Review of last class• Use of arbitrage pricing: if two portfolios give the

same payoff at some future date, then they must be priced the same.

• We derived the put-call parity for European options using this principle:– Put Call Parity: C - P = S - PV(div) - PV(X)– Here, “PV” stands for present value.

• We also observed that buying a butterfly spread must cost you money. If it doesn’t cost you money, then there is an arbitrage.

Pricing a Call on a Single Step Binomial Tree

• We can also use arbitrage pricing to derive the price of the call or put.Here is a simple illustrative example.

• The stock price at maturity can be either one of two prices: u S or d S. eg. if the current stock price, S, is 100, and if u=1.2, d=0.8, then the stock price at maturity can be either 120 or 80.

• In this case, the payoff on a call of strike 100 will be either 20 (if ST=120) or 0 (if ST=80)

• If we can replicate the payoff on this call using the stock and a “bond” (borrowing or lending money), we can price the call.

Example (continued)

• A portfolio of 0.5 S - PV(40) replicates the payoff on the call (Check that the payoff in each of the states, u and d, is exactly the same as the payoff on the call). The value of this portfolio (if the current stock price is 100, r=5% and maturity is 1 year) is 0.5(100) - 40/(1.05) = 11.90

• So the price of this call must also $11.90, as it has the same payoff as the above replicating portfolio.

Rules to figure out the replicating portfolio

• How do you figure out the replicating portfolio?• Rule: if a derivative security can be replicated it can

be hedged perfectly (so that the payoff at maturity is not risky.)

• So if you want to figure out the replicating portfolio, you should figure out how to hedge the option.

• What should the hedge be: if you are long (short) the call, you should be short (long) the stock. If you are long (short) the put, then you should be short (long) the stock.

Creating the hedged portfolio

• Consider the call of the previous example:• Up State (u): uS = 120, Cu=20, Hedge=B• Down State (d): dS = 80, Cd = 0, Hedge=B• Because the hedged portfolio is riskless it should

have the same cash flow (B) every period. Note that this cash flow may be positive or negative.

• In this case, because you are long one call, to construct the hedged portfolio, you should go short some amount (call it “delta”) of the stock. This portfolio should give you the same amount, $B in either state, u or d.

The Example: (continued)• Because we know that the hedged portfolio must

give the same amount in each state, it must be that:

• We can solve these two equations to get delta and B: it follows that B is (-40) and delta = 0.5.

• Note that the hedge ratio (amount of stock required to hedge a call), delta = (Cu - Cd)/(uS-dS).

C S B

C S Bu u

d d

20 120

0 80

( )

( )

Example (cont)• Finally, we can now price the call.• From the above equation, it follows that at time

t=0,

• Which is exactly what we got earlier. Thus, if we know how to hedge the option (i.e., we know the hedge ratio or delta) we can construct the replicating portfolio and price the option.

C S PV B

C S PV B PV

( )

( ) ( . ) ( )0 5 100 40

Another Example: European Put• We can use the same procedure to price a European

put. Consider a put with a strike of 100, maturity of 1 year, r=5%, u=1.20, d=0.8 and the current (zero dividends) stock price is 100.

• Up state: uS = 120, Pu = 0

• Down state: dS=80, Pd = 20

• Hedge ratio: delta = (Pu - Pd)/(uS - dS) = -0.5.

• B = Pu - (-0.5)uS = 0 - (-0.5)(120) = 60.

• P = (delta)(S) + PV(B) = (-0.5)(100)+60/(1+0.05) =$7.14.

• Knowing the call price, we could also have used the put-call parity to derive the price of the put.

Extending the one step binomial tree• As it is not realistic to assume that the stock has

only two possible states at maturity, we need to extend the binomial tree to allow for more states. However, we still want to keep the simplicity of the valuation procedure that we have developed.

• Thus, we generalize the procedure to a multi-step binomial tree, where we assume that we divide the time to maturity into smaller segments, and assume that over each of these smaller segments, the stock can either go up or down.

Valuation by Arbitrage• In a simple world, where the stock price can

be either go up or down, we saw that it is easy to price an option by a replicating strategy.

• Further, we showed how the replicating strategy can be deduced by understanding how to hedge the option.

• For example, for a call we saw that: • C = (delta) S + PV (B), where delta is the

hedge ratio, and B is the cash flow that you would get if you hedged the call with delta.

