# The Black-Scholes Formula

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 ECO OPTIONS AND FUTURES SPRING 2008 The Black-Scholes Formula The Black-Scholes Formula We next examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the geometric Brownian motion assumption for the underlying asset price. Nevertheless they can be interpreted and are easy to use once understood. The payoff to a European call option with strike price K at the maturity date T is c(t ) = max[s(t ) K, 0] where S(T ) is the price of the underlying asset at the maturity date. At maturity if S(T ) > K the option to buy the underlying at K can be exercised and the underlying asset immediately sold for S(T ) to give a net payoff of S(T ) K. Since the option gives only the right and not the obligation to buy the underlying asset, the option to buy the underlying will not be exercised if doing so would lead to a loss, S(T ) K < 0. The Black-Scholes formula for the price of the call option at date t = 0 prior to maturity is given by c(0) = S(0)N(d 1 ) e rt KN(d 2 ) where N(d) is the cumulative probability distribution for a variable that has a standard normal distribution with mean of zero and standard deviation of one. It is the area under the standard normal density function from to d and therefore gives the probability that a random draw from the standard normal distribution will have a value less than or equal to d. We have therefore that 0 N(d) 1 with N( ) = 0,N(0) = 1 and N(+ ) = 1. The 2 term r is the continuously compounded risk-free rate of interest and d 1 and 1

2 2 BLACK-SCHOLES d 2 satisfy d 1 = S(0) ln( K 2 σ2 )T σ T d 2 = d 1 σ S(0) ln( K T = 2 σ2 )T σ T Here σ 2 is the variance of the continuously compounded rate of return on the underlying asset. Likewise the payoff to a European put option with strike price K at the maturity date T is p(t ) = max[k S(T ), 0] as the put option gives the right to sell underlying asset at the strike price of K. The Black-Scholes formula for the price of the put option at date t = 0 prior to maturity is given by p(0) = c(0) + e rt K S(0) = e rt K(1 N(d 2 )) S(0)(1 N(d 1 )) where d 1 and d 2 are defined above. By the symmetry of the standard normal distribution N( d) = (1 N(d)) so the formula for the put option is usually written as p(0) = e rt KN( d 2 ) S(0)N( d 1 ). Assumptions Underlying the Black-Scholes Formula To derive the Black-Scholes formula the following assumptions are required. 1. Markets are perfect: there are no transactions costs or taxes (i.e. there is no bid-ask spread), assets can be short perfectly short sold and are divisible (i.e. we can trade assets in negative and fractional units). 2. There is a risk-free asset and the interest rate on the risk-free asset is unchanging over the life of the option.

3 ECO OPTIONS AND FUTURES 3 3. Trading is continuous. 4. The stock price follows a geometric Brownian motion process. That is d S = µs dt + σs dz where S is the stock price, µ is the expected return on the stock (assumed to be unchanging over the life of the option) and σ is the standard deviation of the return on the stock (assumed to be unchanging over the life of the option) and dz is a Wiener process. We will explain a little bit more about assumptions 3 and 4 and how they relate to the Black-Scholes formula. Continuous Trading We shall consider the effect of continuous trading on the risk-free asset first. The assumption of continuous trading actually has many advantages mathematically and makes the calculation of rates of return over different periods rather easy. The Rate of Return The rate of return is simply the end value less the initial value as a proportion of the initial value. Thus if 100 is invested and at the end value is 120 then the rate of return is = 1 5 or 20%. If the the initial investment is B T 1 at time T 1 and the end value is B T after one more period then the rate of return is r = B T B T 1 B T 1 or equivalently the rate of return r satisfies B T = B T 1 (1 + r). We can generalise this to write the rate of return over T periods say when the initial

4 4 BLACK-SCHOLES investment is B 0 and write the rate of return as r(t ) = B T B 0 B 0 so that r(t ) satisfies B T = B 0 (1 + r(t )). It is important to know the rate of return. However to compare rates of return on different investments with different time horizons it is also important to have a measure of the rate of return per period. One method of making this comparison is to to assume the asset is traded continuously and can be priced by using the continuously compounded rates of return. To explain this we first consider compound returns and then show what happens when the compounding is continuous. Compound Rates of Return Compound interest rates are calculated by assuming that the principal (initial investment) plus interest is re-invested each period. Compounding might be done annually, semi-annually, quarterly, monthly or even daily. Assuming the re-investment is done after each period, the per-period interest rate r on the investment satisfies (1 + r(t )) = (1 + r) T. Now consider dividing up each period into n sub-periods each of length t. This is illustrated in Figure 1. Then if the compounding is done n times per period, the compound interest rate r satisfies (1 + r(t )) = (1 + r n )nt. For example consider a time period of one-year and suppose an investment of 100 that yields 120 after two years (T = 2) has a rate of return r(2) = 0.2. If the interest rate is annualised using annual compounding (n = 1, T =

5 ECO OPTIONS AND FUTURES 5 2), then r = ; with semi-annual compounding (n = 2, T = 2) the annualised interest rate is r = ; with quarterly compounding (n = 4, T = 2) the annualised interest rate is r = etc t t+1 T-1 T t t+δt t+2δt t+3δt t+1=t+nδt Figure 1: Dividing a time interval n sub-periods Continuous Compounding Each period divided into n sub-periods of length t times per period and compounding occurs n times per period. As we divide the period into ever more sub-periods the length of time between compounding be denoted by t = 1 becomes smaller and n grows larger. In this case the compounding n factor r satisfies (1 + r n )nt = (1 + r t) T t. Continuous compounding occurs as t 0 or equivalently as n 0. Let m = 1 r t, then ( (1 + r t) T 1 t = (1 + m )mrt = (1 + 1 ) rt m )m.

