Binomial Model and Option Valuation: Part I ECO 426: Financial Economics Monica Tran-Xuan University at Buffalo Monica Tran-Xuan (UB) Financial Economics 1/ 37 Table of Contents Recap Option Valuation Early Exercise Intrinsic and Time Values Binomial Model Monica Tran-Xuan (UB) Financial Economics 2/ 37 Table of Contents Recap Option Valuation Early Exercise Intrinsic and Time Values Binomial Model Monica Tran-Xuan (UB) Financial Economics 3/ 37 Recap/Intro ▶ Last time we used the concept of replicating portfolio payoffs and the law of one price to derive the put call parity relationship P + S0 = C + Monica Tran-Xuan (UB) Financial Economics K (1 + rf )T 4/ 37 Recap/Intro ▶ Last time we used the concept of replicating portfolio payoffs and the law of one price to derive the put call parity relationship P + S0 = C + K (1 + rf )T ▶ Today, we’ll talk more about option valuation and then use a simple model to price options on a stock Monica Tran-Xuan (UB) Financial Economics 4/ 37 Table of Contents Recap Option Valuation Early Exercise Intrinsic and Time Values Binomial Model Monica Tran-Xuan (UB) Financial Economics 5/ 37 Bounds on the Price of a Call Option ▶ Recall payoff of a call option at expiration is ( ST − K Payoffcall = 0 if ST > K if ST ≤ K or Payoffcall = max{ST − K , 0} Monica Tran-Xuan (UB) Financial Economics 6/ 37 Bounds on the Price of a Call Option ▶ Recall payoff of a call option at expiration is ( ST − K Payoffcall = 0 if ST > K if ST ≤ K or Payoffcall = max{ST − K , 0} ▶ Clearly, a lower bound on the value of a call option at any time is zero: Pcall ≥ 0 Monica Tran-Xuan (UB) Financial Economics 6/ 37 Bounds on the Price of a Call Option ▶ Recall payoff of a call option at expiration is ( ST − K Payoffcall = 0 if ST > K if ST ≤ K or Payoffcall = max{ST − K , 0} ▶ Clearly, a lower bound on the value of a call option at any time is zero: Pcall ≥ 0 ▶ Can we come up with another lower bound? Monica Tran-Xuan (UB) Financial Economics 6/ 37 Bounds on the Price of a Call Option ▶ Suppose that the underlying asset is a stock that pays dividend D just before the expiration date T . ST is the stock price at time T . Monica Tran-Xuan (UB) Financial Economics 7/ 37 Bounds on the Price of a Call Option ▶ Suppose that the underlying asset is a stock that pays dividend D just before the expiration date T . ST is the stock price at time T . ▶ Consider two portfolios: 1. A call option with strike price K on one share of stock that expires at a future time T 2. The stock plus a loan with repayment K + D at T Monica Tran-Xuan (UB) Financial Economics 7/ 37 Bounds on the Price of a Call Option ▶ Suppose that the underlying asset is a stock that pays dividend D just before the expiration date T . ST is the stock price at time T . ▶ Consider two portfolios: 1. A call option with strike price K on one share of stock that expires at a future time T 2. The stock plus a loan with repayment K + D at T ▶ Payoff of the first portfolio (call option) at expiration is Payoff1 = ▶ Payoff of the second portfolio (stock plus loan) at expiration is Payoff2 = Monica Tran-Xuan (UB) Financial Economics 7/ 37 Bounds on the Price of a Call Option ▶ Comparing these payoff, we see that Payoff1 ≥ Payoff2 ∀ST So the payoff of the portfolio 1 (the option) is always at least as large as the payoff of the portfolio 2 (stock plus loan) no matter what ST is (i.e., in all states of the world) Monica Tran-Xuan (UB) Financial Economics 8/ 37 Bounds on the Price of a Call Option ▶ Comparing these payoff, we see that Payoff1 ≥ Payoff2 ∀ST So the payoff of the portfolio 1 (the option) is always at least as large as the payoff of the portfolio 2 (stock plus loan) no matter what ST is (i.e., in all states of the world) ▶ This means that portfolio 1 should be more expensive to establish today than portfolio 2 Monica Tran-Xuan (UB) Financial Economics 8/ 37 Bounds on the Price of a Call Option ▶ Comparing these payoff, we see that Payoff1 ≥ Payoff2 ∀ST So the payoff of the portfolio 1 (the option) is always at least as large as the payoff of the portfolio 2 (stock plus loan) no matter what ST is (i.e., in all states of the world) ▶ This means that portfolio 1 should be more expensive to establish today than portfolio 2 ▶ =⇒ the price of the call today should be greater than or equal to the price of the stock and the loan today Monica Tran-Xuan (UB) Financial Economics 8/ 37 Bounds on the Price of a Call Option ▶ Let S0 represent the price of the stock today and PV (K ) and PV (D) represent the present values of a risk-free payment of K and D respectively at time T Monica Tran-Xuan (UB) Financial Economics 9/ 37 Bounds on the Price of a Call Option ▶ Let S0 represent the price of the stock today and PV (K ) and PV (D) represent the present values of a risk-free payment of K and D respectively at time T ▶ Then we know that Pcall ≥ S0 − PV (K ) − PV (D) Monica Tran-Xuan (UB) Financial Economics 9/ 37 Bounds on the Price of a Call Option ▶ Let S0 represent the price of the stock today and PV (K ) and PV (D) represent the present values of a risk-free payment of K and D respectively at time T ▶ Then we know that Pcall ≥ S0 − PV (K ) − PV (D) ▶ Combining the two lower bounds we have on Pcall gives Pcall ≥ max{0, S0 − PV (K ) − PV (D)} Monica Tran-Xuan (UB) Financial Economics 9/ 37 Bounds on the Price of a Call Option ▶ Let S0 represent the price of the stock today and PV (K ) and PV (D) represent the present values of a risk-free payment of K and D respectively at time T ▶ Then we know that Pcall ≥ S0 − PV (K ) − PV (D) ▶ Combining the two lower bounds we have on Pcall gives Pcall ≥ max{0, S0 − PV (K ) − PV (D)} ▶ This is a stronger lower bound on the price of a call Monica Tran-Xuan (UB) Financial Economics 9/ 37 Bounds on the Price of a Call Option ▶ Let S0 represent the price of the stock today and PV (K ) and PV (D) represent the present values of a risk-free payment of K and D respectively at time T ▶ Then we know that Pcall ≥ S0 − PV (K ) − PV (D) ▶ Combining the two lower bounds we have on Pcall gives Pcall ≥ max{0, S0 − PV (K ) − PV (D)} ▶ This is