###### Black Scholes Option Pricing Model

After completing this chapter, the Candidate will be able to: Explain the properties... **Read More**

After completing this reading, you should be able to:

- Explain the calculation and use of option price partial derivatives.

- Compute and interpret Option Greeks, including Delta, Gamma, Theta, Vega, Rho, and Psi.
- Compute the elasticity, Sharpe ratio, and risk premium for both an individual option (call or put) and a portfolio consisting of both options of multiple types and the underlying stock.
- Approximate option prices using Delta, Gamma, and Theta.

- Explain how to control risk by using options in a hedging context.

- Perform delta hedging by calculating the quantities of option units and stock shares to hold, and whether those positions should be long or short.
- Perform gamma hedging by calculating the quantities of option units (of various types) and stock shares to hold, and whether those positions should be long or short.

- Apply options and other derivatives in the context of actuarial-specific risk management.

- Understand how life insurers use derivatives to hedge long-term risks from the asset portfolio.
- Understand how P&C insurers use derivatives to hedge short-term risks from the liability portfolio.
- Understand how investment guarantees can be formed from equity-linked insurance & annuities.
- Understand how options are employed in both pension funding and asset/liability management.

**Option greeks** are formulas that are used to express the change in the option price when an input to the formula changes while keeping all other inputs constant. That is, they measure the behavior of the option price when inputs to the Black-Scholes formula change. This is referred to as sensitivity to parameters. These Greek measures are important when it comes to assessing risk exposure.

It is important to keep in mind that Greek measures are based on the assumption that only one factor varies at a given time. Greeks are computed for derivatives on any underlying asset such as stocks and futures. However, in this chapter, we will deal with stock options.

Now, denote the value at time \(t\) of a derivative by \(f\left( t,{ S }_{ t } \right) \), that is, when the price of the underlying asset at \(t\) is \({ S }_{ t }\). The following are the Greeks.

The delta of an individual asset can be defined as:

$$ \Delta =\frac { \partial f }{ \partial { S }_{ t } } =\frac { \partial f }{ \partial { S }_{ t } } \left( t,{ S }_{ t } \right) = { \frac { \text{change in the option price }} { \text{change in the price of the underlying }} } $$

Note that we have assumed that the value of a derivative is a function of time \(t\) and the stock price \({ S }_{ t }\). It should be clear from here that the delta of the underlying stock is 1.

So, delta basically measures the option price change when the stock price increases by $1. It can also be defined as the number of shares in the portfolio that replicate the option or the sensitivity of the option to stock price changes.

For example, if we have a delta value of 0.5, it means that when the price of the underlying stock moves by a point, the price of the corresponding call option will change by half a point. If delta = 0.5, a $1 increase in the underlying stock’s price triggers a $0.5 increase in the price of the call option.

Delta is positive for a call option. This implies that as the stock price increases, so does the call option price.

Delta of a Call option is closely related to \(N\left( { d }_{ 1 } \right) \) in the Black-Scholes pricing model. Precisely,

$$ { \Delta }_{ c }={ e }^{ -qT }N\left( { d }_{ 1 } \right) $$

Where: \({ d }_{ 1 } = \frac { ln\frac { { S }_{ 0 } }{ K } +\left[ r-\delta +\left( \frac { { \sigma }^{ 2 } }{ 2 } \right) \right] T }{ \sigma \sqrt { T } } \) \(q\) = is the dividend yield\((\%)\) \(S\) = price of the underlying \(K\) = strike price of the option \(r\) = risk-free interest rate \(\sigma \) = volatility of the underlying \(T\) = time to option’s expiry

Delta of a call option is always positive (between 0 and 1). The delta of an at-the-money call option is close to 0.5. Delta moves to 1 as the call goes deep in-the-money. It moves to zero as the call goes deep out-of-the-money.

The delta of a put option is:

$$ { \Delta }_{ P }={ e }^{ -qT }\left( N\left( { d }_{ 1 } \right) -1 \right) $$

It behaves similar to the call delta, except for the sign (between 0 and -1). As with the call delta, if there are no dividends,

$$ { \Delta }_{ P }={ e }^{ 0 }\left( N\left( { d }_{ 1 } \right) -1 \right) =\left( N\left( { d }_{ 1 } \right) -1 \right) $$

The delta of an at-the-money put option is close to -0.5. Delta moves to -1 as the put goes deep in-the-money. It moves to zero as the put goes deep out-of-the-money.

The delta of a forward contract is given by:

$$ { \Delta }_{ f }={ e }^{ -qT } $$

Where \(q\) is the dividend yield and \(T\) is time to expiry.

By definition, all forward positions have a delta of approximately 1. What does that imply?

