Curriculum Note, Study Session 13, Reading 34

Valuation of Contingent Claims 1. INTRODUCTION A contingent claim is a derivative instrument whose payoff depends on occurrence of a future event. In a contingent claim (unlike forward and futures contracts), one party to the contract receives the right – not the obligation – to buy or sell an underlying asset from another party. The purchase price is fixed over a specific period of time and will eventually expire. Contingent claims include options. • Options derive their value from an underlying asset, which has value. E.g. the payoff on a call (put) option occurs only if the value of the underlying asset is greater (lesser) than an exercise price that is specified at the time the option is created. If this contingency does not occur, the option is worthless. • Like forward, futures, and swaps contracts, option valuation models are based on the principle of no arbitrage. Option valuation models typically use two approaches. 2. 1) Binomial model – based on discrete time. The binomial model is used to value pathdependent options, which are options whose values depend both on the value of the underlying at expiration and how it got there. Such as American options, which can be exercised prior to expiration. (Discussed in detail in Section 3). 2) Black–Scholes–Merton (BSM) model, which is based on continuous time. The BSM model is based on the key assumption that the value of the underlying instrument follows a statistical process called geometric Brownian motion. Geometric Brownian motion implies a lognormal distribution of the return, which implies that the continuously compounded return on the underlying is normally distributed. The BSM model values only path-independent options (i.e. European options), which depend on only the values of their respective underlyings at expiration. PRINCIPLES OF A NO-ARBITRAGE APPROACH TO VALUATION Arbitrage is based on following two fundamental rules as well as law of one price. distribution. 5) Rule #1: Do not use your own money. Rule #2: Do not take any price risk. Key assumptions in Option Valuation1: In this reading, we will make following key assumptions in estimating values of Options: 1) Replicating instruments are identifiable and investable. 2) There are no market frictions, i.e. transaction costs and taxes. 3) Short selling is allowed with full use of proceeds. 4) The underlying instrument follows a known statistical Borrowing and lending at a risk-free interest rate is available. The option payoffs can be replicated with a dynamic portfolio of the underlying instrument and financing. Ø Dynamic Portfolio: A dynamic portfolio is one whose composition changes over time. 1 Throughout this reading, cash outflows are treated as negative and inflows as positive. –––––––––––––––––––––––––––––––––––––– Copyright © FinQuiz.com. All rights reserved. –––––––––––––––––––––––––––––––––––––– FinQuiz Notes – 2 0 2 2 Reading 34 Reading 34 Valuation of Contingent Claims 3. BINOMIAL OPTION VALUATION MODEL The binomial option valuation model is based on the noarbitrage approach to valuation. • • • Value of Call Option at expiration: cT = Max(0,ST – X) Value of Put Option at expiration: pT = Max(0,X – ST) Where, o o o o o St denote the underlying instrument price observed at Time t, where t is expressed as a fraction of a year. E.g. a call option had 60 days to expiration when purchased (T = 60/365), but now only has 35 days to expiration (t = 25/365). ST denotes the underlying instrument price observed at the option expiration date, T. ct denote a European-style call price at Time t and with expiration on Date t = T, where both t and T are expressed in years. Ct denote an American-style call price. X denote the exercise price. If the option values deviate from these expressions, then there will be arbitrage profits available. Since, European options cannot be exercised until expiration, they do not technically have exercise values prior to expiration. Time Value of Options: The time value is always nonnegative for options because of the asymmetry of option payoffs at expiration. For example, for a call option, the upside is unlimited, whereas the downside is limited to zero. At expiration, time value is zero. 3.1 FinQuiz.com One-Period Binomial Model • • Each dot represents a particular outcome at a particular point in time in the binomial lattice. These dots are termed nodes. At the Time 0 node, there are only two possible future paths in the binomial process, an up move and a down move, termed as arcs. At Time 1, there are only two possible outcomes: S+ denotes the outcome when the underlying goes up, and S− denotes the outcome when the underlying goes down. The up factors and down factors are the total returns. The magnitudes of the up and down factors are based on the volatility of the underlying. In general, the higher the volatility, the higher will be the up values and the lower will be the down values. At expiration, Option value is either c++ = Max (0, S++ – X) = Max (0, u2S – X) or c+– = Max (0, S+– – X) = Max (0, udS – X) • The value of a call option is positively related to the value of the underlying. That is, if the underlying goes up (down), value of call option increases (decreases). This implies that in order to hedge position, a trader to be a long position in the underlying. Specifically, the trader buys a certain number of units, ‘h’, of the underlying. The symbol h represents a hedge ratio – as estimated below. Ø The following figure represents a tree for possible outcomes in one-period Binomial Model. Ø The above formula states that Hedge ratio is the value of the call if the underlying goes up minus the value of the call if the underlying goes down divided by the value of the underlying if it goes up minus the value of the underlying if it goes down. Hedge ratio is non-negative because call prices are positively related to changes in the underlying price. Writing One Call Hedge with h Units of the Underlying and Finance: The following table shows payoffs of writing one call hedge with h units of the underlying and finance. Reading 34 Ø Ø