a stronger lower bound on the price of a call ▶ What about an upper bound? Monica Tran-Xuan (UB) Financial Economics 9/ 37 Bounds on the Price of a Call Option ▶ A clear upper bound on the price of a call is S0 Monica Tran-Xuan (UB) Financial Economics 10/ 37 Bounds on the Price of a Call Option ▶ A clear upper bound on the price of a call is S0 ▶ Nobody would pay more than S0 to purchase the right to buy a stock worth S0 today Monica Tran-Xuan (UB) Financial Economics 10/ 37 Bounds on the Price of a Call Option ▶ A clear upper bound on the price of a call is S0 ▶ Nobody would pay more than S0 to purchase the right to buy a stock worth S0 today ▶ We can conclude that max{0, S0 − PV (K ) − PV (D)} ≤ Pcall ≤ S0 Monica Tran-Xuan (UB) Financial Economics 10/ 37 Bounds on the Price of a Call Option Monica Tran-Xuan (UB) Financial Economics 11/ 37 Table of Contents Recap Option Valuation Early Exercise Intrinsic and Time Values Binomial Model Monica Tran-Xuan (UB) Financial Economics 12/ 37 Early Exercise ▶ Question: When (if ever) is it optimal to exercise an American call option on a stock that pays no dividends before expiration? Monica Tran-Xuan (UB) Financial Economics 13/ 37 Early Exercise ▶ Question: When (if ever) is it optimal to exercise an American call option on a stock that pays no dividends before expiration? ▶ Suppose the price of a stock today is S0 , and an investor has bought a call option on the stock with strike price K and maturity date T Monica Tran-Xuan (UB) Financial Economics 13/ 37 Early Exercise ▶ Question: When (if ever) is it optimal to exercise an American call option on a stock that pays no dividends before expiration? ▶ Suppose the price of a stock today is S0 , and an investor has bought a call option on the stock with strike price K and maturity date T ▶ We already showed that Pcall ≥ max{0, S0 − PV (K ) − PV (D)} ▶ Since the stock pays no dividends PV (D) = 0 Monica Tran-Xuan (UB) Financial Economics 13/ 37 Early Exercise ▶ Question: When (if ever) is it optimal to exercise an American call option on a stock that pays no dividends before expiration? ▶ Suppose the price of a stock today is S0 , and an investor has bought a call option on the stock with strike price K and maturity date T ▶ We already showed that Pcall ≥ max{0, S0 − PV (K ) − PV (D)} ▶ Since the stock pays no dividends PV (D) = 0 ▶ So we have Pcall ≥ max{0, S0 − PV (K )} Monica Tran-Xuan (UB) Financial Economics 13/ 37 Early Exercise ▶ We already know that whenever S0 < K the call option is never exercised (since you could just go to the market and buy the stock for S0 ) Monica Tran-Xuan (UB) Financial Economics 14/ 37 Early Exercise ▶ We already know that whenever S0 < K the call option is never exercised (since you could just go to the market and buy the stock for S0 ) ▶ So the only interesting cases are those with S0 ≥ K Monica Tran-Xuan (UB) Financial Economics 14/ 37 Early Exercise ▶ We already know that whenever S0 < K the call option is never exercised (since you could just go to the market and buy the stock for S0 ) ▶ So the only interesting cases are those with S0 ≥ K ▶ Since the investor prefers to have K dollars today than K dollars at date T , we have that K > PV (K ) Monica Tran-Xuan (UB) Financial Economics 14/ 37 Early Exercise ▶ We already know that whenever S0 < K the call option is never exercised (since you could just go to the market and buy the stock for S0 ) ▶ So the only interesting cases are those with S0 ≥ K ▶ Since the investor prefers to have K dollars today than K dollars at date T , we have that K > PV (K ) ▶ Therefore, S0 − PV (K ) > S0 − K Monica Tran-Xuan (UB) Financial Economics 14/ 37 Early Exercise ▶ Since S0 − PV (K ) > S0 − K , and Pcall ≥ max{0, S0 − PV (K )}, this tells us that Pcall ≥ max{0, S0 − PV (K )} ≥ S0 − PV (K ) > S0 − K Monica Tran-Xuan (UB) Financial Economics 15/ 37 Early Exercise ▶ Since S0 − PV (K ) > S0 − K , and Pcall ≥ max{0, S0 − PV (K )}, this tells us that Pcall ≥ max{0, S0 − PV (K )} ≥ S0 − PV (K ) > S0 − K ▶ So Pcall > S0 − K And S0 − K is the payoff from exercising the option today Monica Tran-Xuan (UB) Financial Economics 15/ 37 Early Exercise ▶ Since S0 − PV (K ) > S0 − K , and Pcall ≥ max{0, S0 − PV (K )}, this tells us that Pcall ≥ max{0, S0 − PV (K )} ≥ S0 − PV (K ) > S0 − K ▶ So Pcall > S0 − K And S0 − K is the payoff from exercising the option today ▶ So we can conclude that the investor is better off selling the option for Pcall than exercising it Monica Tran-Xuan (UB) Financial Economics 15/ 37 Early Exercise ▶ So we have shown that call options on stocks that pay no dividends are worth more alive than exercised Monica Tran-Xuan (UB) Financial Economics 16/ 37 Early Exercise ▶ So we have shown that call options on stocks that pay no dividends are worth more alive than exercised ▶ For such calls, the right to exercise before maturity is worthless Monica Tran-Xuan (UB) Financial Economics 16/ 37 Early Exercise ▶ So we have shown that call options on stocks that pay no dividends are worth more alive than exercised ▶ For such calls, the right to exercise before maturity is worthless ▶ =⇒ all else equal, American calls and European call on stocks that pay no dividends are worth the same! Monica Tran-Xuan (UB) Financial Economics 16/ 37 Early Exercise ▶ So we have shown that call options on stocks that pay no dividends are worth more alive than exercised ▶ For such calls, the right to exercise before maturity is worthless ▶ =⇒ all else equal, American calls and European call on stocks that pay no dividends are worth the same! ▶ Note that this conclusion fails to hold if a stock pays dividends: the presence of P(D) in the chain of inequalities would not allow us to conclude that Pcall > S0 − K Monica Tran-Xuan (UB) Financial Economics 16/ 37 Early Exercise ▶ So we have shown that call options on stocks that pay no dividends are worth more alive than exercised ▶ For such calls, the right to exercise before maturity is worthless ▶ =⇒ all else equal, American calls and European call on stocks that pay no dividends are worth the same! ▶ Note that this