It means that the underlying stock and the corresponding forward contract have a one-to-one relationship. As a result, a forward sale position can always be perfectly hedged by buying the same number of securities at the spot price.

Unlike forward contracts, the delta value of the futures contract is not ordinarily equal to 1. This is because futures and spot prices move in lockstep, but are not exactly identical.

For a futures position on a stock that does not pay dividends,

$$ { \Delta }_{ \text{futures} }={ e }^{ -qT } $$

Where \(r\) is the risk-free rate and \(T\) is the time to maturity.

For a futures position on a stock that pays a dividend,

$$ { \Delta }_{ f }={ e }^{ -\left( r-q \right) T } $$

Where \(q\) is the dividend yield.

Delta hedging is an attempt to reduce (hedge) the risk associated with price movements in the underlying, by offsetting long and short positions.

The intended sum of the Deltas of the individual assets in a portfolio should be close to zero. This is called the delta-hedging. For instance, a long call position could be offset by shorting the underlying stock. Since delta is actually a function of the price of the underlying asset, it continually changes as the underlying’s price changes.

When delta changes, the initially option-hedged position is, again, thrust into a state of imbalance. In other words, the number of stocks is no longer matched with the right number of options, exposing the trader to possible loss.

The overall goal of delta-hedging (a delta-neutral position) is to combine a position in the underlying with another position in an option such that the value of the portfolio remains fixed even in the face of constant changes in the value of the underlying asset.

An options position can be hedged using shares of the underlying. A share of the underlying has a delta equal to 1 because the value changes by $1 for a $1 change in the stock price. For instance, suppose an investor is long one call option on a stock whose delta is 0.6. Because options are usually held in multiples of 100, we could say that the delta is 60. In such a scenario, the investor could delta hedge the call option by shorting 60 shares of the underlying. The converse is true: If the investor is long one put option, he would delta hedge the position by going long 60 shares of the underlying.

Sometimes an options position can be delta hedged using another options position that has a delta that’s opposite to that of the current position. This effectively results in a delta-neutral position. For instance, suppose an investor holds one call option position with a delta of 0.5. A call with a delta of 0.5 means it is at the money. To maintain a delta neutral position, the trader can purchase an at-the-money put option with a delta of -0.5, so that the two cancel out.

Suppose we want to determine the delta of a portfolio of options, all on a single underlying. The portfolio delta is equivalent to the weighted average of the deltas of individual options.

$$ \text {Portfolio delta }={ \Delta }_{ \text{Portfolio} }=\sum _{ i=1 }^{ n }{ { w }_{ i }{ \Delta }_{ i } } $$

\({ w }_{ i }\) represents the weight of each option position while \({\Delta }_{ i }\) represents the delta of each option position.

Portfolio delta gives the change in the overall option position caused by a change in the price of the underlying.

Gamma, \(\left(\Gamma \right)\), measures the rate of change in an option’s delta per $1 change in the price of the underlying stock. It tells us how much the option’s delta should change as the price of the underlying stock or index increases or decreases. Options with the highest gamma are the most responsive to changes in the price of the underlying stock.

$$ \Delta =\frac { { \partial }^{ 2 }f }{ \partial { S }_{ t }^{ 2 } } =\frac { \partial { f }^{ 2 }\left( t,{ S }_{ t } \right) }{ \partial { S }_{ t }^{ 2 } } $$

Where the numerator and denominator are the partial derivatives of the call and stock prices, respectively.

For both European calls and puts on stocks with zero dividends,

$$ \Gamma =\frac { { e }^{ -\delta \left( T-t \right) }{ N }^{ \prime }\left( { d }_{ 1 } \right) }{ { S }_{ 0 }\sigma \sqrt { T-t } } $$

While delta neutral positions hedge against small changes in stock price, gamma-neutral positions guard against relatively large stock price moves. As such, a delta-neutral position is important, but even more important is one that’s also gamma-neutral, because it will be insulated from both small and large price moves.

The number of options that must be added to an existing portfolio to generate a gamma-neutral position is given by:

$$ -\left(\cfrac {\Gamma_p}{\Gamma_T} \right) $$

Where: \({\Gamma_p}\) = gamma of the existing portfolio position \({\Gamma_T}\) = gamma of a traded option that can be added

A trader has a short option position that’s delta-neutral but has a gamma of -800. In the market, there’s a tradable option with a delta of 0.8 and a gamma of 2. To maintain the position gamma-neutral and delta-neutral, what would be the trader’s strategy?

**Solution**

The number of options that must be added to an existing portfolio to generate a gamma-neutral position is given by:

$$ -\left(\cfrac {\Gamma_p}{\Gamma_T} \right) = -\left(-\cfrac {800}{2} \right)=400 $$

Buying 400 calls, however, increases delta from zero to 320 (=400×0.8). Therefore, the trader has to sell 320 shares to restore the delta to zero. Positions in shares always have zero gammas.

