BG_FNCE4040_Case3Sol

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Derivative Securities, Fall 2023 FNCE 4040, Case 3 Solutions Estes Park: Dynamic Option Pricing You are an analyst working for Estes Park Associates, a hedge fund that specializes in the tech sector. The partners of the fund believe that the company Iperion is overvalued and they have asked you to look into the pricing of a put option on Iperion with expiration date in three months and a strike price of $ 90. After studying the historical price dynamics of Iperion over the past few years, you estimate that Iperion’s annual volatility σ is 77.3% (very volatile stock!). Given the current stock price of $ 100, you decide to use a three-period binomial model to price Iperion’s put option, where one period corresponds to one month. 1. (*) As a first step, you need to determine the proportional stock movements u and d from Iperion’s annual volatility and the length of a period in your binomial model. Then, write down the evolution of the stock price over the next three months. Solution. Using the formulas to calibrate the Binomial model, u = e σ √ T/n , d = (1 /u ) = e − σ √ T/n , where σ = 0 . 773, T = 3 / 12, and n = 3, we obtain that u = e 0 . 773 √ 1 / 12 = 1 . 25 , d = (1 / 1 . 25) = 0 . 8 . Therefore, every month the stock price either goes up by 25% or down by 20%. This means that the stock price is estimated to evolve as follows: Iperion’s Stock Price today in 1 months in 2 months in 3 months 195.31 156.25 125.00 125.00 100.00 100.00 80.00 80.00 64.00 51.20 2. (*) As a second step, you need to compute the risk-neutral probabilities in each relevant node of the tree. To this end, you take into account that a risk-free bond gives a monthly (not annual) return of 1%, and this rate is expected to remain constant over the next three months. Solution. Since the stock price moves in a proportional way, the risk-neutral probability that the stock goes up in the next month is constant (i.e., it does not depend on which node you will end up in the next two months) and given by p = (1 + r f ) − d u − d = 1 . 01 − 0 . 8 1 . 25 − 0 . 8 = 0 . 4667 ⇒ (1 − p ) = 0 . 5333 . © Buffa-Garc´ ıa, Leeds School of Business Page 1 of 6

Derivative Securities, Fall 2023 FNCE 4040, Case 3 Solutions 3. (**) As a third step, you can finally compute the value of the put option in every period and in every state of the binomial tree. Solution. Solving for the pricing of the put option backward, we start from the expiration date and compute the option payoff, max( S T − 90 , 0), in all four possible states, uing the different stock prices in those states: Iperion’s Put Option Price today in 1 months in 2 months in 3 months 0 ? ? 0 ? ? ? 10 ? 38.8 Moving to the price of the put option in 2 months, there are three possible states. Using risk-neutral pricing, we obtain uu state: P 2 uu = 0 . 4667 × 0 + 0 . 5333 × 0 1 . 01 = 0 , ud (or du ) state: P 2 ud = 0 . 4667 × 0 + 0 . 5333 × 10 1 . 01 = 5 . 28 , dd state: P 2 dd = 0 . 4667 × 10 + 0 . 5333 × 38 . 8 1 . 01 = 25 . 11 . Hence, Iperion’s Put Option Price today in 1 months in 2 months in 3 months 0 0 ? 0 ? 5.28 ? 10 25.11 38.8 Moving to the price of the put option in 1 month, there are two possible states. Using risk- neutral pricing, we obtain u state: P 1 u = 0 . 4667 × 0 + 0 . 5333 × 5 . 28 1 . 01 = 2 . 79 , d state: P 1 d = 0 . 4667 × 5 . 28 + 0 . 5333 × 25 . 11 1 . 01 = 15 . 70 . Hence, © Buffa-Garc´ ıa, Leeds School of Business Page 2 of 6