conclusion fails to hold if a stock pays dividends: the presence of P(D) in the chain of inequalities would not allow us to conclude that Pcall > S0 − K ▶ For a stock paying a dividend, the investor may want to exercise early to get the dividend payment of the stock Monica Tran-Xuan (UB) Financial Economics 16/ 37 Early Exercise ▶ Does this conclusion hold for put option? I.e., are American and European puts worth the same? Monica Tran-Xuan (UB) Financial Economics 17/ 37 Early Exercise ▶ Does this conclusion hold for put option? I.e., are American and European puts worth the same? ▶ Unfortunately not. Suppose the price of a stock today is S0 . An investor has bought a put option on the stock with exercise price K and maturity date T . Monica Tran-Xuan (UB) Financial Economics 17/ 37 Early Exercise ▶ Does this conclusion hold for put option? I.e., are American and European puts worth the same? ▶ Unfortunately not. Suppose the price of a stock today is S0 . An investor has bought a put option on the stock with exercise price K and maturity date T . ▶ Recall that, given the price of the stock today, the payoff from the put would be ( K − S0 if S0 < K 0 if S0 ≥ K Monica Tran-Xuan (UB) Financial Economics 17/ 37 Early Exercise ▶ Suppose at some t < T the firm issuing the stock goes bankrupt and we have St = 0 Monica Tran-Xuan (UB) Financial Economics 18/ 37 Early Exercise ▶ Suppose at some t < T the firm issuing the stock goes bankrupt and we have St = 0 ▶ Then the payoff of the put cannot get any larger, so exercising is optimal at time t < T (since exercising the put later means the investor misses out on the opportunity cost of his money) Monica Tran-Xuan (UB) Financial Economics 18/ 37 Early Exercise ▶ Suppose at some t < T the firm issuing the stock goes bankrupt and we have St = 0 ▶ Then the payoff of the put cannot get any larger, so exercising is optimal at time t < T (since exercising the put later means the investor misses out on the opportunity cost of his money) ▶ Even in the case where St is small (but not zero), the possible gains from not exercising are small while the opportunity cost is large (the time-value of money) Monica Tran-Xuan (UB) Financial Economics 18/ 37 Early Exercise ▶ Suppose at some t < T the firm issuing the stock goes bankrupt and we have St = 0 ▶ Then the payoff of the put cannot get any larger, so exercising is optimal at time t < T (since exercising the put later means the investor misses out on the opportunity cost of his money) ▶ Even in the case where St is small (but not zero), the possible gains from not exercising are small while the opportunity cost is large (the time-value of money) ▶ We can conclude that it could be optimal to exercise a put option before expiration Monica Tran-Xuan (UB) Financial Economics 18/ 37 Table of Contents Recap Option Valuation Early Exercise Intrinsic and Time Values Binomial Model Monica Tran-Xuan (UB) Financial Economics 19/ 37 Intrinsic Value and Time Value ▶ Assume in what follows we are working with call options on a stock that pays no dividends. Monica Tran-Xuan (UB) Financial Economics 20/ 37 Intrinsic Value and Time Value ▶ Assume in what follows we are working with call options on a stock that pays no dividends. ▶ Note that even when a call option with expiration date this week is out of the money, it is not worthless since there exists some non-zero chance that the call will be in the money until maturity. Monica Tran-Xuan (UB) Financial Economics 20/ 37 Intrinsic Value and Time Value ▶ Assume in what follows we are working with call options on a stock that pays no dividends. ▶ Note that even when a call option with expiration date this week is out of the money, it is not worthless since there exists some non-zero chance that the call will be in the money until maturity. Definition: The intrinsic value of a call option with strike price K and current price of the underlying asset S0 is ( S0 − K if S0 ≥ K IVcall = 0 if S0 < K Monica Tran-Xuan (UB) Financial Economics 20/ 37 Intrinsic Value and Time Value ▶ Assume in what follows we are working with call options on a stock that pays no dividends. ▶ Note that even when a call option with expiration date this week is out of the money, it is not worthless since there exists some non-zero chance that the call will be in the money until maturity. Definition: The intrinsic value of a call option with strike price K and current price of the underlying asset S0 is ( S0 − K if S0 ≥ K IVcall = 0 if S0 < K ▶ The intrinsic value of a call option is the payoff of the option upon immediate exercise. Monica Tran-Xuan (UB) Financial Economics 20/ 37 Intrinsic Value and Time Value Definition: The time value of a call option is Pcall − IVcall . The time value of the call option is the part of the value of the option attributable to the fact that the option has not expired yet. Monica Tran-Xuan (UB) Financial Economics 21/ 37 Intrinsic Value and Time Value Definition: The time value of a call option is Pcall − IVcall . The time value of the call option is the part of the value of the option attributable to the fact that the option has not expired yet. Note: Do not confuse this with the time value of money. Monica Tran-Xuan (UB) Financial Economics 21/ 37 Table of Contents Recap Option Valuation Early Exercise Intrinsic and Time Values Binomial Model Monica Tran-Xuan (UB) Financial Economics 22/ 37 Binomial Model ▶ Is there a simple model that allows us to price options? Monica Tran-Xuan (UB) Financial Economics 23/ 37 Binomial Model ▶ Is there a simple model that allows us to price options? ▶ Yes, many of the option pricing models that exist require additional mathematical sophistication, but the important insights from these models can be presented in a much simpler version, the Binomial Model Monica Tran-Xuan (UB) Financial Economics 23/ 37 Binomial Model ▶ Is there a simple model that allows us to price options? ▶ Yes, many of the option pricing models that exist require additional mathematical sophistication, but the important insights from these models can be presented in a much simpler version, the