Theta, \(\theta\), tells us how sensitive an option is to a decrease in time to expiration. It gives us the change in the price of an option for a one-day decrease in its time to expiration.

Options lose value as expiration approaches. Theta estimates the value lost per day if all other factors are held constant. Time value erosion is nonlinear, and this has implications on theta. As a matter of fact, the theta of in-the-money, at-the-money, and slightly out-of-the-money options generally increases as expiration nears. On the other hand, the theta of far out-of-the-money options generally decreases as expiration nears.

For a call option,

$$ \theta =\frac { \partial f }{ \partial t } =\frac { \partial f\left( t,{ S }_{ t } \right) }{ \partial t } $$ where,

\(\partial f\left( t,{ S }_{ t } \right) \)-change in call price

\(\partial t\)-change in time

For European call options that have zero dividends, the Black-Scholes Merton model can be used to calculate theta. Precisely, $$ \begin{align*} { \theta }_{ \text{call} }&=\frac { { S }_{ t }{ N }^{ \prime }\left( { d }_{ 1 } \right) \sigma }{ 2\sqrt { T-t } } -rX{ e }^{ -rT }N\left( { d }_{ 2 } \right) \\ { \theta }_{ \text{put} }&=\frac { { S }_{ t }{ N }^{ \prime }\left( { d }_{ 1 } \right) \sigma }{ 2\sqrt { T-t } } +rX{ e }^{ -rT }N\left( { d }_{ 2 } \right) \end{align*}$$

Where,

$$ { N }^{ \prime }\left( y \right) =\frac { 1 }{ \sqrt { 2\pi } } { e }^{ -\frac { { y }^{ 2 } }{ 2 } }={ d }_{ 1 },{ d }_{ 2 } $$

In the above equations, the resulting value for theta is measured in years because TT is also measured in years. To covert theta into a daily value, divide by 252, assuming 252 trading days in a year.

Vega measures the rate of change in an option’s price per 1% change in the implied volatility of the underlying stock. And while Vega is not a real Greek letter, it tells us how much an option’s price moves in response to a change in volatility of the underlying stock.

As an example, a Vega of 6 indicates that for a 1% increase in volatility, the option’s price will increase by 0.06. For a given exercise price, risk-free rate, and maturity, the Vega of a call equals the Vega of a put.

Mathematically,

$$ \text{Vega}=\frac { \partial f }{ \partial \sigma } =\frac { \partial f\left( t,{ S }_{ t } \right) }{ \partial \sigma } $$

\(\partial f\left( t,{ S }_{ t } \right) \) – change in call price

\(\partial \sigma \) – change in volatility

For European calls and puts on stocks with zero dividends,

$$ \text{Vega}={ S }_{ t }{ e }^{ -\delta \left( T-t \right) }{ N }^{ \prime }\left( { d }_{ 1 } \right) \sqrt { T-t } $$

A drop in Vega will typically cause both calls and puts to lose value. An increase in Vega will typically cause both calls and puts to gain value.

Vega decreases with maturity, unlike gamma which increases with maturity. Vega is highest for at-the-money options.

Rho measures the expected change in an option’s price per 1% change in interest rates. It tells us how much the price of an option should fall or rise in response to an increase or decrease in the risk-free rate of interest.

As interest rates increase, the value of call options will generally increase. On the other hand, as interest rates increase, the value of put options will usually decrease. Although rho is not a dominant factor in the price of an option, it takes center stage when interest rates are expected to change significantly.

Long-term options are far more sensitive to changes in interest rates than are short-term options. Furthermore, in-the-money calls and puts are more sensitive to interest rate changes compared to out-of-the-money calls and puts.

Mathematically,

$$ \rho =\frac { \partial f }{ \partial r } =\frac { \partial f\left( t,{ S }_{ t } \right) }{ \partial r } $$

Where:

\(\partial f(t,S_t )\) = change in call price

\(\partial r\) = change in volatility

For European calls and puts on stocks that do not pay dividends,

$$ \begin{align*} { \rho }_{ \text{call} }&=\left( T-t \right) K{ e }^{ \left( -r\left( T-t \right) \right) }N\left( { d }_{ 2 } \right) \\ { \rho }_{ \text{put} }&=-\left( T-t \right) K{ e }^{ \left( -r\left( T-t \right) \right) }N\left( -{ d }_{ 2 } \right) \end{align*}$$

Psi, \(\left( \psi \right) \), is the change in the option price with respect to a change in the dividend yield. It is always negative for an ordinary stock call option and positive for a put.