Ø Valuation of Contingent Claims At Time 0, the value of the net portfolio should always be zero, else there will be an arbitrage opportunity. If the net portfolio has positive value, then arbitrageurs will write call option, long the “h” underlying units, and then finance his transaction through borrowing. If the net portfolio has negative value, then arbitrageurs will buy call option, short sell the “h” underlying units, and then lend (or invest the proceeds) – pushing the call price up and the underlying price down until the net cash flow at Time 0 is no longer positive. Long a call option = Owning ‘h’ shares of stock partially financed Where, Financed amount = PV (–hS– + c–) or using the per period rate, (–hS– + c–) / (1 + r) Single-period call option valuation using no-arbitrage approach: c = hS + PV(–hS– + c–) or, equivalently, c = hS + PV(–hS+ + c+) FinQuiz.com Practice: Example 1, Reading 34, Curriculum. No-arbitrage single-period put option valuation equation is as follows: p = hS + PV(–hS– + p–) or, equivalently, p = hS + PV (–hS+ + p+) Where, Ø Ø Ø For put options, the hedge ratio is negative because p+ is less than p–. In order to hedge a long put position, the arbitrageur will short sell the underlying (–h = –(p+ – p–)/(S+ – S–) units of the underlying) and lend a portion of the proceeds. Note that since –h is positive, the value –hS results in a positive cash flow at Time Step 0. Please refer to table below. Replicating a Call option: A call option can be replicated with the underlying and financing. Specifically, the call option is equivalent to a leveraged position in the underlying. The trading strategy that will generate the payoffs of taking a long position in a call option within a single-period binomial framework is as follows: Buy h = (c+ – c–)/(S+ – S–) units of the underlying and financing of – PV(–hS– + c–) Please refer to table below: Practice: Example 2, Reading 34, Curriculum. Reading 34 Valuation of Contingent Claims Expectations Approach: The expectations approach results in an identical value as the no-arbitrage approach, but it is usually easier to compute. The formulas are given as follows: c = PV [πc+ + (1 – π) c–] options because it provides the intuition for the fair value of options. Practice: Example 3, Reading 34, Curriculum. Put–call parity: S + p = PV(X) + c and p = PV [πp+ + (1 – π) p–] Where, The value of a put or call option can be found based on put–call parity. E.g. Call option can be expressed as a position in a stock, financing, and a put, i.e. c = S – PV(X) + p Ø Ø Probability of an up move = π = [FV(1) – d]/(u – d) Expected terminal option payoffs: The option values are present value of the expected terminal option payoffs. The expected terminal option payoffs can be expressed as follows: E(c1) = πc+ + (1 – π)c– and E(p1) = πp+ + (1 – π)p– FinQuiz.com Practice: Example 4, Reading 34, Curriculum. 3.2 Two-Period Binomial Model Following figure reflects Two-Period Binomial Lattice as Three One-Period Binomial Lattices. Where c1 and p1 are the values of the options at Time 1. The option values based on the expectations approach can be expressed as follows: c = PVr[E(c1)] and p = PVr[E(p1)] Difference between Expectations approach and discounted cash flow approach to securities valuation: The expectations approach is often regarded as superior method to the discounted cash flow approach because it is based on objective measures as follows. i. The expectation is not based on the investor’s beliefs regarding the future course of the underlying – implying that the probability, π, is objectively determined and not based on the investor’s personal view. This probability is referred to as riskneutral (RN) probability – reason being the expectations approach is not based on assumption regarding risk preferences. ii. In expectations approach, the discount rate is not risk adjusted, rather it is based on the estimated riskfree interest rate. Note: The expectations approach can be applied to European-style options. The no-arbitrage approach can be applied to either European-style or American style Ø Ø For simplicity, it is assumed that the up and down factors are constant throughout the lattice, that is S+– = S–+. For example, assume u = 1.25, d = 0.8, and S0 = 100. Note that S+– = 1.25(0.8)100 = 100 and S–+ = 0.8(1.25)100 = 100. So the middle node at Time 2 is 100 and can be reached from either of two paths. It is important to remember that Option valuation relies on self-financing, dynamic replication. Dynamic replication is obtained by using a portfolio of stock and the financing. The strategy is self-financing because the funds borrowed at Time 1 grew to ‘x’ amount. Call Option Payoffs at Time 2: c++ = Max (0, S++ – X) = Max(0,u2S – X), c+– = Max (0, S+– – X) = Max (0,udS – X), and c– – = Max (0, S– – – X) = Max (0,d2S – X) Reading 34 Valuation of Contingent Claims FinQuiz.com Put Option Payoffs at Time 2: p++ = Max (0, X – S++) = Max (0,X – u2S), p+– = Max (0, X – S+–) = Max (0,X – udS), and p– – = Max (0,X – S– –) = Max (0, X – d2S) Example: Following lattice shows the no-arbitrage approach for solving the two-period binomial call value. Suppose the annual interest rate is 3%, the underlying stock is S = 72, u = 1.356, d = 0.541, and the exercise price is X = 75. The stock does not pay dividends. c = PV [π2c++ + 2π (1 – π) c+– + (1 – π)2c– –] p = PV [π2p++ + 2π (1 – π) p+– + (1 – π)2p– –] The expected terminal option payoffs are expressed as below: E(c2) = π2c++ + 2π (1 – π) c+– + (1 – π)2c– – and E(p2) = π2p++ + 2π (1 – π)p+– + (1 – π)2p– – The two-period binomial option values based on the expectations approach are expressed as: = PVr[Eπ(c2)] and p = PVr[Eπ(p2)] Practice: Example 5, Reading 34, Curriculum. American-style options: American options are options which can be exercised prior to expiration. A nondividend paying call options on stock will not be exercised early because the minimum price of the option exceeds its exercise value. However, this is not true for put options (particularly a deep in the- money put) because the sale proceeds can be invested at the risk-free rate and earn