11 $$

Realized Portfolio Return

$$ \text{R}_{ \text{pr} }={ \left[ { \text{r} }_{ 1 }\times { \text{r} }_{ 2 }\times { \text{r} }_{ 3 }\times \cdots \times { \text{r} }_{ \text{n} } \right] }^{ \cfrac { 1 }{ \text{n} } }-1 $$

\({\text{r}}_{\text{i}}\)=portfolio return at the time i

The continuously compounded return realized over some time of length T is given by:

$$ { \cfrac { 1 }{ \text{T} } \text{ln}\left( \cfrac { { \text{S} }_{ \text{T} } }{ { \text{S} }_{ 0 } } \right) } $$

Example: Realized return

The realized return of a stock initially priced at $50 growing, with volatility, to $87 over five periods would simply be:

$$ \text{Realized return }={ \cfrac { 1 }{5} \text{ln}\left( \cfrac { $87 }{ $50 } \right) }=11.08\% $$

Estimating Historical Volatility

We can calculate historical volatility from daily price data of stock. We simply need to calculate continuously compounded returns per day and then determine the standard deviation.

The continuously compounded return for the day i is calculated as:

$$ \text{ln}\left( \cfrac { { \text{S} }_{ \text{i} } }{ { \text{S} }_{ \text{i}-1 } } \right) $$

The volatility of short periods can be scaled to give the volatility of more extended periods.

For example,

$$ \text{Annual volatility }=\text{daily volatility }\times \sqrt{(\text{ no.of trading days in a year})} $$

Note that this formula is useful throughout the whole FRM part 1 and FRM part 2 exams in estimating volatility.

Conversely,

$$ \text{daily volatility }=\cfrac { \text{annual volatility} }{ \sqrt{\text{ no.of trading days in a year} } } $$

Black-Scholes-Merton Model

The Black-Scholes-Merton model is used to price European options and is undoubtedly the most critical tool for the analysis of derivatives. It is a product of Fischer Black, Myron Scholes, and Robert Merton.

The model takes into account the fact that the investor has the option of investing in an asset earning the risk-free interest rate. The overriding argument is that the option price is purely a function of the volatility of the stock’s price (option premium increases as volatility increases).

Assumptions underlying the Black-Scholes-Merton Option Pricing Model

1. There is no arbitrage.
2. The price of the underlying asset follows a lognormal distribution.
3. The continuous risk-free rate of interest is constant and known with certainty.
4. The volatility of the underlying asset is constant and known.
5. The underlying asset has no cash flow, such as dividends or interest payments.
6. Markets are frictionless – no transaction costs, taxes, or restrictions on short sales.
7. Options can only be exercised at maturity, i.e., they are European-style. The model cannot be used to value American options accurately.

Determining the Value of Zero-dividend European Options using the BSM model

The value of a call option is given by:

$$ { \text{C} }_{ 0 }={ \text{S} }_{ 0 }\times \text{N}\left( { \text{d} }_{ 1 } \right) -\text{K}{ \text{e} }^{ -\text{rT} }\times \text{N}\left( { \text{d} }_{ 2 } \right) $$

The value of a put option is given by:

$$ \text{P}_{ 0 }=\text{K}{ \text{e} }^{ -\text{rT} }\times \text{N}\left( { -\text{d} }_{ 2 } \right) -{ \text{S} }_{ 0 }\times \text{N}\left( -{ \text{d} }_{ 1 } \right) $$

Where:

$$ \begin{align*} { \text{d} }_{ 1 }&=\cfrac { \text{ln}\left( \cfrac { { \text{S} }_{ 0 } }{ \text{K} } \right) +\left[ \text{r}+\left( \cfrac { { \sigma }^{ 2 } }{ 2 } \right) \right] \text{T} }{ \sigma \sqrt { \text{T} } } \\ { \text{d} }_{ 2 }&={ \text{d} }_{ 1 }-\left({ \sigma \sqrt { \text{T} } }\right) \end{align*} $$

T =time to maturity,assuming 365 days per year

\({\text{S}}_{0}\)=asset price

K =exercise price

\({\text{R}}_{\text{f}}^{\text{c}}\)=continuously compounded risk-free rate

\(\sigma\)=volatility of continuously compounded returns on the stock

\(\text{N}({\text{d}}_{\text{i}}) \)=cumulative distribution function for a standardized normal distribution variable

Example: Valuing a call option using the BSM model

Assume \({\text{S}}_{0}\) = $100, K = $90, T = 6 months, r = 10%, and \(\sigma\) = 25%.

