Deeper Understanding, Faster Calc: SOA MFE and CAS Exam 3F

Yufeng Guo

October 4, 2017 ii Contents

1 Introduction to derivatives 1 1.1 What is a ? ...... 1 1.1.1 De…nition of derivatives ...... 2 1.1.2 Major types of derivatives ...... 2 1.1.3 Basic vocabulary ...... 3 1.1.4 Uses of Derivatives ...... 4 1.1.5 Perspectives on Derivatives ...... 5 1.1.6 Financial Engineering and Security Design ...... 5 1.2 The role of …nancial markets ...... 6 1.2.1 Financial Markets and Averages ...... 6 1.2.2 Risk-sharing ...... 6 1.3 Derivatives in practice ...... 6 1.3.1 Growth in derivatives trading ...... 6 1.3.2 How are derivatives used? ...... 6 1.4 Buying and short-selling …nancial assets ...... 7 1.4.1 The lease rate of an asset ...... 7 1.4.2 Risk and scarcity in short-selling ...... 7 1.5 Chapter summary ...... 7 1.6 Review questions ...... 8

2 An Introduction to Forwards and Options 9 2.1 Forward contracts ...... 9 2.1.1 De…nition ...... 9 2.1.2 The payo¤ on a ...... 10 2.1.3 Graphing the payo¤ on a forward contract ...... 10 2.1.4 Comparing a forward and outright purchase ...... 11 2.1.5 Zero-coupon bonds in payo¤ and pro…t diagrams . . . . . 11 2.1.6 Cash settlement vs. delivery ...... 12 2.1.7 Credit risk ...... 12 2.2 Call options ...... 12 2.2.1 terminology ...... 12 2.2.2 Payo¤ and pro…t for a purchased ...... 12 2.2.3 Payo¤ and pro…t for a written call option ...... 12 2.3 Put options ...... 13

iii iv CONTENTS

2.3.1 Payo¤ and pro…t for a purchased ...... 13 2.3.2 Payo¤ and pro…t for a written put option ...... 13 2.4 Summary of forward and option positions ...... 13 2.5 Options are insurance ...... 14 2.6 Example: equity-linked CD ...... 15

3 Insurance, Collars, and Other Strategies 17 3.1 Basic insurance strategies ...... 17 3.1.1 Insuring a long position: ‡oors ...... 17 3.1.2 Insuring a short position: caps ...... 18 3.1.3 Selling insurance ...... 18 3.2 Synthetic forwards ...... 19 3.2.1 Put-call parity ...... 19 3.3 Spreads and collars ...... 20 3.3.1 Bull and bear spreads ...... 20 3.3.2 Box spreads ...... 21 3.3.3 Ratio spreads ...... 21 3.3.4 Collars ...... 22 3.4 Speculating on ...... 22 3.4.1 ...... 22 3.4.2 Butter‡y spread ...... 23 3.4.3 Asymmetric butter‡y spreads ...... 23

5 Financial forwards and futures 25 5.1 Alternative ways to buy a stock ...... 25 5.2 Tailing ...... 25 5.3 Pricing a forward contract ...... 26 5.4 Forward premium ...... 27 5.5 Creating a synthetic forward contract ...... 27 5.5.1 Synthetic forwards in market-making and arbitrage . . . . 29 5.5.2 No-arbitrage bounds with transaction costs ...... 30 5.5.3 Quasi-arbitrage ...... 31 5.6 Futures contracts ...... 32 5.6.1 Role of the clearing house ...... 32 5.6.2 S&P 500 ...... 33 5.6.3 Di¤erence between a forward contract and a futures contract 33 5.6.4 Margins and markings to market ...... 33 5.6.5 Comparing futures and forward prices ...... 36 5.6.6 Arbitrage in practice: S&P 500 index arbitrage ...... 37 5.6.7 Appendix 5.B: Equating forwards and futures ...... 38

