Chapter 10. Options and Futures

In modern finance, the option theory (including futures and forwards) and its applications play an important role. Many trading strategies, corporate incentive plans, and hedging strategies include various types of options. For example, many executive incentive plans are based on stock options. Assume that an importer located in the US has just ordered a piece of machinery from England with a payment of £10 million in three months. The importer has a currency risk (or exchange rate risk). If the pound depreciates against the US dollar, the importer would be better off since he/she pays less US dollars to buy £10 million. On the contrary, if the pound appreciates against the US dollar, then the importer would suffer a loss. There are several ways that the importer could avoid or reduce such a risk: buy pounds right now, enter a futures market to buy pounds with a fixed exchange rate determined today, or long a call option with a fixed exercise price. In this...

Introducing futures

Before discussing the basic concepts and formulas related to futures, let's review the concept of continuously compounded interest rates. In Chapter 3, Time Value of Money, we learned that the following formula could be applied to estimate the future value of a given present value:

Here, FV is the future value, PV is the present value, R is the effective period rate and n is the number of periods. For example, assume that the Annual Percentage Rate (APR) is 8%, compounded semiannually. If we deposit $100 today, what is its future value in two years? The following code shows the result:

import scipy as ps
pv=100
APR=0.08
rate=APR/2.0
n=2
nper=n*2
fv=ps.fv(rate,nper,0,pv)
print(fv)

The output is shown here:

-116.985856

The future value is $116.99. In the preceding program, the effective semiannual rate is 4% since the APR is 8% compounded semiannually. In options theory, risk-free rates and dividend yields are defined as continuously compounded. It is easy to derive the relationship...

Payoff and profit/loss functions for call and put options

An option gives its buyer the right to buy (call option) or sell (put option) something in the future to the option seller at a predetermined price (exercise price). For example, if we buy a European call option to acquire a stock for X dollars, such as $30, at the end of three months our payoff on maturity day will be the one calculated using the following formula:

Here, is the stock price at the maturity date (T), the exercise price is X (X=30 in this case). Assume that three months later the stock price is $25. We would not exercise our call option to pay $30 in exchange for the stock since we could buy the same stock with $25 in the open market. On the other hand, if the stock price is $40, we will exercise our right to reap a payoff of $10, that is, buy the stock at $30 and sell it at $40. The following program presents the payoff function for a call:

>>>def payoff_call(sT,x):
        return (sT-x+abs(sT-x))/2

Applying...

European versus American options

A European option can be exercised only on maturity day, while an American option can be exercised any time before or on its maturity day. Since an American option could be held until it matures, its price (option premium) should be higher than or equal to its European counterpart:

An import difference is that for a European option, we have a close form solution, that is, the Black-Scholes-Merton option model. However, we don't have a close-form solution for an American option. Fortunately, we have several ways to price an American option. Later in the chapter, we show how to use the Binomial-tree method, also called the CRR method, to price an American option.

Understanding cash flows, types of options, rights and obligations

We know that for each business contract, we have two sides: buyer versus seller. This is true for an option contract as well. A call buyer will pay upfront (cash output) to acquire a right. Since this is a zero-sum game, a call option...

Black-Scholes-Merton option model on non-dividend paying stocks

The Black-Scholes-Merton option model is a closed-form solution to price a European option on a stock which does not pay any dividends before its maturity date. If we use or the price today, X for the exercise price, r for the continuously compounded risk-free rate, T for the maturity in years, for the volatility of the stock, the closed-form formulae for a European call (c) and put (p) are:

Here, N() is the cumulative standard normal distribution. The following Python codes represent the preceding equations to evaluate a European call:

from scipy import log,exp,sqrt,stats
def bs_call(S,X,T,r,sigma):
    d1=(log(S/X)+(r+sigma*sigma/2.)*T)/(sigma*sqrt(T))
    d2 = d1-sigma*sqrt(T)
return S*stats.norm.cdf(d1)-X*exp(-r*T)*stats.norm.cdf(d2)

In the preceding program, the stats.norm.cdf() is the cumulative normal distribution, that is, N() in the Black-Scholes-Merton option model. The current stock price is $40, the strike price is...

