MACF1 Introduction to Quantitative Finance Slides - Chapter 1 PDF

MACF1 Introduction to Quantitative Finance Slides - Chapter 1 Y. Lu Concordia University Fall 2022 Y. Lu (2022) MACF1 Fall 2022 1 / 122  Introduction Motivation I...

MACF1 Introduction to Quantitative Finance Slides - Chapter 1 Y. Lu Concordia University Fall 2022 Y. Lu (2022) MACF1 Fall 2022 1 / 122  Introduction Motivation In this chapter, we will introduce financial derivatives. Such products are extensively used by financial institutions to manage their risks. Understanding their behavior requires studying some quantitative financial models. Supplemental reading: Derivative markets, chapters 2, 3,5 Y. Lu (2022) MACF1 Fall 2022 2 / 122  Introducing derivatives Derivatives: a definition Definition A financial derivative (or simply derivative) is a financial contract between two institutions whose value is determined by the value of another asset called the underlying asset. In this course, we still study two main types of derivatives: Forwards & futures, Options. Y. Lu (2022) MACF1 Fall 2022 3 / 122  Introducing derivatives Forward contracts Forward contracts Definition A forward contract is an agreement between two institutions which forces the first to buy a given asset (the underlying asset) from the second at a pre-determined date and price. The first institution (buying) has a long position on the underlying asset. The second institution has a short position on the underlying asset. Pre-determined price: strike price Pre-determined date: maturity date. Y. Lu (2022) MACF1 Fall 2022 4 / 122  Introducing derivatives Forward contracts Underlying asset classes The underlying asset can be from several asset classes, for instance: Equity (stocks, financial indices, ETFs), Fixed income (T-Bills, corporate bonds), Commodities (natural gas, oil, electricity, wheat, metals), Foreign currencies (FX). Y. Lu (2022) MACF1 Fall 2022 5 / 122  Introducing derivatives Forward contracts Forward contract example Example An institution A has agreed with an institution B to buy from the latter 5000 bushels of corn on December 31, 2018 for 25,000$. Strike price: Maturity date: Underlying asset: Holder of the long position: Holder of the short position: Y. Lu (2022) MACF1 Fall 2022 6 / 122  Introducing derivatives Forward contracts Settlement in forward contracts Remark The settlement of a forward contract can be done in two different ways by the short position: Physical: the underlying asset is delivered at maturity. Financial: an amount of money ST $ is paid at maturity. Y. Lu (2022) MACF1 Fall 2022 7 / 122  Introducing derivatives Forward contracts Settlement in forward contracts Remark The settlement of a forward contract can be done in two different ways by the short position: Physical: the underlying asset is delivered at maturity. Financial: an amount of money ST $ is paid at maturity. Financial settlement is much more common (banks typically do not have the expertise to store corns, oil, etc). Y. Lu (2022) MACF1 Fall 2022 7 / 122  Introducing derivatives Forward contracts Payoff Notation: K : strike price, T : maturity date, St : underlying asset price at time t. Definition The forward contract payoff is: Position Payoff Long ST − K Short ? Y. Lu (2022) MACF1 Fall 2022 8 / 122  Introducing derivatives Forward contracts Payoff Figure: Forward contract payoff Y. Lu (2022) MACF1 Fall 2022 9 / 122  Introducing derivatives Forward contracts Strike price determination Question: how is the strike price K determined? This looks like a subjective choice. However, under some assumptions, we will see there is only a single sensible choice. To determine this value, we must first discuss concepts of arbitrage and short sales. Y. Lu (2022) MACF1 Fall 2022 10 / 122  Introducing derivatives Forward contracts Short sale Short sale: consists in selling an asset we don’t currently possess. The asset must therefore temporarily be borrowed from a third party. At the short sale maturity, the asset should be returned to the third party (by being first purchased on the market at that date). If some dividends are paid by the underlying asset during the duration of the short sale: they must also be reimbursed to the third party from which the asset was borrowed (at the time dividends are paid). Y. Lu (2022) MACF1 Fall 2022 11 / 122  Introducing derivatives Forward contracts Short sale cash flow: at the onset Figure: Cash flows at the short sale initiation Y. Lu (2022) MACF1 Fall 2022 12 / 122  Introducing derivatives Forward contracts Short sale: at maturity Figure: Cash flows at the short sale maturity Y. Lu (2022) MACF1 Fall 2022 13 / 122  Introducing derivatives Forward contracts Self-financing portfolio Definition A portfolio is self-financing if there is no cash flow injection in nor withdrawals from the portfolio, except at the initial date where it is set up. In a self-financing portfolio, the purchase of assets is funded directly within the portfolio: no liquidity is injected in nor withdrawn from the portfolio to make these purchases or sales. Y. Lu (2022) MACF1 Fall 2022 14 / 122  Introducing derivatives Forward contracts Arbitrage Definition An arbitrage opportunity is a self-financing portfolio whose initial value is null (V0 = 0) and whose terminal value VT satisfies the two following conditions: P[VT ≥ 0] = 1, P[VT > 0] > 0. An arbitrage opportunity is an opportunity to make a possible profit without taking any risk nor disbursing any amount of money. Y. Lu (2022) MACF1 Fall 2022 15 / 122  Introducing derivatives Forward contracts Zero-valued portfolio example Example of a portfolio with a null value: Borrow 100$ from the bank. Loan value = −100$.1 Purchase 100$ of stocks. Stocks value = 100$. The total portfolio value is 100$ − 100$ = 0$. 