Lectures. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. No tutorials in the first week

Lectures Sergei Fedotov 20912 - Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1

Lecture 1 1 Introduction Elementary economics background What is financial mathematics? The role of SDE s and PDE s 2 Time Value of Money 3 Continuous Model for Stock Price Sergei Fedotov (University of Manchester) 20912 2010 2 / 1

General Information Textbooks: J. Hull, Options, Futures and Other Derivatives, 7th Edition, Prentice-Hall, 2008. P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial Derivatives: A Student Introduction, Cambridge University Press, 1995 Assessment: Test in a week 6: 20% 2 hours examination: 80% Sergei Fedotov (University of Manchester) 20912 2010 3 / 1

Elementary Economics Background This course is concerned with mathematical models for financial markets: Stock Markets, such as NYSE(New York Stock Exchange), London Stock Exchange, etc. Bond Markets, where participants buy and sell debt securities. Futures and Option Markets, where the derivative products are traded. Example: European call option gives the holder the right (not obligation) to buy underlying asset at a prescribed time T for a specified price E. Option market is massive! More money is invested in options than in the underlying securities. The main purpose of this course is to determine the price of options. Why stochastic differential equations (SDE s) and partial differential equations (PDE s)? Sergei Fedotov (University of Manchester) 20912 2010 4 / 1

Time value of money What is the future value V (t) at time t = T of an amount P invested or borrowed today at t = 0? Simple interest rate: V (T) = (1 + rt)p (1) where r > 0 is the simple interest rate, T is expressed in years. Compound interest rate: V (T) = ( 1 + r m) mt P (2) where m is the number interest payments made per annum. Continuous compounding: In the limit m, we obtain V (T) = e rt P (3) since e = lim z ( 1 + 1 z ) z. Throughout this course the interest rate r will be continuously compounded. Sergei Fedotov (University of Manchester) 20912 2010 5 / 1

Simple Model for Stock Price S(t) Let S(t) represent the stock price at time t. How to write an equation for this function? Return (relative measure of change): where S = S(t + δt) S(t) In the limit δt 0 : S S ds S (4) (5) How to model the return? Let us decompose the return into two parts: deterministic and stochastic Sergei Fedotov (University of Manchester) 20912 2010 6 / 1

Modelling of Return Return: ds = µdt + σdw (6) S where µ is a measure of the expected rate of growth of the stock price. In general, µ = µ(s,t). In simpe models µ is taken to be constant (µ = 0.1 0.3). σdw describes the stochastic change in the stock price, where dw stands for W = W (t + t) W (t) as t 0 W (t) is a Wiener process σ is the volatility (σ = 0.2 0.5) Sergei Fedotov (University of Manchester) 20912 2010 7 / 1

Stochastic differential equation for stock price ds = µsdt + σsdw (7) Simple case: volatility σ = 0 ds = µsdt (8) Sergei Fedotov (University of Manchester) 20912 2010 8 / 1

Wiener process Definition. The standard Wiener process W (t) is a Gaussian process such that W (t) has independent increments: if u v s t, then W (t) W (s) and W (v) W (u) are independent W (s + t) W (s) is N(0,t) and W (0) = 0 Clearly EW (t) = 0 and EW 2 = t, where E is the expectation operator. The increment W = W (t + t) W (t) can be written as W = X ( t) 1 2, where X is a random variable with normal distribution with zero mean and unit variance: E W = 0 and E( W ) 2 = t. X N (0,1) Sergei Fedotov (University of Manchester) 20912 2010 9 / 1

Lecture 2 1 Properties of Wiener Process 2 Approximation for Stock Price Equation 3 Itô s Lemma Sergei Fedotov (University of Manchester) 20912 2010 10 / 1

Wiener process The probability density function for W (t) is p(y,t) = 1 ) exp ( y2 2πt 2t and P (a W (t) b) = b a p(y,t)dy Simulations of a Wiener process: Sergei Fedotov (University of Manchester) 20912 2010 11 / 1

Approximation of SDE for small t The increment W = W (t + t) W (t) can be written as W = X ( t) 1 2, where X is a random variable with normal distribution with zero mean and unit variance: X N (0,1) E W = 0 and E( W ) 2 = t. Recall: equation for the stock price is ds = µsdt + σsdw, then S µs t + σsx ( t) 1 2 It means S N ( µs t,σ 2 S 2 t ) Sergei Fedotov (University of Manchester) 20912 2010 12 / 1

Examples Example 1. Consider a stock that has volatility 30% and provides expected return of 15% p.a. Find the increase in stock price for one week if the initial stock price is 100. Answer: S = 0.288 + 4.155X Note: 4.155 is the standard deviation Example 2. Show that the return S S µ t and variance σ 2 t is normally distributed with mean Sergei Fedotov (University of Manchester) 20912 2010 13 / 1

Itô s Lemma We assume that f (S,t) is a smooth function of S and t. Find df if ds = µsdt + σsdw Volatility σ = 0 df = f f t dt + S ds = ( f t Volatility σ 0 Itô s Lemma: + µs f S ) dt ( ) df = f f t + µs S + 1 2 σ2 S 2 2 f dt + σs f S 2 S dw Example. Find the SDE satisfied by f = S 2. Sergei Fedotov (University of Manchester) 20912 2010 14 / 1

Lecture 3 1 Distribution for lns(t) 2 Solution to Stochastic Differential Equation for Stock Price 3 Examples Sergei Fedotov (University of Manchester) 20912 2010 15 / 1

Differential of ln S Example 1. Find the stochastic differential equation (SDE) for by using Itô s Lemma: f = lns ( ) df = f f t + µs S + 1 2 σ2 S 2 2 f dt + σs f S 2 S dw We obtain df = This is a constant coefficient SDE. ) (µ σ2 dt + σdw 2 Integration from 0 to t gives ) f f 0 = (µ σ2 t + σw (t) 2 since W (0) = 0. Sergei Fedotov (University of Manchester) 20912 2010 16 / 1

Normal distribution for lns(t) We obtain for lns(t) lns(t) lns 0 = ) (µ σ2 t + σw (t) 2 where S 0 = S(0) is the initial stock price. ( lns(t) has a normal distribution with mean lns 0 + variance σ 2 t. µ σ2 2 ) t and Example 2. Consider a stock with an initial price of 40, an expected return of 16% and a volatility of 20%. Find the probability distribution of lns in six months. We have lns(t) N ( lns 0 + Answer: ln S(0.5) N (3.759, 0.020) ) ) (µ σ2 T,σ 2 T 2 Sergei Fedotov (University of Manchester) 20912 2010 17 / 1

Probability density function for ln S(t) Recall that if the random variable X has a normal distribution with mean µ and variance σ 2, then the probability density function is ( ) 1 p(x) = exp (x µ)2 2πσ 2 2σ 2 The probability density function of X = lns(t) is ( 1 2πσ 2 t exp (x lns 0 (µ σ 2 /2)t) 2 ) 2σ 2 t Sergei Fedotov (University of Manchester) 20912 2010 18 / 1