Binomial Model ▶ This model will use the concept of replicating portfolio payoffs that we have seen before to find the (no-arbitrage) price of derivatives Monica Tran-Xuan (UB) Financial Economics 23/ 37 Binomial Model ▶ Suppose a stock’s price today is S0 = $100. Consider a call option on the stock that has strike price K = $110 and expiration date 1 year Monica Tran-Xuan (UB) Financial Economics 24/ 37 Binomial Model ▶ Suppose a stock’s price today is S0 = $100. Consider a call option on the stock that has strike price K = $110 and expiration date 1 year ▶ Assume the risk-free rate is 10% Monica Tran-Xuan (UB) Financial Economics 24/ 37 Binomial Model ▶ Suppose a stock’s price today is S0 = $100. Consider a call option on the stock that has strike price K = $110 and expiration date 1 year ▶ Assume the risk-free rate is 10% ▶ Suppose after 1 year there are two possible states of the world: ▶ The stock price goes down to S1 = $90 ▶ The stock price rises to S1 = $120 Monica Tran-Xuan (UB) Financial Economics 24/ 37 Binomial Model ▶ Suppose a stock’s price today is S0 = $100. Consider a call option on the stock that has strike price K = $110 and expiration date 1 year ▶ Assume the risk-free rate is 10% ▶ Suppose after 1 year there are two possible states of the world: ▶ The stock price goes down to S1 = $90 ▶ The stock price rises to S1 = $120 Monica Tran-Xuan (UB) Financial Economics 24/ 37 Binomial Tree ▶ We can draw a binomial tree to represent this stock 120 S0 = 100 90 Monica Tran-Xuan (UB) Financial Economics 25/ 37 Binomial Tree ▶ We can draw a binomial tree to represent this stock 120 S0 = 100 90 ▶ We can also draw the binomial tree for the payoffs of the call option on this stock: 10 C 0 Monica Tran-Xuan (UB) Financial Economics 25/ 37 Binomial Tree ▶ We can draw a binomial tree to represent this stock 120 S0 = 100 90 ▶ We can also draw the binomial tree for the payoffs of the call option on this stock: 10 C 0 ▶ We want to find C , the price of the call option at time 0. Monica Tran-Xuan (UB) Financial Economics 25/ 37 Replicating Portfolios ▶ Compare the payoffs of the call option to that of a portfolio consisting of one share of stock and borrowing $81.82 at the risk-free rate of 10%. The price of establishing this portfolio is 100 − 81.82 − 18.18. The payoffs look like: 120 - (1.1)(81.82) = 30 18.18 90 - (1.1)(81.82) = 0 Monica Tran-Xuan (UB) Financial Economics 26/ 37 Replicating Portfolios ▶ Compare the payoffs of the call option to that of a portfolio consisting of one share of stock and borrowing $81.82 at the risk-free rate of 10%. The price of establishing this portfolio is 100 − 81.82 − 18.18. The payoffs look like: 120 - (1.1)(81.82) = 30 18.18 90 - (1.1)(81.82) = 0 ▶ Note that these payoffs are the same as the payoffs of 3 call options Monica Tran-Xuan (UB) Financial Economics 26/ 37 Replicating Portfolios ▶ Since these two portfolios (the stock plus borrowing at the risk-free rate and the 3 calls) have the same payoffs, the Law of One Price tells us that they must have the same price Monica Tran-Xuan (UB) Financial Economics 27/ 37 Replicating Portfolios ▶ Since these two portfolios (the stock plus borrowing at the risk-free rate and the 3 calls) have the same payoffs, the Law of One Price tells us that they must have the same price ▶ So we know that it must be the case that 3C = 18.18 =⇒ C = 6.06 Monica Tran-Xuan (UB) Financial Economics 27/ 37 Replicating Portfolios ▶ Since these two portfolios (the stock plus borrowing at the risk-free rate and the 3 calls) have the same payoffs, the Law of One Price tells us that they must have the same price ▶ So we know that it must be the case that 3C = 18.18 =⇒ C = 6.06 ▶ Suppose C ̸= 6.06. Where is there an arbitrage opportunity? Monica Tran-Xuan (UB) Financial Economics 27/ 37 Perfect Hedge ▶ Alternative approach: Consider a portfolio consisting of one stock and writing 3 calls: Monica Tran-Xuan (UB) Financial Economics 28/ 37 Perfect Hedge ▶ Alternative approach: Consider a portfolio consisting of one stock and writing 3 calls: ▶ If the price of the stock in year 1 is $120, then the payoff of the call is $10, so the payoff of 3 written calls is −$30. Then the payoff of the whole portfolio is 120 − 30 = $90. Monica Tran-Xuan (UB) Financial Economics 28/ 37 Perfect Hedge ▶ Alternative approach: Consider a portfolio consisting of one stock and writing 3 calls: ▶ If the price of the stock in year 1 is $120, then the payoff of the call is $10, so the payoff of 3 written calls is −$30. Then the payoff of the whole portfolio is 120 − 30 = $90. ▶ If the price of the stock in year 1 is $90, then the payoff of the call is $0, so the payoff of 3 written calls is $0. Then the payoff of the whole portfolio is 90 − 0 = $90. Monica Tran-Xuan (UB) Financial Economics 28/ 37 Perfect Hedge ▶ Alternative approach: Consider a portfolio consisting of one stock and writing 3 calls: ▶ If the price of the stock in year 1 is $120, then the payoff of the call is $10, so the payoff of 3 written calls is −$30. Then the payoff of the whole portfolio is 120 − 30 = $90. ▶ If the price of the stock in year 1 is $90, then the payoff of the call is $0, so the payoff of 3 written calls is $0. Then the payoff of the whole portfolio is 90 − 0 = $90. ▶ This is a risk-free portfolio!! It pays $90 no matter the state of the world (i.e., no matter the final price of the stock). Monica Tran-Xuan (UB) Financial Economics 28/ 37 Perfect Hedge ▶ Since this is a risk-free portfolio paying $90 in one year, it must cost 90 = $81.82 today. (1.10) Monica Tran-Xuan (UB) Financial Economics 29/ 37 Perfect Hedge ▶ Since this is a risk-free portfolio paying $90 in one year, it must cost 90 = $81.82 today. (1.10) ▶ Another way to express the price of this portfolio is in terms of the initial stock price of $100 and the price of call options: 100 − 3C Monica Tran-Xuan (UB) Financial Economics 29/ 37 Perfect Hedge ▶ Since this is a risk-free portfolio paying $90 in one year, it must cost 90 = $81.82 today. (1.10) ▶ Another way to express the price of this portfolio is in terms of the initial stock