Derivatives (Comm 4202) Assignment #3 – Due March 6, 2020 1. It is March 5, 2020. A stock index is traded at a level of 3123, the risk-free rate is 2.53% with continuous compounding, and the dividend yield is 3.08%. What is the futures price for an index futures contract that expires on September 18, 2020? 2. The USD-CAD exchange rate is 1.3283, the 6-month Canadian risk-free rate is 1.73%, and the 6-month U.S. risk-free rate is 0.8%. Both risk-free rates are continuously compounded. From a U.S. investor’s perspective, what is the futures price for a USD-CAD currency futures contract that expires in six months? 3. A stock is expected to pay a dividend of $1 per share in two months and in five months. The stock price is currently traded at $50 per share, and the 2-month, 3-month, 5-month, and 6-month risk-free rates are 3%, 3.025%, 3.05%, and 3.10% per annum with continuous compounding, respectively. (a) What is the futures price on the stock? What is the initial value of the futures contract to a long position holder? (b) If the price of the stock is $48 and the term structure of interest rates is unchanged in three months, what will be the futures price and the value of the short position in the futures contract? 4. A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 5% per annum with continuous compounding. (a) What is the value of a six-month European call option with a strike price of $42? (b) What is the value of a six-month American put option with a strike price of $42? 5. Consider an option on a stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months. 1 (a) What is the price of the option if it is a European call or put based on the Black-Scholes-Merton model? (b) What is the price of the option if it is an American call or put based on a two-period binomial model? 6. The price of a stock index futures contract is at 423.70, and the stock index level is at 420.55. The three-month call option on the index with a strike price of 400 is traded at $26.25, and the three-month put option with the same expiration is traded at $3.25. The three-month continuous compounding risk-free rate is 2.75%. Determine whether the futures and options are priced correctly in relation to each other. If they are not, construct a risk-free portfolio and show how it will earn a rate better than the risk-free rate. 2 Option Valuation Models Derivatives (Comm 4202) Option Pricing Models In the absence of arbitrage, the law of one price says any two financial securities having exactly the same payoffs in all possible future economic scenarios must have the same price. Especially, if a portfolio’s payoff (such as a risk-less bond) has no uncertainty, its cost must be equal to the discounted value of the cash flows at the risk-free rate. 1 A. Binomial Models A one-period binomial model Suppose a stock is currently traded at S . It will either go up to Su or go down to Sd with some probability law. The risk free rate is r with continuous compounding. c Yonggan Zhao, Rowe School of Business, Dalhousie University 1 Option Valuation Models Derivatives (Comm 4202) We want to evaluate a derivative security that will pay fu if the stock goes up or fd if the stock goes down. Below is a diagram of the one-period prices (cash flows) for the stock and the derivative security : Our objective is to construct a riskless portfolio using the stock and the derivative security. Suppose we hold ∆ shares of the stock and short the derivative security. The c Yonggan Zhao, Rowe School of Business, Dalhousie University 2 Option Valuation Models Derivatives (Comm 4202) portfolio’s payoff is  ∆S − f , u u ∆Sd − fd, if the stock price goes up (1) if the stock price goes down To make the portfolio risk-less, we just need set the payoffs equal to each other across the possible states, ∆Su − fu = ∆Sd − fd. That is, ∆= fu − fd . (2) Su − Sd Hence, holding ∆ shares and shorting the derivative simultaneously yield a riskless portfolio. The cost of this portfolio must be ∆S − f = (∆Su − fu)e−rT (discounted at the risk free rate!) where T is the time length in years to the expiration date. Hence, f = ∆S − (∆Su − fu)e−rT c Yonggan Zhao, Rowe School of Business, Dalhousie University (3) 3 Option Valuation Models Derivatives (Comm 4202) Example 1.1. Suppose the current stock price is $20 and it will be either $22 or $18 at the end of three months. Find the value of a call option to buy the stock for $21 in three months. The risk free rate is 12%. Solution: S = 20, Su = 22, Sd = 18, r = 0.12, T = 3/12 = 0.25, K = 21. Option’s payoff is  f If the stock price goes up u = max(22 − 21, 0) = 1, fd = max(18 − 21, 0) = 0, If the stock price goes down Thus, by equation (2) ∆= 1−0 = 0.25. 