# Mathematical Finance: Stochastic Modeling of Asset Prices

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1 Mathematical Finance: Stochastic Modeling of Asset Prices José E. Figueroa-López 1 1 Department of Statistics Purdue University Worcester Polytechnic Institute Department of Mathematical Sciences December 2014

2 Outline 1 A quick glance of mathematical finance Problems Arbitrage-free Pricing of Derivatives Risk-neutral Pricing of Derivatives 2 Modeling stock prices Stylized empirical features of stock prices Stochastic models for asset prices

3 Problems Some classical problems of mathematical finance 1 Mathematical emulation of the empirical features exhibited by the prices of primary assets via probabilistic models 2 Calibration/estimation of the model s parameters 3 Characterization of optimal investment strategies 4 Creation of derivative products 5 Characterization of reasonable pricing procedures for derivative products 6 Implementation of numerical methods for the valuation of derivatives 7 Creation of hedging strategies to offset the potential losses of an investment portfolio of derivatives

4 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 The price of a derivative is deemed reasonable if the whole market" is absent of arbitrage opportunities when such a price process is adopted. 2 What is the market? 1 Primary assets: Stock (risky asset) and Bonds or MMA (checking account for borrowing or lending at some given interest rate) 2 Derivative written on a Stock: Options, Forwards, Contingent Claims, etc. A contract between two parties: the writer or seller and the buyer or derivative holder; the contract stipulates that the writer is obligated to deliver a payoff (X ) to the holder at a specified future time T called the maturity or expiration; the final payoff X depends on the unknown value (S(T )) at expiration of the stock; in exchange, the writer receives a premium (P 0 ) at the starting time of the contract; the premium is called the price of the derivative;

5 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 The price of a derivative is deemed reasonable if the whole market" is absent of arbitrage opportunities when such a price process is adopted. 2 What is the market? 1 Primary assets: Stock (risky asset) and Bonds or MMA (checking account for borrowing or lending at some given interest rate) 2 Derivative written on a Stock: Options, Forwards, Contingent Claims, etc. A contract between two parties: the writer or seller and the buyer or derivative holder; the contract stipulates that the writer is obligated to deliver a payoff (X ) to the holder at a specified future time T called the maturity or expiration; the final payoff X depends on the unknown value (S(T )) at expiration of the stock; in exchange, the writer receives a premium (P 0 ) at the starting time of the contract; the premium is called the price of the derivative;

6 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 The price of a derivative is deemed reasonable if the whole market" is absent of arbitrage opportunities when such a price process is adopted. 2 What is the market? 1 Primary assets: Stock (risky asset) and Bonds or MMA (checking account for borrowing or lending at some given interest rate) 2 Derivative written on a Stock: Options, Forwards, Contingent Claims, etc. A contract between two parties: the writer or seller and the buyer or derivative holder; the contract stipulates that the writer is obligated to deliver a payoff (X ) to the holder at a specified future time T called the maturity or expiration; the final payoff X depends on the unknown value (S(T )) at expiration of the stock; in exchange, the writer receives a premium (P 0 ) at the starting time of the contract; the premium is called the price of the derivative;

7 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 The price of a derivative is deemed reasonable if the whole market" is absent of arbitrage opportunities when such a price process is adopted. 2 What is the market? 1 Primary assets: Stock (risky asset) and Bonds or MMA (checking account for borrowing or lending at some given interest rate) 2 Derivative written on a Stock: Options, Forwards, Contingent Claims, etc. A contract between two parties: the writer or seller and the buyer or derivative holder; the contract stipulates that the writer is obligated to deliver a payoff (X ) to the holder at a specified future time T called the maturity or expiration; the final payoff X depends on the unknown value (S(T )) at expiration of the stock; in exchange, the writer receives a premium (P 0 ) at the starting time of the contract; the premium is called the price of the derivative;