Exact expression for stock price S(t) Definition. The model of a stock ds = µsdt + σsdw is known as a geometric Brownian motion. The random function S(t) can be found from ln(s(t)/s 0 ) = ) (µ σ2 t + σw (t) 2 Stock price at time t: S(t) = S 0 e ) (µ σ2 t+σw(t) 2 Or S(t) = S 0 e ) (µ σ2 t+σ tx 2 where X N (0,1) Sergei Fedotov (University of Manchester) 20912 2010 19 / 1

Lecture 4 1 Financial Derivatives 2 European Call and Put Options 3 Payoff Diagrams, Short Selling and Profit Sergei Fedotov (University of Manchester) 20912 2010 20 / 1

Derivatives Derivative is a financial instrument which value depends on the values of other underlying variables. Other names are financial derivative, derivative security, derivative product. A stock option, for example, is a derivative whose value is dependent on the stock price. Examples: forward contracts, futures, options, swaps, CDS, etc. Options are very attractive to investors, both for speculation and for hedging What is an Option? Definition. European call option gives the holder the right (not obligation) to buy underlying asset at a prescribed time T for a specified (strike) price E. European put option gives its holder the right (not obligation) to sell underlying asset at a prescribed time T for a specified (strike) price E. The question is what does this actually mean? Sergei Fedotov (University of Manchester) 20912 2010 21 / 1

Example. Consider a three-month European call option on a BP share with a strike price E = 15 (T = 0.25). If you enter into this contract you have the right but not the obligation to buy one share for E = 15 in a three months time. Whether you exercise your right depends on the stock price in the market at time T: If the stock price is above 15, say 25, you can buy the share for 15, and sell it immediately for 25, making a profit of 10. If the stock price is below 15, there is no financial sense to buy it. The option is worthless. Sergei Fedotov (University of Manchester) 20912 2010 22 / 1

Payoff Diagrams We denote by C(S,t) the value of European call option and P(S,t) the value of European put option Definition. Payoff Diagram ia a graph of the value of the option position at expiration t = T as a function of the underlying stock price S. Call price at t = T: C(S,T) = max(s E,0) = Put price at t = T: P(S,T) = max(e S,0) = { 0, S E, S E, S > E, { E S, S E, 0, S > E, If a trader thinks that the stock price is on the rise, he can make money by purchase a call option without buying the stock. If a trader believes the stock price is on the decline, he can make money by buying put options. Sergei Fedotov (University of Manchester) 20912 2010 23 / 1

Profit of call option buyer The profit (gain) of a call option holder (buyer) at time T is max(s E,0) C 0 e rt, C 0 initial call price Example: Find the stock price on the exercise date in three months, for a European call option with strike price 10 to give a gain (profit) of 14 if the option is bought for 2.25, financed by a loan with continuously compounded interest rate of 5% Solution: 14 = S(T) 10 2.25 e 0.05 1 4, S(T) = 26.28 For the holder of European put option, the profit at time T is. max(e S,0) P 0 e rt Sergei Fedotov (University of Manchester) 20912 2010 24 / 1

Portfolio and Short Selling Definition. Short selling is the practice of selling assets that have been borrowed from a broker with the intention of buying the same assets back at a later date to return to the broker. This technique is used by investors who try to profit from the falling price of a stock. Definition. Portfolio is the combination of assets, options and bonds. We denote by Π the value of portfolio. Example: Π = 2S + 4C 5P. It means that portfolio consists of long position in two shares, long position in four call options and short position in five put options. Sergei Fedotov (University of Manchester) 20912 2010 25 / 1

Option positions Sergei Fedotov (University of Manchester) 20912 2010 26 / 1

Straddle Straddle is the purchase of a call and a put on the same underlying security with the same maturity time T and strike price E. The value of portfolio is Π = C + P Straddle is effective when an investor is confident that a stock price will change dramatically, but is uncertain of the direction of price move. Example of large profits: S 0 = 40, E = 40, C 0 = 2, P 0 = 2 Ans: 400 Let us find an expected return!! Sergei Fedotov (University of Manchester) 20912 2010 27 / 1

Lecture 5 1 Trading Strategies: Straddle, Bull Spread, etc. 2 Bond and Risk-Free Interest Rate 3 No Arbitrage Principle Sergei Fedotov (University of Manchester) 20912 2010 28 / 1

Portfolio and Short Selling Reminder from previous lecture 4. Definition. Short selling is the practice of selling assets that have been borrowed from a broker with the intention of buying the same assets back at a later date to return to the broker. This technique is used by investors who try to profit from the falling price of a stock. Definition. Portfolio is the combination of assets, options and bonds. We denote by Π the value of portfolio. Examples. Sergei Fedotov (University of Manchester) 20912 2010 29 / 1

Option positions Sergei Fedotov (University of Manchester) 20912 2010 30 / 1

Trading Strategies Involving Options Straddle is the purchase of a call and a put on the same underlying security with the same maturity time T and strike price E. The value of portfolio is Π = C + P Straddle is effective when an investor is confident that a stock price will change dramatically, but is uncertain of the direction of price move. Short Straddle, Π = C P, profits when the underlying security changes little in price before the expiration t = T. Barings Bank was the oldest bank in London until its collapse in 1995. It happened when the bank s trader, Nick Leeson, took short straddle positions and lost 1.3 billion dollars. Sergei Fedotov (University of Manchester) 20912 2010 31 / 1

Bull Spread Bull spread is a strategy that is designed to profit from a moderate rise in the price of the underlying security. Let us set up a portfolio consisting of a long position in call with strike price E 1 and short position in call with E 2 such that E 1 < E 2. The value of this portfolio is Π t = C t (E 1 ) C t (E 2 ). At maturity t = T 0, S E 1, Π T = S E 1, E 1 S < E 2, E 2 E 1, S E 2 The holder of this portfolio benefits when the stock price will be above E 1. Sergei Fedotov (University of Manchester) 20912 2010 32 / 1

Risk-Free Interest Rate We assume the existence of risk-free investment. Examples: US government bond, deposit in a sound bank. We denote by B(t) the value of this investment. Definition. Bond is a contact that yields a known amount F, called the face value, on a known time T, called the maturity date. The authorized issuer (for example, government) owes the holders a debt and is obliged to pay interest (the coupon) and to repay the face value at maturity. Zero-coupon bond involves only a single payment at T. Return db B = rdt, where r is the risk-free interest rate. If B(T) = F, then B(t) = Fe r(t t), where e r(t t) - discount factor Sergei Fedotov (University of Manchester) 20912 2010 33 / 1

