1. Derivatives Inside Black Scholes
Professor André Farber
Solvay Business School
Université Libre de Bruxelles

2. Lessons from the binomial model
Need to model the stock price evolution
Binomial model:
discrete time, discrete variable
volatility captured by u and d
Markov process
Future movements in stock price depend only on where we are, not the history of how we got where we are
Consistent with weak-form market efficiency
Risk neutral valuation
The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate
Remember how we obtained the 1-period valuation formula in the binomial model.
The current stock price is S0. One period late (in t year) it can take two possible values: uS0 and dS0. The values of the derivative security will be fu or fd. To value the derivative, we create a synthetic portfolio composed of  units of the underlying security and M invested in a risk-free bond.  and M are choosen so as to replicate the future value of the derivative. This requires solving the following equations:
uS0 + M ert = fu
 dS0 + M ert = fd
The cost of the synthetic derivative is  S0 + M.
To rule out arbitrage, the value of the derivative should be equal to its synthetic replication. This leads to:
f =  S0 + M
For call option,  is positive (you should buy the asset) and M is negative (you should borrow). Define B as the amount borrowed (B = -M).
The value of the call option is
C =  S0 – B
This is a preview of the famous Black Scholes formula.
Derivatives 08 Inside Black Scholes

3. Black Scholes differential equation: assumptions
S follows the geometric Brownian motion: dS = µS dt +  S dz
Volatility  constant
No dividend payment (until maturity of option)
Continuous market
Perfect capital markets
Short sales possible
No transaction costs, no taxes
Constant interest rate
Consider a derivative asset with value f(S,t)
By how much will f change if S changes by dS?
Answer: Ito’s lemna
Derivatives 08 Inside Black Scholes

4. Ito’s lemna Drift Volatility
Rule to calculate the differential of a variable that is a function of a stochastic process and of time:
Let G(x,t) be a continuous and differentiable function
where x follows a stochastic process dx =a(x,t) dt + b(x,t) dz
Ito’s lemna. G follows a stochastic process:
Ito’s lemna and stochastic calculus were introduced in finance by Robert Merton around A graduate of Columbia and Cal Tech, Merton started working as a research assistant for Paul Samuelson “on how investors should best decide to save or consume when stock prices fluctuate randomly. It was to solve this problem that Merton had started using Ito’s formula” (Dunbar, 2000)
Kiyoshi Ito published his formula in 1951 (Itô, K On a formula concerning stochastic differential, Nagoya Mathematical Journal 3:55-65). As told by Nicholas Dunbar, is was not aware for a long time of his fame in the finance profession.
“Ito was a self-styled pure mathematician, which means he wasn’t interested in applying his work to the real world. Indeed, when invited to a conference celebrating his contribution to economics many years later, Ito seemed bemused at the the fuss, and claimed not to remember deriving the formula in the first place”
Nicholas Dunbar, Inventing Money – The story of Long-Term Capital Management and the legends behind it. Wiley 2000.
Drift
Volatility
Derivatives 08 Inside Black Scholes

5. Ito’s lemna: some intuition
If x is a real variable, applying Taylor:
In ordinary calculus:
In stochastic calculus:
Because, if x follows an Ito process, dx² = b² dt you have to keep it
An approximation
dx², dt², dx dt negligeables
An accessible exposition of Ito’s lemna can be found in Neftci, Salih, An Introduction to the Mathematics of Financial Derivatives, Academic Press 1996.
The above “intuitive” presentation of Ito’s lemna is based on Shimko, David, Finance in Continuous Time – A Primer, Kolb Publishing Company 1992
Use subscripts to denote partial derivative.
dG = Gx dx + Gt dt + ½ Gxx dx²
As: dx = a dt + b dz
and dx² = b² dt
dG = Gx (a dt + b dz) + Gt dt + ½ Gxx (b² dt)
This leads to Ito’s lemna:
dG = (a Gx + Gt + ½ b² Gxx) dt + b Gx dz
Why dx² = b² dt? Rougly, because dX is a normally distributed random variable. As the variance is positive, dX² does not disappear but converges in probability to b² dt.
Derivatives 08 Inside Black Scholes