8 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 The price of a derivative is deemed reasonable if the whole market" is absent of arbitrage opportunities when such a price process is adopted. 2 What is the market? 1 Primary assets: Stock (risky asset) and Bonds or MMA (checking account for borrowing or lending at some given interest rate) 2 Derivative written on a Stock: Options, Forwards, Contingent Claims, etc. A contract between two parties: the writer or seller and the buyer or derivative holder; the contract stipulates that the writer is obligated to deliver a payoff (X ) to the holder at a specified future time T called the maturity or expiration; the final payoff X depends on the unknown value (S(T )) at expiration of the stock; in exchange, the writer receives a premium (P 0 ) at the starting time of the contract; the premium is called the price of the derivative;

9 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 The price of a derivative is deemed reasonable if the whole market" is absent of arbitrage opportunities when such a price process is adopted. 2 What is the market? 1 Primary assets: Stock (risky asset) and Bonds or MMA (checking account for borrowing or lending at some given interest rate) 2 Derivative written on a Stock: Options, Forwards, Contingent Claims, etc. A contract between two parties: the writer or seller and the buyer or derivative holder; the contract stipulates that the writer is obligated to deliver a payoff (X ) to the holder at a specified future time T called the maturity or expiration; the final payoff X depends on the unknown value (S(T )) at expiration of the stock; in exchange, the writer receives a premium (P 0 ) at the starting time of the contract; the premium is called the price of the derivative;

10 Arbitrage-free Pricing of Derivatives Example 1 Forward Contract A Forward is a contract in which one party (the seller) agrees to sell one share of a stock to the counterparty (buyer) at a pre-specified time T for a price F 0,T agreed upon at the begging of the contract. There is no initial premium for the contract. The time T is called the delivery time while the price F 0,T is called the forward price. Key Question: What would a reasonable value for F 0,T be? Answer: F 0,T = S 0 p(0, T ), where S 0 is the price of one share today at time 0 and 1/p(0, T ) is the value of \$ 1 after a time period T.

11 Arbitrage-free Pricing of Derivatives Example 1 Forward Contract A Forward is a contract in which one party (the seller) agrees to sell one share of a stock to the counterparty (buyer) at a pre-specified time T for a price F 0,T agreed upon at the begging of the contract. There is no initial premium for the contract. The time T is called the delivery time while the price F 0,T is called the forward price. Key Question: What would a reasonable value for F 0,T be? Answer: F 0,T = S 0 p(0, T ), where S 0 is the price of one share today at time 0 and 1/p(0, T ) is the value of \$ 1 after a time period T.

12 Arbitrage-free Pricing of Derivatives Example 1 Forward Contract A Forward is a contract in which one party (the seller) agrees to sell one share of a stock to the counterparty (buyer) at a pre-specified time T for a price F 0,T agreed upon at the begging of the contract. There is no initial premium for the contract. The time T is called the delivery time while the price F 0,T is called the forward price. Key Question: What would a reasonable value for F 0,T be? Answer: F 0,T = S 0 p(0, T ), where S 0 is the price of one share today at time 0 and 1/p(0, T ) is the value of \$ 1 after a time period T.

13 Arbitrage-free Pricing of Derivatives Example 2 A put option on a two-state economy Current price per one share of a stock is S(0) = \$100 Put Option = Right to sell one share of the stock at T = 1 for \$100; Key observation: Equivalent to a contingent claim with final payoff X = max{100 S(1), 0} and maturity T = 1 year; A 1-year bond pays an interest of 5% per annum (so, 1/p(0, T ) = 1.05). In T = 1 year, price can go up to S(1) = \$110 per share with probability 1/4 or go down to S(1) = \$95 per share with probability 3/4

14 Arbitrage-free Pricing of Derivatives Example 2 A put option on a two-state economy Current price per one share of a stock is S(0) = \$100 Put Option = Right to sell one share of the stock at T = 1 for \$100; Key observation: Equivalent to a contingent claim with final payoff X = max{100 S(1), 0} and maturity T = 1 year; A 1-year bond pays an interest of 5% per annum (so, 1/p(0, T ) = 1.05). In T = 1 year, price can go up to S(1) = \$110 per share with probability 1/4 or go down to S(1) = \$95 per share with probability 3/4