No Arbitrage Principle The key principle of financial mathematics is No Arbitrage Principle. There are never opportunities to make risk-free profit Arbitrage opportunity arises when a zero initial investment Π 0 = 0 is identified that guarantees non-negative payoff in the future such that Π T > 0 with non-zero probability. Arbitrage opportunities may exist in a real market. But, they cannot last for a long time. Sergei Fedotov (University of Manchester) 20912 2010 34 / 1

Lecture 6 1 No-Arbitrage Principle 2 Put-Call Parity 3 Upper and Lower Bounds on Call Options Sergei Fedotov (University of Manchester) 20912 2010 35 / 1

No Arbitrage Principle The key principle of financial mathematics is No Arbitrage Principle. There are never opportunities to make risk-free profit. Arbitrage opportunity arises when a zero initial investment Π 0 = 0 is identified that guarantees non-negative payoff in the future such that Π T > 0 with non-zero probability. Arbitrage opportunities may exist in a real market. All risk-free portfolios must have the same return: risk-free interest rate. Let Π be the value of a risk-free portfolio, and dπ is its increment during a small period of time dt. Then dπ Π = rdt, where r is the risk-less interest rate. Let Π t be the value of the portfolio at time t. If Π T 0, then Π t 0 for t < T. Sergei Fedotov (University of Manchester) 20912 2010 36 / 1

Put-Call Parity Let us set up portfolio consisting of long one stock, long one put and short one call with the same T and E. The value of this portfolio is Π = S + P C. The payoff for this portfolio is Π T = S + max(e S,0) max(s E,0) = E The payoff is always the same whatever the stock price is at t = T. Using No Arbitrage Principle, we obtain S t + P t C t = Ee r(t t), where C t = C(S t,t) and P t = P (S t,t). This relationship between S t,p t and C t is called Put-Call Parity which represents an example of complete risk elimination. Sergei Fedotov (University of Manchester) 20912 2010 37 / 1

Upper and Lower Bounds on Call Options Put-Call Parity (t = 0): S 0 + P 0 C 0 = Ee rt. It shows that the value of European call option can be found from the value of European put option with the same strike price and maturity: C 0 = P 0 + S 0 Ee rt. Therefore C 0 S 0 Ee rt since P 0 0 S 0 Ee rt is the lower bound for call option. S 0 Ee rt C 0 S 0 Let us illustrate these bounds geometrically. Sergei Fedotov (University of Manchester) 20912 2010 38 / 1

Examples Example 1. Find a lower bound for six months European call option with the strike price 35 when the initial stock price is 40 and the risk-free interest rate is 5% p.a. In this case S 0 = 40, E = 35, T = 0.5, and r = 0.05. The lower bound for the call option price is S 0 E exp ( rt), or 40 35exp ( 0.05 0.5) = 5.864 Example2. Consider the situation where the European call option is 4. Show that there exists an arbitrage opportunity. We establish a zero initial investment Π 0 = 0 by purchasing one call for 4 and the bond for 36 and selling one share for 40. The portfolio is Π = C + B S. Sergei Fedotov (University of Manchester) 20912 2010 39 / 1

Examples At maturity t = T, the portfolio Π = C + B S has the value: { 36.911 S, S 35 Π T = max(s E,0) + 36exp (0.05 0.5) S = 1.911 S > 35 It is clear that Π T > 0, therefore there exists an arbitrage opportunity. Sergei Fedotov (University of Manchester) 20912 2010 40 / 1

Lecture 7 1 Upper and Lower Bounds on Put Options 2 Proof of Put-Call Parity by No-Arbitrage Principle 3 Example on Arbitrage Opportunity Sergei Fedotov (University of Manchester) 20912 2010 41 / 1

Upper and Lower Bounds on Put Option Reminder from lecture 6. Arbitrage opportunity arises when a zero initial investment Π 0 = 0 is identified that guarantees a non-negative payoff in the future such that Π T > 0 with non-zero probability. Put-Call Parity at time t = 0: S 0 + P 0 C 0 = Ee rt. Upper and Lower Bounds on Put Option (exercise sheet 3): Ee rt S 0 P 0 Ee rt Let us illustrate these bounds geometrically. Sergei Fedotov (University of Manchester) 20912 2010 42 / 1

Proof of Put-Call Parity The value of European put option can be found as P 0 = C 0 S 0 + Ee rt. Let us prove this relation by using No-Arbitrage Principle. Assume that P 0 > C 0 S 0 + Ee rt. Then one can make a riskless profit (arbitrage opportunity). We set up the portfolio Π = P S + C + B. At time t = 0 we sell one put option for P 0 (write the put option) sell one share for S 0 (short position) buy one call option for C 0 buy one bond for B 0 = P 0 + S 0 C 0 > Ee rt The balance of all these transactions is zero, that is, Π 0 = 0 Sergei Fedotov (University of Manchester) 20912 2010 43 / 1

Proof of Put-Call Parity At maturity t = T the portfolio Π = P S + C + B has the value Π T = { (E S) S + B0 e rt, S E, S + (S E) + B 0 e rt, S > E, = E + B 0 e rt Since B 0 > Ee rt, we conclude Π T > 0. and Π 0 = 0. This is an arbitrage opportunity. Sergei Fedotov (University of Manchester) 20912 2010 44 / 1

Proof of Put-Call Parity Now we assume that P 0 < C 0 S 0 + Ee rt. We set up the portfolio Π = P + S C B. At time t = 0 we buy one put option for P 0 buy one share for S 0 (long position) sell one call option for C 0 (write the call option) borrow B 0 = P 0 + S 0 C 0 < Ee rt The balance of all these transactions is zero, that is, Π 0 = 0 At maturity t = T we have Π T = E B 0 e rt. Since B 0 < Ee rt, we conclude Π T > 0. This is an arbitrage opportunity!!! Sergei Fedotov (University of Manchester) 20912 2010 45 / 1

Example on Arbitrage Opportunity Three months European call and put options with the exercise price 12 are trading at 3 and 6 respectively. The stock price is 8 and interest rate is 5%. Show that there exists arbitrage opportunity. Solution: The Put-Call Parity P 0 = C 0 S 0 + Ee rt is violated, because 6 < 3 8 + 12e 0.05 1 4 = 6.851 To get arbitrage profit we buy a put option for 6 sell a call option for 3 buy a share for 8 borrow 11 at the interest rate 5%. The balance is zero!! Sergei Fedotov (University of Manchester) 20912 2010 46 / 1

Example: Arbitrage Opportunity The value of the portfolio Π = P + S C B at maturity T = 1 4 is Π T = E B 0 e rt = 12 11e 0.05 1 4 0.862. Combination P + S C gives us 12. We repay the loan 11e 0.05 1 4. The balance 12 11e 0.05 1 4 is an arbitrage profit 0.862. Sergei Fedotov (University of Manchester) 20912 2010 47 / 1

Lecture 8 1 One-Step Binomial Model for Option Price 2 Risk-Neutral Valuation 3 Examples Sergei Fedotov (University of Manchester) 20912 2010 48 / 1