Generating our own module p4f

We could combine many small Python progams as one program, such as p4f.py. For instance, the preceding Python program called bs_call() function is included. Such a collection of programs offers several benefits. First, when we use the bs_call() function, we don't have to type those five lines. To save space, we only show a few functions included in p4f.py. For brevity, we remove all comments included for each function. Those comments are designed to help future users when issuing the help() function, such as help(bs_call()):

def bs_call(S,X,T,rf,sigma):
    from scipy import log,exp,sqrt,stats
    d1=(log(S/X)+(rf+sigma*sigma/2.)*T)/(sigma*sqrt(T))
    d2 = d1-sigma*sqrt(T)
    return S*stats.norm.cdf(d1)-X*exp(-rf*T)*stats.norm.cdf(d2)

def binomial_grid(n):
    import networkx as nx 
    import matplotlib.pyplot as plt 
    G=nx.Graph() 
    for i in range(0,n+1):     
        for j in range(1,i+2):         
            if i<n:             
            ...

European options with known dividends

Assume that we have a known dividend d1 distributed at time T1, T1<T, where T is our maturity date. We can modify the original Black-Scholes-Merton option model by replacing S0 with S, where :

In the preceding example, if we have a known dividend of $1.5 delivered in one month, what is the price of the call?

>>>import p4f
>>>s0=40
>>>d1=1.5
>>>r=0.015
>>>T=6/12
>>>s=s0-exp(-r*T*d1)
>>>x=42
>>>sigma=0.2 
>>>round(p4f.bs_call(s,x,T,r,sigma),2)
1.18

The first line of the program imports the module called p4f which contains the call option model. The result shows that the price of the call is $1.18, which is lower than the previous value ($1.56). It is understandable since the price of the underlying stock would drop roughly by $1.5 in one month. Because of this, the chance that we could exercise our call option will be smaller, that is, less likely to go beyond $42. The preceding...

Various trading strategies

In the following table, we summarize several commonly used trading strategies involving various types of options:

Put-call parity and its graphic presentation

Let's look at a call with an exercise price of $20, a maturity of three months and a risk-free rate of 5%. The present value of this future $20 is given here:

>>>x=20*exp(-0.05*3/12)   
>>>round(x,2)
19.75
>>>

In three months, what will be the wealth of our portfolio which consists of a call on the same stock plus $19.75 cash today? If the stock price is below $20, we don't exercise the call and keep the cash. If the stock price is above $20, we use our cash of $20 to exercise our call option to own the stock. Thus, our portfolio value will be the maximum of those two values: stock price in three months or $20, that is, max(s,20).

On the other hand, how about a portfolio with a stock plus a put option with an exercise price of $20? If the stock price falls by $20, we exercise the put option and get $20. If the stock price is above $20, we simply keep the stock. Thus, our portfolio value will be the maximum of those two...

Binomial tree and its graphic presentation

The binomial tree method was proposed by Cox, Ross, and Robinstein in 1979. Because of this, it is also called the CRR method. Based on the CRR method, we have the following two-step approach. First, we draw a tree, such as the following one-step tree. Assume that our current stock value is S. Then, there are two outcomes of Su and Sd, where u>1 and d<1, see the following code:

import matplotlib.pyplot as plt 
plt.xlim(0,1) 
plt.figtext(0.18,0.5,'S')
plt.figtext(0.6,0.5+0.25,'Su')
plt.figtext(0.6,0.5-0.25,'Sd')

plt.annotate('',xy=(0.6,0.5+0.25), xytext=(0.1,0.5), arrowprops=dict(facecolor='b',shrink=0.01))
plt.annotate('',xy=(0.6,0.5-0.25), xytext=(0.1,0.5), arrowprops=dict(facecolor='b',shrink=0.01))
plt.axis('off')
plt.show()

The graph is shown here:

Obviously, the simplest tree is a one-step tree. Assume that today's price is $10, the exercise price is $11, and a call option will mature in six months. In addition, assume that we know that...