6 6 BLACK-SCHOLES As we let the interval between compounding t go to zero then m. The limit of (1 + 1 m )m as m is well known and is given by the formula ( lim m = e = m m) where e = is the base of the natural logarithm. Thus in the limit the compounding factor is given by (1 + r n )nt = ( (1 + 1 m )m ) rt e rt. This gives a simple method for calculating the continuously compounded rate of return r for any given investment. Suppose the initial outlay is B 0 and the value of the portfolio or investment after T periods is B T. Then since B T = B 0 (1 + r(t )) one has B T B 0 = (1 + r(t )) = e rt so that taking logarithms of both sides ( ) BT ln = rt or rewriting we have r = 1 T ln ( BT B 0 B 0 ) = 1 T (ln B T ln B 0 ). This is known as the continuously compounded rate of return. The Continuously Compounded Rate of Return The continuously compounded rate of return has the property that longer period rates of return can be computed simply by adding shorter continuously compounded rates of return. This is a very convenient feature which makes

7 ECO OPTIONS AND FUTURES 7 using the continuously compounded rates of return especially simple. To see this let r t denote the continuously compounded rate of return from period t to t + 1, that is r t = ln ( Bt+1 where B t is the value of the asset at time t. Let r(t ) denote the continuously compounded rate of return over the period 0 to T, ( ) BT r(t ) = ln = ln B T ln B 0. B 0 B t ) Suppose that T = 2 then we can write this as r(2) = ln B 2 ln B 0 = (ln B 2 ln B 1 ) + (ln B 1 ln B 0 ) = r 1 + r 0. Thus the continuously compounded rate of return over two periods is simply the sum of the two period by period returns. In general for any value of T we can write r(t ) = (ln B T ln B T 1 ) + (ln B T 1 ln B T 2 ) (ln B 2 ln B 1 ) + (ln B 1 ln B 0 ) T 1 = r T 1 + r T r 1 + r 0 = r t. Thus the continuously compounded rate of return over time T is simply the sum of the period by period returns. If r t is constant over time then r(t ) = rt. t=0 A Differential Equation It will be useful to think about the value of the risk-free asset as it changes over time. Let r t denote the rate of return between t and t + 1. Then over any sub-interval of t say between t and t + t, r t satisfies B t+ t = (1 + r t t)b t.

8 8 BLACK-SCHOLES Then taking the limit as t 0 we have B t+ t B t db(t) where B(t) is the price at time t and t dt. Hence we can write or equivalently db(t) = r t B(t)dt db(t) B(t) = r tdt. This is a differential equation. If we assume that r t = r is constant over time (r t = r for each t) then this equation can be solved at by integration to give the asset price at time T ln B T ln B 0 == ln ( BT B 0 ) = rt or B T = B 0 e rt. Geometric Brownian Motion We have assumed so far that the rate of return was known so that we were dealing with a risk-free asset. But for most assets the rate of return is uncertain or stochastic. As the asset value also changes through time the we say that the asset price follows a stochastic process. Fortunately the efficient markets hypothesis provides some strong indication of what properties this stochastic process will have. A Coin Tossing Example To examine the form that uncertain returns may take it is useful to think first of a very simple stochastic process. This we have already seen as the binomial model is itself a stochastic process. As an example consider the case of tossing a fair coin where one unit is won if the coin ends up Heads

9 ECO OPTIONS AND FUTURES 9 and one unit is lost if the coin ends up Tails. An example of the possible payoffs for a particular sequence of Heads and Tails is illustrated in Figure T H H T H H T Figure 2: A Coin Tossing Stochastic Process The important properties of this example are that the distribution of returns are 1) identically distributed at each toss (there is an equal chance of a Head or a Tail); 2) independently distributed (the probability of a Head today is independent of whether there was a Head yesterday); 3) the expected return is the same each period (equal to zero); 4) the variance is constant at each period (equal to one). There are some important implications to note about this process. First let x t denote the winnings on the tth toss. We have x 0 = 0 and E[x 1 ] = 0 where E[x t ] denotes the expected winnings at date t. Then we also have at any date E[x t+1 ] = x t.

10 10 BLACK-SCHOLES Any process with this property is said to be a martingale. Another important property is that the variance of x is increasing proportionately to the number of tosses. In particular letting σt 2 denote the variance of the winnings at the tth toss we have σt 2 = t or in terms of the standard deviation (the square root of the variance σ t = t. This is illustrated in Figure 3. Figure 3 shows all possible winnings through four tosses. The variance of winnings is easily calculated at each toss. For example at the second toss the expected winnings are zero so the variance is given by σ2 2 = 1 4 (2 0) (0 0) ( 2 0)2 = = 2 and so the standard deviation is σ 2 = 2. A Stochastic Process for Asset Prices The efficient markets hypothesis implies that all relevant information is rapidly assimilated into asset prices. Thus asset prices will respond only to new information (news) and since news is essentially unforecastable so to are asset prices. The efficient market hypothesis also implies that it is impossible to consistently make abnormal profits by trading on publicly available information and in particular the past history of asset prices. Thus only the current asset price is relevant in predicting future prices and past prices are irrelevant. This property is know as the Markov property for stock prices. If we add a further assumption that the variability of asset prices is roughly constant over time, then the asset price is said to follow a random walk. This was true of our coin tossing example above. Let u t denote the random rate of return from period t to t + 1. Then S t+1 = (1 + u t )S t.