interest worth more than the time value of the put. Example: Suppose the periodically compounded interest rate is 3%, the non-dividend-paying underlying stock is currently trading at 72, the exercise price is 75, u = 1.356, d = 0.541, and the put option expires in two years. Following lattice reflects “Two-Period Binomial Model for a European-Style Put Option”: Following lattice reflects “Two-Period Binomial Model for an American-Style Put Option”: Put value = p = PV[πp+ + (1 – π)p–] Practice: Example 6, Reading 34, Curriculum. Escrow method: Dividends negatively affect the value of a call option because dividends lower the value of the stock. Most option contracts do not provide protection against dividends. Assuming dividends are perfectly predictable, we can split the underlying instrument into two components: the underlying instrument without the known dividends and the known dividends. For example, the current value of the underlying instrument without dividends can be expressed as follows: Where, γ denotes the present value of dividend payments. ^ symbol is used to denote the underlying instrument without dividends. At expiration, the underlying instrument value is the same,𝑆"# = ST. because it is assumed that any dividends have already been paid. The value of an investment in the stock, however, would be ST + γT, which assumes the dividend payments are reinvested at the risk-free rate. Following lattice reflects “Two-Period Binomial Model for an American-Style Call Option with Dividends” Reading 34 Ø Ø Ø Ø Valuation of Contingent Claims At Time 0, the present value of the US$3 dividend payment is US$2.970297 (= 3/1.01). Therefore, 118.7644 = (100 – 2.970297)1.224 is the stock value without dividends at Time 1, assuming an up move occurs. The exercise value for this call option, including dividends, is 26.7644 [= Max (0,118.7644 + 3 – 95)], whereas the value of the call option per the binomial model is 24.9344. The stock price just before it goes ex-dividend is 118.7644 + 3 = 121.7644, so the option can be exercised for 121.7644 – 95 = 26.7644. If not exercised, the stock drops as it goes exdividend and the option becomes worth 24.9344 at the ex-dividend price. • • FinQuiz.com The rates are expressed in annual compounding. Therefore, at Time 0, the spot rate is (1.0/0.970446) – 1 or 3.04540%. Note that at Time 1, the value in the column labeled “Maturity” reflects time to maturity not calendar time. c++ = Max (0, S++ – X) = Max [0,0.039706 – 0.0325] = 0.007206 c+– = Max (0, S+– – X) = Max [0,0.032542 – 0.0325] = 0.000042 c– – = Max (0, S– – – X) = Max [0,0.022593 – 0.0325] = 0.0 Important to Remember: This example tell us that the American-style call option is worth more than the European-style call option because at Time Step 1 when an up move occurs, the call is exercised early, capturing additional value. For non-dividend paying stocks, the American-style feature has no effect on either the hedge ratio or the option value. American-style put options on non-dividend-paying stock may be (not necessarily always) worth more than the analogous European style put options. p++ = Max (0, X – S++) = Max [0,0.0325 – 0.039706] = 0.0 p+– = Max (0, X – S+–) = Max [0,0.0325 – 0.032542] = 0.0 p– – = Max (0, X – S– –) = Max [0,0.0325 – 0.022593] = 0.009907 At Time Step 1, we have c+ = PV1,2[πc++ + (1 – π)c+–] = 0.962386[0.5(0.007206) + (1 – 0.5)0.000042] = 0.003488 Practice: Example 7, Reading 34, Curriculum. c– = PV1,2[πc+– + (1 – π)c– –] = 0.974627[0.5(0.000042) + (1 – 0.5)0.0] = 0.00002 3.3 p+ = PV1,2[πp++ + (1 – π)p+–] = 0.962386[0.5(0.0) + (1 – 0.5)0.0] = 0.0 • • Interest Rate Options A call option on interest rates will be in the money when the current spot rate > exercise rate. A put option on interest rates will be in the money when the current spot rate < exercise rate. Example: Following is the Two-Year Binomial Interest Rate Lattice by Year. Assume the notional amount of the options is US$1,000,000 and the call and put exercise rate is 3.25% of par and RN probability is 50%. p– = PV1,2[πp+– + (1 – π)p– –] = 0.974627[0.5(0.0) + (1 – 0.5)0.009907] = 0.004828 At Time Step 0, we have c = PVrf,0,1[πc+ + (1 – π)c–] = 0.970446[0.5(0.003488) + (1 – 0.5)0.00002] = 0.00170216 p = PVrf,0,1[πp+ + (1 – π)p–] = 0.970446[0.5(0.0) + (1 – 0.5)0.004828] = 0.00234266 Reading 34 Valuation of Contingent Claims Because the notional amount is US$1,000,000, the call value is = US$1,000,000(0.00170216) = US$1,702.16 and the put value is = US$1,000,000(0.00234266) = US$2,342.66. 3.4 Multiperiod Model 4. 4.2 The two-period model divides the expiration into two periods. The three-period model divides expiration into three periods and so forth. Similarly, the multi-period model divides expiration into multiple periods. Each time step is of equal length, i.e., with a maturity of T, if there are n time steps, then each time step is T/n in length. BLACK–SCHOLES–MERTON OPTION VALUATION MODEL Assumptions of the BSM model The stochastic process (wherein value of instrument evolves over time) chosen by Black, Scholes, and Merton is called geometric Brownian motion (GBM). Assumptions of the BSM model: The standard BSM model assumes a constant growth rate and constant volatility. The specific assumptions of the BSM model are as follows: a) The underlying follows a statistical process called geometric Brownian motion, which implies a lognormal distribution of the return – meaning that the continuously compounded return is normally distributed. b) Geometric Brownian motion implies continuous prices, meaning that the price of underlying instrument does not jump from one value to another; rather, it moves smoothly from value to value. c) The underlying instrument is liquid, i.e. can be easily bought and sold. d) Continuous trading is available, i.e. we can trade at every instant. e) Short selling of the underlying instrument with full use of the proceeds is allowed. f) There are no market frictions, i.e. transaction costs, regulatory constraints, or taxes. g) No-arbitrage opportunities are available in the marketplace. h) The options are European-style, meaning that