6. Lognormal property of stock prices
Suppose: dS=  S dt +  S dz
Using Ito’s lemna: d ln(S) = ( ²) dt +  dz
Consequence:
ln(ST) – ln(S0) = ln(ST/S0)
Continuously compounded return between 0 and T
This is a straightforward application of Ito’s lemna with G(S,t) = ln(S)
GS = 1/S, Gt = 0, GSS = -1/S²
The parameters of the Ito process are:
a = S, b = S
Replace in Ito’s lemna:
dG = (a Gx + Gt + ½ b² Gxx) dt + b Gx dz
to obtain:
d ln(S) = [(S)(1/S) ½ (²S²)(-1/S²)] dt + (S)(1/S) dz
The expression simplifies:
d ln(S) = ( - ½ ²) dt +  dz
ln(ST) is normally distributed so that ST has a lognormal distribution
Derivatives 08 Inside Black Scholes

7. Derivation of PDE (partial differential equation)
Back to the valuation of a derivative f(S,t):
If S changes by dS, using Ito’s lemna:
Note: same Wiener process for S and f
 possibility to create an instantaneously riskless position by combining the underlying asset and the derivative
Composition of riskless portfolio
-1 sell (short) one derivative
fS = ∂f /∂S buy (long) DELTA shares
Value of portfolio: V = - f + fS S
The underlying logic is exactly the same as in the binomial model. Only the math are more elaborate.
To see, remember the pricing formula of a derivative in the binomial model (this formula is reminded in a previous slide):
f =  S0 + M
This presentation is the recipe to create a synthetic derivative in a binomial setting.
Suppose first that >0 (you recipe tells you to buy the underlying security) and M<0 (you borrow to fund the investment)
Now, the formula can be written as:
– f +  S0 = - M
The left hand side of the equality describes the strategy: you short one derivative and you go long on  units of the underlying security. The right hand side of the equality gives you the outcome of this strategy: you invest at the risk-free interest rate (as M<0, -M>0)
Derivatives 08 Inside Black Scholes

8. Here comes the PDE! Using Ito’s lemna This is a riskless portfolio!!!
Its expected return should be equal to the risk free interest rate:
dV = r V dt
This leads to:
This partial differential equation should be satisfied for any derivative on a non dividend paying stock. A closed form solution exist for European options (the Black Scholes formula). For many other derivative securities, closed form solution are not available. One solution is to use numerical techniques (the binomial model is a simple example of this approach). Such techniques might be pretty technical. This is why the derivative industry attracts scientists with degrees in mathematics or physics.
A nice example is Paul Wilmott, the author of Derivatives – The Theory and Practice of Financial Engineering, Wiley 1998 (you can meet on his website His doctorate dealt with the motion of submarines and went on to work on various aspects of fluid mechanics. He then “became aware that there were interesting math problems in the subject of finance”. There was also money to be made.
Nassim Taleb, a former trader, describes how Wall Street managed to meet the increase in the demand for scientists. “It was said that every plane from Moscow had at least its back row full of Russian mathematical physicists en route to Wall Street (they lacked the street smarts to get good seats). One could hire very cheap labor by going to JFK airport with a (mandatory) translator, randomly interviewing those that fitted the stereotype” (in Fooled by Randomness – The Hidden Role of Chance in the Markets and in Life, Texere 2001)
Derivatives 08 Inside Black Scholes

9. Understanding the PDE Assume we are in a risk neutral world
Expected change of the value of derivative security
Change of the value with respect to time
Change of the value with respect to the price of the underlying asset
Change of the value with respect to volatility
Derivatives 08 Inside Black Scholes

10. Black Scholes’ PDE and the binomial model
We have:
BS PDE : f’t + rS f’S + ½ ² f”SS = r f
Binomial model: p fu + (1-p) fd = ert
Use Taylor approximation:
fu = f + (u-1) S f’S + ½ (u–1)² S² f”SS + f’t t
fd = f + (d-1) S f’S + ½ (d–1)² S² f”SS + f’t t
u = 1 + √t + ½ ²t
d = 1 – √t + ½ ²t
ert = 1 + rt
Substituting in the binomial option pricing model leads to the differential equation derived by Black and Scholes
This result is explained in Rubinstein Derivatives: A PowerPlus Picture Book, Vol 1, Part B.
The proof involves a lot of tedious algebra.
Substituting the Taylor approximations in the binomial option pricing model:
f + [pu+(1-p)d]Sf’S – Sf’S + ½[p(u-1)² + (1-p)(d-1)²]S²f”SS + ft = f(1+rt)
But:
p u + (1-p) d = 1 + rt
and:
p (u-1)² + (1-p) (d-1)² = ² t
so that:
f + rt S f’S + ½ ² t S² f”SS + f’t t = f + r f t
Simplify and rearrange to obtain the PDE.
Derivatives 08 Inside Black Scholes