11 ECO OPTIONS AND FUTURES σ 2 = 1 σ 2 = 2 σ 2 = 3 σ 2 = 4 Figure 3: Coin Tossing Example: The Variance is Proportional to Time The return u t is now random because the future asset price is unknown. 1 It can be considered as a random shock or disturbance. Taking natural logarithm of both sides gives ln(1 + u t ) = ln S t+1 ln S t. Let ω t = ln(1 + u t ). The ω t is the continuously compounded rate of return from period t to t + 1. Since u t is random ω t will be random too. We shall however assume 1. ω t are independently and identically distributed 1 This was the case in our binomial model where u t takes on either the value of u in the upstate or d in the down state.

12 12 BLACK-SCHOLES 2. ω t is normally distributed with a mean of ν and variance of σ 2 (both independent of time). 2 With these assumptions we can see how the stock price is distributed at a future date. Suppose we start from a given value S 0, then ln S 1 = ln S 0 + ln(1 + u 0 ) ln S 2 = ln S 1 + ln(1 + u 1 ) = ln S 0 + ln(1 + u 0 ) + ln(1 + u 1 ) ln S 3 = ln S 2 + ln(1 + u 2 ) = ln S 0 + ln(1 + u 0 ) + ln(1 + u 1 ) + ln(1 + u 2 ). =. T 1 T 1 ln S T = ln S 0 + ln(1 + u t ) = ln S 0 + ω t. t=0 t=0 Since ω t is a random variable which is identically and independently distributed and such that the expected value E[ω t ] = ν and variance Var[ω t ] = σ 2 then taking expectations we have that ln S T is normally distributed such that E[ln S T ] = ln S 0 + νt and Var[ln S T ] = σ 2 T. Standard Normal Variable We have seen that ln S T is normally distributed with mean (expected value) of ln S 0 + νt and variance of σ 2 T. It is useful to transform this to a variable which has a standard normal distribution with mean of zero and standard deviation of one. Such a variable is called a standard normal variable. To 2 In fact when we take the limit there is no need to assume normality of the distribution as this will be guaranteed by the Central Limit Theorem.

13 ECO OPTIONS AND FUTURES 13 make this transformation, we subtract the mean and divide by the standard deviation (square root of the variance). Thus ( ) S ln S T ln S 0 νt ln 0 σ S T + νt = T σ T is a standard normal variable. We let N(x) denote the cumulative probability that the standard normal variable is less than or equal to x. A standard normal distribution is drawn in Figure 4. It can be seen that N(0) = 0.5 as the normal distribution is symmetric and half the distribution is to the left of the mean value of zero. It also follows from symmetry that if x > 0, then 1 N(x) = N( x). We will use this property later when we look at the Black-Scholes formula. Figure 4: A Standard Normal Distribution. N(0) = 0.5.

14 14 BLACK-SCHOLES The Expected Rate of Return Unfortunately, the expected rate of return is a rather ill-defined concept. If we rewrite the above equation for ν we have ν = 1 ( [ ( )]) ST E ln. T This is known as the expected continuously compounded rate of return. That is to say we look at the the continuously compounded rates of return and then take expectations. However we can do it another way, which is to compute the expected rates of return and then do our continuous compounding. This would give us µ = 1 T S 0 ( ( [ ])) ST ln E. S 0 Notice that the difference is that the Expectations operator appears inside the logarithm. The value µ is know as the logarithm of expected return or sometimes confusingly as the expected rate of return. The values of ν and µ will be different if the asset is uncertain. Fortunately there is a very simple relationship between ν and µ which is given by 3 µ = ν σ2. The reason why µ exceeds ν can be seen because the distribution of asset prices is lognormal. Lognormal Random Variable Since the logarithm of the asset price is normally distributed the asset price itself is said to be lognormally distributed. In practice when one looks at the empirical evidence asset prices are reasonably closely lognormally distributed. 3 The equation below always holds approximately and holds exactly in the limit.

15 ECO OPTIONS AND FUTURES 15 We have assumed that that ω t = ln(1+u t ) is normally distributed with an expected value of ν and variance σ 2. But 1+u t is a lognormal variable. Since 1 + u t = e ωt we might guess that the expected value of u t is E[u t ] = e ν 1. However this would be WRONG. The expected value of u t is E[u t ] = e ν+ 1 2 σ2 1. The expected value is actually higher than anticipated by half the variance. The reason why can be seen from looking at an example of the lognormal distribution which is drawn in Figure 5. The distribution is skewed and as the variance increases the lognormal distribution will spread out. It cannot spread out too much in a downward direction because the variable is always non-negative. But it can spread out upwards and this tends to increase the mean value. One can likewise show that the expected value of the asset price at time T is E[S T ] = S 0 e (ν+ 1 2 σ2 )T. Hence E[S T ] = s 0 e µt. That is we expect the stock price to grow exponentially at the constant rate µ. However our best forecast would be at a rate of ν and hence we should in some sense always expect less than the expected. Arithmetic and geometric rates of return We now consider µ and ν again. Suppose we have an asset worth 100 and for two successive periods it increases by 20%. Then the value at the end of the first period is 120 and the value at the end of the second period is 144. Now suppose that instead the asset increases in the first period by 30% and in the second period by 10%. The average or arithmetic mean of the return is 20%. However the value of the asset is 130 at the end of first period

16 16 BLACK-SCHOLES Figure 5: A Lognormal Distribution and 143 at the end of the second period. The variability of the return has meant that the asset is worth less after two periods even though the average return is the same. We can calculate the equivalent per period return that would give the same value of 143 after two periods if there were no variance in the returns. That is the value ν that satisfies 143 = 100(1 + ν) 2. This value is known as the geometric mean. It is another measure of the average return over the two periods. Solving this equation gives the geometric mean as ν = or 19.58% per period 4 which is less than the arithmetic rate of return per period. 4 The geometric mean of two numbers a and b is ab. Thus strictly speaking 1 + g = is the geometric mean of 1.1 and 1.3.