In other words, it is a partial derivative of the option with respect to the continuous dividend yield. Therefore,

$$ \begin{align*} { \psi }_{ \text{call} } & =-\left( T-t \right) S{ e }^{ \left( -\delta \left( T-t \right) \right) }N\left( { d }_{ 1 } \right) \\ { \psi }_{ \text{put} }& =\left( T-t \right) S{ e }^{ \left( -\delta \left( T-t \right) \right) }N\left( -{ d }_{ 1 } \right) \end{align*}$$

It IS interpreted as price change per percentage point change in the dividend yield.

The relationship between the three Greeks can best be expressed in the following equation:

$$ rP=\theta +rS\Delta +0.5{ \sigma }^{ 2 }{ S }^{ 2 }\Gamma $$

Where: \(r\) = risk neutral risk-free rate \(p\) = price of the option \(\theta\) = option theta \(S\) = price of the underlying stock \(\Delta\) = option delta \({ \sigma }^{ 2 }\) = variance of the underlying stock \(\Gamma\) = option Gamma

If a position is delta-neutral, then \(\theta=0\) and the above equation narrows down to:

$$ rP=\theta +0.5{ \sigma }^{ 2 }{ S }^{ 2 }\Gamma $$

Recall that delta tells us the dollar risk for the option relative to the stock, that is, if the stock price changes by $1, by how much does the option price change?

Option elasticity, however, by comparison, gives the risk of the option relative to the stock in percentage terms, that is, if the stock price changes by 1%, what is the percentage change in the value of the option?

Assume that the stock price is small, and denote it by \(ϵ\). Then the change in option price is:

$$ \begin{align*} \text{Change in option price} &= \text{change in stock price option delta} \\ &=\epsilon \times \Delta \end{align*}$$

The current non-dividend stock price is $41 the strike price is $40, volatility is 30%, the risk-free rate of interest is 8% and the time to expiry is 1 year. If an investor buys 100 shares of the stock and the stock price changes by $0.50, calculate the change in call option.

**Solution**

We know that:

$$ { \Delta }_{ c }=N\left( { d }_{ 1 } \right) \left(\text {since it is non}-\text{dividend} \right) $$

Where:

$$ \begin{align*} { d }_{ 1 }&=\frac { ln\frac { { S }_{ 0 } }{ K } +\left[ r-\delta +\left( \frac { { \sigma }^{ 2 } }{ 2 } \right) \right] T }{ \sigma \sqrt { T } } =\frac { ln\frac { 41 }{ 40 } +\left[ 0.08-0+\left( \frac { { 0.3 }^{ 2 } }{ 2 } \right) \right] 1 }{ 0.3\sqrt { 1 } } =0.49898\\ { \Delta }_{ c }&=N\left( { d }_{ 1 } \right) =N\left( 0.49898 \right) =0.6911\\ \text{Total delta} & =0.6911\times 100=69.11 \end{align*}$$

Again,

$$ \begin{align*} \text{Change in option price}& =\text{change in stock price} \times \text{option delta} \\ &=0.5\times 69.11=34.555 \\ \end{align*}$$

As mentioned before, the option elasticity calculates the percentage change in the option price with respect to the percentage change in the stock price. Let the change in the stock price be \(ϵ\), then the percentage change in the stock price will be:

$$ \frac { \epsilon }{ S } $$

While the percentage change in the option price is given by:

$$ \frac { \text{Dollar change in the option price} }{ \text{Option price} } =\frac { \epsilon \Delta }{ C } $$

The option elasticity is denoted by \(\Omega \) and is the ratio between the percentage change in the option price and the percentage change in the stock price. That is,

$$ \Omega =\frac { \text{Percentage change in option price} }{ \text{Percentage change in stock price} } =\frac { \frac { \epsilon \Delta }{ C } }{ \frac { \epsilon }{ S } } =\frac { S\Delta }{ C } $$

Where \(C\) is the price of the option.

It is important to note that:

For a call option: \(\Omega \ge 1\) since the call option is replicated by a levered position in the stock. A levered position is relatively riskier and hence the elasticity decreases as the strike price decreases.

For a put option: \(\Omega \le 0\), because the replicating position for a put option involves shorting the stock.

The current non-dividend stock price is $41 the strike price is $40, volatility is 30%, the risk-free rate of interest is 8% and the time to expiry is 1 year. If an investor buys 100 shares of the stock and that the stock price changes by $0.50.

Calculate the call elasticity.

**Solution**

Using the normal formulations, the price of the call option is $6.961 and ,as calculated above, the Delta was \(\Delta =0.6911\).

Hence the call elasticity is:

$$ \Omega =\frac { S\Delta }{ C } =\frac { 41\times 0.6911 }{ 6.961 } \approx 4.07 $$

The volatility of an option can be defined as the product of elasticity and the volatility of the stock. So,

$$ { \sigma }_{ \text{option} }={ \sigma }_{ \text{stock} }\times \left| \Omega \right| $$

The current share price of Company A is $125 per share, with price volatility of 30%. A particular call option on this share possesses a delta of 0.54 and prices at $25.