Derivative Securities, Fall 2023 FNCE 4040, Case 3 Solutions Iperion’s Put Option Price today in 1 months in 2 months in 3 months 0 0 2.79 0 ? 5.28 15.70 10 25.11 38.8 Finally, moving to the price of the put option today, we obtain that P 0 = 0 . 4667 × 2 . 79 + 0 . 5333 × 15 . 70 1 . 01 = 9 . 58 . Hence, Iperion’s Put Option Price today in 1 months in 2 months in 3 months 0 0 2.79 0 9.58 5.28 15.70 10 25.11 38.8 4. (**) To impress your boss, you decide to include in your analysis about the pricing of Iperion’s Put option the dynamic trading strategy that would allow you to replicate the put option over time. This may be useful to the fund in case some arbitrage opportunity were to materialize. So, you need to find out how much Iperion’s stock and risk-free bond to hold today, in one month (for each of the two states), and in two months (for each of the three states). Solution. Proceeding again backwards, consider the replicating portfolios at date 2. • In the uu state, we select the replicating portfolio (∆ 2 uu , B 2 uu ) that matches the two possible payoffs over the next month. Since the payoffs are 0 in both states, the replicating portfolio is ∆ 2 uu = 0 and B 2 uu = 0 (i.e., stay out of the market). So, the value of your portfolio in this state is obviously 0. • In the ud state, we select the replicating portfolio (∆ 2 ud , B 2 ud ) that matches the two possible payoffs over the next month: ∆ 2 ud × 125 + B 2 ud × 1 . 01 = 0 , ∆ 2 ud × 80 + B 2 ud × 1 . 01 = 10 . Solving this system of equations, we obtain ∆ 2 ud = − 0 . 2222 and B 2 ud = 27 . 50. The value of this portfolio in the ud state is − 0 . 2222 × 100 + 27 . 50 = 5 . 28. © Buffa-Garc´ ıa, Leeds School of Business Page 3 of 6

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Derivative Securities, Fall 2023 FNCE 4040, Case 3 Solutions • In the dd state, we select the replicating portfolio (∆ 2 dd , B 2 dd ) that matches the two possible payoffs over the next month: ∆ 2 dd × 80 + B 2 dd × 1 . 01 = 10 , ∆ 2 dd × 51 . 20 + B 2 dd × 1 . 01 = 38 . 80 . Solving this system of equations, we obtain ∆ 2 dd = − 1 and B 2 dd = 89 . 11. The value of this portfolio in the dd state is − 1 × 64 + 89 . 11 = 25 . 11. Now consider the consider the replicating portfolios at date 1. • In the u state we select the replicating portfolio (∆ 1 u , B 1 u ) that matches the two possible payoffs over the next month: ∆ 1 u × 156 . 25 + B 1 u × 1 . 01 = 0 , ∆ 1 u × 100 + B 1 u × 1 . 01 = 5 . 28 . Solving this system of equations, we obtain ∆ 1 u = − 0 . 0939 and B 1 u = 14 . 52. The value of this portfolio in the u state is − 0 . 0939 × 125+14 . 52 = 2 . 79. Note that, in this binomial world, we are able to make sure at date 1 that we have enough money at date 2 to buy the replicating portfolio (at date 2) that synthetically creates the put’s payoffs at date 3. • In the d state we select the replicating portfolio (∆ 1 d , B 1 d ) that matches the two possible payoffs over the next month: ∆ 1 d × 100 + B 1 d × 1 . 01 = 5 . 28 , ∆ 1 d × 64 + B 1 d × 1 . 01 = 25 . 11 . Solving this system of equations, we obtain ∆ 1 d = − 0 . 5508 and B 1 d = 59 . 76. The value of this portfolio in the u state is − 0 . 5508 × 80 + 59 . 76 = 15 . 70. Again, creating this portfolio allows us to replicate, at date 1, the replicating portfolio at date 2. Finally, to find the replicating portfolio at date 0 (today), we solve for (∆ 0 , B 0 ) such that ∆ 0 × 125 + B 0 × 1 . 01 = 2 . 79 , ∆ 0 × 80 + B 0 × 1 . 01 = 15 . 70 , yielding ∆ 0 = − 0 . 2869 and B 0 = 38 . 27. The current value of the replicating portfolio is − 0 . 2869 × 100 + 38 . 27 = 9 . 58. This portfolio will allow you to replicate the replicating portfolio at date 1 that replicates the replicating portfolio at date 2 that replicates the payoffs of the put option at date 3. 5. (***) To conclude your analysis, you want to compute the expected returns of (holding until expiration) the put option throughout its life (i.e., at different points in the binomial tree). For this purpose, you need to estimate the true probability q that the stock moves up over the next month. Having estimated that the monthly expected stock return r S is 2.5%, you can infer the underlying true probability q . Solution. Given a monthly expected stock return r S is 2.5%, we can use the stock price today and the stock prices one month from now to infer the probability q . Indeed, q is such that 100 = q × 125 + (1 − q ) × 80 1 . 025 ⇒ q = 0 . 5 . © Buffa-Garc´ ıa, Leeds School of Business Page 4 of 6