1 The negative amount corresponds to a debt toward the bank. Y. Lu (2022) MACF1 Fall 2022 16 / 122  Introducing derivatives Forward contracts Absence of arbitrage General assumption in derivatives pricing schemes: absence of arbitrage opportunities. Rationale: arbitrage opportunities are unstable and temporary. They are neutralized as soon as investors start exploiting them. How? As soon as investors purchase the arbitrage portfolio, the price of assets from this portfolio will increase (demand increases). The initial arbitrage portfolio value V0 will not be null any more. Y. Lu (2022) MACF1 Fall 2022 17 / 122  Introducing derivatives Forward contracts Underlying asset storability The following assumptions imply that the forward strike price is unique. Assumption There are no arbitrage opportunities on the market. Assumption The underlying asset is storable without cost, i.e. it is possible to purchase/sell it any time and to store without cost for any amount of time. Y. Lu (2022) MACF1 Fall 2022 18 / 122  Introducing derivatives Forward contracts Forward price Let Z (t, T ) be the time-t price of a risk-free zero-coupon bond (or saving account) of maturity T.2 In other words, you save Z (t, T ) at time t to receive 1$ at time T. Theorem Suppose that the underlying asset pays no dividends. Then, the only possible strike price for a forward contract entered at time t is St K=. (1) Z (t, T ) 2 A bond which pays 1$ at the maturity date T with certainty. Y. Lu (2022) MACF1 Fall 2022 19 / 122  Introducing derivatives Forward contracts Two (equivalent) proofs are possible: By looking the forward as a (leveraged) purchase, by differing the payment to T By non-arbitrage arguments Y. Lu (2022) MACF1 Fall 2022 20 / 122  Introducing derivatives Forward contracts Proof 1: forward as a leveraged purchase K is the money to pay at T to receive the underlying at T , worth ST at that time T. Alternatively, we can also pay St immediately. It is clear that K should be larger than St because of the interest. More precisely, K should equal to the amount one receives, if one St saves St at t and takes the saving out at time T , hence K = Z (t,T ). Y. Lu (2022) MACF1 Fall 2022 21 / 122  Introducing derivatives Forward contracts Proof 2 (by non-arbitrage argument) Proof: Case 1: Suppose it is possible to enter a forward contract at time t where St K<. (2) Z (t, T ) Then there exists an arbitrage opportunity which consists in combining the following actions at time t: enter a long position in the forward contract, short one underlying asset share at time t, St long Z (t,T ) units of zero-coupons. Y. Lu (2022) MACF1 Fall 2022 22 / 122  Introducing derivatives Forward contracts Proof cont. Cash flows received by the institution at times t and T : Position Cash inflow at t Cash inflow at T Long on forward 0 ST − K Short on the underlying asset St −ST St St Long on zero-coupon − Z (t,T ) Z (t, T ) Z (t,T ) St Total 0 −K + Z (t,T ) Arbitrage, because Vt = 0 and P[VT > 0] = 1. Y. Lu (2022) MACF1 Fall 2022 23 / 122  Introducing derivatives Forward contracts Proof cont. Case 2: Similarly, if St K>. (3) Z (t, T ) Then replace all the previous long positions by short ones, and all the previous short positions by long positions, we arrive at St K− Z (t, T ) which is positive. Contradiction. Y. Lu (2022) MACF1 Fall 2022 24 / 122  Introducing derivatives Forward contracts Forward price Definition The unique strike price of a forward contract under the assumptions of a storable underlying asset, of the absence of dividends and of absence of St arbitrage is called the forward price and is denoted by Ft,T = Z (t,T ). Y. Lu (2022) MACF1 Fall 2022 25 / 122  Introducing derivatives Forward contracts Storability assumption Remark The storability without fees assumption of the underlying asset is necessary for the previous theorem to hold. Indeed, the proof relies on purchasing the underlying asset at time t and holding on it without fees until time T. If the underlying asset is not storable, it is impossible to implement the St arbitrage strategy presented in the proof when K 6= Z (t,T ). Y. Lu (2022) MACF1 Fall 2022 26 / 122  Introducing derivatives Forward contracts Absence of arbitrage and forward price In summary: the absence of arbitrage assumption is sufficient to determine the forward contract price in a unique way. Y. Lu (2022) MACF1 Fall 2022 27 / 122  Introducing derivatives Forward contracts Assumptions Multiple assumptions were implicitly made in the proof of the previous theorem: the underlying asset does not pay dividends, no transaction costs (frictionless market), no bid-ask spread (perfect liquidity), no taxes, possibility to buy/sell fractions of shares, no storage cost for the underlying asset, no constraints on short sales. no dividend Such assumptions are not realistic in practice. The previous theorem is an approximation of the reality. Y. Lu (2022) MACF1 Fall 2022 28 / 122  Introducing derivatives Forward contracts Forward with an underlying dividend-paying stock In the case of a dividend-paying stock, the strike price will be lower than St Z (t,T ). Indeed, it is equal to: St − FV (Div ) Z (t, T ) FV (Div ) is the accumulated value of the dividend(s) at maturity date T. Y. Lu (2022) MACF1 Fall 2022 29 / 122  Introducing derivatives Forward contracts Example Stock X is currently priced at 30 per share and the company has announced that it will pay a dividend of 0.3 per share after 1 month and 3 months. The annual continuously compounded risk free interest rate is 5%. Calculate the strike price of a forward with maturity: after 2 months after 4 months Y. Lu (2022) MACF1 Fall 2022 30 / 122  Introducing derivatives Forward contracts Solution There is only 1 dividend between now and t + 2months. Thus 2 1 K1 = 30e 0.05 12 − 0.3e 0.05 12 There are two dividends between now and t + 4months. Thus 2 3 1 K2 = 30e 0.05 12 − 0.3e 0.05 12 − 0.3e 0.05 12 Y. Lu (2022) MACF1 Fall 2022 31 / 122  Introducing derivatives Forward contracts Prepaid forward contact A prepaid forward contact is a contract where the strike is paid at the inception of the contract instead of at maturity. Since paying a certain amount A at maturity T is equivalent to paying pre AZ (t, T ) at time t, the time-t prepaid forward price Ft,T for a contract maturing at T is 3 pre Ft,T = Z (t, T )Ft,T. 3 pre St if the underlying asset pays no dividend, Ft,T = Z (t, T ) Z (t,T ) = St. Y. Lu (2022) MACF1 Fall 2022 32 / 122  Introducing derivatives Future contracts Futures contract A futures contract is similar to a forward contract, but with some slight differences: They are more standardized, and negotiated at exchanges. They are settled daily throughout the whole duration of the contract, instead of only at maturity. They are more liquid. There is less (counterparty) credit risk. Y. Lu (2022) MACF1 Fall 2022 33 / 122  Introducing derivatives Future contracts Futures contracts Let us now make the assumption of a constant continuously compounded interest rate r i.e. Z (t, T ) = e −r (T −t). A futures contract has, similarly to a forward contract, an underlying asset S and a maturity date T. Denote by F̄t,T a quantity called the futures price at time t. For now, we consider the quantity as given. The mechanism to determine this quantity is described later. Y. Lu (2022) MACF1 Fall 2022 34 / 122  Introducing derivatives Future contracts Futures contract cash flows