9 Parity and other option relationships 45 9.1 Put-call parity ...... 45 9.1.1 Option on stocks ...... 45 9.1.2 Options on currencies ...... 55 9.1.3 Options on bonds ...... 57 CONTENTS v

9.1.4 Generalized parity and exchange options ...... 57 9.1.5 Comparing options with respect to style, maturity, and strike ...... 63

10 Binomial option pricing: basic concepts 79 10.1 One-period binomial model: simple examples ...... 79 10.2 General one-period binomial model ...... 80 10.2.1 Estimate stock volatility ...... 92 10.2.2 Two or more binomial trees ...... 97 10.2.3 Options on stock index ...... 112 10.2.4 Options on currency ...... 115 10.2.5 Options on futures contracts ...... 119

11 Binomial option pricing: selected topics 127 11.1 Understanding early ...... 127 11.2 Understanding risk-neutral probability ...... 128 11.2.1 Pricing an option using real probabilities ...... 129 11.2.2 Binomial tree and lognormality ...... 136 11.3 Stocks paying discrete dividends ...... 139 11.3.1 Problems with discrete dividend tree ...... 140 11.3.2 Binomial tree using prepaid forward ...... 142

12 Black-Scholes formula 149 12.1 Introduction to the Black-Scholes formula ...... 149 12.1.1 Call and put option price ...... 149 12.1.2 When is the Black-Scholes formula valid? ...... 151 12.2 Derive the Black-Scholes formula ...... 151 12.3 Applying the formula to other assets ...... 165 12.3.1 Black-Scholes formula in terms of prepaid . 165 12.3.2 Options on stocks with discrete dividends ...... 165 12.3.3 Options on currencies ...... 166 12.3.4 Options on futures ...... 167 12.4 Option the ...... 168 12.4.1 Delta ...... 168 12.4.2 Gamma ...... 169 12.4.3 Vega ...... 169 12.4.4 Theta ...... 170 12.4.5 Rho ...... 170 12.4.6 Psi ...... 170 12.4.7 Greek measures for a portfolio ...... 170 12.4.8 Option elasticity and volatility ...... 171 12.4.9 Option risk premium and Sharpe ratio ...... 172 12.4.10 Elasticity and risk premium of a portfolio ...... 173 12.5 Pro…t diagrams before maturity ...... 173 12.5.1 Holding period pro…t ...... 173 12.5.2 ...... 176 vi CONTENTS

12.6 ...... 177 12.6.1 Calculate the implied volatility ...... 177 12.6.2 Volatility skew ...... 178 12.6.3 Using implied volatility ...... 179 12.7 Perpetual American options ...... 179 12.7.1 Perpetual calls and puts ...... 179 12.7.2 Barrier present values ...... 184

13 Market-making and delta-hedging 187 13.1 Delta hedging ...... 187 13.2 Examples of Delta hedging ...... 187 13.3 Textbook Table 13.2 ...... 196 13.4 Textbook Table 13.3 ...... 198 13.5 Mathematics of Delta hedging ...... 199 13.5.1 Delta-Gamma-Theta approximation ...... 199 13.5.2 Understanding the market maker’spro…t ...... 200

14 Exotic options: I 203 14.1 (i.e. average options) ...... 203 14.1.1 Characteristics ...... 203 14.1.2 Examples ...... 204 14.1.3 Geometric average ...... 204 14.1.4 Payo¤ at maturity T ...... 204 14.2 ...... 205 14.2.1 Knock-in option ...... 205 14.2.2 Knock-out option ...... 205 14.2.3 Rebate option ...... 205 14.2.4 Barrier parity ...... 206 14.2.5 Examples ...... 206 14.3 Compound option ...... 207 14.4 Gap option ...... 208 14.4.1 De…nition ...... 208 14.4.2 Pricing formula ...... 209 14.4.3 How to memorize the pricing formula ...... 209 14.5 Exchange option ...... 210

