Option Pricing Approaches Valuation of options FIN 819-lecture 7

Today’s plan Review of what we have learned about options We first discuss a simple business ethics case. We discuss two ways of valuing options Binomial tree (two states) Simple idea Risk-neutral valuation The Black-Scholes formula (infinite number of states) Understanding the intuition How to apply this formula

Business ethics Suppose that you have applied to two jobs: A and B. Now you have received the offer letter for job A and have to make a decision now about whether or not to take job A. But you like job B much more and the decision for job B will be made one week later. What is your decision?

What have we learned in the last lecture? Options Financial and real options European and American options Rights to exercise and obligations to deliver the underlying asset Position diagrams Draw position diagrams for a given portfolio Given position diagrams, figure out the portfolio No arbitrage argument Put-call parity FIN 819

The basic idea behind the binomial tree approach Suppose we want to value a call option on ABC stock with a strike price of K and maturity T. We let C(K,T) be the value of this call option. Remember C(K,T) is the price for the call or the value of the call option at time zero. Let the current price of ABC is S and there are two states when the call option matures: up and down. If the state is up, the stock price for ABC is Su; if the state is down, the price of ABC is Sd. FIN 819

The stock price now and at maturity Su uS S S Sd dS Now maturity Now maturity If we define: u = Su/S and d = Sd/S. Then we have Su=uS and Sd=dS FIN 819

The risk free security The price now and at maturity Rf Here Rf=1+rf 1 FIN 819

The call option payoff Cu=Max(uS-K,0) C(K,T) Cd=Max(dS-K,0) Now maturity FIN 819

Now form a replicating portfolio A portfolio is called the replicating portfolio of an option if the portfolio and the option have exactly the same payoff in each state of future. By using no arbitrage argument, the cost or price of the replication portfolio is the same as the value of the option. FIN 819

Now form a replicating portfolio (continue) Since we have three securities for investment: the stock of ABM, the risk-free security, and the call option, how can we form this portfolio to figure out the price of the call option on ABC? FIN 819

Now form a replicating portfolio (continue) Suppose we buy Δ shares of stock and borrow B dollars from the bank to form a portfolio. What is the payoff for the this portfolio for each state when the option matures? What is the cost of this portfolio? How can we make sure that this portfolio is the replicating portfolio of the option? FIN 819

How can we get a replicating portfolio? Look at the payoffs for the option and the portfolio ΔuS+BRf Portfolio Cu=max(uS-K,0) Option B+ΔS C(K,T) ΔdS+BRf Cd=max(dS-K,0) Now maturity now Maturity FIN 819

Form a replicating portfolio From the payoffs in the previous slide for the call option and the portfolio, to make sure that the portfolio is the replicating portfolio of the option, the option and the portfolio must have exactly the same payoff in each state at the expiration date. That is, ΔuS+BRf = Cu ΔdS+BRf = Cd FIN 819

Form a replicating portfolio Use the following two equations to solve for Δ and B to get the replicating portfolio: ΔuS+BRf = Cu ΔdS+BRf = Cd The solution is FIN 819

To get the value of the call option By no arbitrage argument, the value or the price of the option is the cost of the replicating portfolio, B+ΔS. Can you believe that valuing the option is so simple? Can you summarize the procedure to do it? This procedure walks you through the way of understanding the concept of no arbitrage argument. FIN 819

Summary Using the no arbitrage argument, we can see the cash flows from investing in a call option can be replicated by investing in stocks and risk-free bond. Specifically, we can buy Δ shares of stock and borrow B dollars from the bank. The value of the option is Δ*S+B ( the number of shares *stock price –borrowed money), where B is negative FIN 819

Example of valuing a call Suppose that a call on ABC has a strike price of $55 and maturity of six-month. The current stock price for ABC is $55. At the expiration state, there is a probability of 0.4 that the stock price is $73.33, and there is a probability of 0.6 that the stock price is $41.25. The risk-free rate is 4%. Can you calculate the value of this call option? (the value is $8.32) (u=1.33,d=0.75, Cd=0, Cu= $18.33, Δ=0.57, B=-$23.1) FIN 819

How to value a put using the similar idea We can use the similar idea to value a European put. Before you look at my next two slides, can you do it yourself? Still try to form a replicating portfolio so that the put option and the portfolio have the exactly the same payoff in each state at the expiration date. FIN 819

How can we get a replicating portfolio of a put option? Look at the payoffs for the put option and the portfolio Portfolio ΔuS+BRf Put option Cu=max(K-uS,0) B+ΔS P(K,T) ΔdS+BRf Cd=max(K-dS,0) Now maturity now Maturity FIN 819

Form a replicating portfolio From the payoffs in the previous slide for the put option and the portfolio, to make sure that the portfolio is the replicating portfolio of the option, the put option and the portfolio must have exactly the same payoff in each state at the expiration date. That is, ΔuS+BRf = Cu ΔdS+BRf = Cd FIN 819

Form a replicating portfolio Use the following two equations to solve for Δ and B to get replicating portfolio: ΔuS+BRf = Cu ΔdS+BRf = Cd The solution is FIN 819