Calculate the value of a call option.

$$ \begin{align*} { \text{d} }_{ 1 }&=\cfrac { \text{ln}\left( \cfrac { { \text{S} }_{ 0 } }{ \text{K} } \right) +\left[ \text{r}+\left( \cfrac { { \sigma }^{ 2 } }{ 2 } \right) \right] \text{T} }{ \sigma \sqrt { \text{T} } } \\ &=\cfrac { \text{ln}\left( \cfrac { 100 }{ 90 } \right) +\left[ 0.10+\left( \cfrac { { 0.25 }^{ 2 } }{ 2 } \right) \right] 0.5 }{ 0.25\sqrt { 0.5 } } =\cfrac { 0.1053+0.0656 }{ 0.1768 } =0.9672 \\ \text d_2 & =\text d_1-(\sigma \sqrt{\text T})=0.9672-(0.25 \sqrt{0.5}) =0.7904 \\ \end{align*} $$

From a standard normal probability table, look up N(0.97) = 0.8333 and N(0.79) = 0.7852.

$$ \begin{align*} { \text{C} }_{ 0 }&={ \text{S} }_{ 0 }\times \text{N}\left( { \text{d} }_{ 1 } \right) -\text{K}{ \text{e} }^{ -\text{rT} }\times \text{N}\left( { \text{d} }_{ 2 } \right) \\ &=100\times \text N\left( 0.8340 \right) -90{\text e }^{ -0.10 \times 0.50 }\times \text N\left( 0.7852 \right) =$16.17 \end{align*} $$

Note that the intrinsic value of the option is $10—our answer must be at least that amount.

Exam tips

Tip 1: Given one of either the put value or the call value, you can use the put-call parity to find the other. Precisely,

$$ \begin{align*} { \text{C} }_{ 0 }&=\text{P}_{ 0 }+{ \text{S} }_{ 0 }-\left( \text{K}{ \text{e} }^{ \left( -{ { \text{R} } }_{ { \text{f} } }^{ { \text{c} } }\text{T} \right) } \right) \\ { \text{P} }_{ 0 }&=\text{C}_{ 0 }-{ \text{S} }_{ 0 }+\left( \text{K}{ \text{e} }^{ \left( -{ { \text{R} } }_{ { \text{f} } }^{ { \text{c} } }\text{T} \right) } \right) \end{align*} $$

Tip 2: \(\text{N}(−\text{d}_\text{1})=1−\text{N}(\text{d}_\text{1})\)

Tip 3: As \(\text S_0\) becomes very large, calls (puts) are extremely in-the-money (out-of-the-money)

Tip 4: As \({\text{S}}_{0}\) becomes very small, calls (puts) are extremely out-of-the-money (in-the-money)

Tip 5: Although \(\text{N}(−\text{d}_\text{1})\)and \(\text{N}(−\text{d}_{2})\) can easily be identified from statistical tables; sometimes you’ll be asked to compute \(\text{d}_{2}\) and \(\text{d}_{2}\) without the formulas.