price of $100 and the price of call options: 100 − 3C ▶ So we have that 81.82 = 100 − 3C =⇒ C = 6.06 Monica Tran-Xuan (UB) Financial Economics 29/ 37 Perfect Hedge ▶ Since this is a risk-free portfolio paying $90 in one year, it must cost 90 = $81.82 today. (1.10) ▶ Another way to express the price of this portfolio is in terms of the initial stock price of $100 and the price of call options: 100 − 3C ▶ So we have that 81.82 = 100 − 3C =⇒ C = 6.06 ▶ This second approach exploits the ability to create a perfect hedge, i.e., use a portfolio of shares of stock and options to replicate the payoffs of a risk-free bond Monica Tran-Xuan (UB) Financial Economics 29/ 37 Hedge Ratio ▶ How did we come up with the risk-free portfolio of one share of stock and 3 written calls? Monica Tran-Xuan (UB) Financial Economics 30/ 37 Hedge Ratio ▶ How did we come up with the risk-free portfolio of one share of stock and 3 written calls? ▶ Definition: The Hedge Ratio is definted as H = factors by which the stock price evolves. Monica Tran-Xuan (UB) Financial Economics Cu −Cd , uS0 −dS0 where u and d are the 30/ 37 Hedge Ratio ▶ How did we come up with the risk-free portfolio of one share of stock and 3 written calls? ▶ Definition: The Hedge Ratio is definted as H = factors by which the stock price evolves. Cu −Cd , uS0 −dS0 where u and d are the ▶ In our example, d = 0.9 and u = 1.2. Cu and Cd are the call’s payoff in the two states of the world. Monica Tran-Xuan (UB) Financial Economics 30/ 37 Hedge Ratio ▶ How did we come up with the risk-free portfolio of one share of stock and 3 written calls? ▶ Definition: The Hedge Ratio is definted as H = factors by which the stock price evolves. Cu −Cd , uS0 −dS0 where u and d are the ▶ In our example, d = 0.9 and u = 1.2. Cu and Cd are the call’s payoff in the two states of the world. ▶ In general, our stock tree looks like: uS0 S0 dS0 Monica Tran-Xuan (UB) Financial Economics 30/ 37 Hedge Ratio ▶ And our option tree looks like Cu C Cd Monica Tran-Xuan (UB) Financial Economics 31/ 37 Hedge Ratio ▶ And our option tree looks like Cu C Cd ▶ A portfolio consisting of H shares of stock and one written call will be risk-free. Monica Tran-Xuan (UB) Financial Economics 31/ 37 Hedge Ratio ▶ And our option tree looks like Cu C Cd ▶ A portfolio consisting of H shares of stock and one written call will be risk-free. ▶ In our example, the hedge ratio was H= 10 − 0 1 = 120 − 90 3 So that’s how we chose the portfolio of one share of stock and 3 written calls. Monica Tran-Xuan (UB) Financial Economics 31/ 37 Pricing Procedure Given S0 , u, d, K , and rf , the pricing procedure is: 1. Calculate S1 in the two states, S1 = uS0 and S1 = dS0 . Calculate the payoffs of the option in the two states implied by S1 and K . 2. Find the hedge ratio H. 3. Find the payoff of the portfolio composed of H shares of stock and one written option. 4. Check that this portfolio is risk-free!! 5. Find the present value of this portfolio using the risk-free rate. 6. Set the value above equal to HS0 − Poption 7. Solve for Poption Monica Tran-Xuan (UB) Financial Economics 32/ 37 Example Suppose S0 = 50, u = 1.1, d = 0.9, rf = 0.05, and we want to price a call option on this stock with strike price K = 50. Let’s use our pricing procedure. Monica Tran-Xuan (UB) Financial Economics 33/ 37 Example Suppose S0 = 50, u = 1.1, d = 0.9, rf = 0.05, and we want to price a call option on this stock with strike price K = 50. Let’s use our pricing procedure. ▶ Step 1: Calculate S1 in the two states and the payoffs of the option in the two states: Monica Tran-Xuan (UB) Financial Economics 33/ 37 Example So our stock price tree looks like: uS0 = 55 S0 = 50 dS0 = 45 And our option tree looks like Cu = 5 C Cd = 0 Monica Tran-Xuan (UB) Financial Economics 34/ 37 Example ▶ Step 2: Find the hedge ratio H: Monica Tran-Xuan (UB) Financial Economics 35/ 37 Example ▶ Step 2: Find the hedge ratio H: H= 5−0 1 = 55 − 45 2 ▶ Step 3: Construct a portfolio of H shares of stock and one written option, and find the payoffs of this portfolio in the two states: ▶ If the stock price goes up (state u), our payoff is Monica Tran-Xuan (UB) Financial Economics 35/ 37 Example ▶ Step 2: Find the hedge ratio H: H= 5−0 1 = 55 − 45 2 ▶ Step 3: Construct a portfolio of H shares of stock and one written option, and find the payoffs of this portfolio in the two states: ▶ If the stock price goes up (state u), our payoff is 1 55 − 5 = 22.5 2 ▶ If the stock price goes down (state d), our payoff is Monica Tran-Xuan (UB) Financial Economics 35/ 37 Example ▶ Step 2: Find the hedge ratio H: H= 5−0 1 = 55 − 45 2 ▶ Step 3: Construct a portfolio of H shares of stock and one written option, and find the payoffs of this portfolio in the two states: ▶ If the stock price goes up (state u), our payoff is 1 55 − 5 = 22.5 2 ▶ If the stock price goes down (state d), our payoff is 1 45 − 0 = 22.5 2 ▶ Step 4: Check that it’s risk-free! ✓ Monica Tran-Xuan (UB) Financial Economics 35/ 37 Example ▶ Step 5: Find the present value of this portfolio using the risk-free rate. The present value of a risk-free portfolio paying 22.5 in one period is Monica Tran-Xuan (UB) Financial Economics 36/ 37 Example ▶ Step 5: Find the present value of this portfolio using the risk-free rate. The present value of a risk-free portfolio paying 22.5 in one period is 22.5 = 21.43 1.05 ▶ Step 6: Set this value equal to HS0 − C : Monica Tran-Xuan (UB) Financial Economics 36/ 37 Example ▶ Step 5: Find the present value of this portfolio using the risk-free rate. The present value of a risk-free portfolio paying 22.5 in one period is 22.5 = 21.43 1.05 ▶ Step 6: Set this value equal to HS0 − C : 21.43 = 1 50 − C 2 ▶ Step 7: Solve for C : Monica Tran-Xuan (UB) Financial Economics 36/ 37 Example ▶ Step 5: Find the present value of this portfolio using the risk-free rate. The present value of a risk-free portfolio paying 22.5 in one period is 22.5 = 21.43 1.05 ▶ Step 6: Set this value equal to HS0 − C : 21.43 = 1 50 − C 2 ▶ Step 7: Solve for C : C = 25 − 21.43 = 3.57 And we’re done! We’ve priced this option. Monica Tran-Xuan (UB) Financial Economics 36/ 37 Probabilities/Expected Return ▶ Question: Where did the probabilities that the stock price would go up or down show up in this model/pricing procedure? Monica Tran-Xuan (UB) Financial Economics 37/ 37 Probabilities/Expected Return ▶ Question: Where did the probabilities that the stock price would go up or down show up in this model/pricing procedure? ▶ Answer: They didn’t! Monica Tran-Xuan (UB) Financial Economics 37/ 37 Probabilities/Expected Return ▶ Question: Where did the probabilities that the stock price would go up or down show up in this model/pricing procedure? ▶ Answer: They didn’t! ▶ The probabilities that the stock price increases or decreases don’t matter for pricing derivatives on the stock! Monica Tran-Xuan (UB) Financial Economics 37/ 37 Probabilities/Expected Return ▶ Question: Where did the probabilities that the stock price would go up or down show up in this model/pricing procedure? ▶ Answer: They didn’t! ▶ The probabilities that the stock price increases or decreases don’t matter for pricing derivatives on the stock! ▶ The expected return of the stock doesn’t affect the prices of options on the stock in this model. The prices are derived from a no-arbitrage condition. Monica Tran-Xuan (UB) Financial Economics 37/ 37 Binomial Model and Option Valuation: Part II ECO 426: Financial Economics Monica Tran-Xuan University at Buffalo Monica Tran-Xuan (UB) Financial Economics 1/ 27 Table of Contents Recap Two Step Binomial N-Step Binomial Monica Tran-Xuan (UB) Financial Economics 2/ 27 Table of Contents Recap Two Step Binomial N-Step Binomial Monica Tran-Xuan (UB) Financial Economics 3/ 27 Recap/Intro ▶ Last time we covered using the Hedge Ratio to construct a perfect hedge (risk-free portfolio) and price call options in a one step binomial tree d ▶ Defining H = uSCu −C , we found that a portfolio consisting of being long H 0 −dS0 shares of stock and short one option was risk-free ▶ Since it was risk-free, we can price it as a bond using the risk-free rate ▶ Today we will extend the binomial model to two steps and see how we can price derivatives Monica Tran-Xuan (UB) Financial Economics 4/ 27 Recap/Intro Recall our pricing procedure in the one-step model: 1. Calculate S1 in the two states, S1 = uS0 and S1 = dS0 . Calculate the payoffs of the option in the two states implied by S1 and K . 2. Find the hedge ratio H. 3. Find the payoff of the portfolio composed of H shares of stock and one written option. 4. Check that this portfolio is risk-free!! 5. Find the present value of this portfolio using the risk-free rate. 6. Set the value above equal to HS0 − Poption 7. Solve for Poption Monica Tran-Xuan (UB) Financial Economics 5/ 27 Table of Contents Recap Two Step Binomial N-Step Binomial Monica Tran-Xuan (UB) Financial Economics 6/ 27 Two-Step Binomial Model ▶ Suppose now we want to price a call option on a stock that evolves twice before the expiration of the option, i.e, T = 2 ▶ Suppose u = 1.1, d = 0.95, S0 = 100, and rf = 0.05 per period. We want to price a call option on this stock with strike price K = 110 and maturity T ▶ We want to adjust the pricing procedure to apply in this two-step version Monica Tran-Xuan (UB) Financial Economics 7/ 27 Two-Step Binomial Tree Our stock price tree looks like: uuS0 = 121 uS0 = 110 S0 = 100 udS0 = duS0 = 104.5 dS0 = 95 ddS0 = 90.25 Monica Tran-Xuan (UB) Financial Economics 8/ 27 Two-Step Binomial Tree Our option value will evolve according to: Cuu = 11 Cu Cud = Cdu = 0 Pcall Cd Cdd = 0 Monica Tran-Xuan (UB) Financial Economics 9/ 27 Tree for Cu ▶ We can solve this problem by splitting into three binomial trees and using our pricing procedure to solve each one ▶ First, we solve for Cu using the stock price tree 121 110 104.5 ▶ And option price tree 11 Cu 0 Monica Tran-Xuan (UB) Financial Economics 10/ 27 Finding Cu 1. uuS0 = 121, udS0 = 104.5; Cuu = 11 when uu occurs and Cud = 0 when ud occurs Cuu −Cud 11−0 = 121−104.5 = 23 uuS0 −udS0 Construct portfolio of 32 shares of stock ▶ Payoff if uu: ( 2 )(121) − 11 = 69.66 3 ▶ Payoff if ud: ( 2 )(104.5) − 0 = 69.66 3 2. Hedge ratio is 3. and one written call 4. This portfolio is risk-free! 5. The price of this portfolio at time t = 1 (the previous period) is 6. Then we have that 66.35 = HuS0 − Cu = 2 (110) 3 69.66 1.05 = 66.35 − Cu 7. =⇒ Cu = 6.983 Monica Tran-Xuan (UB) Financial Economics 11/ 27 Tree for Cd ▶ Our second tree is the bottom one, where we need to find Cd . The stock price tree is 104.5 95 90.25 ▶ And option price tree 0 Cd 0 Monica Tran-Xuan (UB) Financial Economics 12/ 27 Finding Cd We once again use our pricing procedure to find Cd : 1. duS0 = 104.5, ddS0 = 90.25; Cdu = 0 when du occurs and Cdd = 0 when dd occurs 2. The hedge ratio is H = Cdu −Cdd duS0 −ddS0 = 0−0 104.5−90.25 =0 3. Construct portfolio of 0 shares of stock and one written call: ▶ Payoff is (0)(104.5) − 0 = 0 when du occurs ▶ Payoff is (0)(90.25) − 0 = 0 when dd occurs 4. This portfolio is risk-free! 5. Price of this portfolio at time t = 1 is 0 1.05 =0 6. Then we have that 0 = HdS0 − Cd = (0)(95) − Cd 7. =⇒ Cd = 0 Monica Tran-Xuan (UB) Financial Economics 13/ 27 Tree for Pcall ▶ Lastly, we plug in Cu and Cd and can solve a one-step binomial tree for Pcall ▶ Stock price tree is 110 S0 = 100 95 ▶ And option price tree Cu = 6.983 Pcall Cd = 0 Monica Tran-Xuan (UB) Financial Economics 14/ 27 Finding Pcall Using our pricing procedure one final time: 1. uS0 = 110, dS0 = 95, Cu = 6.983 and Cd = 0 2. The hedge ratio is H = Cu −Cd uS0 −dS0 = 6.983−0 110−95 = 0.4655 3. Construct portfolio of 0.4655 shares of stock and one written call: ▶ Payoff is (0.4655)(110) − 6.983 = 44.22 when u occurs ▶ Payoff is (0.4655)(95) − 0 = 44.22 when d occurs 4. This portfolio is risk-free! 5. Price of this portfolio at time t = 0 is 44.22 1.05 = 42.11 6. Then we have that 42.11 = (0.4655)(S0 ) − Pcall = (0.4655)(100) − Pcall 7. =⇒ Pcall = 4.44 Monica Tran-Xuan (UB) Financial Economics 15/ 27 Two-Step Binomial Model ▶ So far, we have used this model to price call options ▶ Question: Can we use this model to price other derivatives? ▶ Answer: Yes! This model can be used to price any derivative of the asset price ▶ Any asset whose value is a function of the underlying asset price can be priced this way ▶ Example: Put option Monica Tran-Xuan (UB) Financial Economics 16/ 27 Two-Step Binomial Model: Put Option ▶ Suppose T = 2, S0 = 100, u = 1.2, d = 0.9, r = 0.05, and we want to price a put option on this stock with K = 110. ▶ Our stock price tree is 144 120 S0 = 100 108 90 81 Monica Tran-Xuan (UB) Financial Economics 17/ 27 Two-Step Binomial Model: Put Option ▶ And our option tree is: Puu = 0 Pu Pput Pdu = Pud = 2 Pd Pdd = 29 Monica Tran-Xuan (UB) Financial Economics 18/ 27 Two-Step Binomial Model: Put Option ▶ Just like with the call, we need to work backward and find Pu and Pd , and then use those to find Pput ▶ Pricing procedure is exactly the same! Monica Tran-Xuan (UB) Financial Economics 19/ 27 Finding Pu 1. uuS0 = 144, udS0 = 108, Puu = 0, Pud = 2 2. Hedge ratio is H= Puu − Pud 0−2 = = −0.055 uuS0 − udS0 144 − 108 ▶ Note: Hedge ratio is negative for put option. Why? 3. Construct portfolio of −0.0556 shares of stock and one written put option: ▶ Payoff if uu: (−0.0556)(144) − 0 = −8.00 ▶ Payoff if ud: (−0.0556)(108) − 2 = −8.00 4. It’s risk-free! 5. Price at time t = 1 is −8 (1.05) = −7.62 6. Then we have is (−0.0556)(120) − Pu = −7.62 7. =⇒ Pu = 0.948 Monica Tran-Xuan (UB) Financial Economics 20/ 27 Finding Pd 1. ddS0 = 81, duS0 = 108, Pdd = 29, Pdu = 2 2. Hedge ratio is Pdu − Pdd 2 − 29 −27 = = = −1 duS0 − ddS0 108 − 81 27 3. Construct portfolio of −1 shares of stock and one written put option: H= ▶ Payoff if dd: (−1)(81) − 29 = −110 ▶ Payoff if du: (−1)(108) − 2 = −110 4. It’s risk-free! 5. Price at time t = 1 is −110 1.05 = −104.76 6. Then (−1)(90) − Pd = −104.76 7. =⇒ Pd = 14.76 Monica Tran-Xuan (UB) Financial Economics 21/ 27 Option Tree Plugging in Pu and Pd to the option tree gives Puu = 0 Pu = 0.948 Pput Pdu = Pud = 2 Pd = 14.76 Pdd = 29 Monica Tran-Xuan (UB) Financial Economics 22/ 27 Finding Pput 1. uS0 = 120, dS0 = 90, Pu = 0.948, Pd = 14.76 2. Hedge ratio is Pu − Pd 0.948 − 14.76 = = −0.4604 uS0 − dS0 120 − 90 3. Construct portfolio of −0.4604 shares of stock and one written option: ▶ Payoff if u: (−0.4604)(120) − 0.948 = −56.196 ▶ Payoff if d: (−0.4604)(90) − 14.76 = −56.196 4. It’s risk-free! 5. Price at time t = 0 is −56.196 1.05 = −53.52 6. Then we have that −53.52 = (−0.4604)(100) − Pput 7. =⇒ Pput = 7.48 Monica Tran-Xuan (UB) Financial Economics 23/ 27 Table of Contents Recap Two Step Binomial N-Step Binomial Monica Tran-Xuan (UB) Financial Economics 24/ 27 N-Step Binomial ▶ Can we add more steps to this model and still price derivatives? ▶ Yes, but too much to do by hand ▶ For example, 3 step binomial model has 6 one-step trees; 4 step has 10 one-step trees ▶ But could write computer program to do it ▶ As you add more and more steps, binomial model will approximate Black-Scholes model for calls and puts (next time) Monica Tran-Xuan (UB) Financial Economics 25/ 27 N-Step Binomial ▶ Suppose you want to construct a binomial model with N steps for an option with maturity T (say a year) on a call option with strike price K ▶ Step size is then ∆t = T /N ▶ Given up and down factors u and d, current stock price S0 , and per period risk-free rate rf ▶ Can also calculate u and d from underlying price volatility σ; then per period q ) volatility is σ( T N ▶ u and d factors are q u=e d =e σ −σ T N q T N ▶ Want to build algorithm to solve this binomial tree Monica Tran-Xuan (UB) Financial Economics 26/ 27 N-Step Binomial ▶ Pricing procedure is the same ▶ Find all final option values at the end of the tree Cn,N = max{u n d N−n S0 − K , 0} for a final node that has seen n “up” steps ▶ Work backward using pricing procedure to construct previous nodes Monica Tran-Xuan (UB) Financial Economics 27/ 27 Black-Scholes ECO 426: Financial Economics Monica Tran-Xuan University at Buffalo Monica Tran-Xuan (UB) Financial Economics 1/ 24 Table of Contents Introduction Black-Scholes Model Monica Tran-Xuan (UB) Financial Economics 2/ 24 Table of Contents Introduction Black-Scholes Model Monica Tran-Xuan (UB) Financial Economics 3/ 24 Introduction ▶ Binomial model is flexible but can be tedious with many periods ▶ Option pricing formula would be easier to use ▶ In the 70’s, Economists Fischer Black, Myron Scholes, and Robert Merton developed what is now known as the Black-Scholes formula for call options ▶ Led to a boom in options trading ▶ Idea is exactly the same as the binomial model; continuously hedge the option so as to eliminate risk Monica Tran-Xuan (UB) Financial Economics 4/ 24 Table of Contents Introduction Black-Scholes Model Monica Tran-Xuan (UB) Financial Economics 5/ 24 Model Assumptions ▶ To derive the formula, some additional assumptions must be made: ▶ The risk-free rate is constant over the life of the option ▶ The stock price volatility is constant over the life of the option ▶ Stock pays no dividends until after the expiration of the option ▶ Stock price process is continuous (i.e., no “Jumps”) ▶ With these assumptions, as ∆t → 0 (time between subperiods in the binomial tree), the distribution of the stock price at expiration converges to the lognormal distribution Monica Tran-Xuan (UB) Financial Economics 6/ 24 Notation ▶ St is the price of the underlying asset at time t ▶ t is time denoted in years, T is the expiration date of the option (often t = 0 is right now) ▶ C (St , t) is the price of a European call option with underlying stock price St at time t ▶ Ex: What is C (ST , T )? ▶ K is the strike price of the option ▶ r is the (annualized) risk-free rate ▶ µ is the (annualized) expected return of the stock ▶ σ is the (annualized) volatility (or standard deviation) of the stock returns ▶ N(·) is the standard normal cumulative distribution function ▶ That is, the probability that a random draw from a standard normal N(0, 1) distribution is less than x is N(x) Monica Tran-Xuan (UB) Financial Economics 7/ 24 Standard Normal Distribution Monica Tran-Xuan (UB) Financial Economics 8/ 24 Stock Price Process ▶ From the Black-Scholes