17 ECO OPTIONS AND FUTURES 17 There is a simple relationship between the arithmetic mean return, the geometric mean return and the variance of the return. Let µ 1 = µ + σ be the rate of return in the first period and let µ 2 = µ σ be the rate of return in the second period. Here the average rate of return is 1(µ µ 2 ) = µ and the variance of the two rates is σ 2. The geometric rate of return µ satisfies (1 + ν) 2 = (1 + µ 1 )(1 + µ 2 ). Substituting and expanding this gives 1 + 2ν + ν 2 = (1 + µ + σ)(1 + µ σ) = (1 + µ) 2 σ 2 = 1 + 2µ + µ 2 σ 2 or ν = µ 1 2 σ (µ2 ν 2 ). Since rates of return are typically less than one, the square of the return is even smaller and hence the difference between two squared percentage terms is smaller still. Hence we have the approximation ν µ 1 2 σ2 or geometric mean arithmetic mean 1 2 variance. This approximation will be better the smaller are the interest rates and the smaller is the variance. In the example µ = 0.2 and σ = 0.1 so 1 2 σ2 = and µ 1 2 σ2 = which is close to the actual geometric mean of Thus the difference between µ and ν is that ν is the geometric rate of return and µ is the arithmetic rate of return. It is quite usual to use the arithmetic rate and therefore to write that the expected value of the logarithm of the stock price satisfies E[ln S T ] = ln S 0 + (µ 12 σ2 ) T and Var[ln S T ] = σ 2 T.

18 18 BLACK-SCHOLES The Continuous Trading Limit 5 This section examines what happens in the limit as the stock price can change continuously and randomly. This section is mathematically more demanding and therefore can largely be ignored. The important result is that with continuous trading the asset price follows a stochastic differential equation given by d S(t) = µs(t) dt + σs(s) dz. where dz is a Weiener process which is described in the next section. This process for stock prices is known as geometric Brownian motion and shows that the stock price consists of two components. Firstly a deterministic component which shows the stock price growing at a rate of µ and secondly a random or disturbance component with a volatility of σ. A Wiener Process Consider a variable z which takes on values at discrete points in time t = 0, 1,..., T and suppose that z evolves according to the following rule: z t+1 = z t + ɛ; W 0 fixed where ɛ is a random drawing from a standardized normal distribution, that is with mean of zero and variance of one. The draws are assumed to be independently distributed. This represents a random walk where on average z remains unchanged each period but where the standard deviation of the realized value is one each period. At date t = 0, we have E[z T ] = z 0 and the variance V ar[z T ] = T as the draws are independent. Now divide the periods into n subperiods each of length t. To keep the process equivalent the variance in the shock must also be reduced so that 5 This section can be considered optional.

19 ECO OPTIONS AND FUTURES 19 the standard deviation is t. The resulting process is known as a Wiener process. The Wiener process has two important properties: Property 1 The change in z over a small interval of time satisfies: z t+h = z t + ɛ t. Then as of time t = 0, it is still the case that E[z T ] = z 0 and the variance V ar[z T ] = T. This relation may be written z(t + t) = ɛ t where z(t + t) = z t+ t z t. This has an expected value of zero and standard deviation of t. Property 2 The values of z for any two different short intervals of time are independent. It follows from this that z(t ) z(0) = N ɛ i t i where N = T/ t is the number of time intervals between 0 and T. Hence we have E[Z(T )] = z(0) and Var[z(T )] = N t = T or the standard deviation of z(t ) is T. Now consider what happens in the limit as t 0, that is as the length of the interval becomes an infinitesimal dt. We replace z(t + t) by dz(t) which has a mean of zero and standard deviation of dt. This continuous time stochastic process is also known as Brownian Motion after its use in

20 20 BLACK-SCHOLES physics to describe the motion of particles subject to a large number of small molecular shocks. This process is easily generalized to allow for a non-zero mean and arbitrary standard deviation. A generalized Wiener process for a variable x is defined in terms of dz(t) as follows dx = a dt + b dz where a and b are constants. This formula for the change in the value of x consists of two components, a deterministic component a dt and a stochastic component b dz(t). The deterministic component is dx = a dt or dx = a dt which shows that x = x 0 + at so that a is simply the trend term for x. Thus the increase in the value of x over one time period is a. The stochastic component b dz(t) adds noise or variability to the path for x. The amount of variability added is b times the Wiener process. Since the Wiener process has a standard deviation of one the generalized process has a standard deviation of b. The Asset Price Process Remember that we have ln S t+1 ln S t = ω t where ω t is are independent random variables with mean ν and standard deviation of σ. The continuous time version of this equation is therefore d ln S(t) = ν dt + σ dz where z is a standard Wiener process. The right-hand-side of the equation is just a random variable that is evolving through time. The term ν is called the drift parameter and the standard deviation of the continuously compounded rate of return is Var[r(t)] = σ t and the term σ is referred to as the volatility of the asset return.