One-Step Binomial Model Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d ( u > 1;d < 1). At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. The purpose is to find the current price C 0 of a European call option. Sergei Fedotov (University of Manchester) 20912 2010 49 / 1

Riskless Portfolio Now, we set up a portfolio consisting of a long position in shares and short position in one call Π = S C Let us find the number of shares that makes the portfolio Π riskless. The value of portfolio when stock moves up is S 0 u C u The value of portfolio when stock moves down is S 0 d C d If portfolio Π = S C is risk-free, then S 0 u C u = S 0 d C d Sergei Fedotov (University of Manchester) 20912 2010 50 / 1

No-Arbitrage Argument The number of shares is = Cu C d S 0 (u d). Because portfolio is riskless for this, the current value Π 0 can be found by discounting: Π 0 = ( S 0 u C u )e rt, where r is the interest rate. On the other hand, the cost of setting up the portfolio is Π 0 = S 0 C 0. Therefore S 0 C 0 = ( S 0 u C u )e rt. Finally, the current call option price is where = Cu C d S 0 (u d) C 0 = S 0 ( S 0 u C u ) e rt, (No-Arbitrage Argument). Sergei Fedotov (University of Manchester) 20912 2010 51 / 1

Risk-Neutral Valuation Alternatively where (Risk-Neutral Valuation) C 0 = e rt (pc u + (1 p)c d ), p = ert d u d. It is natural to interpret the variable 0 p 1 as the probability of an up movement in the stock price, and the variable 1 p as the probability of a down movement. Fair price of a call option C 0 is equal to the expected value of its future payoff discounted at the risk-free interest rate. For a put option P 0 we have the same result P 0 = e rt (pp u + (1 p)p d ). Sergei Fedotov (University of Manchester) 20912 2010 52 / 1

Example A stock price is currently $40. At the end of three months it will be either $44 or $36. The risk-free interest rate is 12%. What is the value of three-month European call option with a strike price of $42? Use no-arbitrage arguments and risk-neutral valuation. In this case S 0 = 40, u = 1.1, d = 0.9, r = 0.12, T = 0.25, C u = 2, C d = 0. No-arbitrage arguments: the number of shares and the value of call option = C u C d S 0 u S 0 d = 2 0 40 (1.1 0.9) = 0.25 C 0 = S 0 (S 0 u C u ) e rt = 40 0.25 (40 1.1 0.25 2) e 0.12 0.25 = 1.266 Sergei Fedotov (University of Manchester) 20912 2010 53 / 1

Example Risk-neutral valuation: one can find the probability p and the value of call option p = ert d u d = e0.12 0.25 0.9 = 0.6523 1.1 0.9 C 0 = e rt [pc u + (1 p)c d ] = e 0.12 0.25 [0.6523 2 + 0] = 1.266 Sergei Fedotov (University of Manchester) 20912 2010 54 / 1

Lecture 9 1 Risk-Neutral Valuation 2 Risk-Neutral World 3 Two-Steps Binomial Tree Sergei Fedotov (University of Manchester) 20912 2010 55 / 1

Risk-Neutral Valuation Reminder from Lecture 8. Call option price: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. No-Arbitrage Principle: d < ert < u. In particular, if d > e rt then there exists an arbitrage opportunity. We could make money by taking out a bank loan B 0 = S 0 at time t = 0 and buying the stock for S 0. We interpret the variable 0 p 1 as the probability of an up movement in the stock price. This formula is known as a risk-neutral valuation. The probability of up q or down movement 1 q in the stock price plays no role whatsoever! Why??? Sergei Fedotov (University of Manchester) 20912 2010 56 / 1

Risk Neutral Valuation Let us find the expected stock price at t = T: E p [S T ] = ps 0 u + (1 p)s 0 d = ert d u d S 0u + (1 ert d u d )S 0d = S 0 e rt. This shows that stock price grows on average at the risk-free interest rate r. Since the expected return is r, this is a risk-neutral world. In Real World: E [S T ] = S 0 e µt. In Risk-Neutral World: E p [S T ] = S 0 e rt Risk-Neutral Valuation: C 0 = e rt E p [C T ] The option price is the expected payoff in a risk-neutral world, discounted at risk-free rate r. Sergei Fedotov (University of Manchester) 20912 2010 57 / 1

Two-Step Binomial Tree Now the stock price changes twice, each time by either a factor of u > 1 or d < 1. We assume that the length of the time step is t such that T = 2 t. After two time steps the stock price will be S 0 u 2, S 0 ud or S 0 d 2. The call option expires after two time steps producing payoffs C uu, C ud and C dd respectively. Sergei Fedotov (University of Manchester) 20912 2010 58 / 1

Option Price The purpose is to calculate the option price C 0 at the initial node of the tree. We apply the risk-neutral valuation backward in time: C u = e r t (pc uu + (1 p)c ud ), C d = e r t (pc ud + (1 p)c dd ). Current option price: C 0 = e r t (pc u + (1 p)c d ). Sergei Fedotov (University of Manchester) 20912 2010 59 / 1

Option Price Substitution gives C 0 = e 2r t ( p 2 C uu + 2p(1 p)c ud + (1 p) 2 C dd ), where p 2, 2p(1 p) and (1 p) 2 are the probabilities in a risk-neutral world that the upper, middle, and lower final nodes are reached. Finally, the current call option price is C 0 = e rt E p [C T ], T = 2 t. The current put option price can be found in the same way: P 0 = e 2r t ( p 2 P uu + 2p(1 p)p ud + (1 p) 2 P dd ) or P 0 = e rt E p [P T ]. Sergei Fedotov (University of Manchester) 20912 2010 60 / 1

Two-Step Binomial Tree Example Consider six months European put with a strike price of 32 on a stock with current price 40. There are two time steps and in each time step the stock price either moves up by 20% or moves down by 20%. Risk-free interest rate is 10%. Find the current option price. We have u = 1.2, d = 0.8, t = 0.25, and r = 0.1. Risk-neutral probability p = er t d u d = e0.1 0.25 0.8 1.2 0.8 = 0.5633. The possible stock prices at final nodes are 40 (1.2) 2 = 57.6, 40 1.2 0.8 = 38.4, and 40 (0.8) 2 = 25.6. We obtain P uu = 0, P ud = 0 and P dd = 32 25.6 = 6.4. Thus the value of put( option is ) P 0 = e 2 0.1 0.25 0 + 0 + (1 0.5633) 2 6.4 = 1.1610. Sergei Fedotov (University of Manchester) 20912 2010 61 / 1

Lecture 10 1 Binomial Model for Stock Price 2 Option Pricing on Binomial Tree 3 Matching Volatility σ with u and d Sergei Fedotov (University of Manchester) 20912 2010 62 / 1