assumption, the price process for the underlying asset is what’s known as a geometric Brownian motion: dS = µ dt + σ dZ S where dZ is a standard Brownian motion (basically, a simple random walk, i.e., Zt − Zs ∼ N(0, t − s) and E [dZ ] = 0 ▶ Stock return has expected return µ and variance σ 2 Monica Tran-Xuan (UB) Financial Economics 9/ 24 Black-Scholes Equation The partial differential equation that the Black-Scholes formula solves is ∂C ∂C 1 ∂2C = rC − rS + σ2 S 2 ∂t 2 ∂S 2 ∂S ▶ RHS of this equation is equivalent to our risk-free portfolio of long 1 option and short H = ∂C shares of stock. ∂S ▶ LHS is the time decay value of the option plus a term representing the convexity of the derivative value to the underlying value (i.e., option payoff is bounded below by 0 but can gain a lot of upside with high volatility). Equation states that the time decay term and the convexity term exactly offset each other so the result is a risk-free return. Monica Tran-Xuan (UB) Financial Economics 10/ 24 Black-Scholes Formula The Black-Scholes formula for the price of a European call option is C (St , t) = St N(d1 ) − e −r (T −t) KN(d2 ) where ln d1 = St K
  2 + r + σ2 (T − t) p σ (T − t) d2 = d1 − σ Monica Tran-Xuan (UB) Financial Economics p (T − t) 11/ 24 Intuition Behind the Formula ▶ Can view N(d) terms as risk-adjusted probabilities that the call option will expire in the money ▶ For example, suppose N(d) = 0 (or very close to 0). Looking at the formula on the previous slide, it’s clear that the options will be worthless, i.e., C (St , t) = St N(d1 ) − e −r (T −t) KN(d2 ) = St (0) − e −r (T −t) K (0) = 0 ▶ So if the option will almost certainly expire out of the money, it won’t be worth anything (or very little) Monica Tran-Xuan (UB) Financial Economics 12/ 24 Intuition Behind the Formula ▶ Next, suppose N(d) = 1 (or very close to 1). From the formula we see C (St , t) = St − e −r (T −t) K ▶ So the value of the option if it will almost certainly expire in the money is the stock price minus the present value of the strike price ▶ For middle values of N(d) between 0 and 1, formula can be viewed as the present value of the call’s payoff adjusted for the probability that the option will expire in the money Monica Tran-Xuan (UB) Financial Economics 13/ 24 Intuition Behind the Formula
▶ Why are these N terms risk-adjusted probabilities? Key term is ln SKt , which is roughly the percentage by which the option is in or out of the money ▶ Example: Suppose stock price St = 105, and strike price K = 100
▶ This option is 5% in the money, and ln SKt = 0.049 ▶ Other terms adjust for volatility in remaining time to expiration and the return on a risk-free investment Monica Tran-Xuan (UB) Financial Economics 14/ 24 Black-Scholes for Put Options ▶ Black-Scholes is easily extended to European put option using put-call parity: P(St , t) = e −r (T −t) K − St + C (St , t) = e −r (T −t) K (1 − N(d2 )) − St (1 − N(d1 )) = e −r (T −t) KN(−d2 ) − St N(−d1 ) Monica Tran-Xuan (UB) Financial Economics 15/ 24 Example Suppose you want to value a call option with ▶ S0 = 100 ▶ K = 95 ▶ r = 0.10 (10% per year) ▶ T = 0.25 (3 months or one-quarter of a year) ▶ Since t = 0 here, T − t (time to expiration) is also 0.25 ▶ σ = 0.50 (std dev of 50% per year) It’s pretty straightforward to apply Black-Scholes Monica Tran-Xuan (UB) Financial Economics 16/ 24 Example First calculate: d1 = ln(100/95) + (0.10 + (0.52 )/2)(0.25) √ = 0.43 (0.5) 0.25 √ d2 = 0.43 − (0.5) 0.25 = 0.18 Next find N(d1 ) and N(d2 ), using either a z-score table or computer (NORM.S.DIST(·,True) in Excel 2010). Here we have N(0.43) = 0.6664 N(0.18) = 0.5714 Monica Tran-Xuan (UB) Financial Economics 17/ 24 Example Thus the value of the call option is C = 100(0.6664) − e −(0.10)(0.25) (95)(0.5714) = 66.64 − 52.94 = 13.70 Monica Tran-Xuan (UB) Financial Economics 18/ 24 Using Black-Scholes ▶ Suppose you find what appears to be a mispriced option, is there any reason not to bet heavily on it? ▶ Yes ▶ Do the assumptions of the model hold? ▶ Are my parameter estimates accurate? Monica Tran-Xuan (UB) Financial Economics 19/ 24 Are My Parameters Right? ▶ Most are simply directly observable (i.e., current stock price, strike price, time to maturity) ▶ EXCEPT volatility, which is harder to estimate ▶ Volatility must be estimated from historical data or model ▶ Inaccurate volatility estimate means there could always be a discrepancy between an option’s price and the Black-Scholes estimate Monica Tran-Xuan (UB) Financial Economics 20/ 24 Using Black-Scholes to Estimate Volatility ▶ Instead of using Black-Scholes to price options, can observe the option price in the market and use the Black-Scholes formula to back out what the “implied” volatility of the stock return is ▶ Example: CBOE’s VIX index ▶ Also can look at implied volatility of options on the same stock ▶ Find interesting results: volatility smile ▶ Deep out of the money put options have much higher implied volatility than at the money put options on the same stock ▶ What does this imply about stock price process? Probably includes jumps! Monica Tran-Xuan (UB) Financial Economics 21/ 24 Implied Volatility of S&P 500 Monica Tran-Xuan (UB) Financial Economics 22/ 24 The Greeks ▶ “The Greeks” are partial derivatives of the Black-Scholes formula ▶ They measure the sensitivity of the option value to changes in various parameter values while holding the other parameters fixed Monica Tran-Xuan (UB) Financial Economics 23/ 24 The Greeks ▶ Delta is the sensitivity of the option value to changes in the underlying stock price: ▶ This is the same as the Hedge Ratio! H= Cu − Cd uS0 − dS0 ▶ Positive for calls and negative for puts ▶ Gamma is the sensitivity of Delta to changes in the stock price ▶ Vega is the sensitivity of the option value to changes in the underlying volatility ▶ Theta is the sensitivity of the option value to the passage of time, the “time decay” of the option ▶ Rho is the sensitivity of the option value to changes in interest rates (usually small) Monica Tran-Xuan (UB) Financial Economics 24/ 24 
Purchase answer to see full attachment