Financial Derivatives Section 5

Financial Derivatives Section 5 The Black and Scholes Model Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 1 / 33

Outline 1 A Process for the Price of the Underlying Assets The Model A Simple Idea for Volatility Estimation 2 Itô s Lemma 3 The Black-Scholes Model The Black-Scholes-Merton partial differential equation The risk-neutral valuation The Black-Scholes pricing formula Binomial model and B-S model M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 2 / 33

Building a Model for the Price of the Underlying Assets Constant expected rate of return We assume that the expected rate of return of an underlying asset price remains constant and is equal to a number µ. In other words, we assume: [ ] 1 S(t + τ) S(t) τ E = µ S(t) This means that on average (or when there is no random noise) it holds that: S(t) S(t) = µ τ S(t) = µs(t) τ, for every time t > 0 and time interval τ, where S(t) denotes the difference S(t + τ) S(t). For t dt, this implies that on average: ds(t) S(t) = µdt ds(t) = µs(t)dt. Hence for every t > 0, on average we have S(t) = S(0)e µt. This is the solution of the equation S (t) = µs(t). M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 4 / 33

Building a Model for the Price of the Underlying Assets cont d Including the volatility Of course, underlying assets do exhibit variability. This is included in the model in the following way: S(t) S(t) = µ τ + σε t τ where ε t N(0, 1) (normal distribution with mean 0 and variance 1). This means that [ ] 1 S(t + τ) S(t) τ Var = σ 2 S(t) which implies that the model assumes that the variance of the rate of return is equal to a constant σ 2. The parameter σ is called the volatility of the underlying asset price. M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 5 / 33

Building a Model for the Price of the Underlying Assets cont d Letting τ become very small The price model can be written as S(t) S(t) = µ τ + σε t τ S(t) = µs(t) τ + σs(t)ε t τ. When we consider the changes of the price to happen in any moment of time (continuously), we basically assume that τ dt. Then, the model is usually written as: ds(t) = µs(t)dt + σs(t)dz(t) ( ) where dt stands for a very small (tiny) time change and z(t) is a random variable that follows the Normal Distribution with mean zero and variance equal to dt. The model ( ) is the famous Black-Scholes(-Merton-Samuelson) model for underlying assets prices without dividend. M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 6 / 33

Interpretation of the B-S Model The model implies that the rate of return of the price follows the Normal distribution: S(t) N(µ τ, σ 2 τ). S(t) The parameter µ, can be considered as the expected continuously compounded return (per annum) earned by the underlying asset price. The fact that we consider it constant is a big assumption. The B-S model for the asset prices is keeping with the weak-form of efficient market hypothesis. Equation ( ) is a stochastic differential equation and for the solution we need a tool that is called Ito s Lemma. The solution is ( S(t) = S(0)e µ σ2 2 ) t+σz(t) ( ) which is called the Geometric Brownian Motion (GBM). Notice how the volatility comes into the model... M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 7 / 33

Log-normality of the Asset Prices Equation ( ) implies that for every time t > 0 or ln ln ( ) S(t) = S(0) ( ) S(t) = S(0) ) (µ σ2 t + σdz(t) 2 ) (µ σ2 t + σε t t 2 which means that the logarithmic return of the asset follows normal distribution: ( ) ) ) S(t) ln(s(t)) ln(s(0)) = ln N ((µ σ2 t, σ 2 t, S(0) 2 or equivalently, ( ln (S(t)) N ln (S(0)) + ) ) (µ σ2 t, σ 2 t. 2 M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 8 / 33

Lognormality of the Asset Prices, cont d The asset price at time t obeys the Log-normal distribution, with E[S(t)] = S(0)e µt and ( ) Var(S(t)) = S 2 (0)e 2µt e σ2t 1.

Lognormality of the Asset Prices, cont d M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 10 / 33

A Way to Estimate the Volatility One way to estimate the volatility of the stock price is the following: 1 Choose the frequency of stock price data (every hour, every day etc) (denote this time length by τ). 2 Choose the number of observations of stock prices that is going to be used for the estimation (denote this number by n). 3 If S i denotes the stock price at the end of the i-th interval, calculate the series: ( ) Si u i = ln, for i = 1, 2,..., n. S i 1 4 Then estimate the standard deviation of u i s by: s = 1 n (u i u) n 1 2 where u is the mean of u i s. i=1 5 Estimate the volatility of the stock price by: ˆσ = s τ. 6 Standard error= ˆσ 2n.

