hedging with sblack-scholes-2 - by jan römanjanroman.dhis.org/stud/i2010/hedge/hedging.pdf ·...

Hedging with sBlack-Scholes-2

Group2

1.Eren Keskus 2.Zhao Seven Qi 3.Zhang Zhuozhuo, 4.Tipweerachat Thanyapat

[Pick the date] Hedging with Black-Scholes-2]

Abstract

Since we know the securities price change all the time, a

hedging is one of the methods help to manage this uncertainty. The

Black–Scholes model takes an important part in the pricing of options.

Study in both the theory and practice both hedge and Black schools

model is essential to assist us more understanding how they work and

how to apply them in the fluctuating financial circumstances.


Content

Abstract 2

Introduction 4

1.Hedging with options 4

1.1 Option 4

1.2 Delta Hedging 6

1.3 Delta Neutral 10

1.4 Implied Volatility 12

2.Related Theory 14

2.1 Black Scholes Model 14

2.2 Monte Carlo Simulation 15

3. Application 16

3.1 Case Study Description 16

3.2 Solution 17

A) Strike 100, Put in the Money 18

• Delta Hedged Portfolio 18

o Without transaction cost 18

o With transaction cost 19

• Unhedge Portfolio 20

B) Strike 70, Call in the Money 21

• Delta Hedged Portfolio 21

o Without transaction cost 21

o With transaction cost 22

• Unhedge Portfolio 23

3.3 Outcome 24

3.4 Comparisons Hedged Portfolios with Unhedged 24

3.5 Remark 25

4. Conclusion 26

5. Bibliography 27


Introduction

In financial world, they try to manage the uncertainty impacting

to their investment by building both new models and instruments.

However, we can take that tool when we understand how they work and

what factors impact. The more we know about that, the more we can

decide the effective strategy suitable.

1. Hedging with options

One of the methods to reduce and cancel out uncertainties we

encountered, calling that hedging. This does not prevent only a negative

result from happening, but if it does occur and you are properly hedged,

the impact of the event is reduced. Thus, hedging occurs almost

everywhere, and we see it every day.

There are several specific financial instruments to undertake

this, including insurance policies, forward contracts, swaps, options,

many types of over-the-counter and derivative products, and perhaps

most popularly, futures contracts.

Hedging techniques generally involve the use of complicated

financial instruments known as derivatives, the two most common of

which are options and futures. These tool help you develop trading

strategies where a loss in one investment is offset by a gain in a

derivative.

1.1 Option

Options plays an important role as using to the instruments for

manage their uncertainties since they are derivatives, their value depend

on the underlying assets. Options give their holders the right, but not the

obligation, to buy or sell underlying securities at a fixed strike price at

maturity date or their expiration date. An option to buy is known as a call;

an option to sell is a put.

Options can provide investors with leverage—the opportunity to increase the effect of an investment. As the option helps holders to get a chance when the price of underlying go in a way that option is in the


money position (call option: price of securities rise above the strike price, put option: price of stocks below the strike price) However, if the price of the stock doesn’t go in the other side before the expiration date, the option will expire worthless.

Options also offer the implements for betting against a stock.

Buying a put, an investor can sell the underlying stock at the exercise price, no matter how low the market price falls. Additional, if the price increases, the option expires valueless. Therefore, buying an option may be observed as a very risky investment, with the whole amount of the investment conditional on loss if the stock price does not move as expected. It may be possible, however, to sell the option at a loss prior to expiration to cancel out the investment status. Options as insurance

There are some relatively simple options strategies that a conservative investor can use to protect portfolio positions.

Buying “protective” put. Suppose that you have a stock holding that has appreciated, and you want to protect your profits should the market turn against you. You could set a floor on the stock price by purchasing a put.

Writing “covered” calls. A high- risk investment occurs when issuer write the call option without the underlying stock. On the other hand, you have own the stock, however, all your risk is the loss of above your strike price.

Creating an equity “collar”. Mixing a put to cover a call in the reasonable cost the investor is willing to hold it. The premium expected for the call goes toward paying the premium for the put. Therefore, a floor and ceiling are established.

Writing puts conservatively. For the declining market in short time before moving upward, a put price may below than the market price, the issuer should prepare to buy the stock in money market. In the other hand, the put price decrease, the issuer will obtain the stock at your target price. Of course, if the price drops quickly, you’ll still be required to pay the exercise price, so this method is to be careful with caution in volatile markets. However the issuer can add a equity option for develop the layer of complexity to investing.

Therefore, for protection the portfolio it will be good if the

investors turn to study in how the derivative instruments work or have the

assistance or consult to give some suggestion how to hedge their

portfolio. That help them to get more payoff beyond they expect.