22 − 18 and the option’s value (or price) is, by equation (3) f = 0.25 ∗ 20 − (0.25 ∗ 22 − 1)e−0.12∗0.25 = 0.633 c Yonggan Zhao, Rowe School of Business, Dalhousie University 4 Option Valuation Models B. Derivatives (Comm 4202) The hedging portfolio The replicating portfolio of the risk free asset is ∆ = 0.25 shares and a short position in the derivative security. It can be deduced that the value of the derivative security can be replicated using the stock and the risk-free security. To replicate the derivative security, one needs to hold ∆ shares and a position in the risk free security: B = −(∆S − f ) = −(∆Su − fu)e−rT The portfolio composed of ∆ shares of the underlying asset and a loan of B in the risk-free asset is termed the hedging portfolio of the derivative security. The value of the derivative security can be represented as f = ∆S + B. ∆ and B are called the hedging portfolio weights. In Example 1.1, the hedging portfolio for the call option consists of ∆ = 0.25 shares and a position of B = −(0.25 ∗ 22 − 1)e−0.12∗0.25 = −4.3670 in the c Yonggan Zhao, Rowe School of Business, Dalhousie University 5 Option Valuation Models Derivatives (Comm 4202) risk-free asset. The value of the call option is then the cost of the hedging portfolio: f = ∆S + B = 0.25 ∗ 20 − 4.3670 = $0.6330. C. Risk-neutral valuation It is surprising that the probabilities characterizing the likelihood of the market directional movement never appear in the pricing mechanism. Actually, these important quantities are embedded in the dynamics of the underlying stock prices. We usually value a security on the bases of expectation of the cash flows. Substitute the ∆ given by equation (2) into equation (3) and simplify, f = e−rT [pfu + (1 − p)fd] where p= erT − d u−d c Yonggan Zhao, Rowe School of Business, Dalhousie University (4) . 6 Option Valuation Models Derivatives (Comm 4202) The quantity p must satisfy 0 ≤ p ≤ 1 in the absence of arbitrage, so p and 1 − p form a probability measure and are called risk neutral probabilities for the two future market states, up and down. With this definition, the valuation formula for the derivatives security given by equation (4) can be interpreted as the expected payoff discounted at the risk-free rate (Thrilling!!!). Remark. The assumption of arbitrage free ensures u ≥ erT ≥ d. Verify! Example 1.2. Suppose the current stock price is $100, and it is known that at the end of three months it will be either up by 10% or down by 8%. The risk-free rate is 5% per annum. Find the value of a put option to sell the stock for $105 in three months. Calculate the hedging portfolio weights. Solution: Su = 100 ∗ 1.1 = 110 and d = 100 ∗ 0.92 = 92. The risk neutral probability c Yonggan Zhao, Rowe School of Business, Dalhousie University 7 Option Valuation Models Derivatives (Comm 4202) for the upstate is p= 100 ∗ e0.05∗0.25 − 92 110 − 92 So, the price of the put option is = 0.5143 f = e−0.05∗0.25(0.5143∗max(105−110, 0)+0.4857∗max(105−92, 0)) = $6.24 The hedging portfolio for the put option consists of  ∆ = max(105−110,0)−max(105−92,0) = −0.7222 110−92 B = −(∆Su − fu)e−rT = −(−0.7222 ∗ 110 − 0)e−0.05∗0.25 = $78.4552 which means shorting 0.7222 shares and investing $78.4552 at the risk free rate. We can now verify the valuation of the put option using the hedging portfolio f = −0.7222 ∗ 100 + 78.46 = $6.24. Note that, under the risk neutral probability measure, the expected returns on both the underlying stock price and the option are equal to the risk free rate. This is the fundamental principle of risk neutral valuation. c Yonggan Zhao, Rowe School of Business, Dalhousie University 8 Option Valuation