15 Arbitrage-free Pricing of Derivatives Example 2 A put option on a two-state economy Current price per one share of a stock is S(0) = \$100 Put Option = Right to sell one share of the stock at T = 1 for \$100; Key observation: Equivalent to a contingent claim with final payoff X = max{100 S(1), 0} and maturity T = 1 year; A 1-year bond pays an interest of 5% per annum (so, 1/p(0, T ) = 1.05). In T = 1 year, price can go up to S(1) = \$110 per share with probability 1/4 or go down to S(1) = \$95 per share with probability 3/4

16 Arbitrage-free Pricing of Derivatives Example 2 A put option on a two-state economy Current price per one share of a stock is S(0) = \$100 Put Option = Right to sell one share of the stock at T = 1 for \$100; Key observation: Equivalent to a contingent claim with final payoff X = max{100 S(1), 0} and maturity T = 1 year; A 1-year bond pays an interest of 5% per annum (so, 1/p(0, T ) = 1.05). In T = 1 year, price can go up to S(1) = \$110 per share with probability 1/4 or go down to S(1) = \$95 per share with probability 3/4

17 Arbitrage-free Pricing of Derivatives Example 2 A put option on a two-state economy Current price per one share of a stock is S(0) = \$100 Put Option = Right to sell one share of the stock at T = 1 for \$100; Key observation: Equivalent to a contingent claim with final payoff X = max{100 S(1), 0} and maturity T = 1 year; A 1-year bond pays an interest of 5% per annum (so, 1/p(0, T ) = 1.05). In T = 1 year, price can go up to S(1) = \$110 per share with probability 1/4 or go down to S(1) = \$95 per share with probability 3/4

18 Arbitrage-free Pricing of Derivatives Example 2 A put option on a two-state economy Current price per one share of a stock is S(0) = \$100 Put Option = Right to sell one share of the stock at T = 1 for \$100; Key observation: Equivalent to a contingent claim with final payoff X = max{100 S(1), 0} and maturity T = 1 year; A 1-year bond pays an interest of 5% per annum (so, 1/p(0, T ) = 1.05). In T = 1 year, price can go up to S(1) = \$110 per share with probability 1/4 or go down to S(1) = \$95 per share with probability 3/4

19 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 What should the price P(0) of the derivative be? 2 Key idea: Replicating trading strategy. Consider the following trading strategy at time t = 0: (i) Borrow (19)(200) from the bond market. (3)(21) (ii) Buy 2 3 shares of stock (which costs ) 3 The net necessary investment at time t = 0 is V 0 = 2 3 The resulting wealth at time t = 1 is V 1 = { (100) (19)(200) (3)(21) (19)(200) 110 (1.05) = 10 (3)(21) if stock goes up (19)(200) 95 (1.05) = 0 (3)(21) if stock goes down 3 Conclusion: The arbitrage-free price of the derivative must be P(0) = V 0 =

20 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 What should the price P(0) of the derivative be? 2 Key idea: Replicating trading strategy. Consider the following trading strategy at time t = 0: (i) Borrow (19)(200) from the bond market. (3)(21) (ii) Buy 2 3 shares of stock (which costs ) 3 The net necessary investment at time t = 0 is V 0 = 2 3 The resulting wealth at time t = 1 is V 1 = { (100) (19)(200) (3)(21) (19)(200) 110 (1.05) = 10 (3)(21) if stock goes up (19)(200) 95 (1.05) = 0 (3)(21) if stock goes down 3 Conclusion: The arbitrage-free price of the derivative must be P(0) = V 0 =