Binomial model for the stock price Continuous random model for the stock price: ds = µsdt + σsdw The binomial model for the stock price is a discrete time model: The stock price S changes only at discrete times t, 2 t, 3 t,... The price either moves up S Su or down S Sd with d < e r t < u. The probability of up movement is q. Let us build up a tree of possible stock prices. The tree is called a binomial tree, because the stock price will either move up or down at the end of each time period. Each node represents a possible future stock price. We divide the time to expiration T into several time steps of duration t = T/N, where N is the number of time steps in the tree. Example: Let us sketch the binomial tree for N = 4. Sergei Fedotov (University of Manchester) 20912 2010 63 / 1

Stock Price Movement in the Binomial Model We introduce the following notations: S m n is the n-th possible value of stock price at time-step m t. Then Sn m = u n d m n S0 0, where n = 0,1,2,...,m. S0 0 is the stock price at the time t = 0. Note that u and d are the same at every node in the tree. For example, at the third time-step 3 t, there are four possible stock prices: S 3 0 = d3 S 0 0, S3 1 = ud2 S 0 0, S3 2 = u2 ds 0 0 and S3 3 = u3 S 0 0. At the final time-step N t, there are N + 1 possible values of stock price. Sergei Fedotov (University of Manchester) 20912 2010 64 / 1

Call Option Pricing on Binomial Tree We denote by C m n the n-th possible value of call option at time-step m t. Risk Neutral Valuation (backward in time): C m n = e r t ( pc m+1 n+1 Here 0 n m and p = er t d u d. ) + (1 p)cm+1 n. Final condition: C N n = max ( S N n E,0 ), where n = 0,1,2,...,N, E is the strike price. The current option price C0 0 is the expected payoff in a risk-neutral world, discounted at risk-free rate r: C0 0 = e rt E p [C T ]. Example: N = 4. Sergei Fedotov (University of Manchester) 20912 2010 65 / 1

Matching volatility σ with u and d We assume that the stock price starts at the value S 0 and the time step is t. Let us find the expected stock price, E [S], and the variance of the return, var [ ] S S, for continuous and discrete models. Expected stock price: Continuous model: E [S] = S 0 e µ t. On the binomial tree: E [S] = qs 0 u + (1 q)s 0 d. First equation: qu + (1 q)d = e µ t. Variance of the return: Continuous model: var [ ] S S = σ 2 t (Lecture2) On the binomial tree: var [ ] S S = q(u 1) 2 + (1 q)(d 1) 2 [q(u 1) + (1 q)(d 1)] 2 = qu 2 + (1 q)d 2 [qu + (1 q)d] 2. Recall: var [X] = E [ X 2] [E (X)] 2. Second equation: qu 2 + (1 q)d 2 [qu + (1 q)d] 2 = σ 2 t. Third equation: u = d 1. Sergei Fedotov (University of Manchester) 20912 2010 66 / 1

Matching volatility σ with u and d From the first equation we find q = eµ t d u d. This is the probability of an up movement in the real world. Substituting this probability into the second equation, we obtain Using u = d 1, we get e µ t (u + d) ud e 2µ t = σ 2 t. e µ t (u + 1 u ) 1 e2µ t = σ 2 t. This equation can be reduced to the quadratic equation. (Exercise sheet 4, part 5). One can obtain u e σ t 1 + σ t and d e σ t. These are the values of u and d obtained by Cox, Ross, and Rubinstein in 1979. Recall: e x 1 + x for small x. Sergei Fedotov (University of Manchester) 20912 2010 67 / 1

Lecture 11 1 American Put Option Pricing on Binomial Tree 2 Replicating Portfolio Sergei Fedotov (University of Manchester) 20912 2010 68 / 1

American Option An American Option is one that may be exercised at any time prior to expire (t = T). We should determine when it is best to exercise the option. It is not subjective! It can be determined in a systematic way! The American put option value must be greater than or equal to the payoff function. If P < max(e S,0), then there is obvious arbitrage opportunity. We can buy stock for S and option for P and immediately exercise the option by selling stock for E. E (P + S) > 0 Sergei Fedotov (University of Manchester) 20912 2010 69 / 1

American Put Option Pricing on Binomial Tree We denote by P m n the n-th possible value of put option at time-step m t. European Put Option: P m n = e r t ( pp m+1 n+1 ) + (1 p)pm+1 n. Here 0 n m and the risk-neutral probability p = er t d u d. American Put Option: P m n = max { max(e S m n,0),e r t ( pp m+1 n+1 ) } + (1 p)pm+1 n, where S m n is the n-th possible value of stock price at time-step m t. Final condition: P N n = max ( E S N n,0 ), where n = 0,1,2,...,N, E is the strike price. Sergei Fedotov (University of Manchester) 20912 2010 70 / 1

Example: Evaluation of American Put Option on Two-Step Tree We assume that over each of the next two years the stock price either moves up by 20% or moves down by 20%. The risk-free interest rate is 5%. Find the value of a 2-year American put with a strike price of $52 on a stock whose current price is $50. In this case u = 1.2, d = 0.8, r = 0.05, E = 52. Risk-neutral probability: p = e0.05 0.8 1.2 0.8 = 0.6282 Sergei Fedotov (University of Manchester) 20912 2010 71 / 1

Replicating Portfolio The aim is to calculate the value of call option C 0. Let us establish a portfolio of stocks and bonds in such a way that the payoff of a call option is completely replicated. Final value: Π T = C T = max(s E,0) To prevent risk-free arbitrage opportunity, the current values should be identical. We say that the portfolio replicates the option. The Law of One Price: Π t = C t. Consider replicating portfolio of shares held long and N bonds held short. The value of portfolio: Π = S NB. A pair (,N) is called a trading strategy. How to find (,N) such that Π T = C T and Π 0 = C 0? Sergei Fedotov (University of Manchester) 20912 2010 72 / 1

Example: One-Step Binomial Model. Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d. At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. The value of portfolio: Π = S NB. When stock moves up: S 0 u NB 0 e rt = C u. When stock moves down: S 0 d NB 0 e rt = C d. We have two equations for two unknown variables and N. Current value: C 0 = S 0 NB 0. Prove: C 0 = e rt (pc u + (1 p)c d ), where p = ert d u d. (Exercise sheet 5) Sergei Fedotov (University of Manchester) 20912 2010 73 / 1

Lecture 12 1 Black-Scholes Model 2 Black-Scholes Equation The Black-Scholes model for option pricing was developed by Fischer Black, Myron Scholes in the early 1970s. This model is the most important result in financial mathematics. Sergei Fedotov (University of Manchester) 20912 2010 74 / 1

Black - Scholes model The Black-Scholes model is used to calculate an option price using: stock price S, strike price E, volatility σ, time to expiration T, and risk-free interest rate r. This model involves the following explicit assumptions: The stock price follows a Geometric Brownian motion with constant expected return and volatility: ds = µsdt + σsdw.. No transaction costs. The stock does not pay dividends. One can borrow and lend cash at a constant risk-free interest rate. Securities are perfectly divisible (i.e. one can buy any fraction of a share of stock). No restrictions on short selling. Sergei Fedotov (University of Manchester) 20912 2010 75 / 1