21 ECO OPTIONS AND FUTURES 21 Ito Calculus We have written the process in terms of ln S(t) rather than S(t) itself. This is convenient and shows the connection to the binomial model. However it is usual to think in terms of S(t) itself too. In ordinary calculus we know that d ln S(t) = ds(t) S(t). Thus we might think that ds(t)/s(t) = ν dt + σ dz. WRONG. The correct version is ds(t) (ν S(t) = + 12 ) σ2 dt + σ dz. But this would be This is a special case of Ito s lemma. Ito s lemma shows that for any process of the form dx = a(x, t)dt + b(x, t)dz then the function G(x, t) follows the process ( G G dg = a(x, t) + x t + 1 ) 2 G 2 x 2 b2 (x, t) dt + G b(x, t)dz. x We ll see how to use Ito s lemma. We have d ln S(t) = ν dt + σ dz. Then let ln S(t) = x(t), so s(t ) = G(x, t) = e x. Then upon differentiating G x = ex = S, Hence using Ito s lemma 2 G S 2 = ex = S, G t = 0. d S(t) = (νs(t) σ2 S(t))dt + σs(t) dz or d S(t) = (ν σ2 )S(t) dt + σs(t) dz

22 22 BLACK-SCHOLES Since µ = ν σ2 we can write this as d S(t) = µs(t) dt + σs(s) dz. This process is know as geometric Brownian motion as it is the rate of change which is Brownian motion. Thus sometimes the above equation is written as d S(t) S(t) = µ dt + σ dz. Interpretation of the Black-Scholes Formula After this rather long digression we return to the Black-Scholes formula S(T ) K < 0. The Black-Scholes formula for the price of the call option at date t = 0 prior to maturity is given by where d 1 and d 2 satisfy c(0) = S(0)N(d 1 ) e rt KN(d 2 ) d 1 = S(0) ln( K 2 σ2 )T σ T d 2 = d 1 σ S(0) ln( K T = 2 σ2 )T σ T We need to be able to explain and interpret this formula. Evaluating the Discounted Value of the Future Payoff (Using Risk- Neutral Valuation) Rewrite the Black-Scholes formula as c(0) = e ( rt S(0)e rt N(d 1 ) KN(d 2 ) ). The formula can be interpreted as follows. If the call option is exercised at the maturity date then the holder gets the stock worth S(T ) but has to pay

23 ECO OPTIONS AND FUTURES 23 the strike price K. But this exchange takes place only if the call finishes in the money. Thus S(0)e rt N(d 1 ) represents the future value of the underlying asset conditional on the end stock value S(T ) being greater than the strike price K. The second term in the brackets KN(d 2 ) is the present value of the known payment K times the probability that the strike price will be paid N(d 2 ). The terms inside the brackets are discounted by the risk-free rate r to bring the payments into present value terms. Thus the evaluation inside the brackets is made using the risk-neutral or martingale probabilities and N(d 2 ) represents the probability that the call finishes in the money recall in a risk neutral world. Remember that in a risk-neutral world all assets earn the risk-free rate. we are assuming the the logarithm of the stock price is normally distributed. Thus ν the expected continuously compounded rate of return in a risk neutral world is equal to r 1 2 σ2 where the variance is deducted to calculate the certainty equivalent rate of return. Therefore at time T ln S(T ) is normally distributed with a mean value of ln S(0)+(r 1 2 σ2 )T and a standard deviation of σ T. Thus ln S(T ) (ln S(0)) + (r 1 2 σ2 )T ) σ T is a standard normal variable. The probability that S(T ) < K is therefore given by N( d 2 ) and the probability that S(T ) > K is given by 1 N( d 2 ) = N(d 2 ). It is more complicated to show that S(0)e rt N(d 1 ) is the future value of underlying asset in a risk-neutral world conditional on S(T ) > K but a proof can be found in more advanced textbooks.

24 24 BLACK-SCHOLES The Replicating Portfolio The formula also has another useful interpretation. From our analysis of the binomial model we know that the value of the call is c(0) = S(0) B where is the amount of the underlying asset bought and B is the amount of money borrowed needed to synthesize the call option. From the formula therefore N(d 1 ) is the hedge parameter indicating the number of shares bought and e rt KN(d 2 ) indicates the amount of cash borrowed to part finance the share purchase. Since 0 N(d) 1 the formula shows that the replicating portfolio consists of a fraction of the underlying asset and a positive amount of cash borrowed. The of this formula is also found by partially differentiating the call price formula c(0) S(0) =. It is the slope of the curve relating the option price with the price of the underlying asset. The cost of buying units of the stock and writing one call option is S(0) c(0). This portfolio is said to be delta neutral as a small change in the stock price will not affect the value of the portfolio. Boundary Conditions A boundary condition tells us the price of the put or call at some extreme value. Thus for example if we have a call option which is just about to expire, then the value of the option must just be its intrinsic value max[s(t ) K, 0], as the option has no time value left. If the Black-Scholes formula is correct it must reduce to the intrinsic value when the time to maturity left is very