21 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 What should the price P(0) of the derivative be? 2 Key idea: Replicating trading strategy. Consider the following trading strategy at time t = 0: (i) Borrow (19)(200) from the bond market. (3)(21) (ii) Buy 2 3 shares of stock (which costs ) 3 The net necessary investment at time t = 0 is V 0 = 2 3 The resulting wealth at time t = 1 is V 1 = { (100) (19)(200) (3)(21) (19)(200) 110 (1.05) = 10 (3)(21) if stock goes up (19)(200) 95 (1.05) = 0 (3)(21) if stock goes down 3 Conclusion: The arbitrage-free price of the derivative must be P(0) = V 0 =

22 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 What should the price P(0) of the derivative be? 2 Key idea: Replicating trading strategy. Consider the following trading strategy at time t = 0: (i) Borrow (19)(200) from the bond market. (3)(21) (ii) Buy 2 3 shares of stock (which costs ) 3 The net necessary investment at time t = 0 is V 0 = 2 3 The resulting wealth at time t = 1 is V 1 = { (100) (19)(200) (3)(21) (19)(200) 110 (1.05) = 10 (3)(21) if stock goes up (19)(200) 95 (1.05) = 0 (3)(21) if stock goes down 3 Conclusion: The arbitrage-free price of the derivative must be P(0) = V 0 =

23 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 What should the price P(0) of the derivative be? 2 Key idea: Replicating trading strategy. Consider the following trading strategy at time t = 0: (i) Borrow (19)(200) from the bond market. (3)(21) (ii) Buy 2 3 shares of stock (which costs ) 3 The net necessary investment at time t = 0 is V 0 = 2 3 The resulting wealth at time t = 1 is V 1 = { (100) (19)(200) (3)(21) (19)(200) 110 (1.05) = 10 (3)(21) if stock goes up (19)(200) 95 (1.05) = 0 (3)(21) if stock goes down 3 Conclusion: The arbitrage-free price of the derivative must be P(0) = V 0 =

24 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 What should the price P(0) of the derivative be? 2 Key idea: Replicating trading strategy. Consider the following trading strategy at time t = 0: (i) Borrow (19)(200) from the bond market. (3)(21) (ii) Buy 2 3 shares of stock (which costs ) 3 The net necessary investment at time t = 0 is V 0 = 2 3 The resulting wealth at time t = 1 is V 1 = { (100) (19)(200) (3)(21) (19)(200) 110 (1.05) = 10 (3)(21) if stock goes up (19)(200) 95 (1.05) = 0 (3)(21) if stock goes down 3 Conclusion: The arbitrage-free price of the derivative must be P(0) = V 0 =

25 Arbitrage-free Pricing of Derivatives Arbitrage-free Pricing 1 What should the price P(0) of the derivative be? 2 Key idea: Replicating trading strategy. Consider the following trading strategy at time t = 0: (i) Borrow (19)(200) from the bond market. (3)(21) (ii) Buy 2 3 shares of stock (which costs ) 3 The net necessary investment at time t = 0 is V 0 = 2 3 The resulting wealth at time t = 1 is V 1 = { (100) (19)(200) (3)(21) (19)(200) 110 (1.05) = 10 (3)(21) if stock goes up (19)(200) 95 (1.05) = 0 (3)(21) if stock goes down 3 Conclusion: The arbitrage-free price of the derivative must be P(0) = V 0 =

26 Risk-neutral Pricing of Derivatives Expected Discounted Payoff Interpretation A fundamental question: Is the previous arbitrage-free price consistent with the expected payoff" of the option in present time dollars? 1 The arbitrage-free price P(0) = is such that ; 2 The probabilities weights q u = 2 3 and q d = 1 3 are exactly the needed probabilities to correctly price forward contracts: Arbitrage-free Forward Price: F = S 0 /p(0, T ) = = 105. Present value of forward payoff: ( ) (95 105) = = 0.