early exercise is not allowed. i) The continuously compounded risk-free interest rate is known and constant. j) Borrowing and lending is allowed at the risk-free rate. k) The volatility of the return on the underlying is known and constant. l) If the underlying instrument pays a yield, it is expressed as a continuous known and constant yield at an annualized rate. Practice: Example 9, Reading 34, Curriculum. FinQuiz.com 4.3 BSM model The BSM model is a continuous time version of the discrete time binomial model and therefore, continuously compounded interest rate is used in this model. The volatility (σ) is also expressed in annualized percentage terms. The BSM model for stocks can be expressed as follows: c = SN(d1) – e–rTXN(d2) and p = e–rTXN(–d2) – SN(–d1) Where, Ø Ø N(x) reflects the likelihood of observing values less than x from a random sample of observations taken from the standard normal distribution. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The normal distribution is a symmetric distribution with two parameters, the mean and standard deviation. BSM model for call option is c = PVr[E(cT)] BSM model for put option is p = PVr[E(pT)] Where, E(cT) = SerTN(d1) – XN(d2) and E(pT) = XN(–d2) – SerTN(–d1). The present value term in this context is simply e–rT. Reading 34 Valuation of Contingent Claims FinQuiz.com BSM model can be described as having two components: a stock component and a bond component. Ø Ø For call options, the stock component is SN(d1) and the bond component is e–rTXN(d2). BSM model call value = stock component - bond component For put options, the stock component is SN(–d1) and the bond component is e–rTXN(–d2). BSM model put value = Bond component - Stock component The BSM model can be interpreted as a dynamically managed portfolio of the stock and zero-coupon bonds. Ø For both call and put options, we can represent the initial cost of this replicating strategy as follows: Replicating strategy cost = nSS + nBB Where, o For calls, the equivalent number of underlying shares is nS = N(d1) > 0 and the equivalent number of bonds is nB = –N(d2) < 0. o For puts, the equivalent number of underlying shares is nS = –N(–d1) < 0 and the equivalent number of bonds nB = N(–d2) > 0. o The price of the zero-coupon bond is B = e–rTX. • Ø Important to remember: If n is positive, we are buying the underlying and if n is negative we are selling (short selling) the underlying. The cost of the portfolio will exactly equal either the BSM model call value or the BSM model put value. Ø A call option can be viewed as a leveraged position in the stock or calls because we are simply buying stock with borrowed money because nS > 0 and nB < 0. Ø For call options, –N(d2) implies borrowing money or short selling N(d2) shares of a zero-coupon bond trading at e–rTX. Ø Ø For put options, we are simply buying bonds with the proceeds from short selling the underlying because nS < 0 and nB > 0. A short put can be viewed as an over-leveraged or over-geared position in the stock because the borrowing exceeds 100% of the cost of the underlying. This is because a short position in a put will result in receiving money today and nS > 0 and nB < 0. For put options, N(–d2) implies lending money or buying N(–d2) shares of a zero-coupon bond trading at e–rTX. Comparison between BSM and Binomial Option Valuation Model: The following table summarized difference between BSM and Binomial Valuation model. • If the value of the underlying, S, increases, then the value of N(d1) also increases because S has a positive effect on d1. Thus, the replicating strategy for calls requires continually buying shares in a rising market and selling shares in a falling market. In practical, hedges are imperfect because (i) frequent rebalancing by buying and selling the underlying adds significant costs for the hedger because trading involves transaction costs; (ii) market may move discontinuously (contrary to the BSM model’s assumption mentioned above) which requires continuous hedging adjustments, and (iii) volatility cannot be known in advance. Practice: Example 10, Reading 34, Curriculum. Probability that the call option expires in the money: Probability that the call option expires in the money is denoted as N(d2), and correspondingly, 1 – N(d2) = N(−d2) is the probability that the put option expires in the money. Carry benefits: Carry benefits include dividends for stock options, foreign interest rates for currency options, and coupon payments for bond options. Carry benefits tend to lower the expected future value of the underlying. Carry costs can be treated as negative carry benefits, i.e. storage and insurance costs for agricultural products. Because the BSM model assumes continuous time, these carry benefits can be modelled as a continuous yield, denoted as γc or simply γ. Carry benefit-adjusted BSM model: The carry benefitadjusted BSM model is expressed as follows: c = Se–γTN(d1) – e–rTXN(d2) and p = e–rTXN(–d2) – Se–γTN(–d1) Where, Reading 34 Valuation of Contingent Claims d2 can be expressed as d2 = d1 – 𝜎 √𝑇 Value of a put option = p + Se–γT = c + e–rTX FinQuiz.com Practice: Example 11 & 12, Reading 34, Curriculum. E(cT) = Se(r–γ)TN(d1) – XN(d2) E(pT) = XN(–d2) – Se(r–γ)TN(–d1) The present value term is denoted as e–rT. Foreign exchange options: For foreign exchange options, γ = rf, which is the continuously compounded foreign risk-free interest rate. The carry benefit adjusted BSM model can be described as having two components, a stock component and a bond component. Currency options: In currency options, the underlying instrument is the foreign exchange spot rate. Here, the carry benefit is the interest rate in the foreign country because the foreign currency could be invested in the foreign country’s risk-free instrument. With currency options, the underlying and the exercise price must be quoted in the same currency unit. The volatility in the model is the volatility of the log return of the spot exchange rate. o o o • • For call options, the stock component is Se–γTN(d1) and the bond component is again e–rTXN(d2). For put options, the stock component