What happens? You can see that the formula for calculating the value of a put option is exactly the same as the formula for a call option? Where is the difference? The difference is the calculation of the payoff or cash flows in each state. To get this, please try the valuation of put option in the next slide. FIN 819

Example of valuing a put Suppose that a European put on IBM has a strike price of $55 and maturity of six-month. The current stock price is $55. At the expiration state, there is a probability of 0.5 that the stock price is $73.33, and there is a probability of 0.5 that the stock price is $41.25. The risk-free rate is 4%. Can you calculate the value of this put option? (the value is $7.24) (u=1.33,d=0.75, Cu=0, Cd=$13.75 Δ=-0.43, B=$30.8) FIN 819

Example of valuing a put option (continue) Recall that the value of call option with the same strike price and maturity is $8.32. Can you use this call option value and the put-call parity to calculate the value of the put option? Do you get the same results? ( if not, you have trouble) FIN 819

Can you learn something more? Everybody knows how to set fire by using match. Long, long time ago, our ancestors found that rubbing two rocks will generate heat and thus can yield fire, but why don’t we rub two rocks to generate fire now? It is clumsy, not efficient What have you learned from this example? FIN 819

What can we learn? Using the idea in the last slide, to value a call option, we don’t need to figure out the replicating portfolio by calculating the number of shares and the amount of money to borrow. Instead we can jump to calculate the value of the call option using the way in the next slide. FIN 819

Risk-neutral probability The price of call option is let p=(Rf-d)/(u-d) < 1. Then FIN 819

Risk-neutral probability (continue) Now we can see that the value of the call option is just the expected cash flow discounted by the risk-free rate. For this reason, p is the risk-neutral probability for payoff Cu, and (1-p) is the risk-neutral probability for payoff Cd. In this way, we just directly calculate the risk-neutral probability and payoff in each state. Then using the risk-free rate as a discount rate to discount the expected cash flow to get the value of the call option. FIN 819

Examples for risk-neutral probability Using the risk neutral probability approach to calculate the values of the call and put options in the previous two examples. Call ( u=1.33,d=0.75, Rf=1.02, p=0.47, C1=0.47*18.33, PV(C1) = C1/Rf= $8.37. Put. ( p=0.47, C1=0.53*13.75, PV(C1)=C1/Rf=$7.14) FIN 819

Two-period binomial tree Suppose that we want to value a call option with a strike price of $55 and maturity of six-month. The current stock price is $55. In each three months, there is a probability of 0.3 and 0.7, respectively, that the stock price will go up by 22.6% and fall by 18.4%. The risk-free rate is 4%. Do you know how to value this call? FIN 819

Solution First draw the stock price for each period and option payoff at the expiration 27.67 p Stock price Option 82.67 p 67.43 1-p C(K,T)=? 55 p 1-p 55 1-p 44.88 36.62 Three month Six month Now Three month Sixth month Now FIN 819

Solution Risk-neutral probability is p=(Rf-d)/(u-d) =(1.01-0.816)/(1.226-0.816)=0.473 The probability for the payoff of 27.67 is 0.473*0.473, the probability for other two states are 2*0.473*527, and 0.527*0.527. The expected payoff from the option is 0.473*0.473*27.67= The present value of this payoff is 6.07 So the value of the call option is $6.07 FIN 819

How to calculate u and d In the risk-neutral valuation, it is important to know how to decide the values of u and d, which are used in the calculation of the risk-neutral probability. In practice, if we know the volatility of the stock return of σ, we can calculate u and d as following: Where h the interval as a fraction of year. For example, h=1/4=0.25 if the interval is three month. FIN 819

Example for u and d Using the two-period binomial tree problem in the previous example. If σ is 40.69%, Please calculate u and d? Please calculate the risk-neutral probability p? Please calculate the value of the call option? (u=1.17,d=0.85, p= 0.5) FIN 819

The motivation for the Black-Scholes formula In the real world, there are far more than two possible values for a stock price at the expiration of the options. However, we can get as many possible states as possible if we split the year into smaller periods. If there are n periods, there are n+1 values for a stock price. When n is approaching infinity, the value of a European call option on a non-dividend paying stock converges to the well-known Black-Scholes formula. FIN 819

A three period binomial tree u3S u2dS S ud2S d3S There are three periods. We have four possible values for the stock price FIN 819

The Black-Scholes formula for a call option The Black-Scholes formula for a European call is Where FIN 819

The Black-Scholes formula for a put option The Black-Scholes formula for a European put is Where FIN 819

The Black-Scholes formula (continue) One way to understand the Black-Scholes formula is to find the present value of the payoff of the call option if you are sure that you can exercise the option at maturity, that is, S-exp(-rt)K. Comparing this present value of this payoff to the Black-Scholes formula, we know that N(d1) can be regarded as the probability that the option will be exercised at maturity FIN 819

An example Microsoft sells for $50 per share. Its return volatility is 20% annually. What is the value of a call option on Microsoft with a strike price of $70 and maturing two years from now suppose that the risk-free rate is 8%? What is the value of a put option on Microsoft with a strike price of $70 and maturing in two years? FIN 819

Solution The parameter values are Then FIN 819