27 Risk-neutral Pricing of Derivatives Expected Discounted Payoff Interpretation A fundamental question: Is the previous arbitrage-free price consistent with the expected payoff" of the option in present time dollars? 1 The arbitrage-free price P(0) = is such that ; 2 The probabilities weights q u = 2 3 and q d = 1 3 are exactly the needed probabilities to correctly price forward contracts: Arbitrage-free Forward Price: F = S 0 /p(0, T ) = = 105. Present value of forward payoff: ( ) (95 105) = = 0.

28 Risk-neutral Pricing of Derivatives Expected Discounted Payoff Interpretation A fundamental question: Is the previous arbitrage-free price consistent with the expected payoff" of the option in present time dollars? 1 The arbitrage-free price P(0) = is such that ; 2 The probabilities weights q u = 2 3 and q d = 1 3 are exactly the needed probabilities to correctly price forward contracts: Arbitrage-free Forward Price: F = S 0 /p(0, T ) = = 105. Present value of forward payoff: ( ) (95 105) = = 0.

29 Risk-neutral Pricing of Derivatives Expected Discounted Payoff Interpretation A fundamental question: Is the previous arbitrage-free price consistent with the expected payoff" of the option in present time dollars? 1 The arbitrage-free price P(0) = is such that = 2.5 = ; The probabilities weights q u = 2 3 and q d = 1 3 are exactly the needed probabilities to correctly price forward contracts: Arbitrage-free Forward Price: F = S 0 /p(0, T ) = = 105. Present value of forward payoff: ( ) (95 105) = = 0.

30 Risk-neutral Pricing of Derivatives Expected Discounted Payoff Interpretation A fundamental question: Is the previous arbitrage-free price consistent with the expected payoff" of the option in present time dollars? 1 The arbitrage-free price P(0) = is such that q u q d = ; with q u = 2 3, q d = The probabilities weights q u = 2 3 and q d = 1 3 are exactly the needed probabilities to correctly price forward contracts: Arbitrage-free Forward Price: F = S 0 /p(0, T ) = = 105. Present value of forward payoff: ( ) (95 105) = = 0.

31 Risk-neutral Pricing of Derivatives Expected Discounted Payoff Interpretation A fundamental question: Is the previous arbitrage-free price consistent with the expected payoff" of the option in present time dollars? 1 The arbitrage-free price P(0) = is such that q u q d = ; with q u = 2 3, q d = The probabilities weights q u = 2 3 and q d = 1 3 are exactly the needed probabilities to correctly price forward contracts: Arbitrage-free Forward Price: F = S 0 /p(0, T ) = = 105. Present value of forward payoff: ( ) (95 105) = = 0.

32 Risk-neutral Pricing of Derivatives Risk-neutral Valuation Key Property Under the probabilities weights q u = 2 3 and q d = 1 3, the expected rate of return on the stock coincides with that of the risk-free asset: =.05 Expected return on the stock = risk-free return

33 Risk-neutral Pricing of Derivatives Example 3 Contingent claim in a defaultable market 1 Market: Zero-coupon bond exposed to default risk, with maturity T = 1 Recovery rate 1 δ = 60% Historical default probability p = 1% Current price of defaultable bond is.941 (< 1/1.05 = ). A risk-free" default-free zero-coupon bond with interest rate 5%. 2 Contingent claim on the vulnerable bond: This contract pays 1 dollar if the bond defaults and pays 0 otherwise.

34 Risk-neutral Pricing of Derivatives Pricing and replication 1 Replicating portfolio: Sell 2.5 defaultable bonds (i.e., borrow = dollars) Lend 50/ dollars worth of default-free bonds Time t = 1 value of the portfolio { ( 2.5)(1) + ( 50 )(1.05) = 0 21 V 1 = if no default ( 2.5)(.6) + ( 50 )(1.05) = 1 21 if default Time t = 0 initial investment: V 0 = ( 2.5)(.941) Arbitrage-free price of the option: P(0) = V 0 =.0285.