18 Lognormal distribution 213 18.1 Normal distribution ...... 213 18.2 Lognormal distribution ...... 214 18.3 Lognormal model of stock prices ...... 214 18.4 Lognormal probability calculation ...... 215 18.4.1 Lognormal con…dence interval ...... 216 18.4.2 Conditional expected prices ...... 220 18.4.3 Black-Scholes formula ...... 221 18.5 Estimating the parameters of a lognormal distribution ...... 221 18.6 How are asset prices distributed ...... 223 CONTENTS vii

18.6.1 Histogram ...... 223 18.6.2 Normal probability plots ...... 224 18.7 Sample problems ...... 226

19 Monte Carlo valuation 231 19.1 Example 1 Estimate E (ez) ...... 231 19.2 Example 2 Estimate pi ...... 235 19.3 Example 3 Estimate the price of European call or put options . . 238 19.4 Example 4 Arithmetic and geometric options ...... 241 19.5 E¢ cient Monte Carlo valuation ...... 251 19.5.1 Control variance method ...... 251 19.6 Antithetic variate method ...... 255 19.7 Strati…ed sampling ...... 257 19.7.1 Importance sampling ...... 257 19.8 Sample problems ...... 257

23 Exotic options: II 263 23.1 All-or-nothing options ...... 263

24 Volatility 265

25 Interest rate and bond derivatives 267 25.1 Introduction to interest rate derivatives ...... 267 25.1.1 Bond and interest rate forward ...... 267 25.1.2 Options on bonds and rates ...... 270 25.2 Interest rate derivatives and the Black-Scholes-Merton approach 272 25.2.1 Interest rate is not so simple ...... 272 25.2.2 Equilibrium equation for bond ...... 276 25.3 Continuous-time short-rate models ...... 282 25.3.1 Arithmetic Brownian motion (i.e. Merton model) . . . . . 282 25.3.2 Rendleman-Bartter model ...... 283 25.3.3 Vasicek model ...... 284 25.3.4 CIR model ...... 306 25.3.5 Duration and convexity revisited ...... 311 25.4 Short-rate models and interest rate trees ...... 318 25.4.1 An illustrative tree ...... 318 25.4.2 Black-Derman-Toy model ...... 324 25.5 Market models ...... 329 25.5.1 The ...... 329 viii CONTENTS Preface

This study guide is for the SOA MFE and CAS Exam 3F. Before you start, make sure you have the following items:

1. Derivatives Markets, the 3rd edition.

2. Errata of Derivatives Markets. You can download the errata at http:// derivatives.kellogg.northwestern.edu/errata/errata3e.html#te 3. Download the syllabus from the SOA or CAS website. 4. Download the sample MFE problems and solutions from the SOA website. 5. Download the recent SOA MFE and CAS Exam 3 problems (if any).

Please report any errors to [email protected].

ix Chapter 13

Market-making and delta-hedging

13.1 Delta hedging @V Here is the main idea behind delta hedging. Delta of an option is  = . @S Hence for a small change in stock price, the change of the option value is ap- proximately V1 V0  (S1 S0). Suppose you sell one European call option. If you hold  shares  of stock, you are immunized against a small change of the stock price. If the stock price goes up from S0 to S1, then the call will be more valuable to the buyer and your are exposed to more risk. Suppose the value of the call goes up from V0 to V1 as the stock price goes up from S0 to S1, then your liability will increase by V1 V0. At the same time, the value of your  shares of stock will go up by  (S1 S0). Because V1 V0  (S1 S0), the increase of your liability will be roughly o¤set by the increase  of your asset. However, under delta hedging, you are immunized against only a small change of the stock price (just like immunization by duration matching assets and liabilities is only good for a small change of interest rate). If a big change of stock price knocks o¤ your hedging, you’ll need to rebalance your hedging. However, in the real world, continuously rebalancing the hedging portfolio is impossible. Traders can only do discrete rebalancing.

13.2 Examples of Delta hedging

Make sure you can reproduce the textbook calculation of delta-hedging for 2 days. In addition, make sure you can reproduce the textbook table 13.2 and 13.3 Exam problems may ask you to outline hedging transactions or calculate the hedging pro…t.