The Itô s Lemma This lemma answers questions like the following: If the stock price follows the B-S model, which is the model that the price of a derivative on this stock follows? More precisely, if C(S(t), t) is the price of such a derivative at time t, we want to find how dc(s(t), t) looks like (the dynamics of the derivative price). dc(s(t), t) =(??? )dt+(??? )dz(t) The answer is given by the Itô s lemma. Itô s lemma Suppose that S(t) follows the model ds(t) = µs(t)dt + σs(t)dz(t) and C(S, t) is a twice differentiable function. Then, the dynamics of the stochastic process C(S(t), t) are the following: ( ) ( ) dc(s(t), t) = µs(t) C S + C t + 1 2 σ2 S 2 (t) 2 C S 2 dt + σs(t) C dz(t) S M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 13 / 33

The Itô s Lemma, an example As you have seen, the price of the futures contract when a stock does not pay dividends is given by F (t, T ) = S(t)e r(t t), when there is no dividend payments from the underlying asset. We ask the following: What is the dynamics of F (t, T ) if the stock price follows a GBM: ds(t) = µs(t)dt + σs(t)dz(t)? The answer is given by the Itô s lemma. F S = er(t t), F t df (t, T ) = = rs(t)e r(t t) = rf (t, T ) and 2 C S 2 = 0 ( ) µs(t)e r(t t) rf (t, T ) dt + σs(t)e r(t t) dz(t) df (t, T ) = (µ r)f (t, T )dt + σf (t, T )dz(t). M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 14 / 33

The Itô s Lemma, a second example How the function G(S(t)) = ln(s(t)) evolves? We apply Itô s lemma: Hence, G S = 1 S, 2 G S 2 = 1 G and S 2 t = 0. d ln(s(t)) = Something that we have already seen. ) (µ σ2 dt + σdz(t). 2 M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 15 / 33

The imposed assumptions Throughout this section, we are going to use the following assumptions: 1 The underlying stock price follows a GBM (constant volatility, no jumps). 2 There is no arbitrage opportunities. 3 The underlying asset does not give any dividend until the maturity of the option (we will withdraw this assumption later). 4 The short selling is allowed. 5 There is no transaction costs (no bid/ask spread and no short selling cost). 6 The trading is continuous. 7 The underlying asset is divisible (we can buy and short sell fractions of the asset). 8 The risk-free internet rate refers to continuously discounting and remains constant until the option s maturity: db(t) = rb(t)dt in other words B(t) = B(0)e rt, t (0, T ]. M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 17 / 33

The general ideas The aim of the B-S model is to derive option prices and a way to dynamically hedge the involved risk. The basic idea is described in the following steps: 1 The Black-Scholes-Merton analysis is analogous to the no-arbitrage analysis of the binomial model. We are asking for example how many stocks we should buy/sell in order to hedge the risk of selling a call option. 2 Consider a derivative written on the stock price S(t), with payoff C(T ) = C(S(T ), T ) at maturity. 3 We denote the price of this derivative at time t, by C(t) = C(S(t), t). 4 As in the case of the binomial model, we want to construct a portfolio which consists at time t of (t) shares of the stock and an amount B(t) invested at free-risk interest rate. such that the value of the portfolio to be equal to C(t), at any time t. 5 The outcome is an equation that is satisfied by the function C(t). M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 18 / 33

Derivation of the B-S-M pde Our goal is to find (t) and B(t) such that: (t)s(t) + B(t) = C(S(t), t) for every time t T. We have assumed that: ds(t) = µs(t)dt + σs(t)dz(t) and db(t) = rb(t)dt. The changes of the portfolios in the small period of time give: (t)ds(t) + db(t) = dc(s(t), t) (t)ds(t) + rb(t)dt = dc(s(t), t). How can we find the dynamics of C(S(t), t)? This is exactly where the Itô s lemma helps! ( C C dc(s(t), t) = µs(t) + S t + 1 2 ) C 2 S 2 σ2 S 2 (t) dt + C S σs(t)dz(t). M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 19 / 33

Derivation of the B-S-M pde cont d The last equation can be written as: dc(s(t), t) = C (µs(t)dt + σs(t)dz(t)) + S which is equal to: ( dc(s(t), t) = C S ds(t) + In other words, we have to solve the equation: (t)ds(t) + rb(t)dt = C S ds(t) + C t + 1 2 ( C t + 1 2 ) C 2 S 2 σ2 S 2 (t) dt, ) 2 C S σ 2 S 2 (t) dt 2 ( C t + 1 2 C 2 S 2 σ2 S 2 (t) ) dt. M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 20 / 33