Basic Notation We denote by V (S,t) the value of an option. We use the notations C(S,t) and P(S,t) for call and put when the distinction is important. The aim is to derive the famous Black-Scholes Equation: V t + 1 2 σ2 S 2 2 V V + rs rv = 0. S2 S Now, we set up a portfolio consisting of a long position in one option and a short position in shares. The value is Π = V S. Let us find the number of shares that makes this portfolio riskless. Sergei Fedotov (University of Manchester) 20912 2010 76 / 1

Itô s Lemma and Elimination of Risk The change in the value of this portfolio in the time interval dt: dπ = dv ds, where ds = µsdt + σsdw. Using Itô s Lemma: ( V dv = t + 1 2 σ2 S 2 2 V V + µs S2 S ) dt + σs V S dw, we find ( V dπ = t + 1 ) ( 2 σ2 S 2 2 V V + µs S2 S µs dt + σs V ) S σs dw. The main question is how to eliminate the risk!!! We can eliminate the random component in dπ by choosing = V S. Sergei Fedotov (University of Manchester) 20912 2010 77 / 1

Black-Scholes Equation This choice ( results in a risk-free portfolio Π = V S V S is dπ = V t + 1 2 σ2 S 2 2 V )dt. S 2 whose increment No-Arbitrage Principle: the return from this portfolio must be rdt. dπ Π = rdt or V t + 1 2 σ2 S 2 2 V S 2 V S V S V = r or t + 1 2 σ2 S 2 2 V = r ( V S V ) S 2 S. Thus, we obtain the Black-Scholes PDE: V t + 1 2 σ2 S 2 2 V V + rs rv = 0. S2 S Scholes received the 1997 Nobel Prize in Economics. It was not awarded to Black in 1997, because he died in 1995. Black received a Ph.D. in applied mathematics from Harvard University. Sergei Fedotov (University of Manchester) 20912 2010 78 / 1

The European call value C(S, t) If PDE is of backward type, we must impose a final condition at t = T. For a call option, we have C(S,T) = max(s E,0). Sergei Fedotov (University of Manchester) 20912 2010 79 / 1

Lecture 13 1 Boundary Conditions for Call and Put Options 2 Exact Solution to Black-Scholes Equation C t + 1 2 σ2 S 2 2 C C + rs S2 S rc = 0. Sergei Fedotov (University of Manchester) 20912 2010 80 / 1

Boundary Conditions We use C(S,t) and P(S,t) for call and put option. Boundary conditions are applied for zero stock price S = 0 and S. Boundary conditions for a call option: C(0,t) = 0 and C(S,t) S as S. The call option is likely to be exercised as S Boundary conditions for a put option: P(0,t) = Ee r(t t). We evaluate the present value of E. P(S,t) 0 as S. As stock price S, then put option is unlikely to be exercised. Sergei Fedotov (University of Manchester) 20912 2010 81 / 1

Exact solution The Black-Scholes equation C t + 1 2 σ2 S 2 2 C C + rs S2 S rc = 0 with appropriate final and boundary conditions has the explicit solution: C(S,t) = SN (d 1 ) Ee r(t t) N (d 2 ), where N (x) = 1 2π x e y2 2 dy (cumulative normal distribution) and d 1 = ln(s/e) + ( r + σ 2 /2 ) (T t) σ, d 2 = d 1 σ T t. T t Sergei Fedotov (University of Manchester) 20912 2010 82 / 1

The European call value C(S, t) by K. Rubash Sergei Fedotov (University of Manchester) 20912 2010 83 / 1

Example Calculate the price of a three-month European call option on a stock with a strike price of $25 when the current stock price is $21.6 The volatility is 35% and risk-free interest rate is 1% p.a. In this case S 0 = 21.6,E = 25,T = 0.25,σ = 0.35 and r = 0.01. The value of call option is C 0 = S 0 N (d 1 ) Ee rt N (d 2 ). First, we compute the values of d 1 and d 2 : d 1 = ln(s 0/E)+(r+σ 2 /2)T σ T = 21.6 ln( 25 )+(0.01+(0.35) 2 0.5)) 0.25 0.35 0.7335 0.25 d 2 = d 1 σ T = 0.7335 0.35 0.25 0.9085 Since we obtain N ( 0.7335) 0.2316, N ( 0.9085) 0.1818, C 0 21.6 0.2316 25 e 0.01 0.25 0.1818 = 0.4689 Sergei Fedotov (University of Manchester) 20912 2010 84 / 1

The Limit of Higher Volatility Let us find the limit We know that lim C(S,t). σ C(S,t) = SN (d 1 ) Ee r(t t) N (d 2 ), where d 1 = ln(s/e)+(r+σ2 /2)(T t) σ T t = ln(s/e) σ + r T t T t σ + σ T t 2. In the limit σ, d 1. Since d 2 = d 1 σ T t, in the limit σ, d 2 Thus lim σ N (d 1 ) = 1 and lim σ N(d 2 ) = 0. Therefore lim σ C(S,t) = S. This is an upper bound for call option!!! Sergei Fedotov (University of Manchester) 20912 2010 85 / 1

Lecture 14 1 -Hedging 2 Greek Letters or Greeks Sergei Fedotov (University of Manchester) 20912 2010 86 / 1

Delta for European Call Option Let us show that = C S = N (d 1). First, find the derivative = C S by using the explicit solution for the European call C(S,t) = SN (d 1 ) Ee r(t t) N (d 2 ). = C S = N (d 1) + SN (d 1 ) d 1 S Ee r(t t) N (d 2 ) d 2 S = N (d 1 ) + (SN (d 1 ) Ee r(t t) N (d 2 )) d 1 S. We need to prove See Problem Sheet 6. (SN (d 1 ) Ee r(t t) N (d 2 )) = 0. Sergei Fedotov (University of Manchester) 20912 2010 87 / 1

Delta-Hedging Example Find the value of of a 6-month European call option on a stock with a strike price equal to the current stock price ( t = 0). The interest rate is 6% p.a. The volatility σ = 0.16. Solution: we have = N (d 1 ), where d 1 = ln(s 0/E)+(r+σ 2 /2)T We find σ(t) 1 2 and d 1 = (0.06 + (0.16)2 0.5)) 0.5 0.16 0.5 = N (0.3217) 0.6262 0.3217 Sergei Fedotov (University of Manchester) 20912 2010 88 / 1

Delta for European Put Option Let us find Delta for European put option by using the put-call parity: S + P C = Ee r(t t). Let us differentiate it with respect to S We find 1 + P S C S = 0. Therefore P S = C S 1 = N (d 1) 1. Sergei Fedotov (University of Manchester) 20912 2010 89 / 1