187 188 CHAPTER 13. MARKET-MAKING AND DELTA-HEDGING

The major di¢ culty many candidates face is not knowing how to hedge. They wonder "Should the market-maker buy stocks? Should he sell stocks?" To determine how to hedge a risk, use the following ideas:

The goal of hedging is to break even. If a trader makes money  on option, he must lose money on stock; if he loses money on option, he must make money on stock.

To determine whether a trader should buy stocks or sell stocks,  ask "If the stock price go up (or down), will the trader make money or lose money on the option?"

If a trader loses money on the option as the stock price goes up,  then the trader needs to initially own (i.e. buy) stocks. This way, the value of the trader’s stocks will go up and the trader will make money on his stocks. He can use this pro…t to o¤set his loss in the option.

If a trader makes money on the option as the stock price goes  up, then the trader needs to initially short sell stocks. This way, as the stock price goes up, the trader will lose money on the short sale (because he needs to buy back the stocks at a higher price). His loss in short sale can o¤set his pro…t in option.

Example 13.2.1. The trader sells a call. How can he hedge his risk,?

If the stock price goes up, the call payo¤ is higher and the trader will lose money. To hedge this risk, the trader should buy stocks. This way, if the stock price goes up, the trader makes money in the stocks. This pro…t can be used to o¤set the trader’sloss in the written call. You can also ask the question "If the stock goes down, will the trader make money or lose money on the option?" If the stock goes down, the call payo¤ is lower and the trader will make money. To eat up his pro…t in the option, the trader needs to buy stocks. This way, as the stock price goes down, the value of the trader’s stocks will go down too and the trader will lose money in his stocks. This loss will o¤set the trader’spro…t in the written call.

Example 13.2.2. The trader sells a put. How can he hedge his risk?

If the stock price goes down, the put payo¤ is higher and the trader will make money. To hedge his risk, the trader should short sell stocks. This way, if the stock price goes down, the trader can buy back stocks at lower price, making a pro…t on stocks. This pro…t can be used to o¤set the trader’s loss in the written put. You can also ask the question "If the stock goes up, will the trader make money or lose money on the option?" If the stock goes up, the put payo¤ is lower and the trader will make money on the written put. To eat up his pro…t in the option, the trader needs to short sell stocks. This way, as the stock price 13.2. EXAMPLES OF DELTA HEDGING 189 goes up, the trader needs to buy back stocks at a higher price. The trader will lose money in his stocks. This loss will o¤set the trader’s pro…t in the written put.

Example 13.2.3. The trader buys a call. How can he hedge his risk?

If a trader buys a call, the most he can lose is his premium and there’s no need to hedge. This is di¤erent from selling a call, where the call seller has unlimited loss potential. However, if a trader really wants to hedge his limited risk, he can short sell stocks.

Example 13.2.4. The trader buys a put. How can he hedge his risk?

If a trader buys a put, the most he can lose is his premium and there’s no need to hedge. This is di¤erent from selling a put, where the put seller has a big loss potential. However, if a trader really wants to hedge his limited risk, he can buy stocks.

Example 13.2.5.

Reproduce the textbook example of delta-hedging for 2 days. Here is the recap of the information. At time zero the market maker sells 100 European call options on a stock. The option expires in 91 days.

K = $40  r = 0:08   = 0   = 0:3  The stock price at t = 0 is $40  The stock price at t = 1=365 (one day later) is $40:50  The stock price at t = 2=365 (two days later) is $39:25  The market-maker delta hedges its position daily. Calculate the market- maker’sdaily mark-to-market pro…t

Solution.