How to make it simpler

• We can rewrite the formula as follows :

• C = PV(p Cu + (1-p) Cd)

• p = (R-d)/(u-d), where R = future value of $1 at maturity when invested at the risk-free rate.

• Using R, we can write the call price as:

• C = (p Cu + (1-p) Cd)/R– Normally, in option pricing, we use continuous

compounding. Thus, R= e(rT) where r is the risk-free rate (so that if the continuous time interest rate is 5%, then $1 today is worth e(0.05T) after T periods.)

Example: one step tree• Consider our previous example of a call

with maturity 1 year and exercise price = 100.

• Up state: u=1.2, Su = 120, Cu=20• Down state: d=0.8, Sd = 80, Cd=0• R = 1.05• p = (R-d)/(u-d) = (1.05-0.8)/(1.2-0.8)

=0.625• C = (p Cu + (1-p) Cd)/R = (0.625*20 +

0.375*0)/1.05=$11.90

Another Example: two step

• In a two-step tree, we divide the time to maturity into two halves. For our example, therefore, each sub-period = 1/2 years.

• At t=0.5, we get two possible stock prices: Su and Sd.

• At t=1, we three possible stock prices: Suu, Sud, and Sdd.

• Corresponding to these stock prices, we get three possible option prices at maturity: Cuu, Cud, Cdd.

Example: continued

• To make the example more realistic, we need to choose u and d to reflect the volatility of the stock. Suppose, the volatility (standard deviation) is vol=15% per annum and r=0.05, continous compounded annual rate. Then:

u e e

d u

R e

p R d u d

t

` . .

. ( . )

.

/ .

.

( ) / ( ) .

0 15 0 5

0 05 0 5

11119

1 0 8994

1 0253

0 5924

Example: X=100,T=1,S=100• At Maturity:

• t=1, State uu: Suu = 100(1.1119)(1.1119)=123.63, Cuu = 23.63

• t=1, State ud: Sud = 100, Cud=0

• t=1, State dd: Sdd=80.89, Cud=0.

• Knowing the option price at maturity we can now work backwards to figure out the call price today.

Example: continued

• First Step: From t=1 to t=0.5 • t=0.5, State u: Cu = (p Cuu +

(1-pCud)/R=(0.5924*23.63 + 0.4076*0)/(1.0253)=13.65

• t=0.5, State d: Cd=(pCud + (1-p) Cdd)/R=0• Second Step: From t=0.5 to t=0.• C = (p Cu + (1-p) Cd)/R = (0.5924*13.65 +

0.4076*0)/1.0253= 7.89.

Generalizations• We can extend the tree to as many steps as we

want. The number of calculations increases as you increase the number of steps, but the answer gets more accurate.For almost any option, 200 steps is adequate. For European calls and puts, usually 30 steps will give you a good answer too.

• Note that u,d,R and thus p depends on the number of steps you choose, as they depend on the length of the period in the tree. In general, for n steps,

t T n u e d u

R e

t

r t

/ , , /( )

1

Black Scholes Formula (1/2)

• What is the option price if we take a very large n (n = infinity)?

• It can be shown (with very advanced mathematics) that the European call price for a stock with a dividend yield, d, and stock price, S is equal to:

C Se N d e XN d

d d d T

dT rT

S X r d TT

( ) ( )

,ln( / ) ( / )

1 2

12

2 1

2

Black Scholes Formula (2/2)

• The price of the put can be computed from put-call parity, as P = C – (S – PV(div)) + PV(X).

• In summary, to use the Black-Scholes formula we only need the following information:– Price of underlying asset

– Strike

– Maturity

– Riskfree rate

– Volatility of stock return

Implied Volatility of an Option• Suppose we want to price two calls of nearby

strikes, say, X=100 and X=120, of the same maturity. Under the Black-Scholes assumptions, we should use the same volatility estimate the option prices.

• In practice, when we observe option prices in the market, different volatilities are used to price options of different strikes.

• We can back out the volatility that is used to price the option by equating the market price of the option to the theoretical Black-Scholes price. This estimate of the volatility is called the “implied volatility” of the option.

The Option Smile

• There is a stable relationship between the implied volatilities that the options are priced across different strikes.

• The graph of the implied volatility as a function of the option strike is called the “option smile”

• You can see the current option smile at http://www.impliedvol.com/