25 ECO OPTIONS AND FUTURES 25 small. That is c(0) max[s(0) K, 0] as T 0. 6 boundary conditions are indeed satisfied. We will chck that the Boundary Conditions for a Call Option We shall consider the boundary conditions for the call option. Consider first the boundary condition for the call at expiration when T = 0. To do this consider the formula for the call option as T 0, that is as the time until maturity goes to zero. At maturity c(t ) = max[s(t ) K, 0] so we need to show that as T 0 the formula converges to c(0) = max[s(0) K, 0]. If S(0) > K then ln( S(0) ) > 0 so that as T 0, d K 1 and d 2 +. Thus N(d 1 ) and N(d 2 ) 1. Since e rt 1 as T 0 we have that c(0) S(0) K if S(0) > K. Alternatively if S(0) < K then ln( S(0) ) < 0 so that as T 0, K d 1 and d 2 and hence N(d 1 ) and N(d 2 ) 0. Thus c(0) 0 if S(0) < K. This is precisely as expected. If the option is in the money at maturity, S(0) > K, it is exercised for a profit of S(0) K and if it is out of the money, S(0) < K, the option expires unexercised and valueless. As another example consider what happens as σ 0. In this case the underlying asset becomes riskless so grows at the constant rate of r. Thus the future value of the stock is S(T ) = e rt S(0) and the payoff to the call option at maturity is max[e rt S(0) K, 0]. Thus the value of the call at date t = 0 is max[s(0) e rt K, 0]. To see this from the formula first consider the case where S(0) e rt K > 0 or ln( S(0) ) + rt > 0. Then as σ 0, d K 1 and d 2 + and hence N(d 1 ) and N(d 2 ) 1. Thus c(0) S(0) e rt K. Likewise when S(0) e rt K < 0 or ln( S(0) ) + rt < 0 then d K 1 and d 2 as σ 0. Hence N(d 1 ) and N(d 2 ) 0 and so c(0) 0. Thus combining both conditions c(0) max[e rt S(0) K, 0] as σ 0. 6 In fact the Black-Scholes formula is usually derived by computing the process the call option price must follow and imposing the boundary conditions.

26 26 BLACK-SCHOLES Boundary Conditions for a Put Option For a put option p(t ) = max[k S(T ), 0] so we need to show that as T 0 the formula converges to p(0) = max[k S(0), 0]. If S(0) > K then ln( S(0) ) > 0 so that as T 0, d K 1 and d 2 +. Thus N( d 1 ) and N( d 2 ) 0. Since e rt 1 as T 0 we have that p(0) 0 if S(0) > K. Alternatively if S(0) < K then ln( S(0) ) < 0 so that as T 0, K d 1 and d 2 and hence N( d 1 ) and N( d 2 ) 1. Since e rt 1 as T 0 we have that p(0) K S(0) if S(0) < K This is precisely as expected. If the option is in the money at now (at the maturity date T = 0) and S(0) < K, it is exercised for a profit of K S(0) and if it is out of the money, S(0) > K, the option expires unexercised and valueless. Now consider what happens to the put option as σ 0. In this case the underlying asset becomes riskless so grows at the constant rate of r. Thus the future value of the stock is S(T ) = e rt S(0) and the payoff to the put option at maturity is max[k e rt S(0), 0]. Thus the value of the put at date t = 0 is max[e rt K S(0), 0] (that is discounting by the factor e rt. To see this from the formula first consider the case where e rt K S(0) > 0 or taking logs and rearranging ln( S(0) ) + rt < 0. Then as σ 0, d K 1 and d 2 and hence N( d 1 ) and N( d 2 ) 1. Thus p(0) e rt K S(0). Likewise when e rt K S(0) < 0 or ln( S(0) ) + rt > 0 then d K 1 and d 2 + as σ 0. Hence N( d 1 ) and N( d 2 ) 0 and so p(0) 0. Thus combining both conditions p(0) max[e rt K S(0), 0] as σ 0.

### The Black-Scholes Formula

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

### European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

### One Period Binomial Model

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing

### LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS

LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS Robert S. Pindyck Massachusetts Institute of Technology Cambridge, MA 02142 Robert Pindyck (MIT) LECTURES ON REAL OPTIONS PART II August, 2008 1 / 50

### On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.

### Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13

Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The Black-Scholes-Merton Random Walk Assumption

### CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common

### Black-Scholes Equation for Option Pricing

Black-Scholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there

### Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing

Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common

### The Black-Scholes Model

The Black-Scholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The Black-Scholes Model Options Markets 1 / 19 The Black-Scholes-Merton

### Binomial lattice model for stock prices

Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }

### 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American

### Options and Derivative Pricing. U. Naik-Nimbalkar, Department of Statistics, Savitribai Phule Pune University.

Options and Derivative Pricing U. Naik-Nimbalkar, Department of Statistics, Savitribai Phule Pune University. e-mail: uvnaik@gmail.com The slides are based on the following: References 1. J. Hull. Options,

### Mathematical Finance

Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

### 2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold

Chapter 21 : Options-1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. Black-Scholes

### Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a

Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a variable depend only on the present, and not the history

### Lecture 11: Risk-Neutral Valuation Steven Skiena. skiena

Lecture 11: Risk-Neutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Risk-Neutral Probabilities We can

### Notes on Black-Scholes Option Pricing Formula

. Notes on Black-Scholes Option Pricing Formula by De-Xing Guan March 2006 These notes are a brief introduction to the Black-Scholes formula, which prices the European call options. The essential reading

### The Black-Scholes pricing formulas

The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock

### Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The Black-Scholes-Merton Model

### Exam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D

Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS

### Lecture 6: Option Pricing Using a One-step Binomial Tree. Friday, September 14, 12

Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond

### Monte Carlo Methods in Finance

Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction

### Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative

### The Intuition Behind Option Valuation: A Teaching Note

The Intuition Behind Option Valuation: A Teaching Note Thomas Grossman Haskayne School of Business University of Calgary Steve Powell Tuck School of Business Dartmouth College Kent L Womack Tuck School