The Value of a European Option using the BSM Model on a Dividend-paying Stock

Assume that we have a known dividend d distributed a time \({\text{T}}_{\text{1}},{\text{T1}} < {\text{T}}\) where T is the maturity date. To value calls and puts when there are such dividends, we modify the BSM model by replacing \({\text{S}}_{\text{0}}\)withS, where:

$$ \text{S}={\text{S}}_{0}-{\text{D}} $$

D is the sum of the PV(discounted at \({ { \text{R} } }_{ { \text{f} } }^{ { \text{c} } }\) ) of the dividend payments during the life of the option.

$$ \begin{align*} \text{ For example,with dividends } \text{D}_{1} \text{ and } \text{D}_{2} \text{ at times } {\Delta}\text{t}_{1} \text{ nd } { \Delta}\text{t}_{2}, \\ \text{S}={ \text{S} }_{ 0 }-{ \text{D} }_{ 1 }{ \text{e} }^{ -\left( { { \text{R} } }_{ { \text{f} } }^{ { \text{c} } } \right) \cfrac { \Delta { \text{t} }_{ 1 } }{ \text{m} } }-{ \text{D} }_{ 2 }{ \text{e} }^{ -\left( { { \text{R} } }_{ { \text{f} } }^{ { \text{c} } } \right) \cfrac { \Delta { \text{t} }_{ 2 } }{ \text{m} } } \end{align*} $$

\({\Delta }{\text{t}}_{\text{i}}\) represents the amount of time until the ex-dividend date

m a division factor in bringing the \(\Delta t\) to a full year. e.g. \(\Delta t=2\Delta t=2\) months, m=12 months, so \( \cfrac { \Delta { \text{t} }_{ 1 } }{ \text{m} } =\cfrac {2}{12}=0.1667\) years .

After this, everything else in the computational formulas remains the same, i.e.,

The value of a call option is given by:

$$ { \text{C} }_{ 0 }=\left[ { \text{S} }_{ 0 }\times \text{N}\left( { \text{d} }_{ 1 } \right) \right] -\left| \text{K}\times { \text{e} }^{ \left( -{ { \text{R} } }_{ { \text{f} } }^{ { \text{c} } }\times \text{T} \right) }\times \text{N}\left( { \text{d} }_{ 2 } \right) \right| $$

The value of a put option is given by:

$$ \begin{align*} \text{P}_{ 0 }&=\left[ \text{K}\times { \text{e} }^{ \left( -{ { \text{R} } }_{ { \text{f} } }^{ { \text{c} } }\times \text{T} \right) }\times \left( 1-\text{N}\left( { \text{d} }_{ 2 } \right) \right) \right] -\left[ \text{S}\times \left( 1-\text{N}\left( { \text{d} }_{ 1 } \right) \right) \right] \\ { \text{d} }_{ 1 }&=\cfrac { \text{ln}\left( \cfrac { { \text{S} } }{ \text{K} } \right) +\left[ { { \text{R} } }_{ { \text{f} } }^{ { \text{c} } }+\left( 0.5\times { \sigma }^{ 2 } \right) \right] \text{T} }{ \sigma \sqrt { \text{T} } } \\ { \text{d} }_{ 2 }&={ \text{d} }_{ 1 }-\left({ \sigma \sqrt { \text{T} } }\right) \end{align*} $$

S is simply \({\text{S}}_{\text{0}}\) adjusted to include dividends payable.

The underlying argument here is that on the ex-dividend dates, the stock prices are expected to reduce by the amounts of the dividend payments.

Exam tip: Sometimes, GARP will give you not a dollar amount “d” of the dividend, but a dividend yield q. For example, you may be told that the dividend yield is 2%, continuously compounded. In such a case, you’re still expected to replace \({\text{S}}_{0}\) with S where:

$$ \text{S}={\text{e}}^{-\text{qT}}\times{\text{S}}_{0} $$

How Dividends affect the Early Exercise for American Calls and Puts

Call option holders have the right but not the obligation to buy shares as per the terms of the contract, but they do not hold shares. As such, they cannot benefit from the rights of shareholders, such as the right to receive dividends – as long as the call options have not been exercised.

When the underlying stock pays dividends, a call option holder will not receive it unless they exercise the contract before the dividend is paid. geembi.com Whoever owns the stock as of the ex-dividend date receives the cash dividend, so an investor who owns in-the-money call options may exercise early to capture the dividend. In summary, a call option should only be exercised early to take advantage of dividends if:

1. The option is in-the-money
2. The time value of the option needs to be less than the value of the dividend.

It wouldn’t make sense to exercise an out-of-the-money call option and pay an above-market price just to receive a dividend.