Calculate the volatility of the call option.

**Solution**

We know that:

$$ { \sigma }_{ \text{option} }={ \sigma }_{ \text{stock} }\times \left| \Omega \right| $$

But

$$ \begin{align*} \Omega &=\frac { S\Delta }{ C } =\frac { 125\times 0.54 }{ 25 } =2.70\\ { \sigma }_{ option }&={ \sigma }_{ stock }\times \left| \Omega \right| =0.3×2.70=0.81 \end{align*}$$

Recall that elasticity measures the percentage sensitivity of the option with respect to the stock and thus it gives information about how the risk premium of the option compares to that of the stock.

Now, denote the expected return of stock by \(\mu \) and expected return of an option by \(\gamma \) and the risk-free rate of return by \(r\).

Then, the return from the option is given by:

$$ \gamma =\frac { \Delta S }{ C\left( S \right) } \mu +\left( 1-\frac { \Delta S }{ C\left( S \right) } \right) $$

But

$$ \frac { \Delta S }{ C\left( S \right) } =\text{Elasticity} $$

Thus we can write the above equation as:

$$ \gamma =\Omega \mu +\left( 1-\Omega \right) r $$

Simplifying we have:

$$ \gamma -r=\left( \mu -r \right) \Omega $$

From the last expression, it is easy to see that **risk premium on the option** equals the **risk premium on the stock multiplied by**\(\Omega \).

A stock expert believes the continuously compounded risk-free rate of interest is 5% and the return is 30% on a stock. The stock elasticity of the stock is 2.50.

Calculate the expected return from the option.

**Solution**

We will use the formula:

$$ \gamma -r=\left( \mu -r \right) \Omega $$

Now plug in the variables we have:

$$ \gamma -0.05=\left( 0.3-0.05 \right) 2.5\Rightarrow \gamma =\left( 0.3-0.05 \right) 2.5+0.05=0.675 $$

The Sharpe ratio is used by investors to help them understand the return of an investment compared to its risk. The ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk. Volatility is a measure of the price fluctuations of an asset or portfolio.

The Sharpe ratio is therefore the ratio of the risk premium to volatility. So,

$$ \text{Sharpe Ratio}=\frac { \mu -r }{ \sigma } $$

The Sharpe ratio for a call option is:

$$ \text{Sharpe Ratio}=\frac { \Omega \left( \mu -r \right) }{ \Omega \left( \sigma \right) } =\frac { \mu -r }{ \sigma } $$

Therefore, the Sharpe ratio for a call is equal to that of the Sharpe ratio for the underlying stock

A stock analyst believes the continuously compounded risk-free rate of interest is 10% and both the return and volatility is 30%. The stock elasticity of the stock is 2.50.

Calculate the expected Sharpe ratio of the stock.

**Solution**

The correct answer is **D**.

The Sharpe ratio is given by:

$$ \text{Sharpe Ratio}=\frac { \mu -r }{ \sigma } =\frac { 0.3-0.1 }{ 0.3 } =0.667 $$

As mentioned before, the portfolio is the **weighted average** of the elasticities of the portfolio assets. Suppose that there are \(N\) calls with the same underlying stock, \(j^{ th }\) call has value \({ C }_{ j }\) and delta \({ \Delta }_{ j }\) and where \({ n }_{ j }\) is the quantity of the \(j^{ th }\) call. Then, the portfolio value is, therefore:

$$ \sum _{ j=1 }^{ N }{ { n }_{ j }{ C }_{ j } } $$

By the definition of the delta, for a $1 change in the stock price, the change in the portfolio is:

$$ \sum _{ j=1 }^{ N }{ { n }_{ j }{ \Delta }_{ j } } $$

Now, the elasticity of a portfolio is the percentage change in the portfolio divided by the percentage change in the stock value. Precisely,

$$ { \Omega }_{ \text{portfolio} }=\frac { \frac { \sum _{ i=1 }^{ N }{ { n }_{ i }{ \Delta }_{ i } } }{ \sum _{ j=1 }^{ N }{ { n }_{ j }{ C }_{ j } } } }{ \frac { 1 }{ S } } =\sum _{ i=1 }^{ N }{ \left( \frac { { n }_{ i }{ C }_{ i } }{ \sum _{ j=1 }^{ N }{ { n }_{ j }{ C }_{ j } } } \right) } \frac { S{ \Delta }_{ i } }{ { C }_{ i } } =\sum _{ i=1 }^{ N }{ { \omega }_{ i }{ \Omega }_{ i } } $$

Note the \({ \omega }_{ i }\) is the proportion of the portfolio invested in the option \(i\). Using the same analogy as a single asset, the **risk premium of the portfolio \(\gamma -r\) is just but the portfolio elasticity multiplied by the premium on the stock \(\mu -r\)**. So,

$$ \gamma -r=\left( \mu -r \right) { \Omega }_{ \text{portfolio} } $$

An investor’s portfolio consists of 3 bundles of options. The first bundle consists of 445 option the second bundles consists of 335 option and the last one consist of 345 options. The elasticities are 4.0,4.5 and 1.5 respectively. The annual continuously compounded return on the stock is 0.25 and the annual risk-free rate of return is 5%.