First, let’scalculate the call premium and delta at Day 0, Day 1, and Day 2. I used my Excel spreadsheet to do the following calculation. If you can’t fully match my numbers, it’sOK. 190 CHAPTER 13. MARKET-MAKING AND DELTA-HEDGING

Time t Day 0 (t = 0) Day 1 Day 2 Expiry T (Yrs) 91=365 90=365 89=365 St 40 40:50 39:25 S 1 ln t + 0:08 + 0:32 T 40 2  d1 =   0:208048 0:290291 0:077977 0:3pT N (d1) 0:582404 0:614203 0:531077 d2 = d1 0:3pT 0:058253 0:141322 0:070162 N (d2) 0:523227 0:556192 0:472032 0:08T C = StN (d1) 40e N (d2) 2:7804 3:0621 2:3282 (T t)  = e N (d1) 0:58240 0:61420 0:53108

The above table can be simpli…ed as follows: Time t Day 0 t = 0 Day 1 t = 1=365 Day 2 t = 2=365 Expiry T T0 = 91=365 T1 = 90=365 T2 = 89=365 St S0 = 40 S1 = 40:50 S2 = 39:25 Ct C0 = 100 2:7804 = 278: 04 C1 = 306:21 C2 = 232:82  t 0 = 100 0:58240 = 58: 24 1 = 61:420 2 = 53:108  Beginning of Day 0 Trader #0 goes to work Day 0 t = 0 T0 = 91=365 S0 = 40 C0 = 278:04 0 = 58:240

Trader #0 goes to work. The brokerage …rm (i.e. the employer of Trader #0) gives Trader #0 C0 = $278: 04. This is what the call is worth today. Trader #0 needs to hedge the risk of the written call throughout Day 0. To hedge the risk, Trader #0 buys 0 stocks, costing 0S0 = 58: 24 40 = $ 2329: 6. Since Trader #0 gets $278: 04 from the brokerage …rm, he needs to borrow: 0S0 C0 = 2329: 6 278: 04 = $2051: 56. The trader can borrow $2051: 56 from a bank or use this own money. Either way, this amount is borrowed. The borrowed amount earns a risk free interest rate. Now Trader #0’sportfolio is: Component Value 0 = 58:24 stocks 2487: 51 call liability 278: 04 borrowed amount 2051 : 56 Net position 0

End of Day 0 (or Beginning of Day 1) 13.2. EXAMPLES OF DELTA HEDGING 191

Mark to market without rebalancing the portfolio Day 1 t = 1=365 T1 = 90=365 S1 = 40:50 C1 = 306:21

Method 1 To cancel out his position, Trader #0 can at t = 0

buy a call (we call this the 2nd call) from the market paying C1 = $306:  21. At , the payo¤ of this 2nd call will exactly o¤set the payo¤ of the 1st call. The 1st call is the call sold by the brokerage …rm at t = 0 to the customer who bought the call. For example, if at expiration the stock price is ST = 100, then both calls are exercised. The trader gets ST K = 100 40 = 60 from the 2nd call. The liability of the …rst call is also $60. These two calls cancel each other out.

sell out 0 = 58: 24 stocks for 0S1 = 58: 24 40:5 = $2358: 72   rh 0:08 1=365 pay o¤ the loan. The payment is (0S0 C0) e = 2051: 56e  =  2052: 01

rh At the end of Day 0, the trader’spro…t is C1 + 0S1 (0S0 C0) e = 306: 21 + 2358: 72 2052: 01 = 0:5 Trader #0 hands in $0.5 pro…t to his employer and goes home.

Method 2 We consider the change between Day 0 and Day 1.

In the beginning of Day 0, the trader’sstock is worth 0S0 = 58: 24 (40) =  2329: 6. In the end of Day 0 (or the beginning of Day 1), the trader’sstock is worth 0S1 = 58: 24 (40:5) = 2358: 72. The value of the trader’s 58: 24 stocks goes up by 58: 24 (40:5 40) = 29: 12. This is good for the trader.

In the beginning of Day 0, the call is worth C0 = $278: 04. In the end of  Day 0 (or the beginning of Day 1), the call is worth C1 = 306:21. The call value is the trader’s liability. Now the trader’s liability increases by 306: 21 278: 04 = 28: 17. So the trader has a loss 28: 17 (or a gain of 28:75). Recall under Method 1, the trader has to buy a call for 306: 21 to cancel out the call he sold. So call value increase is bad for the trader.