### Numerical Methods for Option Pricing

Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly

### CAS/CIA Exam 3 Spring 2007 Solutions

CAS/CIA Exam 3 Note: The CAS/CIA Spring 007 Exam 3 contained 16 questions that were relevant for SOA Exam MFE. The solutions to those questions are included in this document. Solution 3 B Chapter 1, PutCall

### More Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options

More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path

### UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014. MFE Midterm. February 2014. Date:

UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014 MFE Midterm February 2014 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book,

### Caput Derivatives: October 30, 2003

Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor

### Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

### Chapter 21 Valuing Options

Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher

### Valuing Options / Volatility

Chapter 5 Valuing Options / Volatility Measures Now that the foundation regarding the basics of futures and options contracts has been set, we now move to discuss the role of volatility in futures and

### Financial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan

Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction

### Option Valuation. Chapter 21

Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price

### 1 Geometric Brownian motion

Copyright c 006 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM

### Contents. iii. MFE/3F Study Manual 9 th edition 10 th printing Copyright 2015 ASM

Contents 1 Put-Call Parity 1 1.1 Review of derivative instruments................................. 1 1.1.1 Forwards........................................... 1 1.1.2 Call and put options....................................

### Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13.

Week 11 The Black-Scholes Model: Hull, Ch. 13. 1 The Black-Scholes Model Objective: To show how the Black-Scholes formula is derived and how it can be used to value options. 2 The Black-Scholes Model 1.

### 1 The Black-Scholes model: extensions and hedging

1 The Black-Scholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes

### Example 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).

Chapter 4 Put-Call Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.

### On Market-Making and Delta-Hedging

On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing What to market makers do? Provide

### Options Pricing. This is sometimes referred to as the intrinsic value of the option.

Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the Put-Call Parity Relationship. I. Preliminary Material Recall the payoff

### Contents. iii. MFE/3F Study Manual 9th edition Copyright 2011 ASM

Contents 1 Put-Call Parity 1 1.1 Review of derivative instruments................................................ 1 1.1.1 Forwards............................................................ 1 1.1.2 Call

### LogNormal stock-price models in Exams MFE/3 and C/4

Making sense of... LogNormal stock-price models in Exams MFE/3 and C/4 James W. Daniel Austin Actuarial Seminars http://www.actuarialseminars.com June 26, 2008 c Copyright 2007 by James W. Daniel; reproduction

### 1. Assume that a (European) call option exists on this stock having on exercise price of \$155.

MØA 155 PROBLEM SET: Binomial Option Pricing Exercise 1. Call option [4] A stock s current price is \$16, and there are two possible prices that may occur next period: \$15 or \$175. The interest rate on

### The Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models

780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond market-maker would delta-hedge, we first need to specify how bonds behave. Suppose we try to model a zero-coupon

### 7: The CRR Market Model

Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The Cox-Ross-Rubinstein

### Lecture 4: The Black-Scholes model

OPTIONS and FUTURES Lecture 4: The Black-Scholes model Philip H. Dybvig Washington University in Saint Louis Black-Scholes option pricing model Lognormal price process Call price Put price Using Black-Scholes

### FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing

FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing I. Put-Call Parity II. One-Period Binomial Option Pricing III. Adding Periods to the Binomial Model IV. Black-Scholes

### Review of Basic Options Concepts and Terminology

Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some

### Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the

### Chapter 9 Parity and Other Option Relationships

Chapter 9 Parity and Other Option Relationships Question 9.1. This problem requires the application of put-call-parity. We have: Question 9.2. P (35, 0.5) C (35, 0.5) e δt S 0 + e rt 35 P (35, 0.5) \$2.27

### Chapter 5 Financial Forwards and Futures

Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment

### MATH3075/3975 Financial Mathematics

MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 We consider the Black-Scholes model M = B, S with the initial stock price S 0 = 9, the continuously compounded interest rate r = 0.01 per

### TABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13

TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:

### Chapter 2: Binomial Methods and the Black-Scholes Formula

Chapter 2: Binomial Methods and the Black-Scholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a call-option C t = C(t), where the

### Put-Call Parity. chris bemis

Put-Call Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain

### Chapter 13 The Black-Scholes-Merton Model

Chapter 13 The Black-Scholes-Merton Model March 3, 009 13.1. The Black-Scholes option pricing model assumes that the probability distribution of the stock price in one year(or at any other future time)

### Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration

CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put:

### Introduction to Options. Derivatives

Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived

### Brownian Motion and Ito s Lemma

Brownian Motion and Ito s Lemma 1 Introduction 2 Geometric Brownian Motion 3 Ito s Product Rule 4 Some Properties of the Stochastic Integral 5 Correlated Stock Prices 6 The Ornstein-Uhlenbeck Process Brownian

### Financial Modeling. Class #06B. Financial Modeling MSS 2012 1

Financial Modeling Class #06B Financial Modeling MSS 2012 1 Class Overview Equity options We will cover three methods of determining an option s price 1. Black-Scholes-Merton formula 2. Binomial trees

### Introduction to Mathematical Finance 2015/16. List of Exercises. Master in Matemática e Aplicações

Introduction to Mathematical Finance 2015/16 List of Exercises Master in Matemática e Aplicações 1 Chapter 1 Basic Concepts Exercise 1.1 Let B(t, T ) denote the cost at time t of a risk-free 1 euro bond,

### A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model

Applied Mathematical Sciences, vol 8, 14, no 143, 715-7135 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/11988/ams144644 A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting

### Option strategies. Stock Price Payoff Profit. The butterfly spread leads to a loss when the final stock price is greater than \$64 or less than \$56.