is Se–γTN(–d1) and the bond component is again e– rTXN(–d2) Important to remember: • If carry benefits increase, they lower the value of the call option and raise the value of the put option. • The carry benefits tend to reduce d1 and d2, and consequently, the probability of being in the money with call options declines as the carry benefit rises. • Dividends influence the dynamically managed portfolio by lowering the number of shares to buy for calls and lowering the number of shares to short sell for puts. Higher dividends will lower the value of d1, thus lowering N(d1). In addition, higher dividends will lower the number of bonds to short sell for calls and lower the number of bonds to buy for puts. BSM call model for a dividend-paying stock: The BSM call model for a dividend-paying stock can be expressed as follows: Se–δTN(d1) – Xe–rTN(d2) Ø The equivalent number of units of stock is nS = e– δTN(d1) > 0 and the equivalent number of units of bonds remains nB = –N(d2) < 0. BSM put model for a dividend-paying stock: The BSM put model for a dividend-paying stock can be expressed as follows: Xe–rTN(–d2) – Se–δTN(–d1) The equivalent number of units of stock is nS = –e–δTN(–d1) < 0 and the equivalent number of units of bonds again remains nB = N(–d2) > 0. BSM model applied to currencies: The BSM model applied to currencies can be described as having two components, a foreign exchange component and a bond component. • For call options, the foreign exchange component • is and the bond component is e– rTXN(d2), where r is the domestic risk-free rate. BSM call model applied to currencies = Foreign exchange component - Bond component For put options, the foreign exchange component is and the bond component is e– rTXN(–d2). BSM put model applied to currencies = Bond component - Foreign exchange component Practice: Example 13, Reading 34, Curriculum. Reading 34 Valuation of Contingent Claims 5. 5.1 BLACK OPTION VALUATION MODEL European Options on Futures Ø Model for European-style futures options is as below: c = e–rT[F0(T)N(d1) – XN(d2)] p = e–rT[XN(–d2) – F0(T)N(–d1)] Where, F0(T) denotes the futures price at Time 0 that expires at Time T, and σ denotes the volatility related to the futures price. Ø FinQuiz.com Futures option put–call parity can be expressed as c = e–rT[F0(T) – X] + p The Black model has two components, a futures component and a bond component. For call options, the futures component is F0(T)e– rTN(d1) and the bond component is again e– rTXN(d2). Black call model = Futures component - Bond component • For put options, the futures component is F0(T)e– rTN(–d1) and the bond component is again e–rTXN(– d2). Black put model = Bond component - Futures component Futures option valuation based on the Black model involves computing the present value of the difference between the futures price and the exercise price. Ø For call options, the futures price is adjusted by N(d1) and the exercise price is adjusted by –N(d2). Ø For put options, the futures price is adjusted by –N(– d1) and the exercise price is adjusted by +N(–d2). • An interest rate put option gains when the reference interest rate falls. For an interest rate call option on three-month Libor with one year to expiration, the underlying interest rate is a forward rate agreement (FRA) rate that expires in one year. The underlying rate of the FRA is a 3-month Libor deposit that is investable in 12 months and matures in 15 months. Interest rates are set in advance, but interest payments are made in arrears, which is referred to as advanced set, settled in arrears. Ø The accrual period in FRAs is based on 30/360 whereas the accrual period based on the option is actual number of days in the contract divided by the actual number of days in the year (identified as ACT/ACT or ACT/365). Example: In a bank deposit, the interest rate is usually set when the deposit is made, say tj–1, but the interest payment is made when the deposit is withdrawn, say tj. The deposit, therefore, has time until maturity = tm = tj – tj– 1. Standard market model: In a standard market model, the prices of interest rate call and put options can be expressed as follows: And Where, Practice: Example 14, Reading 34, Curriculum. 5.2 Interest Rate Options In interest rate options, the underlying instrument is a reference interest rate, i.e. three-month Libor. Ø An interest rate call option gains when the reference interest rate rises. o o FRA(0,tj–1,tm) denote the fixed rate on a FRA at Time 0 that expires at Time tj–1, where the underlying matures at Time tj (= tj–1 + tm), with all times expressed on an annual basis. RX denotes the exercise rate expressed on an annual basis. Reading 34 o o o Valuation of Contingent Claims σ denotes the interest rate volatility. σ is the annualized standard deviation of the continuously compounded percentage change in the underlying FRA rate. Standard market model requires an adjustment when compared with the Black model for the accrual period, that is, FRA(0,tj–1,tm) or the strike rate, RX, are stated on an annual basis, as are interest rates in general. The actual option premium is adjusted for the accrual period. Differences between Black Model and Standard Model: 1) 2) 3) 4) 5) The discount factor is applied to the maturity date of the FRA or tj (= tj–1 + tm), rather than to the option expiration, tj–1. The underlying is an interest rate, specifically a forward rate based on a forward rate agreement or FRA(0,tj–1,tm). It is not a futures price. The exercise price is a rate and reflects an interest rate, not a price. The time to the option expiration, tj–1, is used in the calculation of d1 and d2. Both the forward rate and the exercise rate should be expressed in decimal form rather than as percent (for example, 0.01 and not 1.0). Important to remember: In Black model, a forward or futures price is used as the underlying. In contrast, in BSM model, a spot price is used as the underlying. • • • • Standard market model for calls: c = PV[E(ctj)] FinQuiz.com rate call options are called caplets. Thus, a set of floating-rate loan payments can be hedged with a long position