In the beginning of Day 0, the borrowed amount is 0S0 C0 = 2051: 56.  In the end of Day 0 (or the beginning of Day 1), this borrowed amount rh 0:08 1=365 grows to (0S0 C0) e = 2051: 56e  = 2052: 01. The increase rh 0:08 1=365 (0S0 C0) e 1 = 2051: 56e  2051: 56 = 0:45. This is the interest paid on the amount borrowed. No matter the trader borrows money from a bank or uses his own money, the borrowed money needs to earn a risk free interest rate. 192 CHAPTER 13. MARKET-MAKING AND DELTA-HEDGING

At the end of Day 0, the trader’spro…t is: rh 0S1 0S0 (C1 C0) (0S0 C0) e 1 = 29: 12 28: 17 0:45 = 0:5  You can verify that rh rh 0S1 0S0 (C1 C0) (0S0 C0) e 1 = C1+0S1 (0S0 C0) e  Method 3 On Day 0, the trader owns 0 = 58:240 stocks to hedge the call liability C0 = $278: 04. The trader’snet asset is MV (0) = 0S0 C0 = 58: 24 40 278: 04 = $2051: 56 

In the end of Day 0 (or the beginning of Day 1) before the trader rebalances his portfolio, the trader’sasset is: BR MV (1) = 0S1 C1 = 58: 24 40:5 100 3:0621 = 2052: 51   BR stands for before rebalancing. The trader’spro…t at the end of Day 0 is: BR 0:08 1=365 rh MV (1) MV (0) e  = (0S1 C1) (0S0 C0) e = 2052: 0:08 1=365 51 2051: 56e  = 0:50

Please note that Trader #0 doesn’t need to rebalance the portfolio. The portfolio is rebalanced by the next trader.

Beginning of Day 1 Trader #1 goes to work Day 1 t = 1=365 T1 = 90=365 S1 = 40:50 C1 = 306:21 1 = 61:420

Trader #1 goes to work. He starts from a clean slate. The portfolio is automatically rebalanced since Trader #1 starts from scratch.

The brokerage …rm gives Trader #1 C1 = $306:21. Trader #1 needs to hedge the risk of the written call throughout Day 1. To hedge the risk, Trader #1 buys 1 = 61:420 stocks, costing 1S1 = 61:42 40:5 = $2487: 51. Since Trader #1 gets $306:21 from the brokerage …rm, he needs to borrow: 1S1 C1 = 2487: 51 306:21 = 2181: 3 The trader can borrow $2181: 3 from a bank or use this own money. Either way, this amount is borrowed. The borrowed amount earns a risk free interest rate. Now Trader #1’sportfolio is: 13.2. EXAMPLES OF DELTA HEDGING 193

component value 1 = 61:42 stocks 2487: 51 call liability 306:21 borrowed amount 2181 : 3 Net position 0

One question arises, "What if Trader #1 doesn’t start from a clean slate?" Next, we’ll answer this question. Instead of starting from scratch, Trader #1 can start o¤ with Trader #0’s portfolio. At the end of Day 0, Trader #0 has

0 = 58:24 stocks  rh 0:08 1=365 a borrowed amount (0S0 C0) e = 2051: 56e  = 2052: 01  0.5 pro…t 

At the end of Day 0 or the beginning of Day 1, 1 = 61:420. So Trader #1 buys additional shares: 1 0 = 61:42 58:24 = 3: 18 The cost of these additional shares is (1 0) S1 = 3: 18 40:5 = 128: 79 Since these additional shares are bought at the current market price, Trader #1 can sell these shares at the same price he bought them. This doesn’ta¤ect the mark-to-market pro…t. Trader #1 can borrow 128: 79 to pay for the purchase of 3: 18 stocks. Now Trader #1 has a total of 1 = 61:42 shares worth 1S1 = 61:42 40:5 = 2487: 51  His liability is now C1 = 306:21 The borrowed amount is now 2052: 01 + 128: 79 + 0:5 = 2181: 3. The 0.5 (the pro…t made by Trader #0) is the amount of money Trader #1 borrows from Trader #0.