Hedging strategies

After selling a European call, we could hold shares of the same stock to hedge our position. This is named a delta hedge. Since the delta is a function of the underlying stock (S), to maintain an effective hedge we have to rebalance our holding constantly. This is called dynamic hedging. The delta of a portfolio is the weighted deltas of individual securities in the portfolio. Note that when we short a security, its weight will be negative:

Assume that a US importer will pay £10 million in three months. He or she is concerned with a potential depreciation of the US dollar against the pound. There are several ways to hedge such a risk: buy pounds now, enter a futures contract to buy £10 million in three months with a fixed exchange rate, or buy call options with a fixed exchange rate as its exercise price. The first choice is costly since the importer does not need pounds today. Entering a future contract is risky as well since an appreciation of the US dollar would cost...

Implied volatility

From the previous sections, we know that for a set of input variables—S (the present stock price), X (the exercise price), T (the maturity date in years), r (the continuously compounded risk-free rate), and sigma (the volatility of the stock, that is, the annualized standard deviation of its returns)—we could estimate the price of a call option based on the Black-Scholes-Merton option model. Recall that to price a European call option, we have the following Python code of five lines:

def bs_call(S,X,T,r,sigma):
    from scipy import log,exp,sqrt,stats
d1=(log(S/X)+(r+sigma*sigma/2.)*T)/(sigma*sqrt(T))
d2 = d1-sigma*sqrt(T)
return S*stats.norm.cdf(d1)-X*exp(-r*T)*stats.norm.cdf(d2)

After entering a set of five values, we can estimate the call price as follows:

>>>bs_call(40,40,0.5,0.05,0.25)
3.3040017284767735

On the other hand, if we know S, X, T, r, and c, how can we estimate sigma? Here, sigma is our implied volatility. In other words, if we are given a set values...

Binary-search

To estimate the implied volatility, the logic underlying the earlier methods is to run the Black-Scholes-Merton option model 100 times and choose the sigma value that achieves the smallest difference between the estimated option price and the observed price. Although the logic is easy to understand, such an approach is not efficient since we need to call the Black-Scholes-Merton option model a few hundred times. To estimate a few implied volatilities, such an approach would not pose any problems. However, under two scenarios, such an approach is problematic. First, if we need higher precision, such as sigma=0.25333, or we have to estimate several million implied volatilities, we need to optimize our approach. Let's look at a simple example. Assume that we randomly pick up a value between one and 5,000. How many steps do we need to match this value if we sequentially run a loop from one to 5,000? A binomial search is the log(n) worst-case scenario when linear search is the n...

Retrieving option data from Yahoo! Finance

There are many sources of option data that we can use for our investments, research or teaching. One of them is Yahoo! Finance.

To retrieve option data for IBM, we have the following procedure:

1. Go to http://finance.yahoo.com.

2. Type IBM in the search box.

3. Click on Options in the navigation bar.

The related page is http://finance.yahoo.com/quote/IBM/options?p=IBM. A screenshot of this web page is as follows:

Volatility smile and skewness

Obviously, each stock should possess one value for its volatility. However, when estimating implied volatility, different strike prices might offer us different implied volatilities. More specifically, the implied volatility based on out-of-the-money options, at-the-money options, and in-the-money options might be quite different. Volatility smile is the shape going down then up with the exercise prices, while the volatility skewness is downward or upward sloping. The key is that investors' sentiments and the supply and demand relationship have a fundamental impact on the volatility skewness. Thus, such a smile or skewness provides information on whether investors, such as fund managers, prefer to write calls or puts, as shown in the following code:

import datetime
import pandas as pd
import matplotlib.pyplot as plt
from matplotlib.finance import quotes_historical_yahoo_ochl as getData

# Step 1: input area
infile="c:/temp/callsFeb2014.pkl"
ticker='IBM'
r=0.0003...

References

Please refer to the following articles:

• Black, F., M. Scholes, 1973, The pricing of options and corporate liab ilities, Journal of Political Economy 81,3,637-654, https://www.cs.princeton.edu/courses/archive/fall09/cos323/papers/black_scholes73.pdf

• Cox, J. C., Ross, S. A., Rubinstein, M, 1979, Option pricing: A simplified appro ach, Journal of Financial Economics, 7(3), 229-263, http://www.sciencedirect.com/science/article/pii/0304405X79900151

Appendix A – data case 6: portfolio insurance

Portfolio insurance is a method of hedging a portfolio of stocks against market risk by short selling stock index futures. This hedging technique is frequently used by institutional investors when the market direction is uncertain or volatile. Assume that you manage one of the industry portfolios with a current value of $50 million. If you expect the whole market to be quite volatile in next three months--in other words, the market might go down significantly--what might be our choices at the...

Exercises

1. If the APR is 5% compounded quarterly, what is its equivalent continuously compounded rate?

2. The value of a portfolio is $4.77 million today with a beta of 0.88. If the portfolio manager explains the market will surge in the next three months and s/he intends to increase her/ his portfolio beta from 0.88 to 1.20 in just three months by using S&P500 futures, how many contracts should s/he long or short? If the S&P500 index increases by 70 points what will be her/his gain or loss? How about if the S&P500 falls by 50 points instead?

3. Write a Python program to price a call option.

4. Explain the empty shell method when writing a complex Python program.

5. Explain the logic behind the so-called comment-all-out method when writing a complex Python program.

6. Explain the usage of the return value when we debug a program.

7. When we write the CND (cumulative standard normal distribution), we could define a1, a2, a3, a4, and a5 separately. What are the differences between the following two approaches...

Summary

In this chapter, first we have explained many basic concepts related to portfolio theory, such as covariance,correlation, the formulas on how to calculate variance of a 2-stock portfolio and variance of an n-stock portfolio. After that, we have discussed various risk measures for individual stocks or portfolios, such as Sharpe ratio, Treynor ratio, Sortino ratio, how to minimize portfolio risk based on those measures (ratios), how to setup an objective function, how to choose an efficient portfolio for a given set of stocks, and how to construct an efficient frontier.

In the next chapter, we will discuss one of the most important theory in modern finance: options and futures. We will start from the basic concepts such as payoff functions for a call and for a put. Then we explain the related applications such as various trading strategies, corporate incentive plans, and hedging strategies including different types of options and futures.

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Names	Description	Direction of initial cash-flow	Expectation of future price movement
Bull spread with calls	Buy a call (x1) sell a call (x2) [ x1< x2 ]	Outflow	Rise
Bull spread with puts	Buy a put (x1), sell a put (x2) [ x1< x2 ]	Inflow	Rise
Bear spread with puts	Buy a put (x2), sell a put (x1) [ x1 < x2 ]	Outflow	Fall
Bear spread with calls	Buy a call (x2), sell a call (x1) [x1 < x2 ]	Inflow	Fall
Straddle	Buy a call & sell a put with the same x	Outflow	Rise or fall
Strip	Buy two puts and a call (with the same x)	Outflow	Prob (fall) > prob (rise)
Strap	Buy two calls and one put (with the same x)	Outflow	Prob (rise)> prob(fall)
Strangle	Buy a call (x2) and buy a put (x1) [x1 < x2 ]	Outflow	Rise or fall
Butterfly with calls	Buy two calls (x1,x3) and sell two calls (x2) [x2=(x1+x3)/2]	Outflow...

Put-call parity and its graphic presentation

Let's look at a call with an exercise price of $20, a maturity of three months and a risk-free rate of 5%. The present value of this future $20 is given here:

>>>x=20*exp(-0.05*3/12)   
>>>round(x,2)
19.75
>>>

In three months, what will be the wealth of our portfolio which consists of a call on the same stock plus $19.75 cash today? If the stock price is below $20, we don't exercise the call and keep the cash. If the stock price is above $20, we use our cash of $20 to exercise our call option to own the stock. Thus, our portfolio value will be the maximum of those two values: stock price in three months or $20, that is, max(s,20).