Be careful the downside There is a cost in all hedge, thus before you decide to use

hedging, you must to ask yourself if the benefits received from it justify the expense. One more thing for concerning that, the objective of hedging is not to make money but to protect from losses. The cost of the hedge - whether it is the cost of an option or lost profits from being on the wrong side of a futures contract - cannot be avoided. This is the price you have to pay to avoid uncertainty may happen.

When we compare hedging with insurance, insurance is more precise than hedging. Since insurance help you compensate in your loss or pay in part of additional. For hedging of portfolios is just the protection from loss in the other side. It is so difficult to build the perfect hedge to achieve in practice, although that is the target of the financial managers. What Hedging Means to You

Most of the investors are never using derivative contracts in the market as it is difficult to understand how to them working. However investors can learn to hedge their portfolio suitable in each situation happen. An understanding of hedging will help you to comprehend and analyze these investments.

1.2 Delta Hedging

What Does Delta Hedging Mean?

(http://www.investopedia.com/terms/d/deltahedging.asp)

An options strategy that aims to reduce (hedge) the risk

associated with price movements in the underlying asset by offsetting

long and short positions. For example, a long call position may be delta

hedged by shorting the underlying stock. This strategy is based on the

change in premium (price of option) caused by a change in the price of

the underlying security. The change in premium for each basis-point

change in price of the underlying is the delta and the relationship

between the two movements is the hedge ratio.

Delta measures the sensitivity to changes in the price and is given by

∆= �� for a call option

∆= ��[�� − 1] for a put option


(Lecture notes in Analytical Finance I,Jan R. M. Röman)


Delta (∆) measures the sensitivity of the option prices to the underlying asset price. In the other term, it measures the change of the option price with respect to the underlying asset price. It can be defined by the partial derivative

� =��

(John C. Hull, 2006, p344)

We can apply the delta to compute the optimal number of options to

hedge the portfolio.

Delta is the primary parameter used when the investor monitor the option

risk. We hedge by investing in the underlying asset equal in size to the

option’s delta. To hedge the portfolio, the gain (loss) on the option

position would be offset by the loss (gain) on the stock position.

If F(t, S) is the option value, Nc is the number of options and Ns is the

number of stocks, the total portfolio value is given by:

� = −� ��, � � � �

With a delta hedging:

�� = −�

��, ��

� � ��

As �

�� = 0 ⇒ � = �

#$��,�

#� = �∆�

∆��%&

%'

As ∆�� ⇒ ��

�


Page

10

The delta for the call options is always positive. Using delta hedging for a

short position in a European call option involves keeping a long position

in the underlying asset at any given time. Similarly, using delta hedging

for a long position in a European call option involves maintaining a short

position in the underlying asset at any given time. The delta for the put

options is always negative, which means that a long position in a

European put option should be hedged with a long position in the

underlying asset, and a short position in a European put option should be

hedged with a short position in the underlying asset.

(John C. Hull, 2006, p.346)

It is important to realize the delta changes, the investor’s position

remains delta hedged for only a relatively short period of time. Thus, the

hedge has to be adjusted periodically.

(John C. Hull, 2006, p.345)

1.3 Delta Neutral Trading

Definition

A portfolio consists of positions with offsetting positive and

negative deltas. The deltas balance out to bring the net change of the

position to zero. (http://www.investopedia.com)

An option position which is relatively insensitive to small price

movements of the underlying stock due to having near zero or zero.

Since knowing that the Delta of a stock option is a measure of

how much the option can be expected to gain or lose per $1 move in the

stock price or underlying asset. A Delta-Neutral option position is one in

which there is no gain or loss on the position due to stock price

movement.

A perfectly Delta-Neutral stock position would be long and

short the same number of shares of the same stock at the same time. No

matter what the stock did, the overall position would not gain or lose.


Page

11

The stock investors usually carry these ideas for starting

option trading. They choose the strategies for both the bull or bear

markets to get the benefit from that situation. However the option

changes more than the price of stock, the important task to concern in

part of volatility and remaining of time.

How the option work to get the profit

The options traders earn the profit from options that gain or

lose value due to rising or falling volatility, and/or the trader can profit

from options that gain or lose value due to the passage of time.

A delta hedging is the process of setting or keeping the delta

of a portfolio as close to zero as possible. Nevertheless in practice,

maintaining a zero delta is very complicate since there are risks related

with re-hedging on large movements in the underlying stock's price, and

research point out portfolios that have a tendency to lower cash flows

when re-hedged too frequently.

Creating the position

For constructing the required hedge, a delta hedging may be consisted on buying or selling an amount of the underling that relates to the delta of the portfolio. The portfolio delta can be prepared to sum to zero, and the portfolio is then delta neutral by modifying the amount bought or sold on new positions

Options market makers, or others, may form a delta neutral portfolio using related options instead of the underlying. The portfolio's delta (assuming the same underling asset) is then the sum of all the individual options' deltas. This method can also be used when the underling is difficult to trade, for instance when an underlying stock is hard to borrow and therefore cannot be sold short.