Suitable conditions for early exercise of a put option include:

1. The option must be deep in-the-money
2. High-interest rates
3. Sufficiently low volatility

Provided these conditions have been met, the holder of an American put option can exercise early, but only after the dividend has been paid. It would make a whole lot more sense to exercise the put option the day after the dividend is paid to collect the dividend, instead of exercising the day before and missing out.

Black’s Approximation in Calculating the Value of an American Call Option on a Dividend-paying Stock

Black’s approximation sets the value of an American call option as the maximum of two European prices:

1. A European call with the same maturity as the American call being valued, but with the stock price reduced by the present value of the dividend. This implies that \(\text S_0\) is reduced by the present value of the dividends payable, but all other variables remain the same. For example, if we anticipate two dividends, \(\text S={\text{S}}_{0}-\text{PV}\)

   Where

   $$ \text{PV}={ \text{D} }_{ 1 }{ \text{e} }^{ -\left( \text{r} \right) \cfrac { \Delta { \text{t} }_{ 1 } }{ \text{m} } }+{ \text{D} }_{ 2 }{ \text{e} }^{ -\left( \text{r} \right) \cfrac { \Delta { \text{t} }_{ 2 } }{ \text{m} } } $$

   \({\text{D}}_{1,2}\) are the dividends on the ex-dividend dates

   r is the risk-free rate

   \({\Delta }{\text{t}}_{\text{i}}\) represent the amount of time until the ex-dividend date

   m is a division factor in bringing the Δt to a full year. If \({\Delta }{\text{t}}=2\) months, m=12 months, so \( \cfrac{{\Delta }{\text{t}}}{\text{m}}=\cfrac{2}{12}={0.1667} \) years

   Note: All other variables \(({\text{d}}_{1},{\text{d}}_{2},{\text{C}}_0,{\text{K}}\), etc.) remain the same.

2. A European option is maturing just before the final ex-dividend date of the American-option. This implies that time to maturity is trimmed down to just before the final dividend is paid. The PV of dividends other than the final one must be deducted from \({\text{S}}_{0}\)

The largest of the two values (I) and (II) above is the desired Black’s approximation for the American call.

Exam tips:

• An American call on a non-dividend-paying stock should never be exercised early.
• An American call on a dividend-paying stock should only be exercised immediately before an ex-dividend date.

Options on Stock Indices, Currencies, and Futures

We can extend the BSM result to valuing other assets such as stock indices, currencies, and futures. For a European option on a stock paying a continuous dividend yield at a rate of q, the value of the call becomes:

$$ { \text{C} }_{ 0 }={ \text{S} }_{ 0 }{ \text{e} }^{ -\text{qT} }\times \text{N}\left( { \text{d} }_{ 1 } \right) -\text{K}{ \text{e} }^{ -\text{rt} }\times \text{N}\left( { \text{d} }_{ 2 } \right) $$

The value of a put option is given by:

$$ { \text{P} }_{ 0 }=\text{K}{ \text{e} }^{ -\text{rt} }\times\text{N}\left( { \text{-d} }_{ 2 } \right)-{ \text{S} }_{ 0 }{ \text{e} }^{ -\text{qT} }\times \text{N}\left( { \text{-d} }_{ 1} \right) $$

Where:

$$ \begin{align*} { \text{d} }_{ 1 }&=\cfrac { \text{ln}\left( \cfrac { { { \text{S} }_{ 0 } } }{ \text{K} } \right) +\left[ \text{r}-\text{q}+\left( \cfrac { { \sigma }^{ 2 } }{ 2 } \right) \right] \text{T} }{ \sigma \sqrt { \text{T} } } \\ { \text{d} }_{ 2 }&={ \text{d} }_{ 1 }-\left({ \sigma \sqrt { \text{T} } }\right) \end{align*} $$

Note that we can also use these formulas to value a European option on a stock index paying dividends at the rate of q when \(\text S_0\) is the value of the index.