Derivative Securities, Fall 2023 FNCE 4040, Case 3 Solutions Given the true probability q = 0 . 5 and the option prices in the Binomial tree, the evolution of the option expected return is as follows. • At date 0, the implicit monthly expected return of the put option, r P 0 , is such that 9 . 58 = 3 ( q × (1 − q ) 2 ) × 10 + (1 − q ) 3 × 38 . 80 (1 + r P 0 ) 3 = 0 . 375 × 10 + 0 . 125 × 38 . 80 (1 + r P 0 ) 3 ⇒ r P 0 = − 3 . 5% . Note that there are three paths that take you to the ddu node ( ddu , dud , and udd ), and one single path that led to the ddd state. • At date 1 in the u state, the implicit monthly expected return of the put option, r P 1 u , is such that 2 . 79 = (1 − q ) 2 × 10 (1 + r P 1 u ) 2 = 0 . 25 × 10 (1 + r P 1 u ) 2 ⇒ r P 1 u = − 5 . 3% . • At date 1 in the d state, the implicit monthly expected return of the put option, r P 1 d , is such that 15 . 70 = 2 ( q × (1 − q )) × 10 + (1 − q ) 2 × 38 . 8 (1 + r P 1 d ) 2 = 0 . 5 × 10 + 0 . 25 × 38 . 8 (1 + r P 1 d ) 2 ⇒ r P 1 d = − 3 . 2% . • At date 2 in the uu state, the implicit monthly expected return of the put option r P 2 uu is not defined because there are no states going forward that would give a strictly positive payoff at date 3, thus making the put option price P 2 uu = 0. • At date 2 in the ud state, the implicit monthly expected return of the put option, r P 2 ud , is such that 5 . 28 = (1 − q ) × 10 1 + r P 2 ud = 0 . 5 × 10 1 + r P 2 ud ⇒ r P 2 ud = − 5 . 3% . • At date 2 in the dd state, the implicit monthly expected return of the put option, r P 2 dd , is such that 25 . 11 = q × 10 + (1 − q ) × 38 . 8 1 + r P 2 dd = 0 . 5 × 10 + 0 . 5 × 38 . 8 1 + r P 2 dd ⇒ r P 2 dd = − 2 . 8% . To summarize: Expected Return of Iperion’s Put Option today in 1 months in 2 months NA − 5 . 3% − 3 . 5% − 5 . 3% − 3 . 2% − 2 . 8% The fact the put carries a negative risk-premium is intuitive: the put is equivalent to a short- position in the stock coupled with lending (long position in the risk-free bond), so it should have a negative expected return (as long as the stock has a positive risk premium). © Buffa-Garc´ ıa, Leeds School of Business Page 5 of 6

Derivative Securities, Fall 2023 FNCE 4040, Case 3 Solutions Note that the expected return of the option becomes more negative when the stock price goes up, and less negative when the stock price goes down. This is the case despite the replicating portfolio having a larger short position in the stock when the stock price goes down. The intuition for this seemingly puzzling finding is the fact that it is not just the amount of shorting that matters for the expected returns, but also the amount of lending (the bonds having a positive expected return of 1%). For instance, in the u state at date 1, the replicating portfolio involves a large short position relative to the amount of lending in that state. The punchline is that the risk of options fluctuates with the evolution of the underlying asset in non-trivial ways. © Buffa-Garc´ ıa, Leeds School of Business Page 6 of 6

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