Greeks The option value: V = V (S,t σ,r,t). Greeks represent the sensitivities of options to a change in underlying parameters on which the value of an option is dependent. Delta: = V S measures the rate of change of option value with respect to changes in the underlying stock price Gamma: Γ = 2 V = S 2 S measures the rate of change in with respect to changes in the underlying stock price. Problem Sheet 6: Γ = 2 C S 2 = N (d 1 ) Sσ T t, where N (d 1 ) = 1 2π e d2 1 2. Sergei Fedotov (University of Manchester) 20912 2010 90 / 1

Greeks Vega, V σ, measures sensitivity to volatility σ. One can show that C σ = S T tn (d 1 ). Rho: ρ = V r measures sensitivity to the interest rate r. One can show that ρ = C r = E(T t)e r(t t) N(d 2 ). The Greeks are important tools in financial risk management. Each Greek measures the sensitivity of the value of derivatives or a portfolio to a small change in a given underlying parameter. Sergei Fedotov (University of Manchester) 20912 2010 91 / 1

Lecture 15 1 Black-Scholes Equation and Replicating Portfolio 2 Static and Dynamic Risk-Free Portfolio Sergei Fedotov (University of Manchester) 20912 2010 92 / 1

Replicating Portfolio The aim is to show that the option price V (S,t) satisfies the Black-Scholes equation V t + 1 2 σ2 S 2 2 V V + rs rv = 0. S2 S Consider replicating portfolio of shares held long and N bonds held short. The value of portfolio: Π = S NB. Recall that a pair (,N) is called a trading strategy. How to find (,N) such that Π t = V t? SDE for a stock price S(t) : ds = µsdt + σsdw. Equation for a bond price B (t) : db = rbdt. Sergei Fedotov (University of Manchester) 20912 2010 93 / 1

Derivation of the Black-Scholes Equation By using the Ito s lemma, we find the change in the option value ( V V dv = + µs t S + 1 ) 2 σ2 S 2 2 V S 2 dt + σs V S dw. By using self-financing requirement dπ = ds NdB, we find the change in portfolio value dπ = ds NdB = (µsdt+σsdw) rnbdt = ( µs rnb)dt+ σsdw Equating the last two equations dπ = dv, we obtain = V V S, rnb = t + 1 2 σ2 S 2 2 V. S 2 Since NB = S Π = ( S V S V ), we get the classical Black-Scholes equation V t + 1 2 σ2 S 2 2 V V + rs rv = 0. S2 S Sergei Fedotov (University of Manchester) 20912 2010 94 / 1

Static Risk-free Portfolio Let me remind you a Put-Call Parity. We set up the portfolio consisting of long position in one stock, long position in one put and short position in one call with the same T and E. The value of this portfolio is Π = S + P C. The payoff for this portfolio is Π T = S + max(e S,0) max(s E,0) = E The payoff is always the same whatever the stock price is at t = T. Using No Arbitrage Principle, we obtain S t + P t C t = Ee r(t t), where C t = C(S t,t) and P t = P (S t,t). This is an example of complete risk elimination. Definition: The risk of a portfolio is the variance of the return. Sergei Fedotov (University of Manchester) 20912 2010 95 / 1

Dynamic Risk-Free Portfolio Put-Call Parity is an example of complete risk elimination when we carry out only one transaction in call/put options and underlying security. Let us consider the dynamic risk elimination procedure. We could set up a portfolio consisting of a long position in one call option and a short position in shares. The value is Π = C S. We can eliminate the random component in Π by choosing = C S. This is a -hedging! It requires a continuous rebalancing of a number of shares in the portfolio Π. Sergei Fedotov (University of Manchester) 20912 2010 96 / 1

Lecture 16 1 Option on Dividend-paying Stock 2 American Put Option Sergei Fedotov (University of Manchester) 20912 2010 97 / 1

Option on Dividend-paying Stock We assume that in a time dt the underlying stock pays out a dividend D 0 Sdt, where D 0 is a constant dividend yield. Now, we set up a portfolio consisting of a long position in one call option and a short position in shares. The value is Π = C S. The change in the value of this portfolio in the time interval dt: dπ = dc ds D 0 Sdt. Sergei Fedotov (University of Manchester) 20912 2010 98 / 1

Itô s Lemma and Elimination of Risk Using Itô s Lemma: we find dπ = dc = ( C t + 1 ) 2 σ2 S 2 2 C S 2 dt + C S ds ( C t + 1 2 σ2 S 2 2 C S 2 D 0S C ) dt + C ds ds S S We can eliminate the random component in dπ by choosing = C S. Sergei Fedotov (University of Manchester) 20912 2010 99 / 1

Modified Black-Scholes Equation This choice ( results in a risk-free portfolio ) Π = C S C S is dπ = C t + 1 2 σ2 S 2 2 C D S 2 0 S C S dt. whose increment No-Arbitrage Principle: the return from this portfolio must be rdt. dπ Π = rdt or C t + 1 2 σ2 S 2 2 C S 2 D 0 S C S = r ( C S C S ). Thus, we obtain the modified Black-Scholes PDE: C t + 1 2 σ2 S 2 2 C S 2 + (r D 0)S C S rc = 0. Sergei Fedotov (University of Manchester) 20912 2010 100 / 1

Solution to Modified Black-Scholes Equation Let us find the solution to modified Black-Scholes equation in the form C(S,t) = e D 0(T t) C 1 (S,t). We prove that C 1 (S,t) satisfies the Black-Scholes equation with r replaced by r D 0. If we substitute C(S,t) = e D 0(T t) C 1 (S,t) into the modified Black-Scholes equation, we find the equation for C 1 (S,t) in the form C 1 t + 1 2 σ2 S 2 2 C 1 S 2 + (r D 0) C 1 S (r D 0)C 1 = 0. The auxiliary function C 1 (S,t) is the value of a European Call option with the interest rate r D 0. Sergei Fedotov (University of Manchester) 20912 2010 101 / 1

Solution to Modified Black-Scholes Equation Problem sheet 7: show that the modified Black-Scholes equation has the explicit solution for the European call where C(S,t) = Se D 0(T t) N (d 10 ) Ee r(t t) N (d 20 ), d 10 = ln(s/e) + ( r D 0 + σ 2 /2 ) (T t) σ, d 20 = d 10 σ T t. T t Sergei Fedotov (University of Manchester) 20912 2010 102 / 1

American Put Option Recall that An American Option is one that may be exercised at any time prior to expire (t = T). The American put option value must be greater than or equal to the payoff function. If P < max(e S,0), then there is obvious arbitrage opportunity. Sergei Fedotov (University of Manchester) 20912 2010 103 / 1