Futures contracts imply cash flows occurring throughout the whole duration of the contract, starting from t0 + 1, where t0 is the date of inception of the contract. Thereafter, for all subsequent times t such that t0 ≤ t < T , the holder of a long position in the futures receives F̄t+1,T − F̄t,T at time t + 1 from the counterparty. If this amount is negative, the amount F̄t,T − F̄t+1,T must to paid to the counterparty by the holder of the long position. Called marking to market. Y. Lu (2022) MACF1 Fall 2022 35 / 122  Introducing derivatives Future contracts Futures contract cash flows Moreover, at maturity, the holder of the long position receives the underlying from the short position who will be paid the amount F̄T ,T in exchange. Thus, the holder of the long position will receive in total T X (ST − F̄T ,T ) + (F̄j,T − F̄j−1,T ) = ST − F̄T ,T + (F̄T ,T − F̄t0 ,T ) j=t0 +1 = ST − F̄t0 ,T. between times t and T. Note: it will be shown later on that ST − F̄T ,T = 0. Thus the cash flow at maturity is (F̄T ,T − F̄T −1,T ). We see the analogy with the forward contract which provides a payoff of ST − Ft0 ,T to the long position where Ft0 ,T if the forward price at time t0. Y. Lu (2022) MACF1 Fall 2022 36 / 122  Introducing derivatives Future contracts Accrual of interest of futures contract cash flows However, cash flows generated by a futures contract are not received all at the same time. Interest is accrued on cash flows received before maturity. Thus, the time-T payoff of a long position on a futures entered at time t0 is XT (ST − F̄T ,T ) + (F̄j,T − F̄j−1,T )e r (T −j). j=t0 +1 Y. Lu (2022) MACF1 Fall 2022 37 / 122  Introducing derivatives Future contracts Closing a futures position In order to “close” a position on a futures, it is possible to enter an inverse position on the same contract. at an ulterior time tF. Thereby, cash flows associated with the initial position and the new one will offset each other after time tF. For instance, an entity taking a long position on a futures at time t0 and closing it at time tF will receive the net amounts (F̄j,T − F̄j−1,T ) at respective times j = t0 + 1,... , tF , no cash flow after tF. Y. Lu (2022) MACF1 Fall 2022 38 / 122  Introducing derivatives Future contracts Margin account The margin account of each participant is the account from which money is taken/in which money is deposited by the clearinghouse4 for transfers during the lie of the futures contract. Each trading firm has his own margin account. The margin balance is the current balance of the margin account (after all money transfer due to previous price movements were made). 4 The clearinghouse is the entity responsible for the money transfers between the two parties Y. Lu (2022) MACF1 Fall 2022 39 / 122  Introducing derivatives Future contracts Margin calls In practice, the clearinghouse must ensure a minimal amount of money (called the maintenance margin) is always present in the margin account at all times. If the margin balance of a participant falls below the maintenance margin, the clearinghouse can issue a margin call: Forces the participant to put additional funds in the margin account. If the participant is unwilling/unable to provide such funds, his futures positions are closed. Y. Lu (2022) MACF1 Fall 2022 40 / 122  Introducing derivatives Future contracts Y. Lu (2022) MACF1 Fall 2022 41 / 122  Introducing derivatives Future contracts Futures contract pricing What are the possible values for the futures price which don’t generate arbitrage opportunities? The following shows that under a constant interest rate, the only sensible futures price is the forward price. Theorem Suppose that the continuously compounded rate r is constant, and we can store the underlying asset without cost. Then, the only futures price that does not generate arbitrage opportunities for a futures on the underlying asset S maturing at time T is F̄t,T = Ft,T = St e r (T −t). for all t = 0,... , T. Y. Lu (2022) MACF1 Fall 2022 42 / 122  Introducing derivatives Future contracts Futures contract pricing Proof: We proceed by (backward) induction, for integer values of T. First, we must have F̄T ,T = ST since the only cash flow associated to a long position entered at time T is ST − F̄T ,T. Y. Lu (2022) MACF1 Fall 2022 43 / 122  Introducing derivatives Future contracts Futures contract pricing Then, assume that F̄j,T = St e r (T −j) for all j = t + 1,... , T. We wish to show that this implies F̄t,T = St e r (T −t). Consider the following operations t and t + 1: Enter a long position in the futures at t and close the position at t + 1, Invest e r (T −t−1) St in the bank at time t and withdraw the amount (with interests) at t + 1, Sell short e r (T −t−1) shares of the underlying asset at t and give back the underlying asset at t + 1. Y. Lu (2022) MACF1 Fall 2022 44 / 122  Introducing derivatives Future contracts Futures contract pricing Cash flows associated with these operations are Table: Cash flows related to the previous operations Position Cash inflow at t Cash inflow at t + 1 Long on futures 0 F̄t+1,T − F̄t,T Short on underlying asset e r (T −t−1) St −e r (T −t−1) St+1 Long on bank account −e r (T −t−1) St e r (T −t) St Total 0 F̄t+1,T − F̄t,T −e r (T −t−1) St+1 +e r (T −t) St Y. Lu (2022) MACF1 Fall 2022 45 / 122  Introducing derivatives Future contracts Futures contract pricing The final cash flow is F̄t+1,T − F̄t,T − e r (T −t−1) St+1 + e r (T −t) St = e r (T −t−1) St+1 − F̄t,T − e r (T −t−1) St+1 + e r (T −t) St (by induction hypothesis) = −F̄t,T + e r (T −t) St. This cash flow value is known at time t. Y. Lu (2022) MACF1 Fall 2022 46 / 122  Introducing derivatives Future contracts Futures contract pricing Since the time-t cash flow is null, the cash flow at t + 1 must also be null. Otherwise this would be an arbitrage opportunity. This implies F̄t,T = e r (T −t) St.  