Option strategies Problem 11.20. Three put options on a stock have the same expiration date and strike prices of \$55, \$60, and \$65. The market prices are \$3, \$5, and \$8, respectively. Explain how a butterfly

### Two-State Option Pricing

Rendleman and Bartter [1] present a simple two-state model of option pricing. The states of the world evolve like the branches of a tree. Given the current state, there are two possible states next period.

### 1.2 Solutions to Chapter 1 Exercises

8 CHAPTER. CALCULUS. OPTIONS. PUT CALL PARITY.. Solutions to Chapter Exercises Problem : Compute lnx) dx. Solution: Using integration by parts, we find that lnx) dx = x) lnx) dx = x lnx) xlnx)) dx = x

### Monte Carlo simulations and option pricing

Monte Carlo simulations and option pricing by Bingqian Lu Undergraduate Mathematics Department Pennsylvania State University University Park, PA 16802 Project Supervisor: Professor Anna Mazzucato July,

### Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008

: A Stern School of Business New York University Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 Outline 1 2 3 4 5 6 se notes review the principles underlying option pricing and some of

### Underlying Pays a Continuous Dividend Yield of q

Underlying Pays a Continuous Dividend Yield of q The value of a forward contract at any time prior to T is f = Se qτ Xe rτ. (33) Consider a portfolio of one long forward contract, cash amount Xe rτ, and

### Option pricing. Vinod Kothari

Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate

### ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

### FINANCIAL ECONOMICS OPTION PRICING

OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.

### Geometric Brownian motion makes sense for the call price because call prices cannot be negative. Now Using Ito's lemma we can nd an expression for dc,

12 Option Pricing We are now going to apply our continuous-time methods to the pricing of nonstandard securities. In particular, we will consider the class of derivative securities known as options in

### The Black-Scholes-Merton Approach to Pricing Options

he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining

### This question is a direct application of the Put-Call-Parity [equation (3.1)] of the textbook. Mimicking Table 3.1., we have:

Chapter 3 Insurance, Collars, and Other Strategies Question 3.1 This question is a direct application of the Put-Call-Parity [equation (3.1)] of the textbook. Mimicking Table 3.1., we have: S&R Index S&R

### Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-Normal Distribution

Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-ormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last

### FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the

### The n-period Binomial Model

FIN-0008 FINANCIAL INSTRMENTS SPRING 2008 The n-period Binomial Model Introduction Once we have understood the one period binomial model it is very easy to extend the model to two or more periods so that

### Chapter 3 Insurance, Collars, and Other Strategies

Chapter 3 Insurance, Collars, and Other Strategies Question 3.1. This question is a direct application of the Put-Call-Parity (equation (3.1)) of the textbook. Mimicking Table 3.1., we have: S&R Index

### Black-Scholes Option Pricing Model

Black-Scholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,

### Jung-Soon Hyun and Young-Hee Kim

J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL Jung-Soon Hyun and Young-Hee Kim Abstract. We present two approaches of the stochastic interest

### Lecture 5: Put - Call Parity

Lecture 5: Put - Call Parity Reading: J.C.Hull, Chapter 9 Reminder: basic assumptions 1. There are no arbitrage opportunities, i.e. no party can get a riskless profit. 2. Borrowing and lending are possible

### 1 IEOR 4700: Introduction to stochastic integration

Copyright c 7 by Karl Sigman 1 IEOR 47: Introduction to stochastic integration 1.1 Riemann-Stieltjes integration Recall from calculus how the Riemann integral b a h(t)dt is defined for a continuous function

### Options: Valuation and (No) Arbitrage

Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial

### 1 The Black-Scholes Formula

1 The Black-Scholes Formula In 1973 Fischer Black and Myron Scholes published a formula - the Black-Scholes formula - for computing the theoretical price of a European call option on a stock. Their paper,

### Betting on Volatility: A Delta Hedging Approach. Liang Zhong

Betting on Volatility: A Delta Hedging Approach Liang Zhong Department of Mathematics, KTH, Stockholm, Sweden April, 211 Abstract In the financial market, investors prefer to estimate the stock price

### Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options

Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder

### Chapter 1: The binomial asset pricing model

Chapter 1: The binomial asset pricing model Simone Calogero April 17, 2015 Contents 1 The binomial model 1 2 1+1 dimensional stock markets 4 3 Arbitrage portfolio 8 4 Implementation of the binomial model

### where N is the standard normal distribution function,

The Black-Scholes-Merton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at

### Financial Options: Pricing and Hedging

Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equity-linked securities requires an understanding of financial

### x + 0.9x =0 x x =30

Introduction to derivative securities. Some denitions from nance There are an enormous number of derivative securities being traded in nancial markets. Rather than spending time listing them here, we refer

### Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)

Copyright 2003 Pearson Education, Inc. Slide 08-1 Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared

### Option Pricing. 1 Introduction. Mrinal K. Ghosh

Option Pricing Mrinal K. Ghosh 1 Introduction We first introduce the basic terminology in option pricing. Option: An option is the right, but not the obligation to buy (or sell) an asset under specified

### Pricing Barrier Options under Local Volatility

Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly

### Finance 350: Problem Set 8 Alternative Solutions

Finance 35: Problem Set 8 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. All payoff