in an interest rate cap encompassing a series of interest rate call options. An interest rate floor is a portfolio or strip of interest rate put options in which the expiration of the first underlying corresponds with the expiration of the second option and so forth. The underlying interest rate put options are called floorlets. Thus, a floating-rate bond investment or any other floatingrate lending situation can be hedged with an interest rate floor encompassing a series of interest rate put options. Long an interest rate cap and short an interest rate floor with the same exercise rate is equal to a receive-floating, pay-fixed interest rate swap. When the cap is in the money, the receive-floating counterparty will also receive an identical net payment. When the floor is in the money, the receive-floating counterparty will also pay an identical net payment. Long an interest rate floor and short an interest rate cap with the same exercise rate is equal to a receive-fixed, pay-floating interest rate swap. When the floor is in the money, the receive-fixed counterparty will also receive an identical net payment. When the cap is in the money, the receive-floating counterparty will also pay an identical net payment. If the exercise rate selected in interest rate option is set equal to the swap rate, then the value of the cap must be equal to the value of the floor. When an interest rate swap is initiated, its current value is zero and is known as an at-market swap. When an exercise rate is selected such that the cap equals the floor, then the initial cost of being long a cap and short the floor is also zero. Standard market model for puts: p = PV[E(ptj)] Practice: Example 15, Reading 34, Curriculum. Where, o E(ctj) = (AP)[FRA(0,tj–1,tm)N(d1) – RXN(d2)] o E(ptj) = (AP) [RXN(–d2) – FRA(0,tj–1,tm)N(–d1)] 5.3 Combinations created with interest rate options: • • • If the exercise rate selected in interest rate option is equal to the current FRA rate, then long an interest rate call option and short an interest rate put option is equivalent to a receive-floating, pay-fixed FRA. If the exercise rate selected in interest rate option is equal to the current FRA rate, then long an interest rate put option and short an interest rate call option is equivalent to a receive-fixed, pay-floating FRA. An interest rate cap is a portfolio or strip of interest rate call options in which the expiration of the first underlying corresponds to the expiration of the second option and so forth. The underlying interest Swaptions A swap option or swaption is an option on a swap. It gives the holder the right, but not the obligation, to enter a swap at the pre-agreed swap rate (referred to as the exercise rate). Interest rate swaps can be either receive fixed, pay floating or receive floating, pay fixed. Payer Swaption: A payer swaption is an option on a swap to pay fixed, receive floating. Receiver Swaption: A receiver swaption is an option on a swap to receive fixed, pay floating. Swap payments are advanced set, settled in arrears. Following equation represents the present value of an annuity matching the forward swap payment: Reading 34 Valuation of Contingent Claims • Payer swaption valuation model is expressed as follows: Receiver swaption valuation model is expressed as follows: • • Differences between Swaption Model and Black Model: i. ii. iii. iv. The discount factor is absent in swaption model. The payoff is a series of payments. Thus, the present value of an annuity used here takes into account the option-related discount factor. The underlying is the fixed rate on a forward interest rate swap rather than a futures price,. The exercise price is expressed as an interest rate. Both the forward swap rate and the exercise rate are expressed in decimal form and not as percent (for example, 0.02 and not 2.0). Payer swaption model value is estimated as follows: Receiver swaption model value is estimated as follows: Where, The swaption model can also be described as having two components, a swap component and a bond component. • For payer swaptions, the swap component is (AP)PVA(RFIX)N(d1) and the bond component is (AP)PVA(RX)N(d2). Payer swaption model value = Swap component Bond component For receiver swaptions, the swap component is (AP)PVA(RFIX)N(–d1) and the bond component is (AP)PVA(RX)N(–d2). Receiver swaption model value = Bond component - Swap component Combinations created with Swaptions: • The swaption model requires two adjustments, one for the accrual period and one for the present value of an annuity. FinQuiz.com • Long a receiver swaption and short a payer swaption with the same exercise rate is equivalent to entering a receive-fixed, pay-floating forward swap. Long a payer swaption and short a receiver swaption with the same exercise rate is equivalent to entering a receive-floating, pay-fixed forward swap. If the exercise rate is selected such that the receiver and payer swaptions have the same value, then the exercise rate is equal to the atmarket forward swap rate. A long position in a callable fixed-rate bond can be viewed as being long a straight fixed-rate bond and short a receiver swaption. The receiver swaption buyer will benefit when rates fall and the swaption is exercised. Thus, the embedded call feature is similar to a receiver swaption. Practice: Example 16, Reading 34, Curriculum. Reading 34 Valuation of Contingent Claims 6. 