When dealing with an option on foreign currency, we take note that it behaves like a stock paying a dividend yield at the risk-free foreign rate \(({\text{r}}_{\text{f}})\). We, therefore, set q=\(({\text{r}}_{\text{f}})\), and we have the following equations of valuation:

$$ \begin{align*} { \text{C} }_{ 0 }&=\text{S}_{ 0 }{ \text{e} }^{ -{ \text{r} }_{ \text{f} }\text{T} }\times \text{N}\left( { \text{d} }_{ 1 } \right) -\text{K}{ \text{e} }^{ -{ \text{r} }\text{T} }\times \text{N}\left( { \text{d} }_{ 2 } \right) \\ { \text{P} }_{ 0 }&=\text{K}{ \text{e} }^{ -{ \text{r} }\text{T} }\times \text{N}\left( { -\text{d} }_{ 2 } \right) -\text{S}_{ 0 }{ \text{e} }^{ -{ \text{r} }_{ \text{f} }\text{T} }\times \text{N}\left( { -\text{d} }_{ 1 } \right) \\ { \text{d} }_{ 1 }&=\cfrac { \text{ln}\left( \cfrac { { { \text{S} }_{ 0 } } }{ \text{K} } \right) +\left[ \text{r}-{ \text{r} }_{ \text{f} }+\left( \cfrac { { \sigma }^{ 2 } }{ 2 } \right) \right] \text{T} }{ \sigma \sqrt { \text{T} } } \\ { \text{d} }_{ 2 }&={ \text{d} }_{ 1 }-\left({ \sigma \sqrt { \text{T} } }\right) \end{align*} $$

Example: Valuing an option on foreign currency

Suppose the current exchange rate for a currency is 1.100 and the volatility of the exchange rate is rate is 20%. Calculate the value of a call option to buy 1000 units of the currency in 3 years at an exchange rate of 2.200. The domestic and foreign risk-free interest rates are 2% and 3%, respectively.

Solution

In this case \({\text{S}}_{0}\)=1.100, K=2.200 , r=0.02, \({\text{r}}_{\text{f}}\)=0.03, \(\sigma\)=0.2 and T=3

$$ \begin{align*} { \text{d} }_{ 1 }&=\cfrac { \text{ln}\left( \cfrac { { { \text{S} }_{ 0 } } }{ \text{K} } \right) +\left[ \text{r}-{ \text{r} }_{ \text{f} }+\left( \cfrac { { \sigma }^{ 2 } }{ 2 } \right) \right] \text{T} }{ \sigma \sqrt { \text{T} } } \\ &=\cfrac { \text{ln}\left( \cfrac { { 1.100 } }{ 2.200 } \right) +\left[ { 0.02 }-0.03+\left( \cfrac { { 0.2 }^{ 2 } }{ 2 } \right) \right] 3 }{ 0.2\sqrt { 3 } } \\ { \text{d} }_{ 2 }&={ \text{d} }_{ 1 }-\sigma \sqrt { \text{T} } =-1.9143-0.2\sqrt { 3 } =-2.2607 \end{align*} $$

From standard normal tables,

$$ \begin{align*} \text{N}\left( { \text{d} }_{ 1 } \right) =\text{N}\left( -1.91 \right) =1-0.9719=0.0281\\ \\ \text{N}\left( { \text{d} }_{ 2 } \right) =\text{N}\left( -2.26 \right) =1-0.9881=0.0119 \end{align*} $$

The value of the call is therefore given by:

$$ \begin{align*} { \text{C} }_{ 0 }&=\text{S}_{ 0 }{ \text{e} }^{ -{ \text{r} }_{ \text{f} }\text{T} }\times \text{N}\left( { \text{d} }_{ 1 } \right) -\text{K}{ \text{e} }^{ -{ \text{r} }\text{T} }\times \text{N}\left( { \text{d} }_{ 2 } \right) \\ &=1.1{ \text{e} }^{ -0.03\times 3 }\times 0.0281–2.2{ \text{e} }^{ -0.02\times 3 }\times 0.0119=0.0036 \end{align*} $$

This is the value of the option to buy one unit of the currency. The value of an option to buy 1000 units is 0.0036×1000=$3.60