Derivation of the B-S-M pde cont d One solution of the above equation is when (t) = C S This means that B(t) = C(t) C S S(t) By replacing these findings in the desired equation, we get: rc(t) = rs(t) C C S (t) + t (t) + σ2 S 2 (t) 2 The above equation is written simplier as: 2 C S 2 (t) No stochastic term!!! rc(t) = rs(t)c S (t) + C t (t) + σ2 S 2 (t) 2 C SS (t) This is the famous Black-Scholes-Merton partial differential equation for the derivative prices. If we follow the dynamic hedging of (t), we perfectly replicate the option payoff. Hedging all the risk involved! Exactly what we have succeeded in the binomial model. M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 21 / 33

Discussion of the BSM pde The BSM pde is satisfied by (almost) all the derivatives written on the stock. The BSM pde is going with boundary conditions determined by the derivative payoff. For the call option for example, the boundary condition is C(S(T ), T ) = max{s(t ) K, 0}. The BSM pde can be solved in a closed-form only for European options. For the American ones, numerical methods are used to approximate the solution. A very important notice: r appears in the equation instead of µ. Why? This means that the solution of the pde (or equivalently the derivative value) is independent on µ!! Another important notice: The replication portfolio should be changed frequently since (t) and B(t) change as t and S(t) change. M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 22 / 33

An example of the BSM pde Assume that an investor has a long position in a futures contract on the stock price S(t) with maturity T and strike price K. The value of this contract today is f (t) = S(t) Ke r(t t). Why? Since the future is a derivative on the stock, it should satisfy the BSM pde. Indeed, f t (t) = rke r(t t), f S = 1 and f SS = 0 and hence which holds true. rke r(t t) + rs(t) + 1 2 σ2 S 2 0 = r(s Ke r(t t) ), M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 23 / 33

Risk-neutral valuation As in the case of binomial model, the subjective expected rate of return does not influence the derivative valuation. This is because the perfect replication of derivative payoff is possible and hence any position on a derivative written on the stock could be thought as a position with no risk. In sequel, this implies that for the derivative valuation, we can assume that any security (the stock and the derivatives on this stock) has expected return equal to the risk-free interest rate. In other words, Ê[S(T )] = S(0)e rt or Ê[S(T ) S(t)] = S(t)e r(t t), where Ê is the expectation under the risk-neutral probability. Hence, if we want to find the price of a derivative, we need to calculate: C(0) = e rt Ê[C(T )] or C(t) = e r(t t) Ê[C(T ) S(t)] If we calculate C(t), we can use the formula (t) = C S position on the derivative. to fully hedge the M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 24 / 33

A sketch of the proof of the B-S formula In 1973, Fischer Black and Myron Scholes assumed the GBM model for the stock price and derived a closed form solution for the call option price. We can have a look on a way to derive the B-S formula by following the steps below: (1) As we have seen, according to GBM: E[S(T )] = S(0)e µt. Since the perfect replication of the option payoff is possible, we can assume that we live in a risk-neutral world: Ê[S(T )] = S(0)e rt. (2) Under the risk-neutral probabilities, the stock price follows a log-normal distribution but with different mean: ( ) ) ln (S(T )) N ln (S(0)) + (r σ2 T, σ 2 T. 2 (3) In order to price the call option with strike price K, it would be enough to calculate the expectation: e rt Ê[max{S(T ) K, 0}], where (under the risk-neutral probabilities) S(T ) has the above log-normal distribution.

A sketch of the proof of the B-S formula cont d (4) Note that Ê[max{S(T ) K, 0}] = + (s K)g(s)ds, where g(s) is the K corresponding probability density function of the lognormal distribution. (5) The next step is to define a new variable: x = ln(s) ln(s(0)) + (r σ2 /2)T σ T. (6) The desired expectation becomes: + ln(k) ln(s(0))+rt T σ 2 /2 σ T (e xσ T +ln(s(0))+rt σ2 T 2 K)h(x)dx, where h(x) is the probability density function of the standard normal distribution. M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 26 / 33

The Black & Scholes formula The price of the European call option with strike price K is given by: c(t) = S(t)N(d 1 ) Ke r(t t) N(d 2 ) where d 1 = ln(s(t)/k)+(r+σ2 /2)(T t) σ, T t d 2 = ln(s(t)/k)+(r σ2 /2)(T t) σ = d T t 1 σ T t. Where N(.) is the standardized cumulative Normal distribution. Using the put-call parity, one can directly derive the price of the corresponding European put option: p(t) = Ke r(t t) N( d 2 ) S(t)N( d 1 ) It can be checked that the price c(t) solves the BSM pde. (What a nice exercise!) M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 27 / 33