Almost all successful experts of Delta-Neutral trading take gains early and small losses whenever they have them. They earn the profit overall by having more and larger gains, and fewer and smaller losses


1.4 Implied Volatility

The implied volatility of an option contract is the volatility implied by the market price of the option derived from an option pricing model. In other way we can say that the volatility is used in a particular pricing model, yields a theoretical value for the option equal to the current market price of that option.

In the Black-Scholes model, the volatility of the underlying asset is the only non-directly observable variable. The implied volatility is found to give rise to a skew, smile or frown, depending on the asset or market.

(Lecture notes in Analytical Finance I,Jan R. M. Röman)

The important problem from using the implied volatility takes into the account of the current view of the market, it does not concern about the possible future change in volatility.

To success in the investor perspectives, moreover they apply the volatility mainly for an option valuation, they put the historical price movement.The historical volatility is defined via the standard deviation of the movements in price. For example, we have n observations: ai: a0, a1, ..., an-1. If we define ui = ln(ai/ai-1) we can calculate the standard deviation:

.

The Volatility is then given as S √� where d is the number of trading days in a year ( ) 250 ).


Page

13

2. Related Theory

2.1 Black Scholes Model

An option is a contract that gives the investor the right - but not

the obligation - to buy or sell a specific financial instrument at a specific

price and time. With a European-style option, the contract terms allow

the option to be exercised only on the expiration date. The Black–

Scholes model is the most widely used model in determining the price of

European options.

The model has following assumptions:

• Constant volatility. The most significant assumption is that volatility, a measure of how much a stock can be expected to move in the near-term, is a constant over time. While volatility can be relatively constant in very short term, it is never constant in longer term. Some advanced option valuation models substitute Black-Schole's constant volatility with stochastic-process generated estimates.

• Efficient markets. This assumption of the Black-Scholes model suggests that people cannot consistently predict the direction of the market or an individual stock. The Black-Scholes model assumes stocks move in a manner referred to as a random walk. Random walk means that at any given moment in time, the price of the underlying stock can go up or down with the same probability. The price of a stock in time t+1 is independent from the price in time t.

• No dividends. Another assumption is that the underlying stock does not pay dividends during the option's life. In the real world, most companies pay dividends to their shareholders. The basic Black-Scholes model was later adjusted for dividends, so there is a workaround for this. This assumption relates to the basic Black-Scholes formula. A common way of adjusting the Black-Scholes model for dividends is to subtract the discounted value of a future dividend from the stock price.

• Interest rates constant and known. The same like with the volatility, interest rates are also assumed to be constant in the Black-Scholes model. The Black-Scholes model uses the risk-free rate to represent this constant and known rate. In the real world, there is no such thing as a risk-free rate, but it is possible to use the U.S. Government Treasury Bills 30-day rate since the U. S. government is deemed to be credible enough. However, these treasury rates can change in times of increased volatility.


Page

14

• Lognormally distributed returns. The Black-Scholes model assumes that returns on the underlying stock are normally distributed. This assumption is reasonable in the real world.

• European-style options. The Black-Scholes model assumes European-style options which can only be exercised on the expiration date. American-style options can be exercised at any time during the life of the option, making american options more valuable due to their greater flexibility.

The Black-Scholes use five keys to determine an option's price:

stock price, strike price, volatility, time to expiration, and short-term (risk

free) interest rate.

The formula for calculating the theoretical option price (OP) is as follows:

Where:

The variables are:

S = stock price

X = strike price

t = time remaining until expiration, expressed as a percent of a year

r = current continuously compounded risk-free interest rate

v = annual volatility of stock price (the standard deviation of the short-

term returns over one year)

ln = natural logarithm

N(x) = standard normal cumulative distribution function

e = the exponential function


Page

15

2.2 Monte Carlo Simulation

In our project we intended to make a Monte Carlo simulation

with Black-Scholes model.

Monte Carlo simulation is a technique that converts

uncertainties in input variables of a model into probability distributions.

By combining the distributions and randomly selecting values from them,

it recalculates the simulated model many times and brings out the

probability of the output.


Page

16

3. Application

3.1 Case Study Description

Make a Monte-Carlo simulation with Black-Scholes with parameters given (initial underlying price, strike price, volatility, time to maturity and risk-free interest rate). Simulate the situation where you buy 100 options on an underlying stock and at start, hedge the position with the stock. Each such option has 100 stocks as underlying. So, make the best hedge. During the price movements of the underlying, change the hedge each time you need to buy or sell 100 stocks. Finally, calculate the outcome of this trading strategy. Make 100 000 simulation an present a histogram of the result and calculate the mean value and the variance. Do this for a call option and then for a put option. What happen if you also introduce a trading cost of e.g., 2%.