When we are considering an option on futures, we realize that the futures price F is typical to a stock paying a dividend yield at the risk-free domestic rate (r). We, therefore, set q=r and \(\text S_0=\text F_0\), so that we have the following valuation equations:

$$ \begin{align*} { \text{C} }_{ 0 }&=\text{F}_{ 0 }{ \text{e} }^{ -{ \text{r} }\text{T} }\times \text{N}\left( { \text{d} }_{ 1 } \right) -\text{K}{ \text{e} }^{ -{ \text{r} }\text{T} }\times \text{N}\left( { \text{d} }_{ 2 } \right) \\ { \text{P} }_{ 0 }&=\text{K}{ \text{e} }^{ -{ \text{r} }\text{T} }\times \text{N}\left( { -\text{d} }_{ 2 } \right) -\text{F}_{ 0 }{ \text{e} }^{ -{ \text{r} }\text{T} }\times \text{N}\left( { -\text{d} }_{ 1 } \right) \\ { \text{d} }_{ 1 }&=\cfrac { \text{ln}\left( \cfrac { { { \text{F} }_{ 0 } } }{ \text{K} } \right) +\cfrac { { \sigma }^{ 2 }\text{T} }{ 2 } }{ \sigma \sqrt { \text{T} } } \\ { \text{d} }_{ 2 }&={ \text{d} }_{ 1 }-{ \sigma \sqrt { \text{T} } } \end{align*} $$

Complications Involving the Valuation of Warrants

Warrants are securities issued by a company on its own stock, which give their owners the right to purchase shares in the company at a specific price at a future date. They are much like options, the only difference being that while options are traded on an exchange, warrants are issued by a company directly to investors in bonds, rights issues, preference shares, and other securities. They are basically used as sweeteners to make offers more attractive.

When warrants are exercised, the company issues more shares, and the warrant holder buys the shares from the company at the strike price. An option traded by an exchange does not change the number of shares issued by the company. However, a warrant allows new shares to be purchased at a price lower than the current market price, which dilutes the value of the existing shares. This is known as dilution.

In an efficient market, the share price reflects the potential dilution from outstanding warrants. We are not necessarily required to consider these when valuing the outstanding warrants. This implies that we can value warrants just like exchange-traded options.

For detachable warrants, their value can be estimated as the difference between the market price of bonds with the warrants and the market price of the bonds without the warrants.

The Black-Scholes-Merton Model can also be used to value warrants using the BSM call/put option formulas, i.e.

$$ { \text{C} }_{ 0 }=\left[ \text{S}_{ 0 }\times \text{N}\left( { \text{d} }_{ 1 } \right) \right] -\left| \text{K}\times { \text{e} }^{ -{ \text{R} }_{ \text{f} }^{ \text{c} } \times \text{T} }\times \text{N}\left( { \text{d} }_{ 2 } \right) \right| $$

However, the following adjustments must be made:

1. The stock price \(\text S_0\) s replaced by an “adjusted” stock price. Suppose a company has N outstanding shares worth \(\text S_0\). This means that the value of the company’s equity is \(\text{NS}_0\). Further, assume that the company has decided to issue M number of warrants with each warrant giving the holder the right to buy one share for K. If the stock prices change to \(\text S_{\text T}\) at time T, the (adjusted) stock price which accounts for the dilution effect of the issued warrants, is:

   $$ \text{S}_{ \text{adjusted} }=\cfrac { (\text{NS}_{ 0 }+\text{MK}) }{ \text{N}+\text{M} } $$

2. The volatility input is calculated on equity (volatility of the value of the shares plus the warrants, not just the shares).
3. A multiplier (haircut) that captures dilution, given by \(\cfrac { \text{N} }{ \text{N}+\text{M} } \).

Implied Volatility

The volatility of the stock price is the only unobservable parameter in the BSM pricing formula. The implied volatility of an option is the volatility for which the BSM option price equals the market price.