Y. Lu (2022) MACF1 Fall 2022 47 / 122  Introducing derivatives Future contracts Futures contract pricing When interest rates are random, it is more difficult to determine the possible prices F̄t,T for the futures which do not lead to arbitrage. If the interest rate is positively correlated with the underlying asset, the long futures position cash flows will be positively correlated with the interest rate. ⇒ For a long position, cash flows received will be associated with a larger accumulation of interest than the negative cash flows. The holder of the long position will prefer futures to the forward contract (at a same price), since he will want to receive its gains before maturity to benefit the highest interest income. Y. Lu (2022) MACF1 Fall 2022 48 / 122  Introducing derivatives Future contracts Futures contract pricing Conversely, the holder of a short position would prefer a forward contract to a futures since received cash flows would be associated with a lowest interest accumulation. The futures price should be higher than the forward price in this case: to compensate the short futures position holder. Similarly, if interest rates are negatively correlated to the underlying asset price, the futures price will tend to be inferior to the forward price. Y. Lu (2022) MACF1 Fall 2022 49 / 122  Introducing derivatives Future contracts Illustration of the cash flows (long position holder) F̄3,T F̄2,T F̄T ,T = ST F̄1,T t=0 1 2 3 T (maturity) F̄1,T F̄T −1,T F̄0,T F̄2,T Sum of cash flow is ST − F̄0,T because all intermediate terms cancel out. Y. Lu (2022) MACF1 Fall 2022 50 / 122  Introducing derivatives Standard options Option definition Definition An option is an agreement between two parties giving the first party the right to purchase from (or sell to) the second party an asset at a pre-determined datea and a pre-determined price. a or some date within a pre-determined set of dates Y. Lu (2022) MACF1 Fall 2022 51 / 122  Introducing derivatives Standard options Option definition Definition The first party, the option buyer, holds a long position on the option. The second party is the option seller, which holds a short position on the option. Strike price: pre-determined price. Maturity date: pre-determined date (or last date among dates where the exercise is possible). Call option: option allowing to buy the underlying asset. Put option: option allowing to sell the underlying asset. Y. Lu (2022) MACF1 Fall 2022 52 / 122  Introducing derivatives Standard options Option definition Definition European option: only allows the purchase (or the sale) at a single date. American option: allows the purchase (or the sale) at any date until the maturity. Definition For a call, option is in-the-money if St > K and out-of-the-money if St < K. For a put, option is in-the-money if St < K and out-of-the-money if St > K. If St = K , the option (call or put) is at-the-money. Y. Lu (2022) MACF1 Fall 2022 53 / 122  Introducing derivatives Standard options Option example Example An institution A has purchased an option allowing it to purchase a company X stock from institution B on January 1st, 2025 at the price of 110$. The underlying asset price is currently 100$ on the market. Option type: European call option, Strike price: 110$ (out-of-the-money option), Maturity date: January 1st, 2025, Underlying asset: a company X stock, Long position holder (option buyer): institution A, Short position holder (option seller): institution B. Y. Lu (2022) MACF1 Fall 2022 54 / 122  Introducing derivatives Standard options Option price The option buyer must pay an amount of money to the option seller at the time of the transaction to hold the option.5 Such an amount of money is called the option price, or the option premium. 5 Contrarily to the forward/futures contracts which do not lead to immediate cash flows at the time of the transaction. Y. Lu (2022) MACF1 Fall 2022 55 / 122  Introducing derivatives Standard options Option buyer cash flow Figure: Option buyer cash flow Y. Lu (2022) MACF1 Fall 2022 56 / 122  Introducing derivatives Standard options Call option payoff At the maturity T of the call option, if ST > K , the option buyer will exercise it: he will receive ST − K. Conversely, if ST ≤ K , the option buyer will let the option expire without exercising it and will not receive anything. Definition The payoff for the long position on the European call option is: ( ST − K if ST > K , max(0, ST − K ) = 0 otherwise. Y. Lu (2022) MACF1 Fall 2022 57 / 122  Introducing derivatives Standard options Insurance as a call option Insurance can be viewed as a put option “Underlying” is the claim amount “Strike” is the deductible Y. Lu (2022) MACF1 Fall 2022 58 / 122  Introducing derivatives Standard options Put option payoff At the maturity T of the put option, if K > ST , the option buy will exercise it and will receive K − ST. Conversely, if K ≤ ST , the option buyer will let the option expire without exercising it and will not receive anything. Definition The payoff of the long position on the European put option is: ( K − ST if K > ST , max(0, K − ST ) = 0 otherwise. Y. Lu (2022) MACF1 Fall 2022 59 / 122  Introducing derivatives Standard options American option payoff We can also define the payoff for American options. Definition Let τ be the time at which the option is exercised. The payoff for the long position on the American option is:a ( (Sτ − K )1{τ ≤T } for the call option, (K − Sτ )1{τ ≤T } for the put option. a The indicator 1E is worth 1 if the event E occurs, or zero otherwise. Remark For an American option, the payoff is received at the time of the exercise of the option (at τ ). Y. Lu (2022) MACF1 Fall 2022 60 / 122  Introducing derivatives Standard options Option payoff Figure: Payoff for the option buyer Y. Lu (2022) MACF1 Fall 2022 61 / 122  Introducing derivatives Exotic options Exotic options Exotic options are options that are similar to previously seen standard options, but with a few differences. We will see several examples of exotic options. Note: standard options are often called vanilla options. Y. Lu (2022) MACF1 Fall 2022 62 / 122  Introducing derivatives Exotic options Asian options Asian options (or average options): options whose payoff depends on the average value of the underlying asset price over a pre-definite time period, denoted S (avg ). Here are a few examples: Type Call option Put option Average price max(0, S (avg ) − K ) max(0, K − S (avg ) ) Average strike max(0, ST − S (avg ) ) max(0, S (avg ) − ST ) Y. Lu (2022) MACF1 Fall 2022 63 / 122  Introducing derivatives Exotic options Asian options Two approaches for calculating S (avg ) : arithmetic method or geometric method. Suppose that the (non-negative) underlying asset prices at the average calculation dates are denoted St1 ,... , StN. Then, N 1 X Arithmetic average: S (avg ) = S (ari) = Stn , N n=1 N !1/N Y Geometric average: S (avg ) = S (geo) = Stn. n=1 Y. Lu (2022) MACF1 Fall 2022 64 / 122  Introducing derivatives Exotic options Compound options Compound options: an option on option, i.e. an option whose underlying asset is another option whose maturity is ulterior to the maturity of the compound option. Y. Lu (2022) MACF1 Fall 2022 65 / 122  Introducing derivatives Exotic options Lookback option Lookback option: option whose payoff depends on the extremum (either the minimum or the maximum) of the underlying asset price over some pre-determined time period. Example of payoffs: Standard lookback call: ST − min{St | t ≤ T }, Standard lookback put: max{St | t ≤ T } − ST , Extrema lookback call: max (0, max{St | t ≤ T } − K ) , Extrema lookback put: max (0, K − min{St | t ≤ T }). Y. Lu (2022) MACF1 Fall 2022 66 / 122  Introducing derivatives Exotic options Shout option Shout option: option where the holder can lock a minimum value for a payoff at a time τ < T before maturity. Assume at time τ the holder ’shouts’ i.e. locks the option payoff. Then the time-T payoff is max(0, ST − K , Sτ − K ) (call option), max(0, K − ST , K − Sτ ) (put option). If the holder never locks the payoff before maturity, the payoff is the same than for vanilla options. Y. Lu (2022) MACF1 Fall 2022 67 / 122  Introducing derivatives Exotic options Forward start option Forward start option: option where the strike is unknown at the onset and set at the underlying asset price at a predetermined ulterior date. Let T1 < T be the fixing date. Then the time-T payoff is max(0, ST − ST1 ) (call option), max(0, ST1 − ST ) (put option). Y. Lu (2022) MACF1 Fall 2022 68 / 122  Introducing derivatives Exotic options Barrier options Barrier option: option where the payment of the payoff is conditional on a certain threshold (called the barrier) being reached by the underlying asset price during the life of the option. Knock-in option: provides a payoff only if the barrier has been reached by the underlying asset. Knock-out option: provides a payoff only if the barrier has not been reached by the underlying asset. Example: an option whose payoff is as follows is called a down-and-out barrier call option: max(0, ST − K1 )1{min{St | t≤T }>K2 }. where K1 > K2 are constants. Y. Lu (2022) MACF1 Fall 2022 69 / 122  Introducing derivatives Exotic options Exchange options Exchange option: option allowing to exchange an asset S (1) for another asset S (2). Example: a European call exchange option on asset S (2) with strike asset S (1) has the payoff   (2) (1) max 0, ST − ST. The option pays the time-T shortfall in value of asset S (1) versus asset (2) ST , if any. Y. Lu (2022) MACF1 Fall 2022 70 / 122  Introducing derivatives Exotic options Rainbow options Rainbow option: option whose underlying asset depends on multiple securities. Let’s say we have M risky assets S (1) ,... , S (M). Then one could define   (1) (M) St := f St ,... , St for some function f , such as a weighted average.6 A European rainbow call option would therefore be an option with payoff max (0, ST − K ). Another example: Quanto option: underlying asset is denominated in one currency, and settled in another. 6 In this case the option is also referred to as a basket option. Y. Lu (2022) MACF1 Fall 2022 71 / 122  Introducing derivatives Option strategies and applications Use of derivatives Derivatives can be used for many purposes: Risk management: allowing institutions protecting themselves by offsetting risks they face related to certain possible financial outcomes. Speculation: derivatives can be used as potential profitable investments and allow making bets on market movements. Arbitrage: allowing the exploitation of market price anomalies so as to obtain a profit with a null (or limited) risk. We shall now illustrate several strategies involving options in the context of speculation and risk management. Y. Lu (2022) MACF1 Fall 2022 72 / 122  Introducing derivatives Option strategies and applications Examples of option strategies Here, we assume all options are European and have the same underlying asset and maturity. Example 1: Call bull spread: purchase of a call option with strike K1 , sale of a call option with strike K2 where K2 > K1. Allows speculating on an increase of the underlying asset price. The idea: a call option offers potentially unlimited profit and no loss, but has a high cost We can lower the cost of your strategy if we are willing to reduce your profit should the stock appreciate Maximal profit of a call bull spread: K2 − K1. Y. Lu (2022) MACF1 Fall 2022 73 / 122  Introducing derivatives Option strategies and applications Example cont. Example 2: Put bear spread: purchase of a put option with strike K2 , sale of a put option with strike K1 where K1 < K2. Similarly as a call bull spread, a put bear spread allows speculating on a drop of the underlying asset price. Y. Lu (2022) MACF1 Fall 2022 74 / 122  Introducing derivatives Option strategies and applications Examples cont. Example 3: Butterfly spread: purchase of a call option with strike K1 , sale of two call options with strike K2 , purchase of a call option with strike K3 with K1 < K2 < K3 and K3 − K2 = K2 − K1. Allows speculating on a weak volatility on the market. Y. Lu (2022) MACF1 Fall 2022 75 / 122  Introducing derivatives Option strategies and applications Examples cont. Example 4: Ratio spread: purchase of a call option with strike K1 , sale of m > 1 call options with strike K2 , with K1 < K2. Also allows speculating on a weak volatility of the market. For this strategy, the payoff is however not bounded below. Y. Lu (2022) MACF1 Fall 2022 76 / 122  Introducing derivatives Option strategies and applications Examples cont. Example 5: Strangle: purchase of a put option with strike K1 , purchase of a call option with strike K2 with K1 < K2. Allows speculating on a high volatility of the market. When K1 = K2 , this strategy is called a Straddle. Y. Lu (2022) MACF1 Fall 2022 77 / 122  Introducing derivatives Option strategies and applications Payoff for various option strategies Figure: Payoff for various option strategies Y. Lu (2022) MACF1 Fall 2022 78 / 122  Introducing derivatives Option strategies and applications Protective put Protective put: combined purchase of the underlying asset and of a put option (for protection purposes). I Thus, the combined payoff of the option and of the underlying asset will be greater or equal to the option strike price. Y. Lu (2022) MACF1 Fall 2022 79 / 122  Introducing derivatives Option strategies and applications Collar Collar: combined purchase of a put option with strike K1 and sale of a call option with strike K2 such that K2 > K1. I If combined with a purchase of the underlying asset, the combined payoff will be bounded between K1 et K2. Advantage of a Collar with respect to the Protective put: for a Collar, we can use the premium received from the sale of the call option to fund the purchase of the put option. Y. Lu (2022) MACF1 Fall 2022 80 / 122  Introducing derivatives Option strategies and applications