Now Trader #1’sportfolio is: component value 1 = 61:42 stocks $2487: 51 call liability 306:21 borrowed amount 2181 : 3 Net position 0 The portfolio is the same as the portfolio if Trader #1 starts from scratch. Calculation is simpler and cleaner if we start from scratch.

End of Day 1 (or Beginning of Day 2) Mark to market without rebalancing the portfolio Day 2 t = 2=365 T2 = 89=365 S2 = 39:25 C2 = 232:82 194 CHAPTER 13. MARKET-MAKING AND DELTA-HEDGING

Method 1 To cancel out his position, the trader can

buy a call from the market paying C2 = 232:82. The payo¤ of this call  will exactly o¤set the payo¤ of the call sold by the brokerage …rm to the customer. These two call have the common expiration date T1 = 89=365 and the same payo¤. They will cancel each other out.

sell out 1 = 61:420 stocks for 1S2 = 61:420 39:25 = 2410: 735   rh 0:08 1=365 pay o¤ the loan. The payment is (1S1 C1) e = 2181: 3e  =  2181: 778

The trader’spro…t at the end of Day 1 is rh C2 + 1S2 (1S1 C1) e = 232:82 + 2410: 735 2181: 778 = 3: 863 Trader #1 hands in 3: 863 pro…t to his employer and goes home. Method 2 We consider the change between the beginning of Day 1 and the end of Day 1 (or the beginning of Day 2).

In the beginning of Day 1, the trader’sstock is worth 1S1; In the end of  Day 1 (or the beginning of Day 2), the trader’sstock is worth 1S2. The value of the trader’sstocks goes up by 1S2 1S1 In the beginning of Day 1, the call is worth C1; In the end of Day 1 (or the  beginning of Day 2), the call is worth to C2. The call value is the trader’s liability. Now the trader’sliability increases by C2 C1 In the beginning of Day 1, the borrowed amount is 1S1 C1. In the  end of Day 1 (or the beginning of Day 2), this borrowed amount grows rh rh to (1S1 C1) e . The increase is (1S1 C1) e 1 . This is the interest paid on the amount borrowed. No matter the trader borrows money from a bank or uses his own money, the borrowed money needs to earn a risk free interest rate.

The trader’spro…t at the end of Day 1: rh (1S2 1S1) (C2 C1) (1S1 C1) e 1 rh = C2 + 1S2 (1S1 C1) e = 3: 863  Method 3

In the beginning of Day 1, the trader’snet asset is MV (1) = 1S1 C1 = 61:420 40:5 306:21 = 2181: 3 In the end of Day 1 before the trader rebalances his portfolio, the trader’s asset is: BR MV (2) = 1S2 C2 = 61:420 39:25 232:82 = 2177: 915 The trader’spro…t at the end of Day 1 is: BR rh 0:08 1=365 MV (2) MV (1) e = 2177: 915 2181: 3e  = 3: 863 13.2. EXAMPLES OF DELTA HEDGING 195

You can verify that MV BR (2) MV (1) erh rh = (1S2 1S1) (C2 C1) (1S1 C1) e 1 rh = C2 + 1S2 (1S1 C1) e  Method 3 is often faster. Under this method: Pro…t during a day=Asset at the end of Day before rebalancing the portfolio - Future value of the asset at the beginning of the day

Asset at the end of Day before rebalancing the portfolio = delta at the beginning of the day stock price at the end of the day  BR rh Pro…t at the end of Day t = MV (t + 1) MV (t) e = (tSt+1 Ct+1) rh (tSt Ct) e Next, let’s do additional calculations and calculate the pro…t at the end of Day 2, 3, and 4 (or the pro…t at the beginning of Day 3,4,5)

Day Day 2 Day 3 Day 4 Day 5 Time t t = 2=365 t = 3=365 t = 4=365 t = 5=365 Expiry T T2 = 89=365 T3 = 88=365 T4 = 87=365 T5 = 86=365 St S2 = 39:25 S3 = 38:75 S4 = 40 S5 = 40 Ct C2 = 232:82 C3 = 205: 46 C4 = 271:04 C5 = 269:27 t 2 = 53:108 3 = 49: 564 4 = 58:06 5 = 58:01 Pro…t end of Day 0:40 4: 0 1: 32 Pro…t at the end of Day 2 (or beginning of Day 3).