B-S formula, an example Consider the following inputs for a European call option on a stock price with the following data: S(0) = 9.30, K = 10, r = 1%, σ = 35.32% and T t = 90 trading days. We calculate: d 1 = ln( 9.3 10 )+(0.01+ 0.3532 2 )( 90 252 ) 0.353 90/252 90 = 0.223 and d 2 = d 1 0.3532 252 = 0.434. N( 0.223) = 0.412 and N( 0.434) = 0.332 Hence, and 90 0.01 c(0) = 9.3 0.412 10 e 252 0.332 = 0.52 90 0.01 p(0) = 0.52 + 10 e 252 9.3 = 1.18 M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 28 / 33

The Black & Scholes formula with dividend payments In the case where the stock pays dividend, the option price changes. (Why?) Assume that the amount and the timing of the dividends during the life of the option is known (for short time, this is not an unreasonable assumption). On the day that the dividend is paid, the stock price declines by this exact amount. Denote by D(t) the present value at time t of the dividends paid in period [t, T ]. Under these assumptions the BS formula becomes: c(t) = (S(t) D(t))N(d 1 ) Ke r(t t) N(d 2 ) and where p(t) = Ke r(t t) N( d 2 ) (S(t) D(t))N( d 1 ), d 1 = ln((s(t) D(t))/K) + (r + σ2 /2)(T t) σ T t and d 2 = d 1 σ T t. M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 29 / 33

The Black & Scholes formula with dividend yield When the underline asset is an index, it is much more preferable to consider the dividend payments in the form of a continuous dividend yield q. Then, The payment of a dividend yield at rate q causes the reduction of the price growth of the underling asset. If dividend is paid, the price of the underlying asset grows from S(0) to S(T ), then in the absence of dividend it would grow from S(0) to S(T )e qt. Alternatively, without dividend it would grow from S(0)e qt to S(T ). This is like we continuously discount the dividend payments and reduce the current price accordingly. Under these assumptions the BS formula becomes: and where, c(t) = S(t)e q(t t) N(d 1 ) Ke r(t t) N(d 2 ) p(t) = Ke r(t t) N( d 2 ) S(t)e q(t t) N( d 1 ), d 1 = ln(s(t)/k) + (r q + σ2 /2)(T t) σ T t and d 2 = d 1 σ T t. M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 30 / 33

Discussion on the B-S formula The Black-Scholes portfolio can be constructed only in the case where the underlying asset is tradable. Non-arbitrage relations such as put-call parity and arbitrage bounds are satisfied by the BS formula. The BS formula provides a clear relationship between the option prices and the affecting parameters. For the case of American options, one can use approximation arguments from the BSM pde or from the BS price. There is a number of problems/inconsistencies in the GBM model, which are reflected in the BS formula. The constant volatility is the most problematic one (it has been shown that the volatility changes with time). The validity of the formula is based on the continuous trading, which can only be approximated (the accumulated error of this approximation may be significant). The BS formula tends to overvalue deep OTM calls and undervalue deep ITM calls. M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 31 / 33

The implied volatility Volatility The only parameter in the BS pricing formula that can not be directly observed is the volatility σ. Estimation of volatility An alternative way to estimate the volatility of a particular stock price is to consider the market prices of the options as inputs and use the BS formula to extract the volatility. In other words, we solve the following equation with respect to σ: c(t, σ) = Market call option price at time t. The solutions Since the c(t, σ) is a continuous and increasing function of σ, there is one-to-one correspondence between the marketed call option prices and the implied volatility.

B-S model as a limiting case of binomial model The CRR model, Cox-Ross-Rubinstein Assuming that we have estimated the volatility of a stock price. The CRR binomial model imposes that: Taking the limit u = e σ t and d = e σ t Cox, Ross and Rubinstein (1979) proved that as the time interval t goes to zero, or equivalently as the number of periods of the binomial models goes to infinity, the CRR model for the stock price tends to the GBM. Symmetric random walk Brownian Motion and Exponential random walk Geometric Brownian Motion Note also that the hedging strategy for a derivative in the BS model: (t) = C S (t) can be seen as the limit of the corresponding hedging strategy in the binomial model: (t) = cu c d us(0) ds(0). M. Anthropelos (Un. of Piraeus) B-S Model Spring 2018 33 / 33