Models D. Derivatives (Comm 4202) A two-step binomial model In the one-step model, the valuation approach works for both European and American types of derivatives. However, the valuation process is very different for multiple-period models. Suppose a stock is currently traded at S . It will either go up to Su or go down to Sd for the next two three-month periods. The risk free rate is r with continuous compounding. The diagram for the stock price dynamic: c Yonggan Zhao, Rowe School of Business, Dalhousie University 9 Option Valuation Models Derivatives (Comm 4202) The value of the derivative over time is calculated in the same way as for the one-period model. We first calculate the payoffs, fuu, fud, and fdd, of the derivative in period two, using the contract specific parameters and terms. We then find the values of the derivative, fu and fd, in period 1, using the risk valuation approach:  f −r12 (T −t) [pfuu + (1 − p)fud] u = e fd = e−r12(T −t)[pfud + (1 − p)fdd] where t stands for the time length between now and the end of period 1 and r12 is the forward rate from period 1 to period 2. Finally, we can calculate the value of the derivative: f = e−r1t[pfu + (1 − p)fd], where r1 is the spot rate for time t. A general formula for the two period pricing model (only for European options not for American options) is f = e−r2T [p2fuu + 2p(1 − p)fud + (1 − p)2fdd], c Yonggan Zhao, Rowe School of Business, Dalhousie University 10 Option Valuation Models Derivatives (Comm 4202) where r2 is the spot rate for time T . However, this formula cannot be used for valuing American type of derivatives, because of early exercise opportunity. Example 1.3. Suppose a stock is traded at $30 a share, and it will either increase by 15% or decrease by 10% for the next two three-month periods. The risk free rate is 5% (flat term structure). Find the value of a European call option to sell the stock for $31 in 6 months. Solution:  S u Sd = 30 ∗ 1.15 = 34.5 = 30 ∗ 0.9 = 27     Suu Sud = Sdu    Sdd c Yonggan Zhao, Rowe School of Business, Dalhousie University = 30 ∗ 1.152 = 39.675 = 30 ∗ 1.15 ∗ 0.9 = 31.05 = 30 ∗ 0.92 = 24.3 11 Option Valuation Models Derivatives (Comm 4202) The option’s payoffs in two periods are     fuu = max(39.675 − 31, 0) = 8.675 fud    fdd = max(31.05 − 31, 0) = 0.05 = max(24.30 − 31, 0) = 0 The dynamics of the stock price and the option’s payoff are given below: c Yonggan Zhao, Rowe School of Business, Dalhousie University 12 Option Valuation Models Derivatives (Comm 4202) The risk neutral probability for the up move is p= e0.05∗0.25 − 0.9 = 0.4503. 1.15 − 0.9 Note that, if the term structure is NOT flat, the forward rate must be used for valuation. The values of the option for the up and down moves in period 1 are  f −0.05∗0.25 [0.4503 ∗ 8.675 + 0.5497 ∗ 0.05] = 3.885 u = e fd = e−0.05∗0.25[0.4503 ∗ 0.05 + 0.5497 ∗ 0] = 0.0222 The price of the option now is f = e−0.05∗0.25[0.4503 ∗ 3.885 + 0.5497 ∗ 0.0222] = 1.7379 Using the general formula we also have f = e−0.05∗0.5[0.45032∗8.675+2∗0.4503∗0.5497∗0.05+0.54972∗0] = 1.7379 c Yonggan Zhao, Rowe School of Business, Dalhousie University 13 Option Valuation Models 2 Derivatives (Comm 4202) Valuation of American Options The valuation methodology introduced in the previous section works for European options, but it is not for American options due to early exercise opportunity. However, in terms of valuation procedure, the only difference is that, at each decision point, we need to see whether the value of exercising the option is greater than that of holding it. At a decision point, the value of the option is Vhold if it is not exercised, while the cash flow received on exercise of the option is Vexercise. If Vexercise > Vhold, we then exercise the option. Otherwise, we continue to hold the security. So, the value of the option at this decision point is the greater of the two values. As American calls have the same value as the European counterparts, we show how to valuate an American put option. Example 2.1. Suppose a stock is traded at $30 a share, and it will either increase by 15% or decrease by 10% for the next two three-month periods. The risk c Yonggan Zhao, Rowe