Profit versus payoff For the aforementioned strategies, payoffs of the portfolios are non-negative. However, these are not arbitrage opportunities: their initial cost is non-null. Even if the payoff is positive, the profit is not necessarily positive (because of the initial payments). Y. Lu (2022) MACF1 Fall 2022 81 / 122  Introducing derivatives Option strategies and applications Profit Definition The profit of an option transaction is the future value at the option maturity date of all cash flows generated by the transaction. The interest rate used to compute the future value is the risk-free rate. Y. Lu (2022) MACF1 Fall 2022 82 / 122  Introducing derivatives Option strategies and applications Profit calculation example Example Bob purchases a European call option. Strike: 105$. Maturity: in three years. The underlying asset is worth 100$ today. The option premium C0 is 5$. No dividends. The continuously compounded risk-free rate r is 2% per annum. What is the profit generated by the option purchase if the underlying asset price becomes 120$ in three years? What is the profit is the underlying asset price becomes 100$ in 3 years? Y. Lu (2022) MACF1 Fall 2022 83 / 122  Introducing derivatives Option strategies and applications Profit calculation example Answer: Profit is the future value (FV) of all cash flows at the option maturity date accrued at the risk-free rate: Profit = FV payoff - FV option premium = max(0, ST − K ) − C0 e rT If the underlying asset price becomes 120$, profits are max(0, ST − K ) − C0 e rT = max(0, 120 − 105) − 5e 0.02×3 = 9.69$. If the underlying asset price becomes 100$, profits are max(0, ST − K ) − C0 e rT = max(0, 100 − 105) − 5e 0.02×3 = −5.31$, i.e. Bob incurs a loss even if his payoff is non-negative. Y. Lu (2022) MACF1 Fall 2022 84 / 122  Introducing derivatives Bounds on option prices Arbitrage and options We want to determine the price of an option. Assumption We suppose once again that there exists no arbitrage opportunity on the market. The absence of arbitrage opportunities assumptions allows determining some bounds on option prices. Y. Lu (2022) MACF1 Fall 2022 85 / 122  Introducing derivatives Bounds on option prices Bank account asset Assumption Existence of a risk-free asset in which we can invest or borrow at the continuously compounded risk-free rate rt > 0 at time t. rt can be deterministic or random. The time-t value of this asset is Z t  Bt = exp rs ds. 0 The asset B can be seen as a bank account (or a credit margin when we have a negative number of shares). Because, rt > 0, t > s implies that Bt > Bs. Y. Lu (2022) MACF1 Fall 2022 86 / 122  Introducing derivatives Bounds on option prices Price at maturity versus payoff The absence of arbitrage leads to multiple general results: Theorem Consider an asset whose time-t price is a(t) and whose payoff at maturity T is ΨT. Then, a(T ) = ΨT. The price at maturity of a derivative is equal to its payoff. Y. Lu (2022) MACF1 Fall 2022 87 / 122  Introducing derivatives Bounds on option prices Price monotonicity Theorem (Monotonicity) Consider two derivativesa of the European type having the same maturity T , of respective time-t prices u(t) and a(t), and whose payoffs (u) (a) are respectively ΨT and ΨT. (u) (a) If P[ΨT ≥ ΨT ] = 1, then u(t) ≥ a(t). a or derivatives portfolios Y. Lu (2022) MACF1 Fall 2022 88 / 122  Introducing derivatives Bounds on option prices Price monotonicity Proof: Suppose u(t) < a(t). Then there exists an arbitrage opportunity which consists in performing the following actions: buy asset u and sell asset a at time t, invest the sales proceeds a(t) − u(t) in the risk-free asset.7 The profit of these transactions whose net initial cost is null is (u) (a) (a(t) − u(t)) BBTt + ΨT − ΨT > 0.  7 a(t)−u(t) We purchase Bt shares of asset B at time t. Y. Lu (2022) MACF1 Fall 2022 89 / 122  Introducing derivatives Bounds on option prices Price monotonicity: American options Theorem (Monotonicity) Consider two derivativesa of the American type having the same maturity T where: prices are respectively denoted U(t) and A(t), (U) (A) payoffs are respectively Ψτ and Ψτ if the exercise occurs at time τ. (U) (A) If ∀s ≥ t, P[Ψs ≥ Ψs ] = 1, then U(t) ≥ A(t). a or derivatives portfolios Y. Lu (2022) MACF1 Fall 2022 90 / 122  Introducing derivatives Bounds on option prices Price monotonicity: American options Proof: Suppose U(t) < A(t). Then there exists an arbitrage opportunity which consists in performing the following actions: buy asset U and sell asset A at time t, invest the sales proceeds A(t) − U(t) in the risk-free asset, if option U is exercised by the option buyer at time τ , exercise the option A simultaneously. In this case, invest the difference of payoffs is the risk-free asset. The riskless profit of these transactions whose net initial cost is null is BT BT (A(t) − U(t)) + 1{τ 0 a.s.  Bt Bτ Y. Lu (2022) MACF1 Fall 2022 91 / 122  Introducing derivatives Bounds on option prices Law of one price Previous result have the following direct consequence: Corollary (Law of one price) Consider two derivativesa of the same type (e.g. American or European) having the same maturity date and payoff. Then, their prices are equal. a or two derivatives portfolios Y. Lu (2022) MACF1 Fall 2022 92 / 122  Introducing derivatives Bounds on option prices Notation for options We will now apply previous results to deduce implications on option prices. We use the following notation T is the maturity date, K is the strike price and all options have the same underlying asset: C (t, T , K ) is the time-t price of an American call option, c(t, T , K ) is the time-t price of a European call option P(t, T , K ) is the time-t price of an American put option p(t, T , K ) is the time-t price of a European put option Z (t, T ) is the time-t price of a zero-coupon bond maturing (i.e. which pays 1$) at time T. Y. Lu (2022) MACF1 Fall 2022 93 / 122  Introducing derivatives Bounds on option prices Underlying asset storability Once again, several results we will derive (bounds on option prices) require the following assumption. Assumption We suppose that the underlying asset is storable without cost, i.e. that we can buy it and hold it for any amount of time without having to pay fees. Y. Lu (2022) MACF1 Fall 2022 94 / 122  Introducing derivatives Bounds on option prices Option monotonicity with respect to strike Proposition Let K1 ≤ K2. C (t, T , K1 ) ≥ C (t, T , K2 ), c(t, T , K1 ) ≥ c(t, T , K2 ), P(t, T , K1 ) ≤ P(t, T , K2 ), p(t, T , K1 ) ≤ p(t, T , K2 ). Proof: These result directly result of the previous payoff monotonicity theorems and inequalities ST − K1 ≥ ST − K2 , max(0, ST − K1 ) ≥ max(0, ST − K2 ), K1 − ST ≤ K2 − ST , max(0, K1 − ST ) ≤ max(0, K2 − ST ).  