35 Risk-neutral Pricing of Derivatives Pricing and replication 1 Replicating portfolio: Sell 2.5 defaultable bonds (i.e., borrow = dollars) Lend 50/ dollars worth of default-free bonds Time t = 1 value of the portfolio { ( 2.5)(1) + ( 50 )(1.05) = 0 21 V 1 = if no default ( 2.5)(.6) + ( 50 )(1.05) = 1 21 if default Time t = 0 initial investment: V 0 = ( 2.5)(.941) Arbitrage-free price of the option: P(0) = V 0 =.0285.

36 Risk-neutral Pricing of Derivatives Pricing and replication 1 Replicating portfolio: Sell 2.5 defaultable bonds (i.e., borrow = dollars) Lend 50/ dollars worth of default-free bonds Time t = 1 value of the portfolio { ( 2.5)(1) + ( 50 )(1.05) = 0 21 V 1 = if no default ( 2.5)(.6) + ( 50 )(1.05) = 1 21 if default Time t = 0 initial investment: V 0 = ( 2.5)(.941) Arbitrage-free price of the option: P(0) = V 0 =.0285.

37 Risk-neutral Pricing of Derivatives Pricing and replication 1 Replicating portfolio: Sell 2.5 defaultable bonds (i.e., borrow = dollars) Lend 50/ dollars worth of default-free bonds Time t = 1 value of the portfolio { ( 2.5)(1) + ( 50 )(1.05) = 0 21 V 1 = if no default ( 2.5)(.6) + ( 50 )(1.05) = 1 21 if default Time t = 0 initial investment: V 0 = ( 2.5)(.941) Arbitrage-free price of the option: P(0) = V 0 =.0285.

38 Risk-neutral Pricing of Derivatives Risk-neutral valuation Fundamental question: Is the price consistent with the expected discounted payoff" of the claim? Risk-Neutral Valuation Formula: 1 Calibration: Determine the probability of default q so that all traded assets have the same rate of return: Risk-free rate of return = E Q {Rate of return of the vulnerable bond} 0.05 = Risk-neutral pricing: (1 q) q = q =.03 P(0) = E Q {Discounted payoff} = P(0) =

39 Risk-neutral Pricing of Derivatives Risk-neutral valuation Fundamental question: Is the price consistent with the expected discounted payoff" of the claim? Risk-Neutral Valuation Formula: 1 Calibration: Determine the probability of default q so that all traded assets have the same rate of return: Risk-free rate of return = E Q {Rate of return of the vulnerable bond} 0.05 = Risk-neutral pricing: (1 q) q = q =.03 P(0) = E Q {Discounted payoff} = P(0) =

40 Risk-neutral Pricing of Derivatives Risk-neutral valuation Fundamental question: Is the price consistent with the expected discounted payoff" of the claim? Risk-Neutral Valuation Formula: 1 Calibration: Determine the probability of default q so that all traded assets have the same rate of return: Risk-free rate of return = E Q {Rate of return of the vulnerable bond} 0.05 = Risk-neutral pricing: (1 q) q = q =.03 P(0) = E Q {Discounted payoff} = P(0) =

41 Risk-neutral Pricing of Derivatives Risk-neutral valuation Fundamental question: Is the price consistent with the expected discounted payoff" of the claim? Risk-Neutral Valuation Formula: 1 Calibration: Determine the probability of default q so that all traded assets have the same rate of return: Risk-free rate of return = E Q {Rate of return of the vulnerable bond} 0.05 = Risk-neutral pricing: (1 q) q = q =.03 P(0) = E Q {Discounted payoff} = P(0) =

42 Risk-neutral Pricing of Derivatives Risk-neutral valuation Fundamental question: Is the price consistent with the expected discounted payoff" of the claim? Risk-Neutral Valuation Formula: 1 Calibration: Determine the probability of default q so that all traded assets have the same rate of return: Risk-free rate of return = E Q {Rate of return of the vulnerable bond} 0.05 = Risk-neutral pricing: (1 q) q = q =.03 P(0) = E Q {Discounted payoff} = P(0) =