Asset at the beginning of Day 2: MV (2) = 2S2 C2 = 53:108 39:25  232:82 = 1851: 669 

BR Asset at the end of Day 2 before rebalancing: MV (3) = 2S3 C3 =  53:108 38:75 205: 46 = 1852: 475  Pro…t at the end of Day 2: MV BR (3) MV (2) erh = 1852: 475 1851:  669e0:08=365 = 0:400 1

Pro…t at the end of Day 3 (or beginning of Day 4).

Asset at the beginning of Day 3: MV (3) = 3S3 C3 = 49: 564 38:75  205: 46 = 1715: 145 

BR Asset at the end of Day 3 before rebalancing: MV (4) = 3S4 C4 =  49: 564 40 271:04 = 1711: 52  Pro…t at the end of Day 3: MV BR (4) MV (2) erh = 1711: 52 1715:  145e0:08=365 = 4: 000 96

Pro…t at the end of Day 4 (or beginning of Day 5). 196 CHAPTER 13. MARKET-MAKING AND DELTA-HEDGING

Asset at the beginning of Day 4: MV (4) = 4S4 C4 = 58:06 40  271:04 = 2051: 36 

BR Asset at the end of Day 4 before rebalancing: MV (3) = 4S5 C5 =  58:06 40 269:27 = 2053: 13  Pro…t at the end of Day 4: MV BR (5) MV (4) erh = 2053: 13 2051:  36e0:08=365 = 1: 320 3 You can verify that the pro…t at the end of Day 2, 3, 4 calculated above matches Derivatives Markets Table 13.2. However, in Table 13.2, the pro…t at the end of Day 0 is posted in Day 1 column. Similarly, the pro…t at the end of Day 1 is posted in Day 1 column. So on and so forth.

13.3 Textbook Table 13.2

Next, we want to reproduce the textbook Table 13.2.

Day 0 1 2 3 4 5 stock 40 40:5 39:25 38:75 40 40 call 278:04 306:21 232:82 205:46 271:04 269:27 100*delta 58:24 61:42 53:11 49:56 58:06 58:01 investment 2051:58 2181:3 1851:65 1715:12 2051:35 2051:29 interest 0:45 0:48 0:41 0:38 0:45 capital gain 0:95 3:39 0:81 3:62 1:77 daily pro…t 0:5 3:87 0:4 4:0 1:32 Pro…t at the end of Day t (or the beginning of Day t + 1) can be broken down into two parts: = MV BR (t + 1) MV (t) erh = MV BR (t + 1) MV (t) MV (t) erh + MV (t) = MV BR (t + 1) MV (t) MV (t) erh 1 = MV BR (t + 1) MV (t) + MV (t) erh 1  capital gain at end of Day t Interest earned at the end of Day t De…ne| MV BR{z(t + 1) MV} (t) = CapitalGain| {z earned} at the end of Day t CapitalGain (t) is also equal to: BR MV (t + 1) MV (t) = (tSt+1 Ct+1) (tSt Ct) = t (St+1 St) (Ct+1 Ct) De…ne MV (t) erh 1 as the interest earned at the end of Day t. If the trader invests money at the beginning of Day t, then MV (t) is positive and MV (t) erh 1 is negative. The negative interest earned is just the interest expense incurred by the trader.  De…ne MV (t) as the investment made at the beginning of Day t. If MV (t) is negative, then it means that the trader receives money.