Page

17

3.2 Solution

An investor buys call or put options on a stock in the market. Investors’ portfolio value is 0 at the time 0. Investor borrows money with risk-free rate when it is needed. The outcomes are in SEK discounted to time=0. Above you can find the information of options at time=0 which 15th of October 2010. Investor buys 100 options. The approximately risk free interest rate equal to 2,5%. Day count is bus/252.

At time zero investor portfolio value is zero. Investors pays the premium, borrows and loans money with risk-free interest rate, and has a long or short position in delta at time zero times number of underlyings times number of options.

The Underlyings price is simulated 100000 times. The volatility is constant, implied volatility of the underlying asset at time=0.

Delta will be calculated every day, and in order to keep the portfolio delta-neutral during the options life time, compared with the actual one and updated if neccesary (if one more or less option needs to be hedged).

An Underlying Asset

Swedbank A , the current market price : 94.25 SEK

Here are 2 call options one in the other one out of the money, and 2

put options one in and one out of the money.

Option Name Expiration Date Bid Ask Underlyings Strike

SWEDA1A100 2011-01-21 3.35 3.90 100.00 100.00

SWEDA1M100 2011-01-21 8.50 9.25 100.00 100.00

SWEDA1A100 2011-01-21 24.00 25.50 100.00 70.00

SWEDA1M100 2011-01-21 0.15 0.55 100.00 70.00


Page

18

A) Strike 100, Put in the Money

• Delta Hedged Portfolio

1. Without Transaction Costs


Page

19

2.With Transaction Costs:


Page

20

Unhedged Portfolio


Page

21

B) Strike 70, Call in the Money:

1. Without Transaction Costs


Page

22

2. With Transaction Costs


Page

23

• Unhedged Portfolio


Page

24

3.3 Outcomes

Type Hedged Transaction Costs

Strike Mean

*10^4

Variance

*10^9

Skewness

Call Yes No 100 2.3142 1.0830 0.4674

Call Yes Yes 100 0.55919 1.1648 0.4373

Call No - 100 2.3429 4.6813 2.7335

Put Yes No 100 7.5359 1.0651 0.4658

Put Yes Yes 100 9.5433 1.0716 0.4360

Put No - 100 7.5286 7.8874 0.8553

Call Yes No 70 4.6121 2.0187 -0.0417

Call Yes Yes 70 1.9419 1.6322 -0.0793

Call No - 70 23.590 22.001 0.5965

Put Yes No 70 -0.0442 2.0187 -0.0417

Put Yes Yes 70 0,0001 1.8770 -0.0418

Put No - 70 -0.79355 0.10181 9.2389

3.4 Comparisons Hedged Portfolios with Unhedged

Type Strike Transaction

Costs incl.

Mean Heghed/Mean Unhedged %

Var Heghed/Var Unhedged %

Skewness Heghed/Skewness

Unhedged %

Call 100 Yes 5.1758e+003 24.9810 15.6974

Call 100 No 83.4703 23.3673 17.3384

Put 100 Yes -8.4400e+004 13.5836 50.4031

Put 100 No 108.3012 13.4259 55.4877

Call 70 Yes 42.2051 7.4032 -14.0193

Call 70 No 100.5959 9.0994 -7.5664

Put 70 Yes -2.7114e+004 1.8410e+003 -0.5008

Put 70 No 254.1566 -244.3097 -0.4693


Page

25

3.5 Remarks

Portfolio was tried to be held delta-neutral during the option life, where delta was compared to the actual one every day and updated every time when one more or less option needed to be hedged. One of the parametersof delta, implied volatiliy of underlying asset could not be updated daily. Instead of that it was a constant which is obtained from the information at time=0. This outcomes are due to the constant implied volatility. Better results can be expected in the case of having a dynamic implied volatility.


Page

26

4. Conclusion

The essential part of investment as we know is that risks. The

investors can try to hedge the portfolio by adapt the basic knowledge of

hedging strategies that lead them to the better payoff than without

awareness in portfolios. Whether or not you decide to begin practicing

the intricate uses of derivatives, learning about how hedging works will

help proceed your perceptive of the market, which will always help you

be a better investor.


Page

27

5. Bibliography [1] URL: http://www.investopedia.com/articles/basics/03/080103.asp ,

last visited 2nd October 2010.

[2] URL: http://en.wikipedia.org/wiki/Hedge_%28finance%29 , last visited

2nd October 2010.

[3] John C. Hull, Options, 2006, sixth edition. Futures, and Other

Derivatives, Prentice Hall. ISBN 0-13-149908-4

hedging with sblack-scholes-2 - by jan römanjanroman.dhis.org/stud/i2010/hedge/hedging.pdf ·...

Documents