6.1 OPTION GREEKS AND IMPLIED VOLATILITY Delta Option delta is the change in an option value for a given small change in the value of the underlying stock, holding everything else constant. The option deltas for calls and puts are as follows, respectively Ø Ø Ø Ø Ø Ø Ø FinQuiz.com The delta of long one share of stock is +1.0, and the delta of short one share of stock is –1.0. Delta is a static risk measure because it does not tell us how likely this particular change would be. The range of call delta is 0 and e–δT and the range of put delta is –e–δT and 0. As the stock price increases, the call option goes deeper in the money and the value of N(d1) moves toward 1. As the stock price decreases, the call option goes deeper out of the money and the value of N(d1) moves toward zero. When the option gets closer to maturity, the delta will drift either toward 0 if it is out of the money or drift toward 1 if it is in the money. As the stock price changes and as time to maturity changes, the deltas also changes. Delta neutral portfolio: A delta neutral portfolio refers to setting the portfolio delta to zero. Theoretically, the value of delta neutral portfolio does not change for small changes in the stock instrument. Ø Delta neutral implies that the portfolio delta plus NHDeltaH is equal to zero. The optimal number of hedging units, NH, is Example: Suppose a portfolio consists of 100,000 shares of stock at US$10 per share. In this case, the portfolio delta is 100,000. The delta of the hedging instrument, stock, is +1. Thus, the optimal number of hedging units, NH, is –100,000 (= –100,000/1) or short 100,000 shares. If the portfolio delta is 5,000 and a particular call option with delta of 0.5 is used as the hedging instrument, then to arrive at a delta neutral portfolio, we need to sell 10,000 call options (= –5,000/0.5). Alternatively, if a portfolio of options has a delta of –1,500, then we need to buy 1,500 shares of stock to be delta neutral [= –(– 1,500)/1]. If the hedging instrument is stock, then the delta is +1 per share. Practice: Example 17, Reading 34, Curriculum. Delta approximation Equation: or The delta approximation is fairly accurate for very small changes in the stock. But as the change in the stock increases, the estimation error also increases. The delta approximation is biased low for both a down move and an up move. Where, NH denote the number of units of the hedging instrument; DeltaH denote the delta of the hedging instrument, which could be the underlying stock, call options, or put options. Ø Ø If NH is negative, then we should short the hedging instrument. If NH is positive, then we should go long the hedging instrument. The above chart shows that delta hedging is imperfect and gets worse as the underlying moves further away from its original value of 100. Reading 34 Valuation of Contingent Claims FinQuiz.com Practice: Example 18, Reading 34, Curriculum. 6.2 Gamma Option gamma refers to the change in a given option delta for a given small change in the stock’s value, holding everything else constant. Option gamma is a measure of the curvature in the option price in relationship to the stock price. Gamma approximates the estimation error in delta for options because the option price with respect to the stock is non-linear and delta is a linear approximation. This implies that gamma measures the non-linearity risk. A gamma neutral portfolio implies the gamma is zero. Ø Ø The gamma of a long or short position in one share of stock is 0 because the delta of a share of stock never changes. The delta of stock is always +1 and –1 for a short position in the stock. The gamma for a call and put option are the same and can be expressed as below: Where, n(d1) is the standard normal probability density function. Ø The gamma of a call equals the gamma of a similar put based on put–call parity or c – p = S0 – e– rTX. Note that neither S0 nor e–rTX is a direct function of delta. Hence, the right-hand side of put–call parity has a delta of 1. Ø Gamma is always non-negative. Ø Gamma is largest near at the money. Ø Options deltas do not change substantially for small changes in the stock price if the option is either deep in or deep out of the money. Ø As the stock price changes and as time to expiration changes, the gamma also changes. Ø Buying options (calls or puts) will always increase net gamma. Ø Gamma Risk: It is the risk associated with noncontinuous and un smooth change in stock prices. Important to remember: In delta neutral portfolio strategy, first we need to manage gamma to an acceptable level and then we neutralize the delta is neutralized. This hedging approach is more feasible because options, unlike stocks, have gamma. To alter the portfolio delta, we need to buy or sell stock. Because stock has a positive delta, but zero gamma, the portfolio delta can be brought to its desired level with no impact on the portfolio gamma. Delta-plus-gamma approximation Equation: The call value based on the delta approximation is estimated as below: Call value based on the delta-plus-gamma approximation is expressed as below: Ø The delta approximation and the delta-plus-gamma approximations are fairly accurate for very small changes in the stock. The chart below reflects that the call delta-plus-gamma estimated line is significantly closer to the BSM model call values. We can see that even for fairly large changes in the stock, the delta-plus-gamma approximation is accurate. As the change in the stock increases, the estimation error also increases. The chart also shows that the delta-plus-gamma approximation is biased low for a down move but biased high for an up move. Practice: Example 19, Reading 34, Curriculum. 6.3 Theta Option theta is the change in an option value for a given small change in calendar time, holding everything else constant. In other words, Option theta is the rate at which the option time value declines as the option approaches expiration. Stock theta is zero because stocks do not have an expiration date. Like gamma, Reading 34 Valuation of Contingent Claims theta cannot be adjusted with stock trades. Typically, theta is negative for options. That is, as calendar time passes, expiration time declines and the option value also declines. Time decay: It refers to the gain or loss of an option portfolio in response to the mere passage of calendar time. Particularly with long options positions, often the mere passage of time without any change in other variables, such as the stock, will result is significant losses in value. Ø FinQuiz.com The lower bound of a European-style put option: Zero or the present value of the exercise price less the stock, whichever is greater. The chart given below shows that the call lower bound is 4.88 and the put lower bound is 0. The difference between the call and put can be explained by put–call parity. Please refer to the chart below to assess how the speed of the option value decline increases as time to expiration decreases. 