Y. Lu (2022) MACF1 Fall 2022 95 / 122  Introducing derivatives Bounds on option prices Option price bounds Proposition Assume the underlying asset S not pay dividends. Then, St ≥ C (t, T , K ) ≥ c(t, T , K ) ≥ max(0, St − KZ (t, T )) ≥ 0, K ≥ P(t, T , K ) ≥ p(t, T , K ) ≥ max(0, KZ (t, T ) − St ) ≥ 0. Y. Lu (2022) MACF1 Fall 2022 96 / 122  Introducing derivatives Bounds on option prices Proof for option price bounds First inequality (St ≥ C (t, T , K )): The price monotonicity theorem for American options and Sτ ≥ max(0, Sτ − K ) imply that St ≥ C (t, T , K ). Y. Lu (2022) MACF1 Fall 2022 97 / 122  Introducing derivatives Bounds on option prices Proof for option price bounds Second inequality (C (t, T , K ) ≥ c(t, T , K )): To show that C (t, T , K ) ≥ c(t, T , K ), we notice that if it isn’t the case (i.e. if C (t, T , K ) < c(t, T , K )), there exists an arbitrage opportunity which consists in the following steps: at t, buy the American option and sell the European one, invest c(t, T , K ) − C (t, T , K ) at the risk-free rate, at T , exercise the American option if anf only if the European option is exercised. The riskless profit of these transactions whose net initial cost is null is BT (c(t, T , K ) − C (t, T , K )) > 0. Bt Y. Lu (2022) MACF1 Fall 2022 98 / 122  Introducing derivatives Bounds on option prices Proof for option price bounds Third inequality (c(t, T , K ) ≥ St − KZ (t, T )): To show that c(t, T , K ) ≥ St − KZ (t, T ), we notice that if it isn’t the case, there exists an arbitrage opportunity which consists in the following steps: at t, buy the European option, buy K zero-coupon bond shares and sell short the underlying asset, invest the net proceeds of the sales (St − KZ (t, T ) − c(t, T , K ) > 0) at the risk-free rate in asset Bt , at T , exercise the European option. The riskless profit of these transactions whose net initial cost is null is (St − KZ (t, T ) − c(t, T , K )) BBTt > 0. Y. Lu (2022) MACF1 Fall 2022 99 / 122  Introducing derivatives Bounds on option prices Proof for option price bounds The monotonicity theorem for European options and max(0, ST − K ) ≥ 0 imply that c(t, T , K ) ≥ 0. Combining c(t, T , K ) ≥ 0 and c(t, T , K ) ≥ ST − KZ (t, T ), we obtain c(t, T , K ) ≥ max(0, St − KZ (t, T )). Inequalities for put options: The same rationale that was previously used can apply to show that K ≥ P(t, T , K ) ≥ p(t, T , K ) ≥ max(0, KZ (t, T ) − St ). The proof can be found in the exercise solutions.  Y. Lu (2022) MACF1 Fall 2022 100 / 122  Introducing derivatives Bounds on option prices American option versus European Remark The previous theorem implies than an American option always has a value greater or equal to the corresponding European option with the same maturity and strike price. Y. Lu (2022) MACF1 Fall 2022 101 / 122  Introducing derivatives Bounds on option prices Impact of dividends on bounds Remark Suppose that deterministic discrete dividends Dsi are paid at time si , i = 1,... , N where t < s1 <... < sN < T. Then, lower bounds for European options become N ! X c(t, T , K ) ≥ max 0, St − Dsi Z (t, si ) − KZ (t, T ) , i=1 N ! X p(t, T , K ) ≥ max 0, KZ (t, T ) − St + Dsi Z (t, si ). i=1 Proof: see exercises. Y. Lu (2022) MACF1 Fall 2022 102 / 122  Introducing derivatives Bounds on option prices Exercise of an American option Proposition Suppose that the underlying asset S does not pay dividends until time T. Then C (t, T , K ) = c(t, T , K ) and it is never optimal to exercise an American call option before maturity. Y. Lu (2022) MACF1 Fall 2022 103 / 122  Introducing derivatives Bounds on option prices Exercise of an American option Proof: Suppose that C (t, T , K ) > c(t, T , K ), then we can construct an arbitrage opportunity through the following steps: at t, buy the European option and sell the American one, invest C (t, T , K ) − c(t, T , K ) at the risk-free rate, if the American is exercised by the counterparty before maturity (at a time τ ), sell the European option, use the sale proceeds to pay the American option holder and invest the remainder at the risk-free rate until time T. Y. Lu (2022) MACF1 Fall 2022 104 / 122  Introducing derivatives Bounds on option prices Exercise of an American option If the American option is not exercised before maturity, cash flows of the strategy are given by the following table. The payoff is strictly positive: (C (t, T , K ) − c(t, T , K )) BBTt > 0. Table: Cash flows if the American option is not exercised before maturity Position Inflow at t Inflow at T Short (Amer. option) C (t, T , K ) − max(0, ST − K ) Long (Eur. option) −c(t, T , K ) max(0, ST − K ) Long (bank account) c(t, T , K ) − C (t, T , K ) (C (t, T , K ) − c(t, T , K )) BBTt Total 0 (C (t, T , K ) − c(t, T , K )) BBTt Y. Lu (2022) MACF1 Fall 2022 105 / 122  Introducing derivatives Bounds on option prices Exercise of an American option If the American option is exercised before maturity, cash flows from the strategy are given by the following table. The payoff is strictly positive. Table: Cash flows if the American option is exercised before maturity Position Inflow at t Inflow at τ Inflow at T Short (Amer. option) C (t, T , K ) −(Sτ − K ) 0 Long (Eur. option) −c(t, T , K ) c(τ, T , K ) 0 B Long (bank account) c(t, T , K ) − C (t, T , K ) 0 (C (t, T , K ) − c(t, T , K )) BT t B Long (bank account) 0 Sτ − K − c(τ, T , K ) (c(τ, T , K ) − (Sτ − K )) BT τ B Total 0 0 (C (t, T , K ) − c(t, T , K )) BT t B + (c(τ, T , K ) − (Sτ − K )) BT τ Y. Lu (2022) MACF1 Fall 2022 106 / 122  Introducing derivatives Bounds on option prices Exercise of an American option Since the payoff of the strategy is strictly positive, this is an arbitrage opportunity. The previous arbitrage strategy shows its it impossible to have C (t, T , K ) > c(t, T , K ). Thus, C (t, T , K ) = c(t, T , K ). The possibility to exercise the American call option before maturity has no value.  Y. Lu (2022) MACF1 Fall 2022 107 / 122  Introducing derivatives Bounds on option prices Put-Call Parity The next result provides a relation between the price of a European call and a European call option when the underlying asset, the maturity and the strike are the same: Theorem (Put-Call Parity) Suppose the underlying asset S does not pay dividends. Then, c(t, T , K ) − p(t, T , K ) = St − KZ (t, T ) Y. Lu (2022) MACF1 Fall 2022 108 / 122  Introducing derivatives Bounds on option prices Put-Call Parity Proof: Consider the following two portfolios: Portfolio 1: purchase at European call and sell a European put at t, both options having the same strike K and same maturity T , Portfolio 2: buying the underlying asset and sell short K zero-coupons at time t. The payoff for these two portfolios is ST − K.8 From the Law of one price, these two portfolios have the same price.  