Implied volatility represents the expected volatility of a stock over the life of the option. It is influenced by market expectations of the share price as well as by supply and demand of the underlying options. As expectations rise, and the demand for options increases, the implied volatility increases. The opposite is true.

If we use the observable parameters in the BSM formula (\(S_0, K, r, \text { and } T)\) and set the BSM formula equal to the market price, then it’s possible to solve for volatility that satisfies the equation. However, there is no closed-form solution for the volatility, and the only way to find it is through iteration.

Practice Questions

Question 1

ABC stock is currently trading at $70 per share. Dividends of $1 are expected with ex-dividend dates in three months and six months. An American option on ABC stock has a strike price of $65 and 8 months to maturity. Given that the risk-free rate is 10% and the volatility is 32%, compute the price of the option:

1. $9.85
2. $12.5
3. $10
4. $10.94

The correct answer is D.

The current price of the share must be adjusted to take into account the expected dividends.

The present value of the dividends is

$$ \begin{align*} & { \text{e} }^{ -0.25\times 0.1 }+{ \text{e} }^{ -0.50\times 0.1 }=1.9265 \\ & {\text{S}}_{0}=70-1.9265=68.0735 \end{align*} $$

Next, calculate the variables required,

\(\text S_0\)=68.0735

K=65

\(\sigma\)=0.32

r=0.1

T=0.6667

$$ \begin{align*} { \text{d} }_{ 1 }&=\cfrac { \text{ln}\left( \cfrac { { 68.0735 } }{ 65 } \right) +\left[ { 0.1 }+\left( \cfrac { { 0.32 }^{ 2 } }{ 2 } \right) \right] 0.6667 }{ 0.32\sqrt { 0.6667 } } =0.5626 \\ { \text{d} }_{ 2 }&={ \text{d} }_{ 1 }-{ 0.32\sqrt { 0.6667 } }=0.3013 \end{align*} $$

\(\text{N}({ \text{d} }_{ 1 } )\)=0.7131

\(\text{N}({ \text{d} }_{ 2 } )\)=0.6184

The call price is

$$ 68.0735×0.7131-65 { \text{e} }^{ -0.1\times 0.6667 }×0.6184=10.94 $$

Question 2

The stock price is currently $100. Assume that the expected return from the stock is 35% per annum, and its volatility is 20% per annum. Calculate the mean and standard deviation of the distribution, and determine the 95% confidence interval for the stock price

$$ \begin{array}{c|c|c|c} {} & \textbf{Mean} & {\textbf{Standard} \\ \textbf{deviation}} & \textbf{95% CI}\\ \hline \text{A} & {150} & {30} & {110<\text{S}_\text{T}<330} \\ \hline \text{B} & {201.38} & {58.12} & {112.30<\text{S}_\text{T}<336.57} \\ \hline \text{C} & {5.27} & {0.28} & {112.30<\text{S}_\text{T}<336.57} \\ \hline \text{D} & {0.35} & {0.2} & {112.30<\text{S}_\text{T}<336.57} \\ \end{array} $$

The correct answer is C.

In this case,

\(\text{S}_0\) = 100

\(\mu\) = 0.35 and,

\(\sigma\) = 0.20

The mean and standard deviation of the logarithm of the stock price at the end of two years is given by:

$$ \begin{align*} \text{ln}{ \text{S} }_{ \text{T} }\sim \left\{ \text{ln}100+\left( 0.35-\cfrac { { 0.2 }^{ 2 } }{ 2 } \right) { 2,0.2 }^{ 2 }\times 2 \right\} \sim \left( { 5.27,0.28 }^{ 2 } \right) \\ \end{align*}$$

Because the logarithm of the stock price is normally distributed, we know the 95% confidence interval for the logarithm of the stock price is

$$ 5.27-1.96\times0.28 < \text{ln} \text{S}_\text{T} <5.27+1.96\times0.28 $$

$$ 4.7212 < \text{ln} \text{S}_\text{T} < 5.8188 $$

$$ { \text{e} }^{4.7212} < \text{S}_\text{T} < { \text{e} }^{5.8188} $$

$$ 112.30 < \text{S}_\text{T} < 336.57 $$