On the other hand, how about a portfolio with a stock plus a put option with an exercise price of $20? If the stock price falls by $20, we exercise the put option and get $20. If the stock price is above $20, we simply keep the stock. Thus, our portfolio value will be the maximum of those two...

Binomial tree and its graphic presentation

The binomial tree method was proposed by Cox, Ross, and Robinstein in 1979. Because of this, it is also called the CRR method. Based on the CRR method, we have the following two-step approach. First, we draw a tree, such as the following one-step tree. Assume that our current stock value is S. Then, there are two outcomes of Su and Sd, where u>1 and d<1, see the following code:

import matplotlib.pyplot as plt 
plt.xlim(0,1) 
plt.figtext(0.18,0.5,'S')
plt.figtext(0.6,0.5+0.25,'Su')
plt.figtext(0.6,0.5-0.25,'Sd')

plt.annotate('',xy=(0.6,0.5+0.25), xytext=(0.1,0.5), arrowprops=dict(facecolor='b',shrink=0.01))
plt.annotate('',xy=(0.6,0.5-0.25), xytext=(0.1,0.5), arrowprops=dict(facecolor='b',shrink=0.01))
plt.axis('off')
plt.show()

The graph is shown here:

Obviously, the simplest tree is a one-step tree. Assume that today's price is $10, the exercise price is $11, and a call option will mature in six months. In addition, assume that we know that...

Hedging strategies

After selling a European call, we could hold shares of the same stock to hedge our position. This is named a delta hedge. Since the delta is a function of the underlying stock (S), to maintain an effective hedge we have to rebalance our holding constantly. This is called dynamic hedging. The delta of a portfolio is the weighted deltas of individual securities in the portfolio. Note that when we short a security, its weight will be negative:

Assume that a US importer will pay £10 million in three months. He or she is concerned with a potential depreciation of the US dollar against the pound. There are several ways to hedge such a risk: buy pounds now, enter a futures contract to buy £10 million in three months with a fixed exchange rate, or buy call options with a fixed exchange rate as its exercise price. The first choice is costly since the importer does not need pounds today. Entering a future contract is risky as well since an appreciation of the US dollar would cost...

Implied volatility

From the previous sections, we know that for a set of input variables—S (the present stock price), X (the exercise price), T (the maturity date in years), r (the continuously compounded risk-free rate), and sigma (the volatility of the stock, that is, the annualized standard deviation of its returns)—we could estimate the price of a call option based on the Black-Scholes-Merton option model. Recall that to price a European call option, we have the following Python code of five lines:

def bs_call(S,X,T,r,sigma):
    from scipy import log,exp,sqrt,stats
d1=(log(S/X)+(r+sigma*sigma/2.)*T)/(sigma*sqrt(T))
d2 = d1-sigma*sqrt(T)
return S*stats.norm.cdf(d1)-X*exp(-r*T)*stats.norm.cdf(d2)

After entering a set of five values, we can estimate the call price as follows:

>>>bs_call(40,40,0.5,0.05,0.25)
3.3040017284767735

On the other hand, if we know S, X, T, r, and c, how can we estimate sigma? Here, sigma is our implied volatility. In other words, if we are given a set values...

Binary-search

To estimate the implied volatility, the logic underlying the earlier methods is to run the Black-Scholes-Merton option model 100 times and choose the sigma value that achieves the smallest difference between the estimated option price and the observed price. Although the logic is easy to understand, such an approach is not efficient since we need to call the Black-Scholes-Merton option model a few hundred times. To estimate a few implied volatilities, such an approach would not pose any problems. However, under two scenarios, such an approach is problematic. First, if we need higher precision, such as sigma=0.25333, or we have to estimate several million implied volatilities, we need to optimize our approach. Let's look at a simple example. Assume that we randomly pick up a value between one and 5,000. How many steps do we need to match this value if we sequentially run a loop from one to 5,000? A binomial search is the log(n) worst-case scenario when linear search is the n...