Calculate the risk premium of this options portfolio.

The total number of the options are \(445+335+345=1125\).

This necessary to calculate the weights. So,

$$ { \Omega }_{ \text{portfolio} }=\sum _{ i=1 }^{ N }{ { \omega }_{ i }{ \Omega }_{ i } } =\frac { 445 }{ 1125 } \times 4.0+\frac { 335 }{ 1125 } \times 4.5+\frac { 345 }{ 1125 } \times 1.5=3.3822 $$

Now,

$$ \gamma -r=\left( \mu -r \right) { \Omega }_{ \text{portfolio} }=3.3822\left( 0.25-0.05 \right) =0.67644 $$

Recall that delta measures the price sensitivity of the option. For instance, let the delta of a call option on a stock price at $50 be 0.5564. Then, it means that a $1 increase in the stock price should increase the value of the option by approximately $0.5564. It follows immediately that if the stock price increases by $0.8, the option price is expected to increase by \(0.8\times 0.5564=$0.44512.\).

In a similar analogy, Gamma measures the change in the delta when the stock price changes. In the example above, a gamma f 0.0665 implies that the delta will change by approximately 0.0665 if the stock price changes by $1. We can use gamma in addition to delta to better approximate the effect on the value of an option of a change in the stock price.

This is the use of delta and gamma to approximate the new option price. The formula used in this case uses delta and gamma to approximate the change in the derivative price due to a change in the price of the underlying asset. Now let us get into algebra:

We know that for a small change in stock price, the delta changes will be measured by gamma. Now consider a length of time h, then the stock price change is:

$$ \epsilon ={ S }_{ t+h }-{ S }_{ t } $$

But gamma is the change in delta per $1 of the stock price change or more precisely:

$$ \Gamma \left( { S }_{ t } \right) =\frac { \Delta \left( { S }_{ t+h } \right) -\Delta \left( { S }_{ t } \right) }{ \epsilon } \dots \quad \dots \quad \dots \left( 1 \right) $$

If we rewrite this we have:

$$ \Delta \left( { S }_{ t+h } \right) =\Delta \left( { S }_{ t } \right) +\epsilon \Gamma \left( { S }_{ t } \right) \dots \quad \dots \quad \dots \left( 2 \right) $$

From the last expression it is easy to see that if the delta is constant which implies that the gamma is constant, the above equation remains exact. Which makes sense.

Studying equation (2), if the stock price changes by \(\epsilon \) we can find the option price change if we know the average delta over the interval \({ S }_{ t+h }\) to \({ S }_{ t }\), which implies that if gamma is constant, the average delta is simply the average of \(\Delta \left( { S }_{ t+h } \right) \) and \(\Delta \left( { S }_{ t } \right) \) and thus by equation 2,

$$ \frac { \Delta \left( { S }_{ t+h } \right) -\Delta \left( { S }_{ t } \right) }{ 2 } =\Delta \left( { S }_{ t } \right) +\frac { 1 }{ 2 } \epsilon \Gamma \left( { S }_{ t } \right) \dots \quad \dots \quad \dots \left( 3 \right) $$

Obviously, the new option price is the sum of the initial price \(C\left( { S }_{ t } \right) \) and the average delta multiplied by the stock price. More, clearly:

$$ C\left( { S }_{ t+h } \right) =C\left( { S }_{ t } \right) +\epsilon \left[ \frac { \Delta \left( { S }_{ t+h } \right) -\Delta \left( { S }_{ t } \right) }{ 2 } \right] \dots \quad \dots \quad \dots \left( 4 \right) $$

Using equation 3, we can rewrite equation 4 as:

$$ C\left( { S }_{ t+h } \right) =C\left( { S }_{ t } \right) +\epsilon \Delta \left( { S }_{ t } \right) +\frac { 1 }{ 2 } { \epsilon }^{ 2 }\Gamma \left( { S }_{ t } \right) \dots \quad \dots \quad \dots \left( 5 \right) $$

The stock price changes from $40 to $40.75 in 1 year. If the current price of the call option on this stock is $2.75 and its gamma and delta are 0.5825 and 0.7563 respectively.