43 Modeling stock prices Outline 1 A quick glance of mathematical finance Problems Arbitrage-free Pricing of Derivatives Risk-neutral Pricing of Derivatives 2 Modeling stock prices Stylized empirical features of stock prices Stochastic models for asset prices

44 Modeling stock prices Stylized empirical features of stock prices How does the price of a stock behaves in time?

45 Modeling stock prices Stylized empirical features of stock prices How does the price of a stock behaves in time?

46 Modeling stock prices Stylized empirical features of stock prices How does the price of a stock behaves in time?

47 Modeling stock prices Stylized empirical features of stock prices How does the price of a stock behaves in time?

48 Modeling stock prices Stylized empirical features of stock prices How does the price of a stock behaves in time?

49 Modeling stock prices Stylized empirical features of stock prices Stylized empirical features of stock prices 1 Jiggling or erratic motion, even at fine time scales 2 Sudden" relatively large shifts in the price level (jumps) 3 Volatility clustering effects (intermittency) 4 Log returns exhibiting heavy-tails and high-kurtosis empirical distributions 5 Leverage phenomenon

50 Modeling stock prices Stochastic models for asset prices What are good models for the price process {S t } t 0?

51 Modeling stock prices Stochastic models for asset prices What are good models for the price process {S t } t 0? Black-Scholes model (1973); Samuelson (1965): R (δ) i = log S (i+1)δ = µδ S iδ }{{} drift + δ σ Z } {{ } i, where i.i.d. Z i N (0, 1). noise Jiggling motion, no jump-like changes, no clustering or leverage

52 Modeling stock prices Stochastic models for asset prices What are good models for the price process {S t } t 0? Stochastic volatility Heston model (1993): R (δ) i = log S (i+1)δ S iδ = µδ + δ σ i 1 Z i, σ 2 i = σ 2 i 1 + α(m σ 2 i 1)δ + γ δ σ i 1 Z i, ρ = Corr(Z, Z ). Jiggling motion, clustering and leverage effects, but no jump-like changes

53 Modeling stock prices Stochastic models for asset prices What are good models for the price process {S t } t 0? Stochastic volatility with jumps: R (δ) i where 0 < β < 2 and J (β) i Remarks: = µδ + δ σ i 1 Z i + θδ 1/β J (β) i, σ 2 i = σ 2 i 1 + α(m σ 2 i 1)δ + γ δ σ i 1 Z i. are i.i.d. symmetric with heavy tails: P(J (β) i x) cx β, as x. P(Z i x) e x /x; hence, J i feels like jumps compared to Z i ; The larger β, the lighter the tails, and the smaller J (β) i β is called the index of jump activity. will tend to be;

54 Modeling stock prices Stochastic models for asset prices What are good models for the price process {S t } t 0? Stochastic volatility with jumps: R (δ) i = µδ + δ σ i 1 Z i + θδ 1/β J (β) i, σ 2 i = σ 2 i 1 + α(m σ 2 i 1)δ + γ δ σ i 1 Z i.

55 Modeling stock prices Stochastic models for asset prices What are good models for the price process {S t } t 0? Stochastic volatility with jumps: R (δ) i = µδ + δ σ i 1 Z i + θδ 1/β J (β) i, σ 2 i = σ 2 i 1 + α(m σ 2 i 1)δ + γ δ σ i 1 Z i.

56 Modeling stock prices Stochastic models for asset prices What are good models for the price process {S t } t 0? Stochastic volatility with jumps: R (δ) i = µδ + δ σ i 1 Z i + θδ 1/β J (β) i, σ 2 i = σ 2 i 1 + α(m σ 2 i 1)δ + γ δ σ i 1 Z i.

57 Thank you!!

58 Graphs Empirical distribution of returns Log return during a given time period = log Final price Initial price Back

59 Graphs Dynamics of the price processes Back

60 Graphs Times series of returns Back

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