11. And now, the Black Scholes formulas
Closed form solutions for European options on non dividend paying stocks assuming:
Constant volatility
Constant risk-free interest rate
Call option:
Put option:
N(x) = cumulative probability distribution function for a standardized normal variable
Derivatives 08 Inside Black Scholes

12. Understanding Black Scholes
Remember the call valuation formula derived in the binomial model:
C =  S0 – B
Compare with the BS formula for a call option:
Same structure:
N(d1) is the delta of the option
# shares to buy to create a synthetic call
The rate of change of the option price with respect to the price of the underlying asset (the partial derivative CS)
K e-rT N(d2) is the amount to borrow to create a synthetic call
The BS formula for the European call option is a generalization of the formula that we studied previously for a long forward contract on a non dividend paying stock:
f = S0 – K e-rT
To see this, simply remember if you are certain to exercise the option, a call is in fact a forward contract. For a forward contract:
N(d1) = +1: to replicate you should buy one unit of the underlying asset
N(d2) = +1: the risk neutral probability of exercising the option is 1.
N(d2) = risk-neutral probability that the option will be exercised at maturity
Derivatives 08 Inside Black Scholes

13. A closer look at d1 and d2 S0 / Ke-rt 2 elements determine d1 and d2
A measure of the “moneyness” of the option. The distance between the exercise price and the stock price
S0 / Ke-rt
Note that if the current stock price is equal to the present value of the exercise price, the formulas for
Time adjusted volatility. The volatility of the return on the underlying asset between now and maturity.
Derivatives 08 Inside Black Scholes

14. Example
Stock price S0 = 100
Exercise price K = 100 (at the money option)
Maturity T = 1 year
Interest rate (continuous) r = 5%
Volatility  = 0.15
ln(S0 / K e-rT) = ln(1.0513) = 0.05
√T = 0.15
d1 = (0.05)/(0.15) + (0.5)(0.15) =
In this example, the call option is at the money (the exercise price is equal to the current stock price).
The delta of the option is 0.66 (more precisely, ) which means that an increase of 1 of the stock price (from 100 to 101) would increase the value of the call option from 8.60 to 9.26.
The risk-neutral probability that the option will be exercised is 0.60.
N(d1) =
European call :
100    = 8.60
d2 = – 0.15 =
N(d2) =
Derivatives 08 Inside Black Scholes

15. Relationship between call value and spot price
For call option, time value > 0
Derivatives 08 Inside Black Scholes

16. European put option European call option: C = S0 N(d1) – PV(K) N(d2)
Put-Call Parity: P = C – S0 + PV(K)
European put option: P = S0 [N(d1)-1] + PV(K)[1-N(d2)]
P = - S0 N(-d1) +PV(K) N(-d2)
Delta of call option
Risk-neutral probability of exercising the option = Proba(ST>X)
Delta of put option
Risk-neutral probability of exercising the option = Proba(ST<X)
(Remember: N(x) – 1 = N(-x)
Derivatives 08 Inside Black Scholes

17. Example
Stock price S0 = 100
Exercise price K = 100 (at the money option)
Maturity T = 1 year
Interest rate (continuous) r = 5%
Volatility  = 0.15
N(-d1) = 1 – N(d1) = 1 – =
N(-d2) = 1 – N(d2) = 1 – =
You can check that put-call parity is verified:
C + PV(K) = S + P =
The delta of the put option is call – 1
This easily derived from put call parity by taking the partial derivatives with respect to the price of the underlying asset:
CS + 0 = 1 + PS
If the stock price increases by 1, the value of the call option increases by call In order for the put call to hold (remember, this is a no arbitrage condition), the value of the put option should change by call – 1
In our example, the delta of the put option is
The risk neutral probability of exercising the put option is This is equal to 1 minus the risk neutral probability of exercising the call option.
European put option
- 100 x x = 3.72
Derivatives 08 Inside Black Scholes