American Put Option American put problem can be written as as a free boundary problem. We divide the price axis S into two distinct regions: 0 S < S f (t) and S f (t) < S <, where S f (t) is the exercise boundary. Note that we do not know a priori the value of S f (t). When 0 S < S f (t), the early exercise is optimal: put option value is P(S,t) = E S. When S > S f (t), the early exercise is not optimal, and P(S,t) obeys the Black-Scholes equation. The boundary conditions at S = S f (t) are P (S f (t),t) = max(e S f (t),0), P S (S f (t),t) = 1. Sergei Fedotov (University of Manchester) 20912 2010 104 / 1

Lecture 17 1 Bond Pricing with Known Interest Rates and Dividend Payments 2 Zero-Coupon Bond Pricing Sergei Fedotov (University of Manchester) 20912 2010 105 / 1

Bond Pricing Equation Definition. A bond is a contract that yields a known amount (nominal, principal or face value) on the maturity date, t = T. The bond pays the dividends at fixed times. If there is no dividend payment, the bond is known as a zero-coupon bond. Let us introduce the following notation V (t) is the value of the bond, r(t) is the interest rate, K(t) is the coupon payment. Equation for the bond price is dv dt = r(t)v K(t). The final condition: V (T) = F. Sergei Fedotov (University of Manchester) 20912 2010 106 / 1

Zero-Coupon Bond Pricing Let us consider the case when the dividend payment K(t) = 0. The solution of the equation dv dt = r(t)v with V (T) = F can be written as ( T ) V (t) = F exp r(s)ds. t Let us show this by integration dv V = r(t)dt from t to T. lnv(t) lnv(t) = T t r(s)ds or ln(v (t)/f) = T t r(s)ds. Sergei Fedotov (University of Manchester) 20912 2010 107 / 1

Example A zero coupon bond, V, issued at t = 0, is worth V (1) = 1 at t = 1. Find the bond price V (t) at time t < 1 and V (0), when the continuous interest rate is r(t) = t 2. Solution: T = 1; one can find Therefore 1 t r(s)ds = ( V (t) = exp t ( V (0) = exp 1 ) 3 1 1 t s 2 ds = 1 3 1 3 t3 ) ( t 3 r(s)ds = exp 3 1 ), 3 = 0.7165 Sergei Fedotov (University of Manchester) 20912 2010 108 / 1

Bond Pricing for K(t) > 0 Let us consider the case when the dividend payment K(t) > 0. The solution of the equation dv dt = r(t)v K(t) can be written as ( T ) V (t) = F exp r(s)ds + V 1 (t), ( where V 1 (t) = C (t)exp ) T t r(s)ds. To find C (t) we need to substitute V 1 (t) into the equation t dv 1 dt = r(t)v 1 K(t). (tutorial exercise 8) The explicit solution is ( T V (t) = exp t ) [ T ( T ) ] r(s)ds F + K(y)exp r(s)ds dy. t y Sergei Fedotov (University of Manchester) 20912 2010 109 / 1

Lecture 18 1 Measure of Future Values of Interest Rate 2 Term Structure of Interest Rate (Yield Curve) Sergei Fedotov (University of Manchester) 20912 2010 110 / 1

Measure of Future Value of Interest Rate Let us consider the case when the dividend payment K(t) = 0. Recall that the solution of the equation dv dt = r(t)v with V (T) = F is ( T ) V (t) = F exp r(s)ds. Let us introduce the notation V (t,t) for the bond prices. These prices are quoted at time t for different values of T. Let us differentiate V (t,t) with respect to T : ( V T = F exp ) T t r(s)ds ( r(t)) = V (t,t)r(t), since T T t r(s)ds = r(t). Therefore, r(t) = 1 V V (t,t) T. This is an interest rate at future dates (forward rate). Sergei Fedotov (University of Manchester) 20912 2010 111 / 1 t

Term Structure of Interest Rate (Yield Curve) We define ln(v (t,t)) lnv(t,t) Y (t,t) = T t as a measure of the future values of interest rate, where V (t,t) is taken from financial data. ( ln(f exp T Y (t,t) = ) t r(s)ds ) ln F T t = 1 T t T t r(s)ds is the average value of the interest rate r(t) in the time interval [t,t]. Bond price V (t,t) can be written as V (t,t) = Fe Y (t,t)(t t). Term structure of interest rate (yield curve): ln(v (0,T)) lnv(t,t) Y (0,T) = = 1 T r(s)ds T T 0 is the average value of interest rate in the future. Sergei Fedotov (University of Manchester) 20912 2010 112 / 1

Example Assume that the instantaneous interest rate r(t) is where r 0 and a are constants. Bond price: r(t) = r 0 + at, V (t,t) = Fe T t r(s)ds = Fe T t (r 0 +as)ds = Fe r 0(T t) a 2(T 2 t 2 ). Term structure of interest rate: Y (0,T) = 1 T T 0 r(s)ds = r 0 + at 2. Sergei Fedotov (University of Manchester) 20912 2010 113 / 1

Risk of Default There exists a risk of default of bond V (t) when the principal are not paid to the lender as promised by borrower. How to take this into account? Let us introduce the probability of default p during one year T = 1. Then V (0) e r = V (0) (1 p)e (r+s) + p 0. The investor has no preference between two investments. The positive parameter s is called the spread w.r.t to interest rate r. Let us find it. s = ln(1 p) = p + o(p). Sergei Fedotov (University of Manchester) 20912 2010 114 / 1

Lecture 19 1 Asian Options 2 Derivation of PDE for Option Price Sergei Fedotov (University of Manchester) 20912 2010 115 / 1

Asian Options Definition. Asian option is a contract giving the holder the right to buy/sell an underlying asset for its average price over some prescribed period. Asian call option final condition: We introduce a new variable: ( V (S,T) = max S 1 T ) S(t)dt, 0. T 0 I(t) = t 0 S(t)dt Final condition can be rewritten as V (S,I,T) = max or di dt = S(t). (S IT,0 ). Sergei Fedotov (University of Manchester) 20912 2010 116 / 1

Asian Options Asian option price V (S,I,t) is the function of three variables. Let us derive the equation for this price by using the standard portfolio consisting of a long position in one call option and a short position in shares. The value is Π = V S. The change in the value of this portfolio in the time interval dt: dπ = dv ds. Sergei Fedotov (University of Manchester) 20912 2010 117 / 1

Itô s Lemma and Elimination of Risk Using Itô s Lemma: dv = ( V t + 1 2 σ2 S 2 2 V S 2 ) dt + V V ds + S I di we find dπ = ( V t + 1 ) 2 σ2 S 2 2 V S 2 dt + V V ds ds + S I di. We can eliminate the random component in dπ by choosing = V S. Sergei Fedotov (University of Manchester) 20912 2010 118 / 1

Modified Black-Scholes Equation By using No-Arbitrage Principle, and the equation di = Sdt we can obtain the modified Black-Scholes PDE for the Asian option price: V t + 1 2 σ2 S 2 2 V V + rs S2 S rv + S V I = 0. The value of Asian option must be calculated numerically (no analytical solution!) Sergei Fedotov (University of Manchester) 20912 2010 119 / 1