Risk-Neutral Probabilities

Transcription

1 Debt Instruments an Markets Risk-Neutral Probabilities Concepts Risk-Neutral Probabilities True Probabilities Risk-Neutral Pricing Risk-Neutral Probabilities

2 Debt Instruments an Markets Reaings Tuckman, Chapter 9. No Arbitrage Derivative Pricing Last lecture, we price a erivative by constructing a replicating portfolio from the unerlying zeroes: We starte with a erivative with a payoff at time. The payoff epene on the time price of the zero maturing at time. We moele the ranom future price of the zero an the future payoff of the erivative. We constructe a portfolio of -year an - year zeroes with the same payoff of the erivative by solving simultaneous equations. We then set the price of the erivative equal to the value of the replicating portfolio. Risk-Neutral Probabilities 2

3 Debt Instruments an Markets General Bon Derivative Any security whose time payoff is a function of the time price of the zero maturing at time can be price by no arbitrage. Suppose its payoff is K u in the up state, an K in the own state. Time 0 Time -year zero -year zero N N General portfolio N N K u General erivative? N N K Replicating an Pricing the General Derivative ) Determine the replicating portfolio by solving the equations N N = K u N N = K for the unknown N's. (The two possible K's are known.) 2) Price the replicating portfolio as N N This is the no arbitrage price of the erivative. Risk-Neutral Probabilities 3

4 Debt Instruments an Markets Risk-Neutral Probabilities Finance: The no arbitrage price of the erivative is its replication cost. We know that s some function of the prices an payoffs of the basic unerlying assets. Math: We can use a mathematical evice, riskneutral probabilities, to compute that replication cost more irectly. That s useful when we only nee to know the price, not the other etails of the replicating portfolio. Start with the Prices an Payoffs of the Unerlying Assets In our example, the erivative payoffs were functions of the time price the zero maturing at time. So the unerlying asset is the zero maturing at time an the riskless asset is the zero maturing at time. The prices an payoffs are, in general terms: Time 0 Time u Risk-Neutral Probabilities 4

5 Debt Instruments an Markets Fin the Probabilities that Risk-Neutrally Price the Unerlying Risky Asset Fin the probabilities of the up an own states, p an -p, that make the price of the unerlying asset equal to its expecte future payoff, iscounte back at the riskless rate. I.e., fin the p that solves Price = iscounte expecte future payoff: = [ p u + ( p) ] The solution is p = u Example of p In our example, p = = 76 p = Risk-Neutral Probabilities 5

6 Debt Instruments an Markets Result: These Probabilities Price All Derivatives of this Unerlying Asset Risk-Neutrally If a erivative has payoffs K u in the up state an K in the own state, its replication cost turns out to be equal to Derivative price =. 5[ p Ku + ( p) K 0 I.e., price = iscounte expecte future payoff ] Examples of Risk-Neutral Pricing With the risk-neutral probabilities, the price of an asset is its expecte payoff multiplie by the riskless zero price, or equivalently, iscounte at the riskless rate: call option: put option: ( ) = = ( ) = = Risk-Neutral Probabilities 6

7 Debt Instruments an Markets Examples of Risk-Neutral Pricing... -year zero: ( ) = = year zero (riskless asset): ( ) = = True Probabilities The risk-neutral probabilities are not the same as the true probabilities of the future states. Notice that pricing contingent claims i not involve the true probabilities of the up or own state actually occurring. Let's suppose that the true probabilities are chance the up state occurs an chance the own state occurs. What coul we o with this information? For one, we coul compute the true expecte returns of the ifferent securities over the next 6 months. Risk-Neutral Probabilities 7

8 Debt Instruments an Markets True Expecte Returns Recall that the unannualize return on an asset over a given horizon is future value initial value For the 6-month zero the unannualize return over the next 6 months is = 2.77% with certainty. This will be the return regarless of which state occurs. That's why this asset is riskless for this horizon. Of course, the annualize semi-annually compoune ROR is 5.54%, the quote zero rate. True Expecte Returns... The return on the -year zero over the next 6 months will be either = 2.60% with probability, or = 3.00% with probability The expecte return on the -year zero over the next 6 months is 2.80%. Notice that it is higher than the return of 2.77% on the riskless asset. Risk-Neutral Probabilities 8

9 Debt Instruments an Markets True Expecte Returns... Why might the longer zero have a higher expecte return? Investors have short-term horizons, an islike the price risk of the longer zero Investors require a premium to hol securities that covary positively with long bons (bullish securities) because government bons are in positive net supply Sometimes the reverse coul be true. In general, assets with ifferent risk characteristics have ifferent expecte returns. Their expecte returns also epen on how their payoffs covary with other assets. True Expecte Returns... What is the expecte rate of return on the call over the next 6 months? The possible returns are: 0 = 00% with probability, or = 42% with probability The expecte return on the call is 2%. Risk-Neutral Probabilities 9

10 Debt Instruments an Markets True Expecte Returns... What is the expecte rate of return on the put over the next 6 months? The possible returns are: 2.7 = 78% with probability, or.52 0 = 00% with probability..52 The expecte return on the put is -%. The put is bearish--it insures (heges) the risk of bullish positions. By no arbitrage, if bullish assets have positive risk premia, bearish assets must have negative risk premia. Intuitively, investors must pay up for this insurance. Expecte Returns with Risk- Neutral Probabilities Note that we can rearrange the erivative pricing equation, price = iscounte expecte payoff, as V = V = [ p K u + ( p) K p Ku + ( p) K + r / 2 p Ku + ( p) K V = + r I.e., Expecte return = the riskless rate. (Here return are unnannualize. )Thus, with the riskneutral probabilities, all assets have the same expecte return, equal to the riskless rate. Because of this interpretation, we call them "risk-neutral" probabilities. ], or / 2 Risk-Neutral Probabilities 0

11 Debt Instruments an Markets Risk-Neutral Expecte Returns Using the risk-neutral probabilities to compute expecte (unannualize) returns sets all expecte returns equal to the riskless rate. Asset Unannualize Up Return ("prob"=76) Unannualize Down Payoff ("prob"=0.424) "Expecte" Unannualize Return -Year Zero / = 2.77% -Year Zero / = 2.60% Call 0/ = -00% Put 2.703/.59 - = 78.42% / = 2.77% / = 3.00%.0859/ = 42.39% 0/.59 - = -00% 2.77% 2.77% 2.77% 2.77% Why Does the p that Works for the Unerlying Asset Also Work for All Its Derivatives? ) The expecte return on a portfolio is the average of the expecte returns of the iniviual assets. 2) The risk-neutral probabilities are constructe to make the expecte return on the unerlying risky asset equal to the riskless asset return. (See slie 9.) 3) So uner the risk-neutral probabilities, the expecte return on every portfolio of the unerlying an riskless assets is also that same riskless return. 4) Every erivative of the unerlying can be viewe as a portfolio of the unerlying asset an the riskless asset. (See last lecture.) 5) So the erivative s expecte return must also equal the riskless return uner the risk-neutral probabilities. 6) So the erivative s price must equal its expecte payoff, using the risk-neutral probabilities, iscounte back at the riskless rate. (See slie 20.) Risk-Neutral Probabilities