School of Business, Dalhousie University 14 Option Valuation Models Derivatives (Comm 4202) free rate is 5%. Find the value of an American put option to sell the stock for $31.5 in 6 months. Solution: The model settings for the stock are the same as those in Example 1.3. The put option’s payoffs at expiration date are     fuu = max(31.5 − 39.675, 0) = 0 fud = max(31.5 − 31.05, 0) = 0.45    fdd = max(31.5 − 24.3, 0) = 7.2 The dynamics of the underlying stock price and the option’s payoff are as below c Yonggan Zhao, Rowe School of Business, Dalhousie University 15 Option Valuation Models Derivatives (Comm 4202) In period 1, if the put option is not exercised  f −0.05∗0.25 [0.4503 ∗ 0 + 0.5497 ∗ 0.45] = 0.2443 u,hold = e fd,hold = e−0.05∗0.25[0.4503 ∗ 0.45 + 0.5497 ∗ 7.2] = 4.1088 In period 1, if the put option is exercised  f u,exercise = 31.5 − 34.5 = −3 fd,exercise = 31.5 − 27 = 4.5 Since fu,exercise < fu,hold, the put option is held at the up move in period 1. Since fu,exercise > fu,hold, the option is exercised at the down move in period 1. Hence  f = max(fu,hold, fu,exercise) = 0.2443 fd = max(fd,hold, fd,exercise) = 4.5 u The price of the put option in the beginning is then, if not exercised, fhold = e−0.05∗0.25 ∗ [0.4503 ∗ 0.2443 + 0.5497 ∗ 4.5] = 2.5516 If it is exercised in the beginning, fexercise = max(31.5 − 30, 0) = 1.5 c Yonggan Zhao, Rowe School of Business, Dalhousie University 16 Option Valuation Models Derivatives (Comm 4202) Thus, the option is not exercised now, and the price of the option is f = max(fexercise, fhold) = 2.5516 Remark. An option is not exercised in the beginning, otherwise, there will be an arbitrage opportunity. The previous two-period valuation approach for European options does not give the same solution: e−0.05∗0.5[0.45032 ∗ 0 + 2 ∗ 0.4503 ∗ 0.5497 ∗ 0.45 + 0.54972 ∗ 7.2] = 2.3392 which is less than its American counterpart. 3 The Black-Scholes-Merton Model The setting of the Black-Scholes-Model is quite different from the binomial model. As observed, stock price changes very often with infinitely many possible outcomes – A continuous stochastic process is assumed. Within an arbitrarily small time intervalt to t + dt, stock returns are assumed to follow a c Yonggan Zhao, Rowe School of Business, Dalhousie University 17 Option Valuation Models Derivatives (Comm 4202) log-normal distribution dSt = µdt + σdZt St where St stands for price of the stock at time t and Zt being normally dis√ tributed with mean 0 and standard deviation σ t. µ and σ are called the model drift and diffusion parameters. The stock price St can be written as St = 1 (µ− σ 2 )t+σZt S0 e 2 where S0 is the stock price at time now. As a result, the expected stock return with continuous compounding, ln St/S0, is normally distributed with mean (µ − 12 σ 2)t and standard deviation of √ σ t. σ is usually called the volatility. The expected price and the variance in time t c Yonggan Zhao, Rowe School of Business, Dalhousie University 18 Option Valuation Models Derivatives (Comm 4202) are, respectively, E[St] = S0eµt 2 V [St] = S02e2µt(eσ t − 1) Example 3.1. Suppose the current stock price is S0 = $10, µ = 0.25 and, σ = 0.4. What is expected return with continuous compounding and the volatility of the stock price? What is the expected price and the variance of the stock in 6 months? Solution: The expected return (with continuous compounding) in 6 months E[ln(ST /S0)] = (0.25 − 0.5 ∗ 0.42) ∗ 0.5 = 8.5% and the standard deviation √ V [ln(ST /S0)] = 0.4 0.5 = 11.31% The expected price in 6 months is E[ST ] = 10 ∗ e0.25∗0.5 = 11.3315 c Yonggan Zhao, Rowe School of Business, Dalhousie University 19 Option Valuation Models Derivatives (Comm 4202) and the variance 2 2∗0.25∗0.5 V [ST ] = 10 e A. [e 0.42 ∗0.5 − 1] = 1.0694. Basic concepts under the Black-Scholes-Merton model • The option price and the stock price depend on the same underlying source of uncertainty • We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty. • The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate B. Risk-neutral valuation As in the discrete model case, the probability is adjusted so that the expected returns on the option and the underlying