Retrieving option data from Yahoo! Finance

There are many sources of option data that we can use for our investments, research or teaching. One of them is Yahoo! Finance.

To retrieve option data for IBM, we have the following procedure:

1. Go to http://finance.yahoo.com.

2. Type IBM in the search box.

3. Click on Options in the navigation bar.

The related page is http://finance.yahoo.com/quote/IBM/options?p=IBM. A screenshot of this web page is as follows:

Volatility smile and skewness

Obviously, each stock should possess one value for its volatility. However, when estimating implied volatility, different strike prices might offer us different implied volatilities. More specifically, the implied volatility based on out-of-the-money options, at-the-money options, and in-the-money options might be quite different. Volatility smile is the shape going down then up with the exercise prices, while the volatility skewness is downward or upward sloping. The key is that investors' sentiments and the supply and demand relationship have a fundamental impact on the volatility skewness. Thus, such a smile or skewness provides information on whether investors, such as fund managers, prefer to write calls or puts, as shown in the following code:

import datetime
import pandas as pd
import matplotlib.pyplot as plt
from matplotlib.finance import quotes_historical_yahoo_ochl as getData

# Step 1: input area
infile="c:/temp/callsFeb2014.pkl"
ticker='IBM'
r=0.0003...

References

Please refer to the following articles:

• Black, F., M. Scholes, 1973, The pricing of options and corporate liab ilities, Journal of Political Economy 81,3,637-654, https://www.cs.princeton.edu/courses/archive/fall09/cos323/papers/black_scholes73.pdf

• Cox, J. C., Ross, S. A., Rubinstein, M, 1979, Option pricing: A simplified appro ach, Journal of Financial Economics, 7(3), 229-263, http://www.sciencedirect.com/science/article/pii/0304405X79900151

Appendix A – data case 6: portfolio insurance

Portfolio insurance is a method of hedging a portfolio of stocks against market risk by short selling stock index futures. This hedging technique is frequently used by institutional investors when the market direction is uncertain or volatile. Assume that you manage one of the industry portfolios with a current value of $50 million. If you expect the whole market to be quite volatile in next three months--in other words, the market might go down significantly--what might be our choices at the...

Exercises

1. If the APR is 5% compounded quarterly, what is its equivalent continuously compounded rate?

2. The value of a portfolio is $4.77 million today with a beta of 0.88. If the portfolio manager explains the market will surge in the next three months and s/he intends to increase her/ his portfolio beta from 0.88 to 1.20 in just three months by using S&P500 futures, how many contracts should s/he long or short? If the S&P500 index increases by 70 points what will be her/his gain or loss? How about if the S&P500 falls by 50 points instead?

3. Write a Python program to price a call option.

4. Explain the empty shell method when writing a complex Python program.

5. Explain the logic behind the so-called comment-all-out method when writing a complex Python program.

6. Explain the usage of the return value when we debug a program.

7. When we write the CND (cumulative standard normal distribution), we could define a1, a2, a3, a4, and a5 separately. What are the differences between the following two approaches...

Summary

In this chapter, first we have explained many basic concepts related to portfolio theory, such as covariance,correlation, the formulas on how to calculate variance of a 2-stock portfolio and variance of an n-stock portfolio. After that, we have discussed various risk measures for individual stocks or portfolios, such as Sharpe ratio, Treynor ratio, Sortino ratio, how to minimize portfolio risk based on those measures (ratios), how to setup an objective function, how to choose an efficient portfolio for a given set of stocks, and how to construct an efficient frontier.

In the next chapter, we will discuss one of the most important theory in modern finance: options and futures. We will start from the basic concepts such as payoff functions for a call and for a put. Then we explain the related applications such as various trading strategies, corporate incentive plans, and hedging strategies including different types of options and futures.