6.5 Rho Rho is the change in a given portfolio for a given small change in the risk-free interest rate, holding everything else constant. Thus, rho measures the sensitivity of the portfolio to the risk-free interest rate. Ø 6.4 Vega Vega is the change in a given portfolio for a given small change in volatility, holding everything else constant. Thus, vega measures the sensitivity of a portfolio to changes in the volatility used in the option valuation model. The vega of an option is positive, i.e., an increase in volatility results in an increase in the option value for both calls and puts. Ø Ø Ø Ø Based on put–call parity, the vega of a call is equal to the vega of a similar put. Vega is high when options are at or near the money and are short dated. Volatility is usually only hedged with other options. Volatility is sometimes considered a separate asset class or a separate risk factor. Unlike the delta, gamma, and theta, vega is based on an unobservable parameter, i.e. future volatility. Future volatility is a subjective measure similar to future value. Option’s value is most sensitive to volatility changes. When volatility is low, the option values tend toward their lower bounds. Ø The lower bound of a European-style call option: Zero or the stock less the present value of the exercise price, whichever is greater. Ø Ø Ø The rho of a call is positive because purchasing a call option allows an investor to earn interest on the money that otherwise would have gone to purchasing the stock. The higher the interest rate, the higher the call value. The rho of a put is negative because purchasing a put option rather than selling the stock deprives an investor of the potential interest that would have been earned from the proceeds of selling the stock. The higher the interest rate, the lower the put value. When interest rates are zero, the call and put option values are the same for at-the money options. As interest rates rise, the difference between call and put options increases. Reading 34 Valuation of Contingent Claims The option prices not highly sensitive to changes in interest rates change when compared with changes in volatility and changes in the stock. 6.6 Implied Volatility Implies volatility refers to the volatility estimated from option prices. Implied volatility is a measure of future volatility, whereas historical volatility is a measure of past volatility. The implied volatility can be estimated by using BSM model. The that implied volatility provides us information regarding the perceived uncertainty going forward and thereby allows us to gauge collective opinions of investors on the volatility of the underlying and the demand for options. Ø Ø Ø If the demand for options increases and the noarbitrage approach is not perfectly reflected in market prices (e.g. due to transaction costs) then the option prices increase, and hence, the observed implied volatility also increases. If the implied volatility of a put increases, it indicates that it is more expensive to buy downside protection with a put. Hence, the market price of hedging rises. The original BSM model assumes constant volatility of underlying instrument. However, practically, the implied volatilities vary depending on exercise prices and observe different implied volatilities for calls and puts with the same terms. Implied volatility also varies across time to expiration as well as across exercise prices. Implied volatility is also not constant through calendar time. There are two types of implied volatility: 1) Term structure of volatility: The implied volatility with respect to time to expiration is known as the term structure of volatility. The volatility surface is a three dimensional plot of the implied volatility with respect to both expiration time and exercise prices. 2) Volatility smile: The implied volatility with respect to the exercise price is known as the volatility smile or sometimes skew depending on the particular shape. The volatility smile is a two dimensional plot of the implied volatility with respect to the exercise price. We can trade futures and options on various volatility indexes available in the market in order to manage our vega exposure in other options. In the option markets, volatility can be used by investors as the medium in which to quote options. For example, rather than quote a particular call option as trading for €14.23, we may quote it as 30.00, where 30.00 denotes in percentage points the implied volatility based on a €14.23 option price. Quoting the option price in terms of implied volatility allows us to trade volatility. FinQuiz.com Important to remember: Ignoring rounding errors, there is a one-to-one relationship between the implied volatility and the option price. Uses of Implied Volatility: § § § Implied volatility can be used to assess the relative value of different options, neutralizing the moneyness and time to expiration effects. Implied volatility can be used to revalue existing positions over time. Regulators, banks, compliance officers, and most option traders use implied volatilities to communicate information related to options portfolios because implied volatilities provide the “market consensus” valuation. Example: The Chicago Board Options Exchange S&P 500 Volatility Index, known as the VIX, is a volatility index. The VIX is quoted as a percent and reflects the implied volatility of the S&P 500 over the next 30 days. VIX is often termed the fear index because it is viewed as a measure of market uncertainty. Thus, an increase in the VIX index is regarded as greater investor uncertainty. Example: If a trader thinks that based on the current outlook, the implied volatility of S&P 500 (say 20%) should be 25%, it indicates that volatility is understated by the dealer. In this case, since the S&P 500 call is expected to increase in value. Hence, trader would buy the call. Practice: Example 20 & 21, Reading 34, Curriculum. 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