8 By noticing that max(0, ST − K ) − max(0, K − ST ) = ST − K. Y. Lu (2022) MACF1 Fall 2022 109 / 122  Introducing derivatives Bounds on option prices Relations for American options Following result provide inequalities on American option prices. Proposition Suppose the underlying asset S does not pay dividends. Then, St − K ≤ C (t, T , K ) − P(t, T , K ) ≤ St − KZ (t, T ). Y. Lu (2022) MACF1 Fall 2022 110 / 122  Introducing derivatives Bounds on option prices Relations for American options Proof: First inequality (St − K ≤ C (t, T , K ) − P(t, T , K )): Consider the following two portfolios constructed at time t. The first portfolio contains the underlying asset and a American put option. The payoff of the strategy is ( (K − Su ) + Su = K at time u if the put option is exercised at u < T , max(K , ST ) at time T otherwise. The second portfolio contains a European call option and K $ invested in the risk-free asset. The portfolio value is ( c(u, T , K ) + KB Bt at time u where t ≤ u < T , u max(0, ST − K ) + KB Bt at time T. T Y. Lu (2022) MACF1 Fall 2022 111 / 122  Introducing derivatives Bounds on option prices Relations for American options The first portfolio payoff is always smaller or equal to the second portfolio payoff: ( K ≤ c(u, T , K ) + KB Bt u at time u if t ≤ u < T , max(K , ST ) ≤ max(0, ST − K ) + KB Bt T at time T. Then, the first portfolio has a value that is inferior to the second portfolio value and thus St + P(t, T , K ) ≤ c(t, T , K ) + K. Rearranging terms, we obtain St − K ≤ c(t, T , K ) − P(t, T , K ) ≤ C (t, T , K ) − P(t, T , K ). Y. Lu (2022) MACF1 Fall 2022 112 / 122  Introducing derivatives Bounds on option prices Relations for American options Second inequality (C (t, T , K ) − P(t, T , K ) ≤ St − KZ (t, T )): We have that C (t, T , K ) − P(t, T , K ) = c(t, T , K ) − P(t, T , K ) ≤ c(t, T , K ) − p(t, T , K ) = St − KZ (t, T ). (Put-call parity) Note (see previous propositions): First equality: because it is never optimal to exercise an American call option before maturity when the underlying asset pays no dividends. First inequality: since American option is always worth at least more than a European option.  Y. Lu (2022) MACF1 Fall 2022 113 / 122  Introducing derivatives Bounds on option prices Arbitrage and option prices In summary, the absence of arbitrage assumption implies that options prices must satisfy multiple inequalities. However, these bounds don’t specify a unique price for options. This is different from the forward and futures contracts situation where the absence of arbitrage specified their price in a unique manner. To determine option prices, additional assumptions must be posed. Y. Lu (2022) MACF1 Fall 2022 114 / 122  Introducing derivatives Bounds on option prices Additional bounds on option prices Several additional bounds on option prices are presented in exercises. Y. Lu (2022) MACF1 Fall 2022 115 / 122  Introducing derivatives Bounds on option prices Continuous dividends In some models, it is assumed that the risky asset pays dividends continuously at the continuously compounded rate of δ per period.9 This means that for all t, a dividend St+∆t δ∆t is provided at time t + ∆t where ∆t is an infinitesimal time elapse. 9 In chapter 3 we will sometimes use the notation q instead for the dividend rate (dividend yield). Y. Lu (2022) MACF1 Fall 2022 116 / 122  Introducing derivatives Bounds on option prices Continuous dividends Proposition Assume the holder of one risky asset shares at time 0 continuously reinvests dividends he receives to purchase stocks. At time T the holder will possess e δT shares whose total value will be e δT ST. Corollary To obtain a position worth ST at time T , an investor should purchase e −δT shares of the risky asset at time 0, and hold on it on top of continuously reinvesting the dividends to buy stock shares. Y. Lu (2022) MACF1 Fall 2022 117 / 122  Introducing derivatives Bounds on option prices Continuous dividends Proof: Let’s break down the time interval [0, T ] into N subintervals of length ∆t ≡ T /N. We will at the end let N tend to infinity. At time ∆t , the stock holder will have a position worth approximately   T T S∆t + S∆t δ = S∆t 1 + δ , N N and therefore will hold 1 + δ T
N shares of the stock. Y. Lu (2022) MACF1 Fall 2022 118 / 122  Introducing derivatives Bounds on option prices Continuous dividends Repeating this procedure, after n time steps of length ∆t , the stock share holder will hold approximately T n   1+δ N and therefore will hold at time T a total of T N   1+δ N shares. Letting N tend to infinity (to consider infinitesimal time steps), the exact number of shares at time T held by the stockholder which had one share initially is T N   lim 1 + δ = e δT.  N→∞ N Y. Lu (2022) MACF1 Fall 2022 119 / 122  Introducing derivatives Bounds on option prices Continuous dividends, forward price and put-call parity Suppose the underlying asset S pays continuous dividends at the continuously compounded rate δ. Then, Proposition The forward price is given by St e −δ(T −t) Ft,T =. Z (t, T ) Theorem (Put-Call Parity with continuous dividends) c(t, T , K ) − p(t, T , K ) = St e −δ(T −t) − KZ (t, T ) Proof: omitted (very similar to the no-dividend case). Y. Lu (2022) MACF1 Fall 2022 120 / 122  Introducing derivatives Bounds on option prices Transaction costs and arbitrage In practice, when buying/selling financial products, transaction costs are incurred. These can limit the possibility to implement arbitrage strategies. Positive profits from arbitrage can become negative once transaction costs are accounted for. This was disregarded in the current chapter. Y. Lu (2022) MACF1 Fall 2022 121 / 122  Introducing derivatives Bounds on option prices Types of transaction costs In practice, transaction costs can be direct or indirect. Direct transaction costs: Commissions paid to brokers, Fees paid to the exchange. Indirect transaction costs: Bid-ask spread. I The price at which one can buy and can sell are different. The difference between the buying price (bid) and selling price (ask) is the bid-ask spread. I The difference between the mid price (average between bid and ask prices) and the transaction price can be seen as a transaction cost. Y. Lu (2022) MACF1 Fall 2022 122 / 122