stock are equal to the risk free rate. c Yonggan Zhao, Rowe School of Business, Dalhousie University 20 Option Valuation Models Derivatives (Comm 4202) Thus, we need to adjust the parameter µ to the risk free rate, so that the stock will earn the risk free rate with continuously compounding. The new price dynamics under the risk neutral probability measure is dSt = r dt + σd Zt St and the price of a European option price with maturity T is C = e−rT E[max(ST − K, 0)]. Let X = ln ST /S0, then X is normally distributed with mean (r − 12 σ 2)T and variance σ 2T . Hence, Z ∞ C = e−rT max(S0ex − K, 0)φ(x)dx −∞ where φ(x) is the normal density function: φ(x) = √ 1 2πT σ c Yonggan Zhao, Rowe School of Business, Dalhousie University e− 1 x−(r− σ 2 )T 2 σ2 T !2 . 21 Option Valuation Models C. Derivatives (Comm 4202) The Black-Scholes-Merton formula We just need to calculate the above integral to obtain the BSM pricing formula for a European call option with a strike price K and T years to expire: √ −rT C = S0N (d) − Ke N (d − σ t) where ln(S0/K) + (r + 21 σ 2)T d= √ σ T and Z x 2 1 − y2 e dy N (x) = √ 2π −∞ is the cumulative distribution function of the standard normal random variable. Example 3.2. Suppose the risk free rate r = 0.08 and the stock price is traded at $100 a share. The stock price follows dSt = 0.25 dt + 0.40 dZt. St Calculate a call option that can buy the stock for $100 in 6 months. c Yonggan Zhao, Rowe School of Business, Dalhousie University 22 Option Valuation Models Derivatives (Comm 4202) Solution: ln(100/100) + (0.08 + 0.5 ∗ 0.42) ∗ 0.5 d= = 0.282843 √ 0.4 ∗ 0.5 We use Excel to find the cumulative distribution function for the standard normal random variable, normsdist(x), √ N (0.282843) = 0.611351 and N (0.282843 − 0.4 0.5) = 0.500000 Hence, the price of the call option is c = 100 ∗ 0.611351 − 100 ∗ e−0.08∗0.5 ∗ 0.500000 = 13.095658. D. Hedging portfolios for European options Similar to the binomial model, European options can be completely hedged under BSM setting. √ −rT For calls, long N (d) shares of the stock and take a loan of Ke N (d−σ T ) amount in the risk free asset. c Yonggan Zhao, Rowe School of Business, Dalhousie University 23 Option Valuation Models Derivatives (Comm 4202) −rT For puts, short N (−d) shares and deposit Ke free asset. E. √ N (−(d − σ t)) in the risk Matching σ with u and d The binomial model uses the up and down moves, u and d, while the BlackScholes-Merton model uses stock price volatility. How these quantities are related? The formulas are u=e √ σ ∆t and d = 1/u where ∆t is the length of one time step on the tree. With a dividend-style paying underlying asset at rate q , the risk-neutral probability is then represented as p= e (r−q)∆t √ σ e ∆t −e − √ −σ ∆t √ −σ ∆t e √ The arbitrage-free condition guarantees −σ < (r − q) ∆t < σ . c Yonggan Zhao, Rowe School of Business, Dalhousie University 24 Option Valuation Models Derivatives (Comm 4202) In Example 3.2, r = 5%, q = 2%, ∆t = 0.25, then the stock volatility must √ √ σ > |((r − q) ∆t| = |(0.05 − 0.02) 0.25| = 15%. If σ = 30%, u=e √ 0.30 0.25 and = 1.1682 d = 1/1.1618 = 0.8607 The risk neutral probability for the up move is p= e (0.05−0.02)∗0.25 √ 0.3∗ 0.25 e −e √ −0.3∗ 0.25 √ −0.3∗ 0.25 e = 0.4876 − Example 3.3. Using a two-step binomial model, find the value of the call option in Example 3.2. Solution: As calculated above, the risk neutral probability for the up move is p = 0.4876 The payoff of the call option in period 2 is     f uu = max(S0uu − 18, 0) = 9.2938 f ud = max(S0ud − 18, 0) = 2    f dd = max(S0dd − 18, 0) = 0 c Yonggan Zhao, Rowe School of Business, Dalhousie University 25 Option Valuation Models Derivatives (Comm 4202) Thus  f u fd = e−0.05∗0.25[0.4876 ∗ 9.2938 + 0.5124 ∗ 2] = 5.4874 = e−0.05∗0.25[0.4876 ∗ 2 + 0.5124 ∗ 0] = 0.9631 The price of the option is c = e−0.05∗0.25[0.4876 ∗ 5.4874 + 0.5124 ∗ 0.9631] = 3.1298 Remark 1: Note the differences of the prices obtained using the binomial and the BSM. Remark 2: Sometimes matching with continuous compounding returns may be more efficient. c Yonggan Zhao, Rowe School of Business, Dalhousie University 26 
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