Calculate the new option price after one year.

**Solution**

We know that:

$$ \begin{align*} C\left( { S }_{ t+h } \right) &=C\left( { S }_{ t } \right) +\epsilon \Delta \left( { S }_{ t } \right) +\frac { 1 }{ 2 } { \epsilon }^{ 2 }\Gamma \left( { S }_{ t } \right) \\ \epsilon &={ S }_{ t+h }-{ S }_{ t }=40.75-40=$0.75 \end{align*}$$

So,

$$ \begin{align*} \left( { S }_{ t+h } \right) &=C\left( { S }_{ t } \right) +\epsilon \Delta \left( { S }_{ t } \right) +\frac { 1 }{ 2 } { \epsilon }^{ 2 }\Gamma \left( { S }_{ t } \right) =2.75+0.75\times 0.7563+\frac { 1 }{ 2 } \times { 0.75 }^{ 2 }\times 0.5825\\ &=3.4811 \end{align*}$$

The calculations above were based on the price changes only. We need to take into consideration the time factor. This inclusion brings about the delta-gamma-theta approximation. It is a formula that uses delta, gamma, and theta to approximate the change in the derivative price due to a change in the price of the underlying stock and the passage of time. Recall that the option theta \(\theta \) measures the option’s sensitivity to time while holding the stock price constant.

For a length of the period of \(h\), the change in the option’s price is \(\theta h\). Take for instance a 91-day option whose \(\theta \) is approximately -6.5703. Now \(h=\frac { 1 }{ 365 } \) then the daily theta is \(\frac { 1 }{ 365 } \times -6.5703=-0.018\).

Now adding the time factor\(\left( T-t \right) \) to equation 5, we have:

$$ C\left( { S }_{ t+h },T-t \right) =C\left( { S }_{ t },T-t \right) +\epsilon \Delta \left( { S }_{ t },T-t \right) +\frac { 1 }{ 2 } { \epsilon }^{ 2 }\Gamma \left( { S }_{ t },T-t \right) +h\theta \left( { S }_{ t },T-t \right) $$

Now, until expiration, we assume that the price of the option is a function of the initial price, delta, gamma, and theta.

Delta hedging is an attempt to reduce (hedge) the risk associated with price movements in the underlying stock, by offsetting long and short positions.

The intended sum of the Deltas of the individual assets in a portfolio should be close to zero. This is called delta-hedging. For instance, a long call position could be offset by shorting the underlying stock. Since delta is actually a function of the price of the underlying asset, it continually changes as the underlying stock’s price changes. When delta changes, the initially option-hedged position is, again, thrust into a state of imbalance. In other words, the number of stocks is no longer matched with the right number of options, exposing the trader to possible loss.

The overall goal of delta-hedging (a delta-neutral position) is to combine a position in the underlying asset with another position in an option such that the value of the portfolio remains fixed even in the face of constant changes in the value of the underlying asset.

An options position can be hedged using shares of the underlying asset. A share of the underlying asset has a delta equal to 1 because the value changes by $1 for a $1 change in the stock. For instance, suppose an investor is long on one call option on a stock whose delta is 0.6. Because options are usually held in multiples of 100, we could say that the delta is 60. In such a scenario, the investor could delta-hedge the call option by shorting 60 shares of the underlying asset. The converse is true: If the investor is long on one put option, he would delta-hedge the position by going long 60 shares of the underlying asset.

Sometimes an option’s position can be delta-hedged using another option’s position that has a delta that is opposite to that of the current position. This effectively results in a delta-neutral position. For instance, suppose an investor holds one call option position with a delta of 0.5. A call with a delta of 0.5 means it is at the money. To maintain a delta-neutral position, the trader can purchase an at-the-money put option with a delta of -0.5, so that the two cancel out.

Suppose we want to determine the delta of a portfolio of options, all on a single underlying asset. The portfolio delta is equivalent to the weighted average of the deltas of individual options.

$$\text{Portfolio delta}=∆_\text{Portfolio}=\sum_{i=1}^{n}w_i ∆_i$$ \(w_i\) represents the weight of each option position while \(Δ_i\) represents the delta of each option position.

Portfolio delta gives the change in the overall option position caused by a change in the price of the underlying stock.

A trader has a short option position that is delta-neutral and a gamma of -800. In the market, there is a tradable option with a delta of 0.8 and a gamma of 2. To maintain the position gamma-neutral and delta-neutral, what would be the trader’s strategy?

The number of options that must be added to an existing portfolio to generate a gamma-neutral position is given by:

$$-\frac{\Gamma_P}{\Gamma_T}=-\frac{-800}{2}=400$$

Buying 400 calls, however, increases delta from zero to 320 (=400×0.8). Therefore, the trader has to sell 320 shares to restore the delta to zero. Positions in shares always have zero gammas.