18. Relationship between Put Value and Spot Price
For put option, time value >0 or <0
One important lesson from this figure is that the time value for a European put option can be negative. This is the case if the stock price is low relative to the strike price (the option is “deep in the money”).
To understand this, remember that if you own a put option, you sell the underlying asset if you exercise. You receive the strike price. Now imagine that the option is very, very deep in the money: you are almost certain that you will exercise the option at maturity. But this like selling forward: you are short on a forward contract. The value of your put option is equal to minus the value of the forward contract. Remember that the value of the forward contract is the difference between the spot price and the present value of the exercise price. So, you get:
P = PV(K) – S = (K – S) – [K – PV(K)]
Intrinsic value – Time value
But the time value also incorporate an “insurance value”: the value of the option that you have to exercise or not. This insurance value of always positive which explains why the time value of the put option is positive if the option is at the money or out of the money.
Derivatives 08 Inside Black Scholes

19. Dividend paying stock
If the underlying asset pays a dividend, substract the present value of future dividends from the stock price before using Black Scholes.
If stock pays a continuous dividend yield q, replace stock price S0 by S0e-qT.
Three important applications:
Options on stock indices (q is the continuous dividend yield)
Currency options (q is the foreign risk-free interest rate)
Options on futures contracts (q is the risk-free interest rate)
Derivatives 08 Inside Black Scholes

20. Dividend paying stock: binomial model
t = u = 1.25, d = 0.80 r = 5% q = 3% Derivative: Call K = 100
uS0 eqt with dividends reinvested
uS ex dividend 125
fu 25
S0 100
dS0 eqt with dividends reinvested 82.44
fd 0
dS ex dividend 80
f =  S0 + M
f = [ p fu + (1-p) fd] e-rt =
Replicating portfolio:
A 1-period gives us a clearer view of the impact of dividends for the valuation of a derivative.
We assume that the dividend yield q (with continuous compounding) is known.
Remember that the trick to value a derivative is create a replicating portfolio (a synthetic derivative). In our example, the derivative is a call option. The point is that if you buy shares, you receive a dividend whereas if you buy the call option, you miss the dividend paid before maturity. The dividend shows up in the future value of the replicating portfolio. As a consequence, the numbers of shares to buy to replicate the call option is smaller. This means a lower delta. The difference appears in the formula for the delta of the option. The amount borrowed is not affected by the dividend yield.
The risk neutral probability is higher with dividend than without dividend. This is because the stock price grows at rate r-q rather than r when there is a dividend yield at rate q.
The probability of an up movement should therefore satisfy:
[p uS0 + (1 – p) dS0] eqt = S0 ert
As a consequence, the value of the call option is lower if the dividend yield is positive.
(You can check that setting q = 0 would lead to a value of for the call option instead of with q = 3%)
 uS0 eqt + M ert = fu  M = 25
p = (e(r-q)t – d) / (u – d) =
 dS0 eqt + M ert = fd  M = 0
 = (fu – fd) / (u – d )S0eqt =
Derivatives 08 Inside Black Scholes

21. Black Scholes Merton with constant dividend yield
The partial differential equation: (See Hull 5th ed. Appendix 13A)
Expected growth rate of stock
Call option
Put option
These results were first derived by Merton in 1971 who was later to share the Nobel prize in 1997 with Myron Scholes.
Those of you willing to have a option valuation spreadsheet in their computer should use these formulas. By setting q = 0, you are back to the case of an option on a non dividend paying stock.
Derivatives 08 Inside Black Scholes

22. Options on stock indices
Option contracts are on a multiple times the index ($100 in US)
The most popular underlying US indices are
the Dow Jones Industrial (European) DJX
the S&P 100 (American) OEX
the S&P 500 (European) SPX
Contracts are settled in cash
Example: July 2, 2002 S&P 500 =
SPX September
Strike Call Put
, ,
Source: Wall Street Journal
Options on stock indices are also available on many European stock indices.
Derivatives 08 Inside Black Scholes

23. Options on futures A call option on a futures contract.
Payoff at maturity:
A long position on the underlying futures contract
A cash amount = Futures price – Strike price
Example: a 1-month call option on a 3-month gold futures contract
Strike price = $310 / troy ounce
Size of contract = 100 troy ounces
Suppose futures price = $320 at options maturity
Exercise call option
Long one futures
+ 100 (320 – 310) = $1,000 in cash
Derivatives 08 Inside Black Scholes