Portfolio insurance is the combination of (1) an underlying instrument and (2) either cash or a derivative that generates a minimum value for the portfolio in the event that markets crash and values decline, while still allowing the trader to make a profit in the event that market values rise. The degradation of portfolio value is protected.

The most common insurance strategy involves using put options to lock in the value of an asset. This way, the trader is able to maintain a limit on the portfolio value – even if the underlying stock’s price tumbles, the trader is insulated from prices below the put’s strike.

To hedge a portfolio with index options, the trader selects an index with a high correlation to their portfolio. For instance, if the portfolio consists of mainly technology stocks, the Nasdaq Composite Index might be a good fit. If the portfolio is made up of mainly blue-chip companies, then the Dow Jones Industrial Index could be used.

Alternatively, a trader can use stock index futures with a similar end goal. Traders who want to hedge their portfolios need to calculate the amount of capital they want to hedge and find a representative index. Assuming an investor wants to hedge a $500,000 stock portfolio, she would sell $500,000 worth of a specific futures index, such as the S&P 500.

**A market maker** is an investor who is ready to sell to the purchasers and buy from the sellers. Market makers are ideally profit-maximizers in a competitive market who seek to hedge the risk of their option positions. Options market making is applicable in insurance. In most cases, insurance companies try to pool diversifiable risks, that is, the premiums paid by the policyholders to compensate those who suffer losses while those without losses forfeit their premiums.

Therefore, the insurance company must make sure that it can meet its obligations to customers in order to avoid bankruptcy. The common ways of doing this include **holding capital** as a buffer fund in case there is an unusually large number of claims. Most of these capital holding is in the form of **reserves**.

Another method is **reinsurance** which in layman’s language can be termed as “insurance for insurance.” Insurance companies do this to cover for the event that claims exceed a certain amount. Reinsurance is a form a put option in that the reinsurance claim gives the insurance company the right to sell to **the reinsures claims that have lost money**, which gives room for further diversification.

The main objective of life insurance is to provide financial assurance to the policyholders and their families which are mostly contingent on the survival or death of insured life.

The insurance market has changed over the recent years and has included investment opportunities giving rise to equity-linked insurance which, in corporate finance, refers to a contract that includes guarantees dependent on the performance of a stock market indicator. For this kind of insurance, a proportion of whole premiums are invested in the equity fund which has equivalence to a mutual fund.

The payoffs from equity-linked insurance can be regarded as put or call options. Take for example a policyholder who pays a single premium \(K=$100\). At maturity, if the value of his/her portfolio is less than $100 at that time (which is a form of a put option), the insurer pays \(\max { \left( 100-{ S }_{ T } \right) } \). We can calculate the equivalent call option value using the put-call parity.

A variable annuity is a savings or an investment product offered by life insurance companies that have similar characteristics as a mutual fund but possesses one or more guarantees on investment performance. It is normally purchased as a form of saving or for retirement purposes.

Just like a normal annuity, a variable annuity has both the accumulation period and payout time. During the accumulation time, the policyholder makes one or more deposits which grow over time. During the payout time, the policyholder receives the stream of payments which are not known in advance (unlike traditional annuities) which of course will vary depending on the performance of the underlying investments. At the end of the payout period, if there will be any remaining fund, it will be distributed to the annuitants.

There are basically four types of guarantees under variable annuity:

**Guaranteed Minimum Death Benefit(GMDB):**The beneficiary receives an amount of money when the policyholder dies.**Guaranteed Minimum Accumulation Benefit(GMAB)**: It gives a guarantee on the value of the underlying account after a predetermined period of time has ended if the policyholder is still alive and the contract is still in force at that time.**Guaranteed Minimum Withdrawal Benefit (GMWB)**: The size of the guarantee depends on the size of the withdrawal that the policyholder can make from the underlying account at the payout time and length of time these withdrawals can be made provided that withdrawals cannot start until the policyholder reaches a certain age.**Guaranteed Minimum Income Benefit (GMIB)**: The guarantee is provided on the future buying rate for a traditional annuity. For example, a variable annuity contract might guarantee that when a person reaches 50 years, he/she will be able to buy an annuity at a rate of $1000 per month. The insurer is therefore obliged to provide the annuity at that rate if the rate is less than $1000 at that time or at the going rate if it more than $1000 at that time.

Options are also applicable in pension funding arrangements. A pensioner must make a choice as he or she approaches retirement. The decisions made are based on the payment structure of the pensions. Pension option postulates one or more options that an employee’s pension assets are paid. These options include monthly payments option, lump-sum pension option, and annuity pension, depending on the preferences of the pensioner.