24. Option on futures: binomial model
uF0 → fu
Futures price F0
dF0 →fd
Replicating portfolio:  futures + cash
 (uF0 – F0) + M ert = fu
 (dF0 – F0) + M ert = fd
Consider the following example (see Hull 4th ed. 12.7)
The current futures prices is F0 = 30. Over the next month (t = 1/12), it can move up to 33 (u = 1.10) or down to 28 (d = ). The risk-free interest rate is 6%
Consider a one-month call option on the futures with a strike price K = 29. The two possible payoff for this option are fu = 4 and fd = 0.
To replicate the call option, you take a position on  futures contracts and M in a cash account. Note that taking a position on the futures contracts does not cost anything now. The only cost of the replicating strategy is the amount to invest in the cash account.
One month later, at the maturity of the option, the futures contract if marked to market. The cash flow is  times the change in the futures price. In addition, you get the future value of the amount invested in the cash account.
In our example, you should go long on 0.80 futures contract ( = 0.80) and invest in the cash account.
f = M
Derivatives 08 Inside Black Scholes

25. Options on futures versus options on dividend paying stock
Compare now the formulas obtained for the option on futures and for an option on a dividend paying stock:
Futures
Dividend paying stock
The reason for treating the futures price in the same way as a stock providing a dividend at a dividend yield equal to the risk-free interest rate can also be understood by comparing the BS partial differential equation when the underlying asset is a stock providing a continuous dividend yield and when the underlying asset is a futures price.
If the stock price follows the process
dS = S dS + S dz
the PDE to be satisfied by a derivative security is:
ft + (r-q) S fS + ½ ² S² fSS = r f
If the futures price follows the process
dF = F dF + F dz
ft + ½ ² F² fSS = r f
The term involving fS has disappeared. This is consistent with setting r = q in the PDE when the underlying asset is a stock price.
See Hull 4th ed. Appendix Chap. 12 for details.
Futures prices behave in the same way as a stock paying a continuous dividend yield at the risk-free interest rate r
Derivatives 08 Inside Black Scholes

26. Assumption: futures price has lognormal distribution
Black’s model
Assumption: futures price has lognormal distribution
The importance of Black’s model goes well beyond the valuation of European option on futures contracts. It can be used whenever the distribution of the variable underlying the derivative is lognormal.
One important application, to be analyzed in more details later, is the valuation of options on bonds and interest rates.
Derivatives 08 Inside Black Scholes

27. Implied volatility – Call option
When calculating the implied volatility, you start from an observed option price. Remember that the only variable requiring to be estimated. Other variable (the stock price, the exercise price, the time to maturity, the interest rate and the dividend yield) can be observed.
Assume that the Black-Scholes-Merton formula correctly price the option. Starting from the market price, you can obtain the volatility to use in the formula to obtain the market price as the output of the BSM formula.
This is easily done in Excel using the Goal Seek. You simply tell Excel to recalculate the value of the option by changing the volatility until the answer is equal to the market price.(For more details on methods to calculate the implied volatility see Chriss, N., Black Scholes and Beyond, McGraw-Hill 1997)
This is illustrated in the slide using a market price of 30 for a 3-month call option with S = , K = 1,005, r = 1.86%, q = 2%
The result is an implied volatility equal to 23.73%
This figure also illustrates an important property of the call price: the value of a call option is an increasing function of volatility. The higher the volatility, the higher the value of the option.
Derivatives 08 Inside Black Scholes

28. Implied volatility – Put option
Here we illustrate the calculation of the implied volatility using the market price for a put option.
Note that the value of the put option is also an increasing function of volatility. This should be expected. If volatility changes, put call parity still has to hold. As the stock price and the present value of the strike price are not sensitive to changes in the volatility, the change in the put value should be same as the change in the call value.
The example is the figure is based on a put option having the same characteristics as the call option used in the previous slide: same underlying asset, same exercise price, same maturity, same interest rate, same yield to maturity.
Normally, in a Black Scholes setting, the implied volatility should be identical for European options on the same underlying asset and the same expiration date.
This is clearly not the case here. The implied volatility for the put option is 16.29% and the volatility for the call option is 23.73%.
We have a problem. Either the market is wrong or the Black-Scholes formula is wrong.
Derivatives 08 Inside Black Scholes

29. Smile
If we expand the analysis to cover a larger number of options, all with the same maturity but with different striking prices, we observe systematic deviation from Black-Scholes. Remember that all implied volatilities should be identical.
These deviations are observed in many different cases: they are referred to as the volatility smile. This name refers to shape of the relationship between the strike price as a percentage of the spot price and the implied volatility which often takes the following shape:
Implied volatility
Derivatives 08 Inside Black Scholes
Strike/spot
See Hull 5th ed. Chap 15 for more details